*Note: The contents of these manuscripts are the IP of the authors, and may not be reproduced, presented to funding agencies or private foundations or in other presentations without attribution.*

## Quantum thermodynamics of nonequilibrium processes in lattice gauge theories

A key objective in nuclear and high-energy physics is to describe nonequilibrium dynamics of matter, e.g., in the early universe and in particle colliders, starting from the Standard Model. Classical-computing methods, via the framework of lattice gauge theory, have experienced limited success in this mission. Quantum simulation of lattice gauge theories holds promise for overcoming computational limitations. Because of local constraints (Gauss’s laws), lattice gauge theories have an intricate Hilbert-space structure. This structure complicates the definition of thermodynamic properties of systems coupled to reservoirs during equilibrium and nonequilibrium processes. We show how to define thermodynamic quantities such as work and heat using strong-coupling thermodynamics, a framework that has recently burgeoned within the field of quantum thermodynamics. Our definitions suit instantaneous quenches, simple nonequilibrium processes undertaken in quantum simulators. To illustrate our framework, we compute the work and heat exchanged during a quench in a $Z_2$ lattice gauge theory coupled to matter in 1+1 dimensions. The thermodynamic quantities, as functions of the quench parameter, evidence an expected phase transition.

Generally, we derive a simple relation between a quantum many-body system’s entanglement Hamiltonian, measurable with quantum-information-processing tools, and the Hamiltonian of mean force, used to define strong-coupling thermodynamic quantities.

## Qu8its for Quantum Simulations of Lattice Quantum Chromodynamics

We explore the utility of d=8 qudits, qu8its, for quantum simulations of the dynamics of 1+1D SU(3) lattice quantum chromodynamics, including a mapping for arbitrary numbers of flavors and lattice size and a re-organization of the Hamiltonian for efficient time-evolution. Recent advances in parallel gate applications, along with the shorter application times of single-qudit operations compared with two-qudit operations, lead to significant projected advantages in quantum simulation fidelities and circuit depths using qu8its rather than qubits. The number of two-qudit entangling gates required for time evolution using qu8its is found to be more than a factor of five fewer than for qubits. We anticipate that the developments presented in this work will enable improved quantum simulations to be performed using emerging quantum hardware.

## Real-time Dynamics of the Schwinger Model as an Open Quantum System with Neural Density Operators

Ab-initio simulations of multiple heavy quarks propagating in a Quark-Gluon Plasma are computationally difficult to perform due to the large dimension of the space of density matrices. This work develops machine learning algorithms to overcome this difficulty by approximating exact quantum states with neural network parametrisations, specifically Neural Density Operators. As a proof of principle demonstration in a QCD-like theory, the approach is applied to solve the Lindblad master equation in the 1+1D lattice Schwinger Model as an open quantum system. Neural Density Operators enable the study of in-medium dynamics on large lattice volumes, where multiple-string interactions and their effects on string-breaking and recombination phenomena can be studied. Thermal properties of the system at equilibrium can also be probed with these methods by variationally constructing the steady state of the Lindblad master equation. Scaling of this approach with system size is studied, and numerical demonstrations on up to 32 spatial lattice sites and with up to 3 interacting strings are performed.

## Comparative Study of Quarkonium Transport in Hot QCD Matter

This document summarizes the efforts of the EMMI Rapid Reaction Task Force on “Suppression and (re)generation of quarkonium in heavy-ion collisions at the LHC”, centered around their 2019 and 2022 meetings. It provides a review of existing experimental results and theoretical approaches, including lattice QCD calculations and semiclassical and quantum approaches for the dynamical evolution of quarkonia in the quark-gluon plasma as probed in high-energy heavy-ion collisions. The key ingredients of the transport models are itemized to facilitate comparisons of calculated quantities such as reaction rates, binding energies, and nuclear modification factors. A diagnostic assessment of the various results is attempted and coupled with an outlook for the future.

## Classical and Quantum Computing of Shear Viscosity for 2+1D SU(2) Gauge Theory

We perform a nonperturbative calculation of the shear viscosity for 2+1-dimensional SU(2) gauge theory by using the lattice Hamiltonian formulation. The retarded Green’s function of the stress-energy tensor is calculated from real time evolution via exact diagonalization of the lattice Hamiltonian with a local Hilbert space truncation and the shear viscosity is obtained via the Kubo formula. When taking the continuum limit, we account for the renormalization group flow of the coupling but no additional operator renormalization. We find the ratio of the shear viscosity and the entropy density eta/s is consistent with a well-known holographic result 1/(4 pi) at several temperatures on a 4×4 hexagonal lattice with the local electric representation truncated at j_max=1/2. We also find the ratio of the spectral function and frequency rho^(xy)(omega)/omega exhibits a peak structure when the frequency is small. Both exact diagonalization method and simple matrix product state classical simulation method beyond j_max=1/2 on bigger lattices require exponentially growing resources. So we develop a quantum computing method to calculate the retarded Green’s function and analyze various systematics of the calculation including j_max truncation and finite size effects and Trotter errors. We test our quantum circuit on both the Quantinuum emulator and the IBM simulator for a small lattice and obtain results consistent with the classical computing ones.

## Scattering wave packets of hadrons in gauge theories: Preparation on a quantum computer

Quantum simulation holds promise of enabling a complete description of high-energy scattering processes rooted in gauge theories of the Standard Model. A first step in such simulations is preparation of interacting hadronic wave packets. To create the wave packets, one typically resorts to adiabatic evolution to bridge between wave packets in the free theory and those in the interacting theory, rendering the simulation resource intensive. In this work, we construct a wave-packet creation operator directly in the interacting theory to circumvent adiabatic evolution, taking advantage of resource-efficient schemes for ground-state preparation, such as variational quantum eigensolvers. By means of an ansatz for bound mesonic excitations in confining gauge theories, which is subsequently optimized using classical or quantum methods, we show that interacting mesonic wave packets can be created efficiently and accurately using digital quantum algorithms that we develop. Specifically, we obtain high-fidelity mesonic wave packets in the Z_2 and U(1) lattice gauge theories coupled to fermionic matter in 1+1 dimensions. Our method is applicable to both perturbative and non-perturbative regimes of couplings. The wave-packet creation circuit for the case of the Z_2 lattice gauge theory is built and implemented on the Quantinuum H1-1 trapped-ion quantum computer using 13 qubits and over 300 entangling gates. The fidelities agree well with classical benchmark calculations after employing a simple symmetry-based noise-mitigation technique. This work serves as a step toward quantum computing scattering processes in quantum chromodynamics.

## Entanglement Entropy of 2+1-Dimensional SU(2) Lattice Gauge Theory

We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2+1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum 2+1-dimensional SU(2) gauge theory is a “fast” scrambler.

## Quantum Simulations of Hadron Dynamics in the Schwinger Model using 112 Qubits

Hadron wavepackets are prepared and time evolved in the Schwinger model using 112 qubits of IBM’s 133-qubit Heron quantum computer ibm_torino. The initialization of the hadron wavepacket is performed in two steps. First, the vacuum is prepared across the whole lattice using the recently developed SC-ADAPT-VQE algorithm and workflow. SC-ADAPT-VQE is then extended to the preparation of localized states, and used to establish a hadron wavepacket on top of the vacuum. This is done by adaptively constructing low-depth circuits that maximize the overlap with an adiabatically prepared hadron wavepacket. Due to the localized nature of the wavepacket, these circuits can be determined on a sequence of small lattices using classical computers, and then robustly scaled to prepare wavepackets on large lattices for simulations using quantum computers. Time evolution is implemented with a second-order Trotterization. To reduce both the required qubit connectivity and circuit depth, an approximate quasi-local interaction is introduced. This approximation is made possible by the emergence of confinement at long distances, and converges exponentially with increasing distance of the interactions. Using multiple error-mitigation strategies, up to 14 Trotter steps of time evolution are performed, employing 13,858 two-qubit gates (with a CNOT depth of 370). The propagation of hadrons is clearly identified, with results that compare favorably with Matrix Product State simulations. Prospectsfor a near-term quantum advantage in simulations of hadron scattering are discussed.

## Recent Developments in Quarkonium as an Open Quantum System in Quark-Gluon Plasma

We review recent progress in understanding quarkonium dynamics inside the quark-gluon plasma as an open quantum system with a focus on the definition and nonperturbative calculations of relevant transport coefficients and generalized gluon distributions.

## Quarkonium transport in weakly and strongly coupled plasmas

We report on progress in the nonperturbative understanding of quarkonium dynamics inside a thermal plasma. The time evolution of small-size quarkonium is governed by two-point correlation functions of chromoelectric fields dressed with an adjoint Wilson line, known in this context as generalized gluon distributions (GGDs). The GGDs have been calculated in both weakly and strongly coupled plasmas by using perturbative and holographic methods. Strikingly, the results of our calculations for a strongly coupled plasma indicate that the quarkonium dissociation and recombination rates vanish in the transport descriptions that assume quarkonium undergoes Markovian dynamics. However, this does not imply that the dynamics is trivial. As a starting point to explore the phenomenological consequences of the result at strong coupling, we show a calculation of the $\Upsilon(1S)$ formation probability in time-dependent perturbation theory. This is a first step towards the development of a transport formalism that includes non-Markovian effects, which, depending on how close the as of yet undetermined nonperturbative QCD result of the GGDs is to the strongly coupled $\mathcal{N}=4$ SYM result, could very well dominate over the Markovian ones in quark-gluon plasma produced at RHIC and the LHC.

## Testing Eigenstate Thermalization Hypothesis for Non-Abelian Gauge Theories

We report on progress in full quantum understanding of thermalization in non-Abelian gauge theories. Specifically, we test the eigenstate thermalization hypothesis for (2+1)-dimensional SU(2) lattice gauge theory.

## Time Scales in Many-Body Fast Neutrino Flavor Conversion

Time scales associated with many-body fast neutrino flavor conversions in core-collapse supernova are explored in the context of an effective two-flavor “light-bulb” model. We present a preliminary study of time scales obtained from linear stability analyses and from distributions of Loschmidt echo crossing times (intimately connected to dynamical phase transitions) determined by time evolution with the exact many-body Hamiltonian. Starting from a tensor-product initial state describing systems of N neutrinos, with N/2 electron-type and N/2 heavy-type, with uniform distributions of axially-symmetric directions, the Loschmidt echo crossing times are found to exhibit two distinct time scales that are exponentially separated. The second peak structure at longer times, effectively absent for N=4, develops with increasing N. When re-scaled in terms of log t, the distributions are found to become increasingly well described by the sum of two stable distributions. The distribution of Loschmidt echo crossing times differs somewhat from the results of the linear stability analysis, which exhibits a peak at finite frequency and a second peak at zero frequency. The exact analysis suggests that the zero-frequency instability manifests itself as a modest flavor-conversion time scale.

## Quantum Computing for Nuclear Physics

Future quantum computers are anticipated to be able to perform simulations of quantum many-body systems and quantum field theories that lie beyond the capabilities of classical computation. This will lead to new insights and predictions for systems ranging from dense non equilibrium matter, to low-energy nuclear structure and reactions, to high energy collisions. I present an overview of digital quantum simulations in nuclear physics, with select examples relevant for studies of quark matter.

*Conference proceedings for Quark Matter 2023, 3–9 Sept 2023, Houston, Texas, USA.*

## Generalized Gluon Distribution for Quarkonium Dynamics in Strongly Coupled N=4 Yang-Mills Theory

We study the generalized gluon distribution that governs the dynamics of quarkonium inside a non-Abelian thermal plasma characterizing its dissociation and recombination rates. This gluon distribution can be written in terms of a correlation function of two chromoelectric fields connected by an adjoint Wilson line. We formulate and calculate this object in N=4 supersymmetric Yang-Mills theory at strong coupling using the AdS/CFT correspondence, allowing for a nonzero center-of-mass velocity v of the heavy quark pair relative to the medium. The effect of a moving medium on the dynamics of the heavy quark pair is described by the simple substitution T with Sqrt[gamma] T, in agreement with previous calculations of other observables at strong coupling, where T is the temperature of the plasma in its rest frame, and gamma = (1 – v^2)^{-1/2} is the Lorentz boost factor. Such a velocity dependence can be important when the quarkonium momentum is larger than its mass. Contrary to general expectations for open quantum systems weakly coupled with large thermal environments, the contributions to the transition rates that are usually thought of as the leading ones in Markovian descriptions vanish in this strongly coupled plasma. This calls for new theoretical developments to assess the effects of strongly coupled non-Abelian plasmas on in-medium quarkonium dynamics. Finally, we compare our results with those from weakly coupled QCD, and find that the QCD result moves toward the N=4 strongly coupled result as the coupling constant is increased within the regime of applicability of perturbation theory. This behavior makes it even more pressing to develop a non-Markovian description of quarkonium in-medium dynamics.

## Generalized Ginsparg-Wilson relations

We give a general derivation of Ginsparg-Wilson relations for both Dirac and Majorana fermions in any dimension. These relations encode continuous and discrete chiral, parity and time reversal anomalies and will apply to the various classes of free fermion topological insulators and superconductors (in the framework of a relativistic quantum field theory in Euclidian spacetime). We show how to formulate the exact symmetries of the lattice action and the relevant index theorems for the anomalies.

## Eigenstate Thermalization in 2+1 dimensional SU(2) Lattice Gauge Theory

We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the Eigenstate Thermalization Hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to j=0,1/2 , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space (j <= 7/2) that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for RMT behavior and then analyse the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (b) and (c) agree with ETH predictions while for case (a) deviations are found to be large for one of the analyzed observables. To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary.

## Liouvillian Dynamics of the Open Schwinger Model: String Breaking and Kinetic Dissipation in a Thermal Medium

Understanding the dynamics of bound state formation is one of the fundamental questions in confining quantum field theories such as Quantum Chromodynamics (QCD). One hadronization mechanism that has garnered significant attention is the breaking of a string initially connecting a fermion and an anti-fermion. Deepening our understanding of real-time string-breaking dynamics with simpler, lower dimensional models like the Schwinger model can improve our understanding of the hadronization process in QCD and other confining systems found in condensed matter and statistical systems. In this paper, we consider the string-breaking dynamics within the Schwinger model and investigate its modification inside a thermal medium, treating the Schwinger model as an open quantum system coupled to a thermal environment. Within the regime of weak coupling between the system and environment, the real-time evolution of the system can be described by a Lindblad evolution equation. We analyze the Liouvillian gaps of this Lindblad equation and the time dependence of the system’s von Neumann entropy. We observe that the late-time relaxation rate decreases as the environment correlation length increases. Moreover, when the environment correlation length is infinite, the system exhibits two steady states, one in each of the sectors with definite charge-conjugation-parity (CP) quantum numbers. For parameter regimes where an initial string breaks in vacuum, we observe a delay of the string breaking in the medium, due to kinetic dissipation effects. Conversely, in regimes where an initial string remains intact in vacuum time evolution, we observe string breaking (melting) in the thermal medium. We further discuss how the Liouvillian dynamics of the open Schwinger model can be simulated on quantum computers and provide an estimate of the associated Trotter errors.

## Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits

The vacuum of the lattice Schwinger model is prepared on up to 100 qubits of IBM’s Eagle-processor quantum computers. A new algorithm to prepare the ground state of a gapped translationally-invariant system on a quantum computer is presented, which we call Scalable Circuits ADAPT-VQE (SC ADAPT-VQE). This algorithm uses the exponential decay of correlations between distant regions of the ground state, together with ADAPT-VQE, to construct quantum circuits for state preparation that can be scaled to arbitrarily large systems. SC-ADAPT-VQE is applied to the Schwinger model, and shown to be systematically improvable, with an accuracy that converges exponentially with circuit depth. Both the structure of the circuits and the deviations of prepared wavefunctions are found to become independent of the number of spatial sites, L. This allows for a controlled extrapolation of the circuits, determined using small or modest-sized systems, to arbitrarily large L. The circuits for the Schwinger model are determined on lattices up to L=14 (28 qubits) with the qiskit classical simulator, and subsequently scaled up to prepare the L=50 (100 qubits) vacuum on IBM’s 127 superconducting-qubit quantum computers ibm_brisbane and ibm_cusco . After applying an improved error-mitigation technique, which we call Operator Decoherence Renormalization, the chiral condensate and charge-charge correlators obtained from the quantum computers are found to be in good agreement with classical Matrix Product State simulations.

## Optimization of Algorithmic Errors in Analog Quantum Simulations

Analog quantum simulation is emerging as a powerful tool for uncovering classically unreachable physics such as many-body real-time dynamics. We study a class of errors for approximate time evolution algorithms in Heisenberg-type systems on analog quantum devices described by the Ising Hamiltonian. A general framework for quantifying these errors is introduced and applied to several proposed time evolution methods, including Trotter-like methods and Floquet-engineered constant-field approaches. This analysis explains the interplay of different error sources arising from approximations to the propagator of a physical theory. The limitations placed on the accuracy of time evolution methods by current devices are discussed. Characterization of the error scaling provides a way to extend the presented Hamiltonian engineering methods to take advantage of forthcoming device capabilities.

## A new basis for Hamiltonian SU(2) simulations

Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely-used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories.

## Quarkonium transport in strongly coupled plasmas

Suppression of open heavy flavors and quarkonia in heavy-ion collisions is among the most informative probes of the quark-gluon plasma. Interpreting the full wealth of data obtained from the collision events requires a precise understanding of the evolution of heavy quarks and quarkonia as they propagate through the nearly thermal and strongly coupled plasma. Systematic theoretical studies of quarkonium time evolution in the QGP in the regime where the temperature of the QGP is much smaller than the inverse of quarkonium size have only been carried out in the past few years.

Such calculations require the evaluation of a gauge-invariant correlator of chromoelectric fields dressed with Wilson lines, which is similar to, but different from, the correlation used to define the well-known heavy quark diffusion coefficient. The origin of this difference has been explained previously. Here we show a calculation of the analogous correlator in strongly coupled N=4 SYM using the AdS/CFT correspondence at a finite temperature. While it resembles the open heavy quark case, it has some crucial differences that highlight the relevance of quantum color correlations. We will also discuss the results for the quarkonium transport coefficients obtained from this correlator, thereby establishing the first analytic results at strong coupling in this context.

## Quantum Simulation of Lattice QCD with Improved Hamiltonians

Quantum simulations of lattice gauge theories are anticipated to directly probe the real time dynamics of QCD, but scale unfavorably with the required truncation of the gauge fields. Improved Hamiltonians are derived to correct for the effects of gauge field truncations on the SU(3) Kogut-Susskind Hamiltonian. It is shown in 1+1D that this enables low chromo-electric field truncations to quantitatively reproduce features of the untruncated theory over a range of couplings and quark masses. In 3+1D, an improved Hamiltonian is derived for lattice QCD with staggered massless fermions. It is shown in the strong coupling limit that the spectrum qualitatively reproduces aspects of two flavor QCD and simulations of a small system are performed on IBM’s Perth quantum processor.

## Simple Hamiltonian for Quantum Simulation of Strongly Coupled 2+1D SU(2) Lattice Gauge Theory on a Honeycomb Lattice

We find a simple spin Hamiltonian to describe physical states of 2+1D SU(2) lattice gauge theory on a honeycomb lattice with a truncation of the electric field representation at j_max=1/2. The simple spin Hamiltonian only contains local products of Pauli matrices, even though Gauss’s law has been completely integrated out.

## Quantum Imaginary Time Propagation algorithm for preparing thermal states

Calculations at finite temperatures are fundamental in different scientific fields, from nuclear physics to condensed matter. Evolution in imaginary time is a prominent classical technique for preparing thermal states of quantum systems. We propose a new quantum algorithm that prepares thermal states based on the quantum imaginary time propagation method, using a diluted operator with ancilla qubits to overcome the non-unitarity nature of the imaginary time operator. The presented method is the first that allows us to obtain the correct thermal density matrix on a general quantum processors for a generic Hamiltonian. We prove its reliability in the actual quantum hardware computing thermal properties for two and three neutron systems.

## Multi-Body Entanglement and Information Rearrangement in Nuclear Many-Body Systems

The present work examines how effective-model-space calculations of nuclear many-body systems are able to rearrange and converge multi-particle entanglement, and information more generally. The generalized Lipkin-Meshkov-Glick (LMG) model is considered here as a demonstration, to motivate and provide insight for future developments of entanglement-driven descriptions of nuclei. The effective approach is based on a truncation of the Hilbert space to an effective model space, together with a variational rotation of the qubits (spins), which constitute the relevant elementary degrees of freedom in this model. The non-commutivity of the rotation and truncation allow for an exponential improvement of the energy convergence throughout much of the model space, compared with truncations without rotations, as we have shown previously in the reduced LMG model. Our analysis is extended to examine measures of correlations and entanglement, and to quantify their convergence with increasing cut-off. We focus on one- and two-spin entanglement entropies, mutual information, as well as n-tangles for n=2, 4 to estimate multi-body entanglement. The effective description is found to provide “minimal” entropies and mutual information of the rotated spins, while, at the same time, being able to recover the exact results to a large extent with low cut-offs. Naive truncations of the bare Hamiltonian, on the other hand, artificially underestimate these measures. The n-tangles in the present model provide a basis-independent measures of n-particle entanglement. While these are found to be more difficult to capture with the EMS description, the improvement in convergence, compared to truncations of the bare Hamiltonian, is significantly more dramatic. Finally, we investigate how spin squeezing, which is particularly sensitive to entanglement details of the system, is reproduced by quantum simulations of the LMG model we previously performed using IBM’s quantum computers. We conclude that the low-energy effective model space techniques, that successfully provide predictive capabilities for low-lying observables in many-body systems, exhibit analogous efficacy for quantum correlations and multi-body entanglement in the LMG model, motivating future studies in nuclear many-body systems that are closer to nature. We anticipate our results have import for effective field theories relevant to

high-energy physics and nuclear physics more broadly.

## Quantum simulation of fundamental particles and forces

Key static and dynamic properties of matter — from creation in the Big Bang to evolution into subatomic and astrophysical environments — arise from the underlying fundamental quantum fields of the standard model and their effective descriptions. However, the simulation of these properties lies beyond the capabilities of classical computation alone. Advances in quantum technologies have improved control over quantum entanglement and coherence to the point at which robust simulations of quantum fields are anticipated in the foreseeable future. In this Perspective article, we discuss the emerging area of quantum simulations of standard-model physics, outlining the challenges and opportunities for progress in the context of nuclear and high-energy physics.

## Real Time Quarkonium Transport Coefficients in Open Quantum Systems from Euclidean QCD

Recent open quantum system studies showed that quarkonium time evolution inside the quark-gluon plasma is determined by transport coefficients that are defined in terms of a gauge invariant correlator of two chromoelectric field operators connected by an adjoint Wilson line. We study the Euclidean version of the correlator for quarkonium evolution and discuss the extraction of the transport coefficients from this Euclidean correlator, highlighting its difference from other problems that also require reconstructing a spectral function, such as the calculation of the heavy quark diffusion coefficient. Along the way, we explain why the transport coefficient gamma(adj) differs from gamma(fund) at finite temperature at O(g^4), in spite of the fact that their corresponding spectral functions differ only by a temperature-independent term at the same order. We then discuss how to evaluate the Euclidean correlator via lattice QCD methods, with a focus on reducing the uncertainty caused by infrared renormalons in determining the renormalization factor nonperturbatively.

## High-energy Collision of Quarks and Hadrons in the Schwinger Model: From Tensor-Networks to Circuit-QED

With the aim of studying nonperturbative, out-of-equilibrium dynamics of high-energy particle collisions on quantum simulators, we investigate the scattering dynamics of lattice quantum electrodynamics in 1+1 dimensions. Working in the bosonized formulation of the model, an analog circuit-QED implementation is proposed that is native to the platform, hence requires minimal ingredients and approximations, and enables practical schemes for particle wavepacket preparation and evolution. Furthermore, working in the thermodynamic limit, uniform-matrix-product-state tensor networks are used to construct multi-particle wavepacket states, evolve them in time, and detect outgoing particles post collision. This facilitates the numerical simulation of scattering experiments in both confined and deconfined regimes of the model at different energies, giving rise to rich phenomenology, including inelastic production of quark and meson states, meson disintegration, and dynamical string formation and breaking. Elastic and inelastic scattering cross sections are obtained, together with time-resolved momentum and position distributions of the outgoing particles. This study highlights the role of classical and quantum simulation in enhancing our understanding of scattering processes in quantum field theories in real time.

## Quantum Simulations of SO(5) Many-Fermion Systems using Qudits

The structure and dynamics of many-body systems are the result of a delicate interplay between underlying interactions. Fermionic pairing plays a central role in various physical systems and can lead to collective phenomena such as superconductivity and superfluidity. We explore the potential utility of quantum computers with arrays of qudits in simulating interacting fermionic systems, when the qudits can be naturally mapped to the relevant degrees of freedom. The Agassi model of fermions is based on an underlying $so(5)$ algebra, and the systems it describes can be partitioned into pairs of modes with five basis states, which naturally embed in arrays of $d=5$ qudits (qu5its). Classical noiseless simulations of the time evolution of systems of fermions embedded in up to twelve qu5its are performed using Google’s {\tt cirq} software. The resource requirements of the qu5it circuits are analyzed and compared with two different mappings to qubit systems, a physics-aware Jordan-Wigner mapping and a state-to- state mapping. We find advantages in using qudits, specifically in lowering the required quantum resources and reducing anticipated errors that take the simulation out of the physical space. A previously unrecognized sign problem has been identified from Trotterization errors in time evolving high-energy excitations. This has implications for quantum simulations in high energy and nuclear physics, specifically of fragmentation and highly inelastic, multi-channel processes.

## Randomized measurement protocols for lattice gauge theories

Randomized measurement protocols, including classical shadows, entanglement tomography, and randomized benchmarking are powerful techniques to estimate observables, perform state tomography, or extract the entanglement properties of quantum states. While unraveling the intricate structure of quantum states is generally difficult and resource-intensive, quantum systems in nature are often tightly constrained by symmetries. This can be leveraged by the symmetry-conscious randomized measurement schemes we propose, yielding clear advantages over symmetry-blind randomization such as reducing measurement costs, enabling symmetry-based error mitigation in experiments, allowing differentiated measurement of (lattice) gauge theory entanglement structure, and, potentially, the verification of topologically ordered states in existing and near-term experiments. Crucially, unlike symmetry-blind randomized measurement protocols, these latter tasks can be performed without relearning symmetries via full reconstruction of the density matrix.

## Chromoelectric field correlator for quarkonium transport in the strongly coupled N=4 Yang-Mills plasma from AdS/CFT

Previous studies have shown that a gauge-invariant correlation function of two chromoelectric fields connected by a straight timelike adjoint Wilson line encodes crucial information about quark-gluon plasma (QGP) that determines the dynamics of small-sized quarkonium in the medium. Motivated by the successes of holographic calculations to describe strongly coupled QGP, we calculate the analog gauge-invariant correlation function in strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills theory at finite temperature by using the AdS/CFT correspondence. Our results indicate that the transition processes between bound and unbound quarkonium states are suppressed in strongly coupled plasmas, and moreover, the leading contributions to these transition processes vanish in both the quantum Brownian motion and quantum optical limits of open quantum system approaches to quarkonia.

## SU(2) Non-Abelian Gauge Theory on a Plaquette Chain Obeys Eigenstate Thermalization Hypothesis

We test the eigenstate thermalization hypothesis (ETH) for 2+1 dimensional SU(2) lattice gauge theory. By considering the theory on a chain of plaquettes and truncating basis states for link variables at j=1/2, we can map it onto an Ising chain and numerically exactly diagonalize the Hamiltonian for reasonably large lattice sizes. We find energy level repulsion in momentum sectors with no remaining discrete symmetries. We study two local observables made up of Wilson loops and calculate their matrix elements in the energy eigenbasis, which are shown consistent with the ETH. Our study implies a subset of states in the physical Hilbert space of Quantum Chromodynamics (QCD) obeys the ETH.

## White paper on Quantum Information Science and Technology for Nuclear Physics. Input into U.S. Long-Range Planning, 2023.

In preparation for the 2023 NSAC Long Range Plan (LRP), members of the Nuclear Science community gathered to discuss the current state of, and plans for further leveraging opportunities in, QIST in NP research at the *Quantum Information Science for U.S. Nuclear Physics Long Range Planning *workshop [4], held in Santa Fe, New Mexico on January 31—Feb 1, 2023. The workshop, jointly-sponsored by Los Alamos National Laboratory (LANL) and the InQubator for Quantum Simulation (IQuS), included 45 in-person participants and 53 remote attendees. The outcome of the workshop identified strategic plans and requirements for the next 5-10 years to advance quantum sensing and quantum simulations within NP, and to develop a diverse quantum-ready workforce. The plans include resolutions endorsed by the participants to address the compelling scientific opportunities at the intersections of NP and QIST. These endorsements are aligned with similar affirmations by the LRP Computational Nuclear Physics and AI/ML Workshop, the Nuclear Structure, Reactions, and Astrophysics LRP Town Hall, and the Fundamental Symmetries, Neutrons, and Neutrinos LRP Town Hall communities.

## Gravitational Wave Backgrounds from Colliding ECOs

Long baseline atom interferometers offer an exciting opportunity to explore mid-frequency gravitational waves. In this work we survey the landscape of possible contributions to the total “gravitational wave background” in this frequency band and advocate for targeting this observable. Such an approach is complimentary to searches for resolved mergers from individual sources and may have much to reveal about the Universe. We find that the inspiral phases of stellar-mass compact binaries cumulatively produce a signal well within reach of the proposed AION-km and AEDGE experiments. Hypothetical populations of dark sector exotic compact objects, harbouring just a tiny fraction of the dark energy density, could also generate signatures unique to mid- and low-frequency gravitational wave detectors, providing a novel means to probe complexity in the dark sector.

## Demonstration of a quantum-classical coprocessing protocol for simulating nuclear reactions

Quantum computers hold great promise for arriving at exact simulations of nuclear dynamical processes (e.g., scattering and reactions) that are paramount to the study of nuclear matter at the limit of stability and to explaining the formation of chemical elements in stars. However, quantum simulations of the unitary (real) time dynamics of fermionic many-body systems require a currently prohibitive number of reliable and long-lived qubits. We propose a co-processing algorithm for the simulation of real-time dynamics in which the time evolution of the spatial coordinates is carried out on a classical processor, while the evolution of the spin degrees of freedom is carried out on a quantum processor. This hybrid algorithm is demonstrated in a quantum simulation of the scattering of two neutrons performed on the Lawrence Berkeley National Laboratory’s Advanced Quantum Testbed. We show that, after implementation of error mitigation strategies to improve the accuracy of the algorithm in addition to the use of either circuit compression techniques or tomography as methods to elucidate the onset of decoherence, this initial demonstration validates the principle of the proposed co-processing scheme. We anticipate that a generalization of this present scheme will open the way for (real-time) path integral simulations of nuclear scattering.

## Quantum Simulations in Effective Model Spaces (I): Hamiltonian Learning-VQE using Digital Quantum Computers and Application to the Lipkin-Meshkov-Glick Model

The utility of effective model spaces in quantum simulations of non-relativistic quantum many-body systems is explored in the context of the Lipkin-Meshkov-Glick model of interacting fermions. We introduce an iterative hybrid-classical-quantum algorithm, Hamiltonian learning variational quantum eigensolver (HL-VQE), that simultaneously optimizes an effective Hamiltonian, thereby rearranging entanglement into the effective model space, and the associated ground-state wavefunction. HL-VQE is found to provide an exponential improvement in Lipkin-Meshkov-Glick model calculations, compared to a naive truncation without Hamiltonian learning, throughout a significant fraction of the Hilbert space. Quantum simulations are performed to demonstrate the HL-VQE algorithm, using an efficient mapping where the number of qubits scales with the log of the size of the effective model space, rather than the particle number, allowing for the description of large systems with small quantum circuits. Implementations on IBM’s QExperience quantum computers and simulators for 1- and 2-qubit effective model spaces are shown to provide accurate and precise results, reproducing classical predictions. This work constitutes a step in the development of entanglement-driven quantum algorithms for the description of nuclear systems, that leverages the potential of noisy intermediate-scale quantum (NISQ) devices.

## Gauge Invariance of Non-Abelian Field Strength Correlators: the Axial Gauge Puzzle

Many transport coefficients of the quark-gluon plasma and nuclear structure functions can be written as gauge invariant correlation functions of non-Abelian field strengths dressed with Wilson lines. We discuss the applicability of axial gauge n.A=0 to calculate them. In particular, we address issues that appear when one attempts to trivialize the Wilson lines in the correlation functions by gauge-fixing. We find it is always impossible to completely remove the gauge fields n.A in Wilson lines that extend to infinity in the n-direction by means of gauge transformations. We show how the obstruction appears in an explicit example of a perturbative calculation, and we also explain it more generally from the perspective of the path integral that defines the theory. Our results explain why the two correlators that define the heavy quark and quarkonium transport coefficients, which are seemingly equal in axial gauge, are actually different physical quantities of the quark-gluon plasma and have different values. Furthermore, our findings provide insights into the difference between two inequivalent gluon parton distribution functions.

## Preparation for Quantum Simulation of the 1+1D O(3) Non-linear σ-Model using Cold Atoms

The 1+1D O(3) non-linear σ-model is a model system for future quantum lattice simulations of other

asymptotically-free theories, such as non-Abelian gauge theories. We find that utilizing dimensional reduction

can make efficient use of two-dimensional layouts presently available on cold atom quantum simulators. A

new definition of the renormalized coupling is introduced, which is applicable to systems with open boundary

conditions and can be measured using analog quantum simulators. Monte Carlo and tensor network calculations

are performed to determine the quantum resources required to reproduce perturbative short-distance observables.

In particular, we show that a rectangular array of 48 Rydberg atoms with existing quantum hardware capabilities

should be able to adiabatically prepare low-energy states of the perturbatively-matched theory. These states can

then be used to simulate non-perturbative observables in the continuum limit that lie beyond the reach of classical

computers.

## Quantum Simulation of Light-Front QCD for Jet Quenching in Nuclear Environments

We develop a framework to simulate jet quenching in nuclear environments on a quantum computer. The formulation is based on the light-front Hamiltonian dynamics of QCD. The Hamiltonian consists of three parts relevant for jet quenching studies: kinetic, diffusion and splitting terms. In the basis made up of $n$-particle states in momentum space, the kinetic Hamiltonian is diagonal. Matrices representing the diffusion and splitting parts are sparse. The diffusion part of the Hamiltonian depends on classical background gauge fields, which need to be sampled classically before constructing quantum circuits for the time evolution. The cost of the sampling scales linearly with the time length of the evolution and the momentum grid volume. The framework automatically keeps track of quantum interference and thus it can be applied to study the Landau-Pomeranchuk-Migdal effect in cases with more than two splittings, which is beyond the scope of state-of-the-art analyses, no matter whether the medium is static or expanding, thin or thick, hot or cold. We apply this framework to study a toy model and gluon in-medium radiation on a small lattice. The Landau-Pomeranchuk-Migdal effect that suppresses the total radiation probability is observed in the quantum simulation results of both the toy model and the gluon case.

## Faster spectral density calculation using energy moments

Accurate predictions of inclusive scattering cross sections in the linear response regime require efficient and controllable methods to calculate the spectral density in a strongly-correlated many-body system. In this work we reformulate the recently proposed Gaussian Integral Transform technique in terms of Fourier moments of the system Hamiltonian which can be computed efficiently on a quantum computer. One of the main advantages of this framework is that it allows for an important reduction of the computational cost by exploiting previous knowledge about the energy moments of the spectral density. For a simple model of medium mass nucleus like ⁴⁰Ca and target energy resolution of 1 MeV we find an expected speed-up of ≈ 125 times for the calculation of the giant dipole response and of ≈ 50 times for the simulation of quasi-elastic electron scattering at typical momentum transfers.

## Multi-Neutrino Entanglement and Correlations in Dense Neutrino Systems

The time-evolution of multi-neutrino entanglement and correlations are studied in two-flavor collective neutrino oscillations, relevant for dense neutrino environments, building upon previous works. Specifically, simulations performed of systems with up to 12 neutrinos using Quantinuum’s H1-1 20 qubit trapped-ion quantum computer are used to compute n-tangles, and two- and three-body correlations, probing beyond mean-field descriptions. n-tangle re-scalings are found to converge for large system sizes.

## State Preparation in the Heisenberg Model through Adiabatic Spiraling

An adiabatic state preparation technique, called the adiabatic spiral, is proposed for the Heisenberg model. This technique is suitable for implementation on a number of quantum simulation platforms such as Rydberg atoms, trapped ions, or superconducting qubits. Classical simulations of small systems suggest that it can be successfully implemented in the near future. A comparison to Trotterized time evolution is performed and it is shown that the adiabatic spiral is able to outperform Trotterized adiabatics.

## Quantum computation of dynamical quantum phase transitions and entanglement tomography in a lattice gauge theory

Strongly-coupled gauge theories far from equilibrium may exhibit unique features that could illuminate the physics of early universe and of hadron and ion colliders. Studying real-time phenomena has proven challenging with classical-simulation methods, but is a natural application of quantum-simulation devices. To demonstrate this prospect, we quantum compute non-equal time correlation functions and perform entanglement tomography of non-equilibrium states of a simple lattice gauge theory, the Schwinger model, using a trapped-ion quantum computer by IonQ Inc. As an ideal target for near-term devices, a recently-predicted dynamical quantum phase transition in this model is studied by preparing, quenching, and tracking the subsequent non-equilibrium dynamics in three ways: i) overlap echos signaling dynamical transitions, ii) non-equal time correlation functions with an underlying topological nature, and iii) the entanglement structure of non-equilibrium states, including Entanglement Hamiltonians. These results constitute the first observation of a dynamical quantum phase transition in a lattice gauge theory on a quantum computer, and are a first step toward investigating topological phenomena in nuclear and high-energy physics using quantum technologies.

## Lattice regularization of Theta-vacua in Grassmannian nonlinear sigma models: Anomalies, sign-problems and qubit models

Anomalies are a powerful way to gain insight into possible lattice regularizations of a quantum field theory. In this work, we consider lattice regularizations of a class of the toy-models of QCD: the 1+1-dimensional asymptotically-free Grassmanian nonlinear sigma models with a theta term. We argue that the continuum anomaly for a given symmetry can be matched by a manifestly symmetric lattice regularization only if (i) the symmetry action is offsite, or (ii) if the continuum anomaly is reproduced exactly on the lattice. Using the Grassmanian nonlinear sigma models as a case study, we provide examples of lattice regularizations in which both possibilities are realized. For possibility (i), we generalize recent work for the O(3) NLSM with an arbitrary theta term, where it was regulated using model of qubits with a small extra dimension, solving a sign problem present in conventional formulations of theta vacua. We argue that Grassmannian NLSM can be obtained similarly from SU(N) antiferromagnets with a well-defined continuum limit, reproducing both the IR physics of theta vacua and the UV physics of asymptotic freedom. These results enable the application of new classical algorithms to lattice Monte Carlo studies of these quantum field theories, and provide a viable realization suited for their quantum simulation. On the other hand, we show that, perhaps surprisingly, the conventional lattice regularization of theta vacua due to Berg and Luscher reproduces the anomaly exactly on the lattice, providing a realization of the second possibility.

## Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (II) Single-Baryon Beta-Decay in Real Time

A framework for quantum simulations of real-time weak decays of hadrons and nuclei in a 2-flavor lattice theory in one spatial dimension is presented. A single generation of the Standard Model is mapped to spin operators via the Jordan-Wigner transformation, and both quantum chromodynamics and flavor-changing weak interactions are included in the dynamics, the latter through four-Fermi effective operators. This mapping requires 16 qubits per spatial lattice site. Quantum circuits which implement time evolution in this lattice theory are developed and run on Quantinuum’s H1 1 20-qubit trapped ion system to simulate the beta-decay of a single baryon on one lattice site. Simulations of the real-time evolution of a single baryon, including initial state preparation, are performed for both one and two Trotter time steps. We comment on the potential intrinsic error-correction properties of this type of lattice theory.

## Branching States as The Emergent Structure of a Quantum Universe

Quantum Darwinism builds on decoherence theory to explain the emergence of classical behavior within a quantum universe. We demonstrate that the differential geometric underpinnings of quantum mechanics provide a uniquely informative window into the structure of correlations needed to validate quantum Darwinism. This leads us to two crucial insights about the emergence of classical phenomenology, centered around the nullity of quantum discord. First, we show that the so-called branching structure of the joint state of system and environment is the only one compatible with zero discord. Second, we prove that for small, but nonzero discord, the structure of the globally pure state is arbitrarily close to the branching form. These provide strong evidence that this class of branching states is the only one compatible with the emergence of classical phenomenology, as described in quantum Darwinism.

## Simulating Heisenberg interactions in the Ising model with strong drive fields

The time-evolution of an Ising model with large driving fields over discrete time intervals is shown to be reproduced by an effective XXZ-Heisenberg model at leading order in the inverse field strength. For specific orientations of the drive field, the dynamics of the XXX-Heisenberg model is reproduced. These approximate equivalences, valid above a critical driving field strength set by dynamical phase transitions in the Ising model, are expected to enable quantum devices that natively evolve qubits according to the Ising model to simulate more complex systems.

## Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (I) Axial Gauge

The resource requirements for quantum simulations of 1+1 dimensional quantum chromodynamics are estimated. When formulated in axial gauge and with two flavors of quarks, this system requires 12 qubits per spatial site once Gauss’s law has been enforced to uniquely constrain the gauge fields. Classical computations and D-Wave’s quantum annealer Advantage are used to determine the hadronic spectrum, enabling a decomposition of the masses and a study of quark entanglement. We identify color “edge” states, resulting from open boundary conditions, that are confined within a screening length to the end of the lattice. Quantum circuits for the time evolution of SU(Nc) gauge theory with Nf flavors of quarks are developed and used to determine the resources required for large-scale quantum simulations. IBM’s 7-qubit quantum computers ibmq_jakarta and ibm_perth were used to compute the trivial vacuum-to-vacuum and trivial vacuum-to-qr qr bar probabilities in Nf=1 QCD with one spatial site from one Trotter step of the time-evolution operator.

## Quantum Monte Carlo in Configuration Space with Three-Nucleon Forces

Neutron matter, through its connection to neutron stars as well as systems like cold atom gases, is one of the most interesting yet accessible systems in nuclear physics. The Configuration-Interaction Monte Carlo (CIMC) method is a stochastic many-body technique allowing to tackle strongly coupled systems. In contrast to other Quantum Monte Carlo methods employed in nuclear physics, the CIMC method can be formulated directly in momentum space allowing for an efficient use of non-local interactions.

In this work we extend CIMC method to include three-nucleon interactions through the normal-ordered two-body approximation. We present results for the equation of state of neutron matter in line with other many-body calculations with low resolution chiral interactions, and provide predictions for the momentum distribution and the static structure factor.

## Snowmass Computational Frontier: Topical Group Report on Quantum Computing

Quantum computing will play a pivotal role in the High Energy Physics (HEP) science program over the early parts of the Twenty First Century, both as a major expansion of our capabilities across the Computational Frontier, and in synthesis with quantum sensing and quantum networks. This report outlines how Quantum Information Science (QIS) and HEP are deeply intertwined endeavors that benefit enormously from a strong engagement together.

A number of problems ranging from simulation of quantum field theories, to fully leveraging the most sensitive sensor suites for new particle searches, and even data analysis will run into limiting bottlenecks if constrained to our current computing paradigms. Easy early access to quantum computers is needed to build a deeper understanding of these connections and the opportunities for advancement. In turn, HEP brings crucial expertise to the national quantum ecosystem in quantum domain knowledge, materials science, cryogenic and fast electronics, and massive-scale project management. The role of quantum technologies across the entire economy and society at large is expected to grow rapidly over the next decade, so it is important to establish the role of HEP in the efforts surrounding QIS. Fully delivering on the promise of quantum technologies and the goals of the HEP science program requires robust support. It is important to invest in the co-design opportunities afforded by the broader quantum computing ecosystem by leveraging HEP strengths in QIS for materials, electronics, and theory with the goal of designing quantum computers tailored to HEP science.

## Quantum Simulation for High-energy Physics

It is for the first time that quantum simulation for high-energy physics (HEP) is studied in the U.S. decadal particle-physics community planning, and in fact until recently, this was not considered a mainstream topic in the community. This fact speaks of a remarkable rate of growth of this subfield over the past few years, stimulated by the impressive advancements in quantum-information sciences (QIS) and associated technologies over the past decade, and the significant investment in this area by the government and private sectors in the U.S. and other countries. High-energy physicists have quickly identified problems of importance to our understanding of nature at the most fundamental level, from tiniest distances to cosmological extents, that are intractable with classical computers but may benefit from quantum advantage. They have initiated, and continue to carry out, a vigorous program in theory, algorithm, and hardware co-design for simulations of relevance to the HEP mission. This community whitepaper is an attempt to bring this exciting and yet challenging area of research to the spotlight, and to elaborate on what the promises, requirements, challenges, and potential solutions are over the next decade and beyond.

This whitepaper is prepared for the topical groups CompF6 (Quantum computing), TF05 (Lattice Gauge Theory), and TF10 (Quantum Information Science) within the Computational Frontier and Theory Frontier of the U.S. Community Study on the Future of Particle Physics (Snowmass 2021).

## From Asymptotic Freedom to Theta Vacua: Qubit Embeddings of the O(3) Nonlinear Sigma Model

Conventional lattice formulations of $\theta$ vacua in the $1+1$-dimensional $O(3)$ nonlinear sigma model suffer from a sign problem. Here, we construct the first sign-problem-free regularization for \emph{arbitrary} $\theta$. Using efficient lattice \ac{MC} computations, we demonstrate how a Hamiltonian model of spin-$\tfrac12$ degrees of freedom on a 2-dimensional spatial lattice reproduces both the infrared sector for arbitrary $\theta$, as well as the ultraviolet physics of asymptotic freedom. Furthermore, as a model of qubits on a two-dimensional square lattice with only nearest neighbor interactions, it is naturally suited for studying the physics of $\theta$ vacua and asymptotic freedom on near-term quantum devices. Our construction generalizes to $\theta$-vacua in all $\CP(N-1)$ models, solving a long standing sign problem.

## Some Conceptual Aspects of Operator Design for Quantum Simulations of Non-Abelian Lattice Gauge Theories

In the Kogut-Susskind formulation of lattice gauge theories, a set of quantum numbers resides at the ends of each link to characterize the vertex-local gauge field. We discuss the role of these quantum numbers in propagating correlations and generating entanglement that ensures each vertex remains gauge invariant, despite time evolution induced by operators with (only) partial access to each vertex Hilbert space. Applied to recent proposals for eliminating vertex-local Hilbert spaces in quantum simulation, we describe how the required entanglement is generated via delocalization of the time evolution operator with nearest-neighbor controls. These hybridizations, organized with qudits or qubits, exchange classical operator pre-processing for reductions in resource requirements that extend throughout the lattice volume.

*(Contribution to proceedings of the 2021 Quantum Simulation for Strong Interactions (QuaSi) Workshops at the InQubator for Quantum Simulation (IQuS))*

## Entanglement and correlations in fast collective neutrino flavor oscillations

Collective neutrino oscillations play a crucial role in transporting lepton flavor in astrophysical settings like supernovae and neutron star binary merger remnants, which are characterized by large neutrino densities. In these settings, simulations in the mean-field approximation show that neutrino-neutrino interactions can overtake vacuum oscillations and give rise to fast collective flavor evolution on time-scales t ~ μ^{-1}, with μ proportional to the local neutrino density. In this work, we study the full out-of-equilibrium flavor dynamics in simple multi-angle geometries displaying fast oscillations. Focusing on simple initial conditions, we analyze the production of pair correlations and entanglement in the complete many-body-dynamics as a function of the number N of neutrinos in the system.

Similarly to simpler geometries with only two neutrino beams, we identify three regimes: stable configurations with vanishing flavor oscillations, marginally unstable configurations with evolution occurring at long time scales τ~μ^{-1}√N and unstable configurations showing flavor evolution on short time scales τ~μ^{-1}log(N). We present evidence that these fast collective modes are generated by the same dynamical phase transition which leads to the slow bipolar oscillations, establishing a connection between these two phenomena and explaining the difference in their time scales.

We conclude by discussing a semi-classical approximation which reproduces the entanglement entropy at short to medium time scales and can be potentially useful in situations with more complicated geometries where classical simulation methods starts to become inefficient.

## Large-charge conformal dimensions at the O(N) Wilson-Fisher fixed point

Recent work using a large-charge effective field theory (EFT) for the O(N) Wilson-Fisher conformal field theory has shown that the anomalous dimensions of large-charge operators can be expressed in terms of a few low-energy constants (LECs) of the EFT. By performing lattice Monte Carlo computations at the O(N) Wilson-Fisher critical point, we compute the anomalous dimensions of large-charge operators up to charge Q=10, and extract the leading and subleading LECs of the O(N) large-charge EFT for N=2,4,6,8. To alleviate the signal-to-noise ratio problem present in the large-charge sector of conventional lattice formulations of the O(N) theory, we employ a recently developed qubit formulation of the O(N) nonlinear sigma models with a worm algorithm. This enables us to test the validity of the large-charge expansion, and the recent predictions of large-N expansion for the coefficients of the large-charge EFT.

## Qubit regularized O(N) nonlinear sigma models

Motivated by the prospect of quantum simulation of quantum field theories, we formulate the O(N) nonlinear sigma model as a “qubit” model with an (N+1)-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in 2+1 dimensions, where the continuum physics of the nontrivial O(N) Wilson-Fisher fixed point is reproduced. We compute the critical exponents nu and eta for the O(N) qubit models up to N=8, and find excellent agreement with known results in literature from various analytic and numerical techniques for the O(N) Wilson-Fisher universality class. Our models are suited for studying O(N) nonlinear sigma models on quantum computers up to N=8 in d=2,3 spatial dimensions.

## Basic Elements for Simulations of Standard Model Physics with Quantum Annealers: Multigrid and Clock States

We explore the potential of D-Wave’s quantum annealers for computing basic components required for quantum simulations of Standard Model physics. By implementing a basic multigrid (including zooming) and specializing Feynman-clock algorithms, D-Wave’s Advantage is used to study harmonic and anharmonic oscillators relevant for lattice scalar field theories and effective field theories, the time evolution of a single plaquette of SU(3) Yang-Mills lattice gauge field theory, and the dynamics of flavor entanglement in four neutrino systems.

## Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions

In this work we provide a new approach for approximating an ordered operator exponential using

an ordinary operator exponential that acts on the Hilbert space of the simulation as well as a finite-

dimensional clock register. We show that as the number of qubits used for the clock grows, the error in

the ordered operator exponential vanishes, as well as the entanglement between the clock register and

the register of the state being acted upon. Hence, the clock is a convenient device that allows us to

translate results for simulating time-independent systems to the time-dependent case. As an application,

we provide a new family of multiproduct formulas (MPFs) for time-dependent Hamiltonians that yield

both commutator scaling and poly-logarithmic error scaling. This in turn, means that this method

outperforms existing methods for simulating physically-local, time-dependent Hamiltonians. Finally, we

apply the formalism to show how qubitization can be generalized to the time-dependent case and show

that such a translation can be practical if the second derivative of the Hamiltonian is sufficiently small.

## Nematic Confined Phases in the U(1) Quantum Link Model on a Triangular Lattice: An Opportunity for Near-Term Quantum Computations of String Dynamics on a Chip

The U(1) quantum link model on the triangular lattice has two rotation-symmetry-breaking nematic confined phases. Static external charges are connected by confining strings consisting of individual strands with fractionalized electric flux. The two phases are separated by a weak first order phase transition with an emergent almost exact SO(2) symmetry. We construct a quantum circuit on a chip to facilitate near-term quantum computations of the non-trivial string dynamics.

## Preparation of the SU(3) Lattice Yang-Mills Vacuum with Variational Quantum Methods

Studying QCD and other gauge theories on quantum hardware requires the preparation of physically interesting states. The Variational Quantum Eigensolver (VQE) provides a way of performing vacuum state preparation on quantum hardware. In this work, VQE is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Bayesian optimization and gradient descent were investigated for performing the classical optimization. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using domain decomposition and a stitching procedure analogous to the Density Matrix Renormalization Group (DMRG). Small examples are performed on IBM’s superconducting Manila processor.

## Classical and Quantum Evolution in a Simple Coherent Neutrino Problem

The extraordinary neutrino flux produced in extreme astrophysical environments like the early universe, core-collapse supernovae and neutron star mergers may produce coherent quantum neutrino oscillations on macroscopic length scales. The Hamiltonian describing this evolution can be mapped into quantum spin models with all-to-all couplings arising from neutrino-neutrino forward scattering. To date many studies of these oscillations have been performed in a mean-field limit where the neutrinos time evolve in a product state.

We examine a simple two-beam model evolving from an initial product state and compare the mean-field and many-body evolution. The symmetries in this model allow us to solve the real-time evolution for the quantum many-body system for hundreds or thousands of spins, far beyond what would be possible in a more general case with an exponential number (2^{N}) of quantum states. We compare mean-field and many-body solutions for different initial product states and ratios of one- and two-body couplings, and find that in all cases in the limit of infinite spins the mean-field (product state) and many-body solutions coincide for simple observables. This agreement can be understood as a consequence of the spectrum of the Hamiltonian and the initial energy distribution of the product states. We explore quantum information measures like entanglement entropy and purity of the many-body solutions, finding intriguing relationships between the quantum information measures and the dynamical behavior of simple physical observables.

## Causality and dimensionality in geometric scattering

The scattering matrix which describes low-energy,

non-relativistic scattering of two species of spin-1/2 fermions

interacting via finite-range potentials can be obtained from a

geometric action principle in which space and time do not appear

explicitly arXiv:2011.01278. In the case of zero-range

forces, constraints imposed by

causality –requiring that the scattered wave not be emitted before

the particles have interacted– translate into non-trivial

geometric constraints on scattering trajectories in the geometric

picture. The dependence of scattering on the number of spatial dimensions

is also considered; by varying from three to two spatial dimensions, the

dependence on spatial dimensionality in the geometric picture is

found to be encoded in the phase of the harmonic potential that appears in

the geometric action.

## UV/IR symmetries of the S-matrix and RG flow

The low energy S-matrix which describes non-relativistic two-body

scattering arising from finite-range forces has UV/IR symmetries

that are hidden in the corresponding effective field theory (EFT) action.

It is shown that the S-matrix symmetries are manifest as geometric properties

of the RG flow of the coefficients of local operators in the EFT action.

## Symmetries of the nucleon-nucleon S-matrix and effective field theory expansions

The s-wave nucleon-nucleon (NN) scattering matrix

(S-matrix) exhibits UV/IR symmetries which are hidden in the

effective field theory (EFT) action and scattering amplitudes, and

which explain some interesting generic features of the phase

shifts. These symmetries offer clarifying interpretations of

existing pionless EFT expansions, and suggest starting points for

novel expansions. The leading-order (LO) S-matrix obtained in the

pionless EFT with scattering lengths treated exactly is shown to

have a UV/IR symmetry which leaves the sum of s-wave phase shifts

invariant. A new scheme, which treats effective range corrections

exactly, and which possesses a distinct UV/IR symmetry at LO, is

developed up to NLO (next-to-LO) and compared with data.

## Quantum Error Correction with Gauge Symmetries

Quantum simulations of Lattice Gauge Theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss’ law constraint to correct one-qubit errors for a Z_{2} or truncated U(1) LGT in 1+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss’ law, our circuits are fault tolerant and the overall error correction scheme outperforms a naive application of the [5,1,3] code. The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.

## Entanglement Structures in Quantum Field Theories: Negativity Cores and Bound Entanglement in the Vacuum

The many-body entanglement between two finite (size-d) disjoint vacuum regions of noninteracting lattice scalar field theory in one spatial dimension, i.e., a (dA×dB)mixed Gaussian continuous variable system, is locally transformed into a tensor-product *core* of (1A×1B)mixed entangled pairs. Accessible entanglement within these core pairs exhibits an exponential hierarchy and as such identifies the structure of dominant region modes from which vacuum entanglement could be extracted into a spatially separated pair of quantum detectors. Beyond the core, the remaining modes of the *halo* are determined to be AB separable in isolation, as well as separable from the core. However, state preparation protocols that distribute entanglement in the form of (1A×1B)mixed core pairs are found to require additional entanglement in the halo that is obscured by classical correlations. This inaccessible (bound) halo entanglement is found to mirror the accessible entanglement, but with a step behavior as the continuum is approached. It remains possible that alternate initialization protocols that do not utilize the exponential hierarchy of core-pair entanglement may require less inaccessible entanglement. Entanglement consolidation is expected to persist in higher dimensions and may aid classical and quantum simulations of asymptotically free gauge field theories, such as quantum chromodynamics.

## Nuclear two point correlation functions on a quantum computer

The calculation of dynamic response functions are expected to be an early application benefiting from rapidly developing quantum hardware resources. The ability to calculate real-time quantities of strongly-correlated quantum systems is one of the most exciting applications that can easily reach beyond the capabilities of traditional classical hardware. Response functions of fermionic systems at moderate momenta and energies corresponding roughly to the Fermi energy of the system are a potential early application because the relevant operators are nearly local and the energies can be resolved in moderately short real time, reducing the spatial resolution and gate depth required.

This is particularly the case in quasielastic electron and neutrino scattering from nuclei, a topic of great interest in the nuclear and particle physics communities and directly related to experiments designed to probe neutrino properties. In this work we use current hardware to calculate response functions for a highly simplified nuclear model through calculations of a 2-point real time correlation function for a Fermi-Hubbard model in two dimensions with three distinguishable nucleons on four lattice sites on current quantum hardware, and evaluate current error mitigation strategies.

## Spectral density reconstruction with Chebyshev polynomials

Accurate calculations of the spectral density in a strongly correlated quantum many body system are of fundamental importance to study many-particle dynamics in the linear response regime. Typical examples are the calculation of inclusive and semi-exclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter.

Integral transform techniques have played an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned.

In order to circumvent this problem, a quantum algorithm based on an appropriate expansion in Chebyshev polynomials was recently proposed. In the present work we build on this idea. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigours error estimates while allowing for efficient simulations on classical computers. We apply our idea to simple model response functions and comment on the applicability of the method to study realistic systems using scalable nuclear many-body methods.

## Hybridized Methods for Quantum Simulation in the Interaction Picture

Conventional methods of quantum simulation involve trade-offs that limit their applicability to specific contexts where their use is optimal. This paper demonstrates how different simulation methods can be hybridized to improve performance for interaction picture simulations over known algorithms. These approaches show asymptotic improvements over the individual methods that comprise them and further make interaction picture simulation methods practical in the near term. Physical applications of these hybridized methods yields a gate complexity scaling as log²Λ in the electric cutoff Λ for the Schwinger Model and independent of the electron density for collective neutrino oscillations, outperforming the scaling for all current algorithms with these parameters. For the general problem of Hamiltonian simulation subject to dynamical constraints, these methods yield a query complexity independent of the penalty parameter λ used to impose an energy cost on time-evolution into an unphysical subspace.

## Hierarchical Qubit Maps and Hierarchical Quantum Error Correction

We consider hierarchically implemented quantum error correction (HI-QEC) in which the fidelities of logical qubits are differentially optimized to enhance the capabilities of quantum devices in scientific applications. By employing qubit representations that propagate hierarchies in simulated systems to those in logical qubit noise sensitivities, heterogeneity in the distribution of physical qubits among logical qubits can be systematically structured. For concreteness, we estimate HI-QEC’s impact on surface code resources in computing low-energy observables to a fixed precision, finding up to ~60% reductions in qubit requirements possible in early error corrected simulations. This heterogeneous distribution of physical-to-logical qubits is identified as another element that can be optimized in the co-design process of quantum simulations of Standard Model physics.

## Entanglement minimization in hadronic scattering with pions

Recent work conjectured that

entanglement is minimized in low-energy hadronic scattering

processes. It was shown that the minimization of the entanglement

power (EP) of the low-energy baryon-baryon S-matrix implies novel

spin-flavor symmetries that are distinct from large-N_c QCD

predictions and are confirmed by high-precision lattice QCD

simulations. Here the conjecture of minimal entanglement is

investigated for scattering processes involving pions and

nucleons. The EP of the S-matrix is constructed for the pi-pi

and pi-N systems, and the consequences of minimization of

entanglement are discussed and compared with large-N_c QCD

expectations.

## Standard Model Physics and the Digital Quantum Revolution: Thoughts about the Interface

Remarkable advances in isolating, controlling and entangling quantum systems are transforming what was once a curious feature of quantum mechanics into a vehicle for disruptive scientific and technological progress. Pursuing the vision articulated by Feynman, a concerted effort across many areas of research and development is introducing prototypical digital quantum devices into the computing ecosystem available to domain scientists. Through interactions with these early quantum devices, the abstract vision of exploring classically-intractable quantum systems is evolving toward becoming a tangible reality. Beyond catalyzing these technological advances, entanglement is enabling parallel progress as a diagnostic for quantum correlations and as an organizational tool, both guiding improved understanding of quantum many-body systems and quantum field theories defining and emerging from the Standard Model. From the perspective of three domain science theorists, this article compiles “thoughts about the interface” on entanglement, complexity, and quantum simulation in an effort to contextualize recent NISQ-era progress with the scientific objectives of nuclear and high-energy physics.

## Quantum Machine Learning with SQUID

In this work we present the Scaled QUantum IDentifier (SQUID), an open-source framework for exploring hybrid Quantum-Classical algorithms for classification problems. The classical infrastructure is based on PyTorch and we provide a standardized design to implement a variety of quantum models with the capability of back-propagation for efficient training. We present the structure of our framework and provide examples of using SQUID in a standard binary classification problem from the popular MNIST dataset. In particular we highlight the implications for scalability for gradient based optimization of quantum models on the choice of output for variational quantum models.

## Entanglement Spheres and a UV-IR connection in Effective Field Theories

Disjoint regions of the latticized, massless scalar field vacuum become separable at large distances beyond the entanglement sphere, a distance that extends to infinity in the continuum limit. Through numerical calculations in one-, two- and three-dimensions, the radius of an entanglement sphere is found to be determined by the highest momentum mode of the field supported across the diameter, d, of two identical regions. As a result, the long-distance behavior of the entanglement is determined by the short-distance structure of the field. Effective eld theories (EFTs), describing a system up to a given momentum scale Lambda, are expected to share this feature, with regions of the EFT vacuum separable (or dependent on the UV-completion) beyond a distance proportional to Λ. The smallest non-zero value of the entanglement negativity supported by the field at large distances is conjectured to be N_{N}~exp(-Λ d), independent of the number of spatial dimensions. This phenomenon may be manifest in perturbative QCD processes.

## Dynamical Phase Transitions in models of Collective Neutrino Oscillations

Collective neutrino oscillations can potentially play an important role in transporting lepton flavor in astrophysical scenarios where the neutrino density is large, typical examples are the early universe and supernova explosions. It has been argued in the past that simple models of the neutrino Hamiltonian designed to describe forward scattering can support substantial flavor evolution on very short time scales t≈log(N)/(GFρ), with N the number of neutrinos, GF the Fermi constant and ρ the neutrino density. This finding is in tension with results for a similar but exactly solvable model for which t≈√N/(GFρ) instead. In this work we provide a coherent explanation of this tension in terms of Dynamical Phase Transitions (DPT) and study the possible impact that a DPT could have in more realistic models of neutrino oscillations and their mean-field approximation.

## Simulation of Collective Neutrino Oscillations on a Quantum Computer

In astrophysical scenarios with large neutrino density, like supernovae and the early universe, the presence of neutrino-neutrino interactions can give rise to collective flavor oscillations in the out-of-equilibrium collective dynamics of a neutrino cloud. The role of quantum correlations in these phenomena is not yet well understood, in large part due to complications in solving for the real-time evolution of the strongly coupled many-body system. Future fault-tolerant quantum computers hold the promise to overcome much of these limitations and provide direct access to the correlated neutrino dynamic. In this work, we present the first simulation of a small system of interacting neutrinos using current generation quantum devices. We introduce a strategy to overcome limitations in the natural connectivity of the qubits and use it to track the evolution of entanglement in real-time. The results show the critical importance of error-mitigation techniques to extract meaningful results for entanglement measures using noisy, near term, quantum devices.

## Entanglement and Many-Body Effects in Collective Neutrino Oscillations

Collective neutrino oscillations play a crucial role in transporting lepton flavor in astrophysical settings, such as supernovae, where the neutrino density is large. In this regime, neutrino-neutrino interactions are important and simulations in mean-field approximations show evidence for collective oscillations occurring at time scales much larger than those associated with vacuum oscillations. In this work, we study the out-of-equilibrium dynamics of a corresponding spin model using Matrix Product States and show how collective bipolar oscillations can be triggered by quantum fluctuations if appropriate initial conditions are present. The origin of these flavor oscillations, absent in the mean-field, can be traced to the presence of a dynamical phase transition, which drastically modifies the real-time evolution of the entanglement entropy. We find entanglement entropies scaling at most logarithmically in the system size, suggesting that classical tensor network methods could be efficient in describing collective neutrino dynamics more generally.

## A Trailhead for Quantum Simulation of SU(3) Yang-Mills Lattice Gauge Theory in the Local Multiplet Basis

Maintaining local interactions in the quantum simulation of gauge field theories relegates most states in the Hilbert space to be unphysical—theoretically benign, but experimentally difficult to avoid. Reformulations of the gauge fields can modify the ratio of physical to gauge-variant states often through classically preprocessing the Hilbert space and modifying the representation of the field on qubit degrees of freedom. This paper considers the implications of representing SU(3) Yang-Mills gauge theory on a lattice of irreducible representations in both a global basis of projected global quantum numbers and a local basis in which controlled-plaquette operators support efficient time evolution. Classically integrating over the internal gauge space at each vertex (e.g., color isospin and color hypercharge) significantly reduces both the qubit requirements and the dimensionality of the unphysical Hilbert space. Initiating tuning procedures that may inform future calculations at scale, the time evolution of one- and two-plaquettes are implemented on one of IBM’s superconducting quantum devices, and early benchmark quantities are identified. The potential advantages of qudit environments, with either constrained 2D hexagonal or 1D nearest-neighbor internal state connectivity, are discussed for future large-scale calculations.

Editors Suggestion in Physical Review D.

## Preparation of Excited States for Nuclear Dynamics on a Quantum Computer

We study two different methods to prepare excited states on a quantum computer, a key initial step to study dynamics within linear response theory. The first method uses unitary evolution for a short time*T*=O(√(1−*F))* to approximate the action of an excitation operator *O* with fidelity *F*and success probability *P*≈1−*F*. The second method probabilistically applies the excitation operator using the Linear Combination of Unitaries (LCU) algorithm. We benchmark these techniques on emulated and real quantum devices, using a toy model for thermal neutron-proton capture. Despite its larger memory footprint, the LCU-based method is efficient even on current generation noisy devices and can be implemented at a lower gate cost than a naive analysis would suggest. These findings show that quantum techniques designed to achieve good asymptotic scaling on fault tolerant quantum devices might also provide practical benefits on devices with limited connectivity and gate fidelity.

## Geometry and entanglement in the scattering matrix

A formulation of nucleon-nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix — that appears at leading-order in the EFT description — possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.

## An Algorithm for Quantum Computation of Particle Decays

A quantum algorithm is developed to calculate decay rates and cross sections using quantum resources that scale polynomially in the system size assuming similar scaling for state preparation and time evolution. This is done by computing finite-volume one- and two-particle Green’s functions on the quantum hardware. Particle decay rates and two particle scattering cross sections are extracted from the imaginary parts of the Green’s function. A 0 + 1 dimensional implementation of this method is demonstrated on IBM’s superconducting quantum hardware for the decay of a heavy scalar particle to a pair of light scalars.

## Spectral Density Estimation with the Gaussian Integral Transform

The spectral density operator *ρ*(*ω*)=*δ*(*ω*−*H*) plays a central role in linear response theory as it’s expectation value, the dynamical response function, can be used to compute scattering cross-sections. In this work, we describe a near optimal quantum algorithm providing an approximation to the spectral density with energy resolution Δ and error *ϵ* using O(√(log(1/ϵ)(log(1/Δ)+log(1/ϵ)))/Δ) operations. This is achieved without using expensive approximations to the time-evolution operator but exploiting instead qubitization to implement an approximate Gaussian Integral Transform (GIT) of the spectral density. We also describe appropriate error metrics to assess the quality of spectral function approximations more generally.

## Geometric Quantum Information Structure in Quantum Fields and their Lattice Simulation

An upper limit to distillable entanglement between two disconnected regions of massless noninteracting scalar field theory has an exponential decay defined by a geometric decay constant. When regulated at short distances with a spatial lattice, this entanglement abruptly vanishes beyond a dimensionless separation, defining a negativity sphere. In two spatial dimensions, we determine this geometric decay constant between a pair of disks and the growth of the negativity sphere toward the continuum through a series of lattice calculations. Making the connection to quantum field theories in three-spatial dimensions, assuming such quantum information scales appear also in quantum chromodynamics (QCD), a new relative scale may be present in effective field theories describing the low-energy dynamics of nucleons and nuclei. We highlight potential impacts of the distillable entanglement structure on effective field theories, lattice QCD calculations and future quantum simulations.

## Quantum Algorithms for Simulating the Lattice Schwinger Model

The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on *N*/2 physical sites with coupling constant *x*^{-1/2} and electric field cutoff *x*^{-1/2}Λ can be simulated on a quantum computer for time 2*xT* using a number of *T*-gates or CNOTs in *Õ*(*N*^{3/2} *T*^{3/2} *x*^{1/2} Λ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable—the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.

## Entanglement Rearrangement in Self-Consistent Nuclear Structure Calculations

Entanglement properties of 4He and 6He are investigated using nuclear many-body calculations, specifically the single-nucleon entanglement entropy, and the two-nucleon mutual information and negativity. Nuclear wavefunctions are obtained by performing active-space no-core configuration interaction calculations using a two-body nucleon-nucleon interaction derived from chiral effective field theory. Entanglement measures within single-particle bases, the harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT) and variational natural (VNAT) bases, are found to exhibit different degrees of complexity. Entanglement in both nuclei is found to be more localized within NAT and VNAT bases than within a HO basis for the optimal HO parameters, and, as anticipated, a core-valence (tensor product) structure emerges from the full six-body calculation of 6He. The two-nucleon mutual information shows that the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active and inactive spaces. We conclude that measures of one- and two-nucleon entanglement are useful in analyzing the structure of nuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developing more efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-body systems more generally.

## Loop, String, and Hadron Dynamics in SU(2) Hamiltonian Lattice Gauge Theories

The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory—a loop-string-hadron (LSH) formulation—that describes dynamics directly in terms of its loop, string, and hadron degrees of freedom, while alleviating several disadvantages of quantum-simulating the Kogut-Susskind formulation. This LSH formulation transcends the local loop formulation of d+1-dimensional lattice gauge theories by incorporating staggered quarks, furnishing the algebra of gauge-singlet operators, and being used to reconstruct dynamics between states that have Gauss’s law built in to them. LSH operators are then factored into products of “normalized” ladder operators and diagonal matrices, priming them for classical or quantum information processing. Self-contained expressions of the Hamiltonian are given up to d=3. The LSH formalism uses makes little use of structures specific to SU(2) and its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).

## Systematically Localizable Operators for Quantum Simulations of Quantum Field Theories

Correlations and measures of entanglement in ground state wavefunctions of relativistic quantum field theories are spatially localized over length scales set by the mass of the lightest particle. We utilize this localization to design digital quantum circuits for preparing the ground states of free lattice scalar quantum field theories. Controlled rotations that are exponentially localized in their position-space extent are found to provide exponentially convergent wavefunction fidelity. These angles scale with the classical two-point correlation function, as opposed to the more localized mutual information or the hyper-localized negativity. We anticipate that further investigations will uncover quantum circuit designs with controlled rotations dictated by the measures of entanglement. This work is expected to impact quantum simulations of systems of importance to nuclear physics, high-energy physics, and basic energy sciences research.

## Fixed-Point Quantum Circuits for Quantum Field Theories

Renormalization group ideas and effective operators are used to efficiently determine localized unitaries for preparing the ground states of non-interacting scalar field theories on digital quantum devices. With these methods, classically computed ground states in a small spatial volume can be used to determine operators for preparing the ground state in a beyond-classical quantum register, even for interacting scalar field theories. Due to the exponential decay of correlation functions and the double exponential suppression of digitization artifacts, the derived quantum circuits are expected to be relevant already for near-term quantum devices.

## Quantum Computer Systems for Scientific Discovery

The great promise of quantum computers comes with the dual challenges of building them and finding their useful applications. We argue that these two challenges should be considered together, by co-designing full-stack quantum computer systems along with their applications in order to hasten their development and potential for scientific discovery. In this context, we identify scientific and community needs, opportunities, a sampling of a few use case studies, and significant challenges for the development of quantum computers for science over the next 2–10 years. This document is written by a community of university, national laboratory, and industrial researchers in the field of Quantum Information Science and Technology, and is based on a summary from a U.S. National Science Foundation workshop on Quantum Computing held on October 21–22, 2019 in Alexandria, VA.

## Quantum Computing for Neutrino-nucleus Scattering

Neutrino-nucleus cross section uncertainties are expected to be a dominant systematic in future accelerator neutrino experiments. The cross sections are determined by the linear response of the nucleus to the weak interactions of the neutrino, and are dominated by energy and distance scales of the order of the separation between nucleons in the nucleus. These response functions are potentially an important early physics application of quantum computers. Here we present an analysis of the resources required and their expected scaling for scattering cross section calculations. We also examine simple small-scale neutrino-nucleus models on modern quantum hardware. In this paper, we use variational methods to obtain the ground state of a three nucleon system (the triton) and then implement the relevant time evolution. In order to tame the errors in present-day NISQ devices, we explore the use of different error-mitigation techniques to increase the fidelity of the calculations.

## SU(2) Non-Abelian Gauge Field Theory in One Dimension on Digital Quantum Computers

An improved mapping of one-dimensional SU(2) non-Abelian gauge theory onto qubit degrees of freedom is presented. This new mapping allows for a reduced unphysical Hilbert space. Insensitivity to interactions within this unphysical space is exploited to design more efficient quantum circuits. Local gauge symmetry is used to analytically incorporate the angular momentum alignment, leading to qubit registers encoding the total angular momentum on each link. Results of multi-plaquette calculations on IBM’s quantum hardware are presented.

## Short-depth circuits for efficient expectation value estimation

The evaluation of expectation values *Tr*[*ρO*] for some pure state *ρ* and Hermitian operator *O* is of central importance in a variety of quantum algorithms. Near optimal techniques developed in the past require a number of measurements *N* approaching the Heisenberg limit *N*=O(1/*ϵ*) as a function of target accuracy *ϵ*. The use of Quantum Phase Estimation requires however long circuit depths *C*=O(1/*ϵ*) making their implementation difficult on near term noisy devices. The more direct strategy of Operator Averaging is usually preferred as it can be performed using *N*=O(1/*ϵ*^{2}) measurements and no additional gates besides those needed for the state preparation. In this work we use a simple but realistic model to describe the bound state of a neutron and a proton (the deuteron) and show that the latter strategy can require an overly large number of measurement in order to achieve a reasonably small relative target accuracy *ϵr*. We propose to overcome this problem using a single step of QPE and classical post-processing. This approach leads to a circuit depth *C*=O(*ϵ*^{m}) (with *m*≥0) and to a number of measurements *N*=O(1/*ϵ*^{2+n}) for 0<*n*≤1. We provide detailed descriptions of two implementations of our strategy for *n*=1 and *n*≈0.5 and derive appropriate conditions that a particular problem instance has to satisfy in order for our method to provide an advantage.

## Minimally-Entangled State Preparation of Localized Wavefunctions on Quantum Computers

Initializing a single site of a lattice scalar field theory into an arbitrary state requires O(2^{nQ}) entangling gates on a quantum computer with n_{Q }qubits per site. It is conceivable that, instead, initializing to functions that are good approximations to states may have utility in reducing the number of required entangling gates. In the case of a single site of a non-interacting scalar field theory, initializing to a symmetric exponential wavefunction requires n_{Q }− 1 entangling gates, compared with the 2^{nQ−1 }+ n_{Q }− 3 + δ_{nQ,1 }required for a symmetric Gaussian wavefunction. In this work, we explore the initialization of 1-site (n_{Q }= 4), 2-site (n_{Q }= 3) and 3-site (n_{Q }= 3) non- interacting scalar field theories with symmetric exponential wavefunctions using IBM’s quantum simulators and quantum devices (Poughkeepsie and Tokyo). With the digitizations obtainable with n_{Q }= 3, 4, these tensor-product wavefunctions are found to have large overlap with a Gaussian wavefunction, and provide a suitable low-noise initialization for subsequent quantum simulations. In performing these simulations, we have employed a workflow that interleaves calibrations to mitigate systematic errors in production. The calibrations allow tolerance cuts on gate performance including the fidelity of the symmetrizing Hadamard gate, both in vacuum (|0⟩^{⊗nQ }) and in medium (n_{Q }− 1 qubits initialized to an exponential function). The results obtained in this work are relevant to systems beyond scalar field theories, such as the deuteron radial wavefunction, 2- and 3-dimensional cartesian-space wavefunctions, and non-relativistic multi-nucleon systems built on a localized eigenbasis.

## Solving Gauss’s Law on Digital Quantum Computers with Loop-String-Hadron Digitization

We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (arXiv:1912.06133) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss’s law) exactly. We first discuss the structure of the LSH Hilbert space in *d* spatial dimensions, its truncation, and its digitization with qubits. Error detection and mitigation in gauge theory simulations would benefit from physicality “oracles,” so we decompose circuits that flag gauge invariant wavefunctions. We then analyze the logical qubit costs and entangling gate counts involved with the protocols. The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how that is digitized as well as the spatial dimension. The numerous other clear benefits encourage future studies into applying this framework.

## Entanglement Suppression and Emergent Symmetries of Strong Interactions

Entanglement suppression in the strong interaction S-matrix is shown to be correlated with approximate spin-flavor symmetries that are observed in low-energy baryon interactions, the Wigner SU(4) symmetry for two flavors and an SU(16) symmetry for three flavors. We conjecture that dynamical entanglement suppression is a property of the strong interactions in the infrared, giving rise to these emergent symmetries and providing powerful constraints on the nature of nuclear and hypernuclear forces in dense matter.

## Oracles for Gauss’s Law on Digital Quantum Computers

Formulating a lattice gauge theory using only physical degrees of freedom generically leads to non-local interactions. A local Hamiltonian is desirable for quantum simulation, and this is possible by treating the Hilbert space as a subspace of a much larger Hilbert space in which Gauss’s law is not automatic. Digital quantum simulations of this local formulation will wander into unphysical sectors due to errors from Trotterization or from quantum noise. In this work, oracles are constructed that use local Gauss law constraints to projectively distinguish physical and unphysical wave functions in Abelian lattice gauge theories. Such oracles can be used to detect errors that break Gauss’s law.

## Simulations of Subatomic Many-Body Physics on a Quantum Frequency Processor

Simulating complex many-body quantum phenomena is a major scientific impetus behind the development of quantum computing, and a range of technologies are being explored to address such systems. We present the results of the largest photonics-based simulation to date, applied in the context of subatomic physics. Using an all-optical quantum frequency processor, the ground-state energies of light nuclei including the triton (3H), 3He, and the alpha particle (4He) are computed. Complementing these calculations and utilizing a 68-dimensional Hilbert space, our photonic simulator is used to perform sub-nucleon calculations of the two-body and three-body forces between heavy mesons in the Schwinger model. This work is a first step in simulating subatomic many-body physics on quantum frequency processors—augmenting classical computations that bridge scales from quarks to nuclei.[Image (left) was created by Pavel Lougovski.]

## Digitization of Scalar Fields for NISQ-Era Quantum Computing

Qubit, operator and gate resources required for the digitization of lattice λϕ4 scalar field theories onto quantum computers in the NISQ era are considered, building upon the foundational work by Jordan, Lee and Preskill. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spetzouris, Amundson and Harnik building on the work of Somma, provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan, Lee and Preskill, and eigenstates of a harmonic oscillator, to describe 0+1- and 1+1-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in Lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in λϕ4, but are found to be inferior to a complete digitization-improvement of the Hamiltonian using a Quantum Fourier Transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as |log|log|ϵdig|||∼nQ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as |log|ϵlatt||∼NQ (total number of qubits in the simulation). We find that fewer than nQ=10 qubits per spatial lattice site are sufficient to reduce digitization errors below noise levels expected in NISQ-era quantum devices for both localized and delocalized field-space wavefunctions. For localized wavefunctions, nQ=4 qubits are likely to be sufficient for calculations requiring modest precision.

## Gauss’s Law, Duality, and the Hamiltonian Formulation of U(1) Lattice Gauge Theory

Quantum computers have the potential to explore the vast Hilbert space of entangled states that play an important role in the behavior of strongly interacting matter. This opportunity motivates reconsidering the Hamiltonian formulation of gauge theories, with a suitable truncation scheme to render the Hilbert space finite-dimensional. Conventional formulations lead to a Hilbert space largely spanned by unphysical states; given the current inability to perform large scale quantum computations, we examine here how one might restrict wave function evolution entirely or mostly to the physical subspace. We consider such constructions for the simplest of these theories containing dynamical gauge bosons—U(1) lattice gauge theory without matter in d=2,3 spatial dimensions—and find that electric-magnetic duality naturally plays an important role. We conclude that this approach is likely to significantly reduce computational overhead in d=2 by a reduction of variables and by allowing one to regulate magnetic fluctuations instead of electric. The former advantage does not exist in d=3, but the latter might be important for asymptotically-free gauge theories.

## Linear Response on a Quantum Computer

The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly-interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei. We present a general algorithm for quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.

## Quantum-classical computation of Schwinger model dynamics using quantum computers

We present a quantum-classical algorithm to study the dynamics of the two-spatial-site Schwinger model on IBM’s quantum computers. Using rotational symmetries, total charge, and parity, the number of qubits needed to perform computation is reduced by a factor of ∼5, removing exponentially-large unphysical sectors from the Hilbert space. Our work opens an avenue for exploration of other lattice quantum field theories, such as quantum chromodynamics, where classical computation is used to find symmetry sectors in which the quantum computer evaluates the dynamics of quantum fluctuations.

## Ground States via Spectral Combing on a Quantum Computer

A new method is proposed for determining the ground state wave function of a quantum many-body system on a quantum computer, without requiring an initial trial wave function that has good overlap with the true ground state. The technique of Spectral Combing involves entangling an arbitrary initial wave function with a set of auxiliary qubits governed by a time dependent Hamiltonian, resonantly transferring energy out of the initial state through a plethora of avoided level crossings into the auxiliary system. The number of avoided level crossings grows exponentially with the number of qubits required to represent the Hamiltonian, so that the efficiency of the algorithm does not rely on any particular energy gap being large. We give an explicit construction of the quantum gates required for the realization of this procedure and explore the results of classical simulations of the algorithm on a small quantum computer with up to 8 qubits. We show that for certain systems and comparable results, Spectral Combing requires fewer quantum gates to realize than the Quantum Adiabatic Algorithm.