Causality and dimensionality in geometric scattering

Silas Beane, Roland Farrell |

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

Silas Beane, Roland Farrell |

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

Silas Beane, Roland Farrell |

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

Abhishek Rajput, Alessandro Roggero, Nathan Wiebe |

Quantum simulations of Lattice Gauge Theories (LGT) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. While this has the significant advantage of simplifying the implementation of the time-evolution operator, algorithmic approximations or errors on a quantum device can break gauge invariance by allowing unwanted transitions into the unphysical sector. Recent works have proposed to detect errors that break gauge invariance by means of quantum oracles that check Gauss’s Law. These approaches however are not able to correct errors and are not fault-tolerant. We present in this work a simple fault-tolerant procedure that combines Gauss’s Law with bit and phase flip error correction codes to detect and correct phase and bit-flip errors for a Z2 or truncated U(1) LGT in 1+1 dimensions with a link flux cutoff of 1. For a pure gauge theory on 2N links with periodic boundary conditions, we reduce the space requirement for error correction by exploiting gauge invariance and reach a physical-to-logical qubit ratio of 4.5, below what is possible using the perfect 5-qubit code on each logical qubit alone. The construction also applies to the case where static charges are present and can even accommodate dynamical fermions using a simple extension of our results. The constructions outlined can stimulate further developments in fault-tolerant error correction procedures for LGT systems with larger cutoffs, higher space-time dimensions, and possibly different symmetry groups.

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

Natalie Klco, Douglas Beck, Martin Savage |

We present numerical evidence for the presence of bound entanglement, in addition to distillable entanglement, between disjoint regions of the one-dimensional non-interacting scalar field vacuum. To reveal the entanglement structure, we construct a local unitary operation that transforms the high-body entanglement of latticized field regions into a tensor-product core of mixed (1 x 1) pairs exhibiting an exponential negativity hierarchy and a separable halo with non-zero entanglement. This separability-obscured entanglement (SOE) is driven by non-simultaneous separability within the full mixed state, rendering unentangled descriptions of the halo incompatible with a classically connected core. We quantify the halo SOE and find it to mirror the full negativity as a function of region separation, and conjecture that SOE provides a physical framework encompassing bound entanglement. Similar entanglement structures are expected to persist in higher dimension and in more complex theories relevant to high-energy and nuclear physics, and may provide a language for describing the dynamics of information in transitioning from quarks and gluons to hadrons.

Nuclear two point correlation functions on a quantum computer

Alessandro Baroni, Joseph Carlson, Rajan Gupta, Andy Li, Gabriel Perdue, Alessandro Roggero | arXiv:2111.02982

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

Alessandro Roggero, Joanna Sobczyk | arXiv:2110.02108

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

Abhishek Rajput, Alessandro Roggero, Nathan Wiebe | arXiv:2109.03308

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

Natalie Klco, Martin Savage | arxiv:2109.01953 [quant-ph]

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

Silas Beane, Roland Farrell | arxiv:2108.00646 [hep-ph]

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