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arXiv:1811.03629 · Physical Review A

Digitizing gauge fields: Lattice Monte Carlo results for future quantum computers

Daniel C. Hackett, Kiel Howe, Ciaran Hughes, William I. Jay, Ethan T. Neil, James N. Simone

Published Jun 28, 2019• 61 citations
arXivHTMLPDF

Abstract

In the near-future noisy intermediate-scale quantum era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we use ensembles of SU(2) lattice gauge fields generated with standard classical computing to explore the effects of digitizing elements of the gauge group to a finite set. We consider several methods for digitizing the group, finding the best performance from an action-preserving projection onto a mesh. Working in $(3+1)$ dimensions, we find that using $\ensuremath{\sim}7$ (qu)bits to represent each $\mathrm{SU}(2)$ gauge link induces a digitization error of the order of $10%$ in short-distance observables and $2%$ in long-distance observables. Promisingly, our results indicate that each $\mathrm{SU}(2)$ gauge link can be represented by $O(10)$ (qu)bits, from which we estimate that a ${16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)$ lattice could be simulated with no more than $O({10}^{5})$ (qu)bits. Our results on digitization are also of interest as a form of lossy compression that could be used in high-performance classical computing to alleviate communications bottlenecks.

DOI
10.1103/physreva.99.062341

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In the near-future noisy intermediate-scale quantum era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we use ensembles of SU(2) lattice gauge fields generated with standard classical computing to explore the effects of digitizing elements of the gauge group to a finite set. We consider several methods for digitizing the group, finding the best performance from an action-preserving projection onto a mesh. Working in (3+1)(3+1)(3+1) dimensions, we find that using \ensuremath∼7\ensuremath{\sim}7\ensuremath∼7 (qu)bits to represent each SU(2)\mathrm{SU}(2)SU(2) gauge link induces a digitization error of the order of 1010%10 in short-distance observables and 22%2 in long-distance observables. Promisingly, our results indicate that each SU(2)\mathrm{SU}(2)SU(2) gauge link can be represented by O(10)O(10)O(10) (qu)bits, from which we estimate that a 163SU(2){16}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{SU}(2)163SU(2) lattice could be simulated with no more than O(105)O({10}^{5})O(105) (qu)bits. Our results on digitization are also of interest as a form of lossy compression that could be used in high-performance classical computing to alleviate communications bottlenecks.
Simulations of gauge theories on quantum computers require the digitization of continuous field variables. Digitization schemes that use the minimum amount of qubits are desirable. We present a practical scheme for digitizing SU(3)SU(3)SU(3) gauge theories via its discrete subgroup S(1080)S(1080)S(1080). The S(1080)S(1080)S(1080) standard Wilson action cannot be used since a phase transition occurs as the coupling is decreased, well before the scaling regime. We propose a modified action that allows simulations in the scaling window and carry out classical Monte Carlo calculations down to lattice spacings of order $a\ensuremath{\approx}0.08\text{ }\text{ }\mathrm{fm}$. We compute a set of observables with subpercent precision at multiple lattice spacings and show that the continuum extrapolated value agrees with the full SU(3)SU(3)SU(3) results. This suggests that this digitization scheme provides sufficient precision for noisy intermediate-scale quantum era QCD simulations.
With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.
With the long term perspective of using quantum computers and tensor networks for lattice gauge theory simulations, an efficient method of digitizing gauge group elements is needed. We thus present our results for a handful of discretization approaches for the non-trivial example of SU(2), such as its finite subgroups, as well as different classes of finite subsets. We focus our attention on a freezing transition observed towards weak couplings. A generalized version of the Fibonacci spiral appears to be particularly efficient and close to optimal.
Qubit, operator, and gate resources required for the digitization of lattice $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan et al. [Quantum Inf. Comput. 14, 1014 (2014); Science 336, 1130 (2012)], with a focus towards noisy intermediate-scale quantum devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin et al. [Phys. Rev. A 98, 042312 (2018)] building on the work of Somma [Quantum Inf. Comput. 16, 1125 (2016)], provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan et al., and eigenstates of a harmonic oscillator, to describe (0+10+10+1)- and (d+1d+1d+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 $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$, 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 $|log|log|{\ensuremath{\epsilon}}_{\mathrm{dig}}|||\ensuremath{\sim}{n}_{Q}$ (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 $|log|{\ensuremath{\epsilon}}_{\mathrm{latt}}||\ensuremath{\sim}{N}_{Q}$ (total number of qubits in the simulation). For localized (delocalized) field-space wave functions, it is found that ${n}_{Q}\ensuremath{\sim}4(7)$ qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices. Only classical computing resources have been used to obtain the results presented in this work.
This research paper provides a comprehensive review of quantum simulation methods for lattice gauge theories during the current era of noisy intermediate-scale quantum computing. It focuses on the fundamental framework used to understand strong interactions in high-energy physics and explores how quantum devices can overcome the limitations of traditional computational methods, such as the sign problem found in classical simulations. The manuscript is built around three primary pillars that are critical for near-term progress. The first pillar involves encoding strategies, describing the movement away from simple mapping techniques toward more advanced, compact methods that significantly decrease the number of required quantum bits by focusing only on physically relevant states and exploiting local gauge constraints. The second pillar analyzes specific variational quantum algorithms that use flexible circuit structures to find low-energy states, comparing different approaches based on their circuit depth and suitability for current hardware. The third pillar evaluates various error mitigation techniques used to extract accurate information from imperfect quantum hardware, such as methods that amplify and then remove noise or those that use physical symmetries to detect and discard errors. Additionally, the paper compares different hardware technologies—including those based on superconducting circuits, trapped atoms, and ions—to weigh their respective benefits, such as gate speed and connectivity, against their drawbacks. It also introduces emerging hybrid approaches that combine quantum circuits with artificial neural networks to improve simulation power and state representation. Ultimately, the review identifies pathways toward achieving practical advantages in the field and discusses the challenges that remain for scaling these systems to larger sizes.
Matter under extreme conditions behaves as a non-equilibrium, strongly-interacting quantum system. Such systems are in the non-perturbative regime of quantum chromodynamics (QCD), where their time evolution is classically intractable. Quantum computing offers a promising alternative, but present-day devices are limited in qubit count and circuit depth, so efficient formulations and algorithms are key to scaling toward experimentally relevant QCD simulations. A scalable framework is established for simulating energy loss and hadronization in 1+1-dimensional SU(2) lattice gauge theory, in which a heavy quark traverses a lattice of light quarks. Classical lattice QCD methods are translated into physics-informed circuits to prepare the interacting ground state. The evolution is run on IBM's quantum processor ibm_pittsburgh, where the charge distribution and energy loss are measured. A further study develops the quantum simulation of the fully gauge-fixed SU(2) Hamiltonian in a mixed basis, a formulation that is expected to remain efficient at all gauge couplings. Digitizing the continuous gauge fields with as few as three qubits per plaquette reproduces low-energy observables with 10−310^{-3}10−3 precision. Scaling to larger systems is limited by resource overhead, which motivates truncations that preserve the relevant observables. Sequency Hierarchy Truncation (SeqHT) exploits the hierarchy in basis-operator contributions. Demonstrated on the adiabatic preparation of the λφ4λφ^4λφ4 ground state on ibm_sherbrooke, SeqHT reduces circuit depth by ~30%, suppressing hardware noise in the extracted observables. All simulations in this dissertation are validated on quantum hardware against classical benchmarks. These studies advance quantum simulations of real-time QCD dynamics.