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arXiv:2006.16248 · PRX Quantum

Faster Digital Quantum Simulation by Symmetry Protection

Minh C. Tran, Yuan Su, Daniel Carney, Jacob M. Taylor

Published Feb 12, 2021• 95 citations
arXivHTMLPDF

Abstract

Simulating the dynamics of quantum systems is an important application of quantum computers and has seen a variety of implementations on current hardware.We show that by introducing quantum gates implementing unitary transformations generated by the symmetries of the system, one can induce destructive interference between the errors from different steps of the simulation, effectively giving faster quantum simulation by symmetry protection.We derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection.In particular, when the symmetry transformations are chosen as powers of a unitary, the error of the algorithm is approximately projected to the so-called quantum Zeno subspaces.We prove a bound on this approximation error, exponentially improving a recent result of Burgarth, Facchi, Gramegna, and Pascazio.We apply the symmetry-protection technique to the simulations of the XXZ Heisenberg interactions with local disorder and the Schwinger model in quantum field theory.For both systems, the technique can reduce the simulation error by several orders of magnitude over the unprotected simulation.Finally, we provide numerical evidence suggesting that the technique can also protect simulation against other types of coherent, temporally correlated errors, such as the 1/f noise commonly found in solid-state experiments.

DOI
10.1103/prxquantum.2.010323

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Simulating the dynamics of quantum systems is an important application of quantum computers and has seen a variety of implementations on current hardware.We show that by introducing quantum gates implementing unitary transformations generated by the symmetries of the system, one can induce destructive interference between the errors from different steps of the simulation, effectively giving faster quantum simulation by symmetry protection.We derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection.In particular, when the symmetry transformations are chosen as powers of a unitary, the error of the algorithm is approximately projected to the so-called quantum Zeno subspaces.We prove a bound on this approximation error, exponentially improving a recent result of Burgarth, Facchi, Gramegna, and Pascazio.We apply the symmetry-protection technique to the simulations of the XXZ Heisenberg interactions with local disorder and the Schwinger model in quantum field theory.For both systems, the technique can reduce the simulation error by several orders of magnitude over the unprotected simulation.Finally, we provide numerical evidence suggesting that the technique can also protect simulation against other types of coherent, temporally correlated errors, such as the 1/f noise commonly found in solid-state experiments.
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Quantum-simulator hardware promises new insights into problems from particle and nuclear physics.A major challenge is to reproduce gauge invariance, as violations of this quintessential property of lattice gauge theories can have dramatic consequences, e.g., the generation of a photon mass in quantum electrodynamics.Here, we introduce an experimentally friendly method to protect gauge invariance in U(1) lattice gauge theories against coherent errors in a controllable way.Our method employs only single-body energy-penalty terms, thus enabling practical implementations.As we derive analytically, some sets of penalty coefficients render undesired gauge sectors inaccessible by unitary dynamics for exponentially long times.Further, for few-body error terms, we show numerically that this is achieved with resources exhibiting little dependence on system size.These findings constitute an exponential improvement over previously known results from energy-gap protection or perturbative treatments.In our method, the gaugeinvariant subspace is protected by an emergent global symmetry, meaning it can be immediately applied to other symmetries.In our numerical benchmarks for continuous-time and digital quantum simulations, gauge protection holds for all calculated evolution times (up to t > 10 10 /J for continuous time, with J the relevant energy scale).Crucially, our gauge-protection technique is simpler to realize than the associated ideal gauge theory, and can thus be readily implemented in current ultracold-atom analog simulators as well as digital noisy intermediate-scale quantum devices.
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Symmetry-protected subspaces in quantum systems are subspaces demarcated by a conserved quantity of that quantum system. We introduce two classical algorithms, which efficiently compute and elucidate features of these subspaces. The first algorithm explores the entire symmetry-protected subspace of an initial state in time complexity linear to the size of the subspace by closing local basis state-to-basis state transitions. The second algorithm determines, with bounded error, if a given measurement outcome of a dynamically-generated state is within the symmetry-protected subspace of the state in which the dynamical system is initialized. This software is paired with the paper of the same name at https://arxiv.org/pdf/2302.08586.pdf.