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arXiv:2305.19568

Digital quantum simulator for the time-dependent Dirac equation using discrete-time quantum walks

Shigetora Miyashita, Takahiko Satoh, Michihiko Sugawara, Naphan Benchasattabuse, Ken M. Nakanishi, Michal Hajdušek, Hyensoo Choi, Rodney Van Meter

Published May 31, 2023• 0 citations
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Abstract

We introduce a quantum algorithm for simulating the time-dependent Dirac equation in 3+1 dimensions using discrete-time quantum walks. Thus far, promising quantum algorithms have been proposed to simulate quantum dynamics in non-relativistic regimes efficiently. However, only some studies have attempted to simulate relativistic dynamics due to its theoretical and computational difficulty. By leveraging the convergence of discrete-time quantum walks to the Dirac equation, we develop a quantum spectral method that approximates smooth solutions with exponential convergence. This mitigates errors in implementing potential functions and reduces the overall gate complexity that depends on errors. We demonstrate that our approach does not require additional operations compared to the asymptotic gate complexity of non-relativistic real-space algorithms. Our findings indicate that simulating relativistic dynamics is achievable with quantum computers and can provide insights into relativistic quantum physics and chemistry.

DOI
10.48550/arxiv.2305.19568

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We introduce a quantum algorithm for simulating the time-dependent Dirac equation in 3+1 dimensions using discrete-time quantum walks. Thus far, promising quantum algorithms have been proposed to simulate quantum dynamics in non-relativistic regimes efficiently. However, only some studies have attempted to simulate relativistic dynamics due to its theoretical and computational difficulty. By leveraging the convergence of discrete-time quantum walks to the Dirac equation, we develop a quantum spectral method that approximates smooth solutions with exponential convergence. This mitigates errors in implementing potential functions and reduces the overall gate complexity that depends on errors. We demonstrate that our approach does not require additional operations compared to the asymptotic gate complexity of non-relativistic real-space algorithms. Our findings indicate that simulating relativistic dynamics is achievable with quantum computers and can provide insights into relativistic quantum physics and chemistry.
The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann–Wigner equations and symmetric hyperbolic systems in general. We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation. We do so by adapting a powerful method from standard numerical analysis, which is of general interest to the field of quantum simulation. At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme. At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator-splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speed-up over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
In this thesis a study of a special type of quantum cellular automata called Dirac quantum walks (DQWs) is presented. At the continuous limit, these DQWs coincide with the Dirac equation. A new systematic method to build DQWs, with particular focus on non-square or cube shaped grids, is constructed and is put to test on several examples of increasing difficulty. Ultimately, a quantum walk simulating the Dirac equation on a spatial polar grid is constructed with a discrete analogue to the conservation of angular momentum. Cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated.
Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d -dimensional Schrödinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log⁡(g′/ϵ))) , where ϵ is the discretization error, g′ controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g′ from poly(g′/ϵ) to poly(log⁡(g′/ϵ)) and polynomially improves the dependence on T and d , while maintaining best known performance with respect to η . For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log⁡(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log⁡(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.
In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined as states smoothly peaked in some wave-vector eigenstate of the walk—is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.