Abstract

Quantum computers can efficiently simulate many-body systems. As a widely used Hamiltonian simulation tool, the Trotter-Suzuki scheme splits the evolution into the number of Trotter steps $N$ and approximates the evolution of each step by a product of exponentials of each individual term of the total Hamiltonian. The algorithmic error due to the approximation can be reduced by increasing $N$, which however requires a longer circuit and hence inevitably introduces more physical errors. In this work, we first study such a trade-off and numerically find the optimal number of Trotter steps $N_{\textrm{opt}}$ given a physical error model in a near-term quantum hardware. Practically, physical errors can be suppressed using recently proposed error mitigation methods. We then extend physical error mitigation methods to suppress the algorithmic error in Hamiltonian simulation. By exploiting the simulation results with different numbers of Trotter steps $N\le N_{\textrm{opt}}$, we can infer the exact simulation result within a higher accuracy and hence mitigate algorithmic errors. We numerically test our scheme with a five qubit system and show significant improvements in the simulation accuracy by applying both physical and algorithmic error mitigations.

Highlights

  • It is hard to simulate quantum systems using a classical computer, as the computational cost increases exponentially with the system size

  • We extend physical error mitigation methods to suppress the algorithmic error in Hamiltonian simulation

  • Such a problem can be resolved by quantum simulation, as proposed by Feynman in 1982 [1], saying “let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws.”

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Summary

INTRODUCTION

It is hard to simulate quantum systems using a classical computer, as the computational cost increases exponentially with the system size. The latest methods [6,7,9] have been significantly improved, the simulation accuracy is still limited by finite resources, such as short circuit depth, finite system runtime, and large physical errors in the system. To study such a limitation, we focus on the the Trotterization method [10], introduced for quantum simulation by Lloyd [2]. [12], in Sec. II we first find the optimal number of Trotter steps Nopt for error-prone quantum simulations. We numerically show the optimal number of Trotter steps Nopt under an inhomogeneous Pauli error model and apply both physical and algorithmic error mitigation methods to significantly increase the simulation accuracy. VI, we conclude our work and discuss its possible extension in general quantum information processing

THE OPTIMAL NUMBER OF TROTTER STEPS FOR NOISY QUANTUM SIMULATION
ERROR MITIGATION AND EXTRAPOLATION TECHNIQUE
Richardson extrapolation
Exponential extrapolation
ERROR MITIGATION FOR ALGORITHMIC ERRORS
NUMERICAL SIMULATION
Inhomogeneous Pauli error after each gate No physical error
DISCUSSION
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