Mitigating algorithmic errors in a Hamiltonian simulation

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 $N_{\mathrm{opt}}$ 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 $N\le N_{\mathrm{opt}}$, 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.

Figure 1

Schematic of the five-qubit Hamiltonian.

Figure 2

The optimal number of Trotter steps $N$ and the continuity of the measurement as a function of the inverse of the number of Trotter steps ${\varepsilon }_N=1/N$. (a) The trace distance between the ideal state and the simulated state in the presence of noise. The blue curve denotes the trace distance without physical error, which goes to zero with an infinite large number of Trotter steps. The black curve denotes the trace distance with the inhomogeneous Pauli error after each gate and the optimal number of Trotter steps is 25. (b) The expectation value of the observable versus ${\varepsilon }_N=1/N$. Here $N$ is the number of Trotter steps. The horizontal axis corresponds to the number of Trotter steps 10–200. From the simulation result, we can confirm the continuity of the function.

Figure 3

The error ${\delta }^2$ against the total runtime resource $M$ for different cases with no algorithmic error mitigation, linear algorithmic error extrapolation, and three-point algorithmic error extrapolation. We consider physical errors to the circuit and set the total evolution time as $t=0.5$. The number of Trotter steps $N=25$, 20, and 15; $N=25$ and 15; and $N=25$ are used for the three-point, linear extrapolation, and no error mitigation cases, respectively. The blue line denotes the no error mitigation case, the black line denotes the linear algorithmic error extrapolation case, and the red line denotes the three-point algorithmic error extrapolation case. (a) No physical error mitigation is applied. (b) Linear extrapolation is applied to suppress physical errors. (c) Exponential extrapolation is applied to suppress physical errors. Here, we set the ratio $r$ of the two error rates to be 2 for both linear and exponential extrapolation.