Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

TL;DR

This work provides an improved, rigorous analysis of Richardson extrapolation and polynomial interpolation for the task of calculating time-evolved expectation values, and provides a more accurate characterisation of the algorithmic error mitigation techniques currently proposed to reduce Trotter error.

Abstract

Product formulae are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by use of these formulae. This work provides an improved, rigorous analysis of these techniques for the task of calculating time-evolved expectation values. We demonstrate that, to achieve error $ε$ in a simulation of time $T$ using a $p^\text{th}$-order product formula with extrapolation, circuits depths of $O\left(T^{1+1/p} \textrm{polylog}(1/ε)\right)$ are sufficient -- an exponential improvement in the precision over product formulae alone. Furthermore, we achieve commutator scaling, improve the complexity with $T$, and do not require fractional implementations of Trotter steps. Our results provide a more accurate characterisation of the algorithmic error mitigation techniques currently proposed to reduce Trotter error.

Figures (4)

• Figure 1: Schematic of the polynomial interpolation procedure. The dotted purple line is the true value of the Trotter-evolved expectation value, $\langle \tilde{O}(T,s)\rangle$, and the blue line is the interpolating polynomial $P_{m-1} f(s)$ for $m=8$. The red points are estimates of the time-evolved expectation value obtained via product formula evolution with measurement, which are then used to construct the polynomial. The final estimator of the expectation value is given by $P_{m-1} f(0)$.
• Figure 2: Flowchart indicating the incoherent (Method 1) and coherent (Method 2) schemes for performing extrapolation of the time-evolved expectation values using product formulae, where the specified time and final error are $T$ and $\epsilon$ respectively. For each time-step size, we need to make a measurement of the observable's expectation to precision $\epsilon$. While the incoherent scheme is simpler and requires shorter circuit depths, the coherent scheme achieves a quadratic speedup from using quantum amplitude estimation. Regardless of the measurement protocol, the algorithm concludes with the same classical calculation of the acquired data.
• Figure 3: Left: Error comparison between the observable measured on the time-evolved state using Trotterisation vs. the extrapolated error for different maximum numbers of Trotter steps on a system of $6$ qubits. The initial number of steps corresponds to $\sim (\Lambda T)^{3/2}$. We see the plot levels out at the bottom due to floating point precision. Right: The same as left, but where the extrapolation procedures are performed with a measurement error of $10^{-6}$ for each measurement.
• Figure 4: A comparison in the performance of Richardson extrapolation as a function of the Richardson extrapolation degree for different simulation times.

Theorems & Definitions (31)

• Theorem 1: Trotter Extrapolation Resource Counts (Informal)
• Lemma 2: Effective Hamiltonian Error Series
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5: Well-conditioned Richardson Extrapolation low2019well
• proof
• Lemma 6: Richardson Extrapolation Error