Contrasting Statistical Phase Estimation with the Variational Quantum Eigensolver in the era of Early Fault Tolerant Quantum Computation

TL;DR

This paper compares statistical phase estimation and VQE for ground state energy estimation in early fault-tolerant quantum computing, highlighting conditions where SPE outperforms VQE under low error rates.

Contribution

It introduces the Gaussian Fitting algorithm and systematically compares SPE and VQE using noisy simulations within the STAR architecture.

Findings

SPE is more advantageous than VQE at low physical error rates.

The Gaussian Fitting algorithm offers an alternative post-processing method.

Simulations focus on 4-qubit H2 Hamiltonian in the STO-3G basis.

Abstract

In this review, we give an overview of the proposed applications in the early-FTQC (EFTQC) era. Starting from the error correction architecture for EFTQC device, we first review the recently developed space-time efficient analogue rotation (STAR) architecture \cite{akahoshiPartiallyFaultTolerantQuantum2024}, which is a partially fault-tolerant error correction architecture. Then, we review the requirements of an EFTQC algorithm. In particular, the class of ground state energy estimation (GSEE) algorithm known as the statistical phase estimation algorithm (SPE) is studied. We especially cast our attention on two SPE-type algorithms, the step-function filter-based variant by Lin and Tong (LT22) \cite{Lin:2021rwb} and Gaussian Filter \cite{Wang:2022gxu}. Based on the latter, we introduce the Gaussian Fitting algorithm, which uses an alternative post-processing procedure compared to \cite{Wang:2022gxu}. Finally, we systematically simulate the aforementioned algorithms and Variational Quantum Eigensolver (VQE) using the 1-uCJ ansatz with different shot counts. Most importantly, we perform noisy simulations based on the STAR architecture. We find that for estimating the ground state energy of the 4-qubit $H_2$ Hamiltonian in the STO-3G basis, SPE becomes more advantageous over VQE when the physical error rate is sufficiently low.