Statistical Quantum Phase Estimation: Extensions and Practical Considerations

TL;DR

This paper enhances the statistical quantum phase estimation framework by introducing practical improvements such as handling negative Pauli weights, using changepoint detection, and exploiting symmetry to reduce computational costs, making it more feasible for near-term quantum devices.

Contribution

The authors generalize the random compilation for negative weights, apply changepoint detection for GSE, and utilize symmetry to optimize the SQPE method for realistic quantum computing scenarios.

Findings

Reduced number of circuit runs by a factor of 2 using symmetry.

Generalized random compilation to include negative Pauli weights.

Numerical validation via quantum simulator in Qiskit.

Abstract

We present several refinements and extensions of the statistical quantum phase estimation (SQPE) framework to address some of its key practical limitations, improving its applicability to realistic cases. Recently, a family of statistical approaches for QPE have been proposed where each run uses only a few ancillae and shorter circuits than standard QPE and thus is better suited for early fault-tolerant quantum computers that are qubit-and depth-limited. SQPE method within that family estimates the cumulative distribution function (CDF) associated with spectral density of the Hamiltonian for a given trial state by using its Fourier approximation and then identifies the first jump discontinuity of the CDF to determine the ground state energy (GSE) of the Hamiltonian. It relies on random compilation procedure based on linear combination of unitaries (LCU) decomposition of the Hamiltonian assuming positive Pauli weights and requires a good estimate of lower bound on the overlap between the trial and true ground state, both of which may be difficult to achieve in practice. We address these limitations by generalizing the random compilation procedure for negative Pauli weights and employing a changepoint detection method for determining GSE which does not rely on an estimate of this overlap. We also show that by exploiting symmetry of the Fourier series one can reduce number of circuit runs/samples by a factor of 2x while keeping the GSE estimation accuracy the same. We illustrate these new developments numerically via a quantum simulator in Qiskit.