Efficient measurement schemes for the Monte Carlo projective quantum eigensolver

The Monte Carlo Projective Quantum Eigensolver (MC-PQE) is a hybrid quantum-classical algorithm designed to estimate ground-state energies by using a quantum Monte Carlo-inspired stochastic process. While MC-PQE is more measurement-efficient than the standard Variational Quantum Eigensolver (VQE), it still faces significant measurement overhead due to the need to evaluate numerous asymmetric expectation values. This paper investigates whether techniques developed for VQE—specifically Hamiltonian grouping and classical shadow tomography—can be adapted to MC-PQE to further reduce this overhead.

The authors evaluate two primary Hamiltonian grouping strategies: Qubit-wise Commutativity (QWC) and Sorted Insertion (SI), which groups Pauli terms into fully commuting sets. They also propose several measurement allocation procedures (Pure Variance, Shift-weighted, Even-variance, and Even-determinant) to distribute quantum shots optimally across the various Hamiltonian and overlap terms required by the MC-PQE algorithm.

The study finds that Sorted Insertion (SI) significantly outperforms Qubit-wise Commutativity (QWC) in reducing the number of required Hamiltonian groups. When combined with tailored measurement allocation, this leads to a 5-10x reduction in the standard error of the energy estimators for the same total number of quantum measurements. The authors observe that the instantaneous Monte Carlo wavefunction provides an excellent source for estimating variances, allowing for dynamic and effective shot allocation without requiring expensive external heuristics.

Regarding classical shadow tomography, the authors find that while it offers a path to reduce circuit complexity, it currently underperforms compared to direct Hamiltonian measurement for the systems studied. The requirement to resample the Clifford group at each step of the MC-PQE iteration to maintain an unbiased estimator makes the shadow approach computationally expensive in the current simulation framework.

Reducing the measurement overhead is critical for the practical application of hybrid quantum algorithms in chemistry. By showing that standard VQE measurement optimization techniques can be successfully adapted to the projective framework of MC-PQE, this work provides a clear pathway to more efficient ground-state energy estimations. The results suggest that for current and near-term quantum devices, direct Hamiltonian grouping remains the most reliable and efficient strategy for MC-PQE.