Efficient measurement schemes for the Monte Carlo projective quantum eigensolver

This paper introduces and analyzes efficient measurement schemes for the Monte Carlo Projective Quantum Eigensolver (MC-PQE), presenting a method that significantly reduces measurement cost compared to conventional variational quantum algorithms like VQE. The research focuses on adapting existing techniques, such as operator term grouping and classical shadow tomography, to the asymmetric expectation values required by MC-PQE, demonstrating substantial reductions in standard error for molecular systems up to 12 qubits.

Background Theory of Measurement Techniques

The paper reviews two primary measurement techniques used to reduce sampling overhead in hybrid quantum algorithms: operator term grouping and classical shadow tomography. Hamiltonian grouping aims to measure contributions from multiple Pauli operators simultaneously by forming groups of mutually commuting operators. Two approaches are detailed:

1. Qubit-wise Commutativity (QWC): This is the simplest technique, relying on the fact that Pauli matrices commute with themselves and with the identity operator on each qubit. The paper notes that QWC can yield one large collection of diagonal terms for typical Hamiltonians, resulting in O(n) collections for an O(n4) Hamiltonian.

2. Sorted Insertion (SI): This approach improves upon QWC by considering whether the overall Pauli strings commute and grouping strings with even numbers of anticommuting single-qubit Paulis together. The paper states that the optimal number of collections required to cover all Pauli strings is 2n + 1.

Classical shadow tomography estimates multiple observables simultaneously by applying a unitary operator from an ensemble and storing the resulting quantities. The expectation value map is expressed as a linear map of the initial density matrix, and if this map can be inverted, it allows for the reconstruction of the original density matrix or operator expectation values. The paper notes that for O(n4) terms in a molecular Hamiltonian, this approach would require only log(n) shots to obtain a fixed additive error in all operators considered.

Monte Carlo Projective Quantum Eigensolver (MC-PQE) Dynamics

The MC-PQE algorithm is an iterative form derived from the imaginary time evolution equation, where the wavefunction is parametrized by a unitary ansatz, such as the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz: the wavefunction is given by ΨUCCSD⟩ = e τΦ0⟩. The evolution involves updating walkers on each parameter based on measurements of the residual term:

1. Quantum computer measures the residual: ri(β) = ⟨Φi∆β(Hˆ − S)Ψ(β)⟩ = ∆β(Hi − Sci).

2. Populations are updated iteratively: Ni(β + ∆β) = Ni(β) − N0(β)ri.

3. The projected energy is computed by the estimator: Eproj(β) = ⟨Φ0Hˆ Ψ(β)⟩ ⟨Φ0Ψ(β) = H0 c0.

4. The shift S is updated to maintain a constant population, which functions as an independent estimator for the ground state energy.

Variance Propagation and Measurement Allocation

The variance of the estimators is complex in MC-PQE because the final estimators are not simple averages of measured quantities. The variance of the projected energy estimator is given by: σ squared E¯proj = σ squared H¯0 H¯ 2 0 + σ squared c¯0 c¯ 2 0. The variance of the shift estimator is more intricate, depending on the noise in population ratios. To reduce this noise, measurement allocation schemes are proposed:

1. Pure Variance (PV): Allocates measurements based on individual variances: mhiG = M/p Var(hiG) / P G p Var(hiG) + p Var(ci).

2. Shift-weighted (SW): Allocates measurements to minimize the variance of the shift: mhiG = M/p Var(hiG) / P G p Var(hiG) + S/p Var(ci).

3. Even-variance (EV): Allocates measurements evenly between different Hi and ci, then by variance within each Hi.

Performance Analysis and Optimal Schemes

The paper evaluates these schemes for various molecular systems, finding that full commuting groups generated by sorted insertion reduce measurement overhead significantly compared to QWC. Furthermore, the analysis shows that distributions that take into account the magnitude of the shift out-perform those that do not, with the ED-SW approach providing a good compromise for both S and Eproj for all systems considered. The study concludes that "reducing the variances of both the overlap and Hamiltonian terms is crucial to reduce the noise in the MC-PQE estimators and therefore the measurement overhead.

Based on the provided scientific paper, here are specific improvements for AI systems and what those improved systems can achieve:

) 1. Implement Measurement Allocation Schemes in Quantum Monte Carlo Projective Eigensolver (MC-PQE) Residuals:

The AI system should be updated to dynamically select between measurement allocation strategies (Pure Variance (PV), Shift-weighted (SW), Even-variance (EV), or Even-determinant (ED)) for the quantum measurements required to estimate the residuals of the MC-PQE algorithm.

2. Improved Hamiltonian Grouping Strategy:

The system should transition from simple Qubit-wise Commutativity (QWC) grouping to a more sophisticated method like Sorted Insertion (SI) for Hamiltonian term grouping when preparing the measurement schedule, as SI has been shown to reduce the number of required Pauli groups by up to 10 times compared to QWC.

3. Enhanced Variance Estimation for Energy Estimators:

The system must utilize the instantaneous Monte Carlo wavefunction and its associated variance (instead of relying solely on the initial Hartree–Fock wavefunction) when computing measurement allocation parameters. This allows for a more accurate estimation of the noise in both the projected energy estimator and, critically, the shift estimator.

4. Application of Shift-Weighted Measurement Allocation:

The AI system should prioritize using a Shift-weighted (SW) allocation strategy when estimating the ground state energy shift, as this method has been shown to outperform others in minimizing variance for both energy and shift estimators across various molecular systems tested.

5. Dynamic Recomputation of Measurement Allocation:

The system should be designed to recompute the optimal measurement allocation scheme during the steady-state phase of MC-PQE (once the shift begins varying). This dynamic adjustment is crucial because it allows the measurement strategy to adapt to the evolving noise profile as the simulation approaches convergence, leading to significantly reduced noise in instantaneous estimators.

6. Integration of Advanced Observables via Partial Shadow Tomography:

For complex molecular systems, the AI system can leverage partial shadow tomography techniques (using Clifford unitaries sampled from a pool) to estimate residuals directly, potentially bypassing the need for high-order excitation amplitudes (like quadruple excitations) required by standard UCCSD residual calculations.

These improvements will result in an AI system capable of performing highly accurate and efficient quantum simulations:

1. A quantum chemistry simulation engine that can calculate molecular ground state energies with significantly lower measurement overhead compared to standard VQE or PQE methods.

2. The ability to achieve sub-milliHartree standard errors for energy estimations on systems up to 12 qubits, particularly for challenging molecules like those in the LiH, BeH2, and H2O series studied.

3. A more robust and reliable ground state energy estimator that is less sensitive to the initial choice of wavefunction (e.g., using FCI or correlated wavefunctions instead of just HF) due to improved variance handling during measurement allocation.

4. An optimized computational workflow that automatically selects the most efficient measurement strategy (like ED-SW) for the specific molecular system being simulated, ensuring the best possible trade-off between circuit complexity and final accuracy.