Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates (2201.01471v3)

Abstract: Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, a common strategy is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate sampling of the observable's expectation value. We consider several partitioning schemes based on grouping of commuting multi-qubit Pauli products with the goal of minimizing the number of measurements. Three main directions are explored: 1) grouping commuting operators using the greedy approach, 2) involving non-local unitary transformations for measuring, and 3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two new measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.

Deterministic Improvements in Quantum Measurements for Variational Quantum Algorithms

The paper "Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates," by Tzu-Ching Yen and colleagues, investigates strategies to optimize the measurement process in variational quantum algorithms (VQAs). The inefficiencies in measuring observables, especially complex molecular Hamiltonians, present significant challenges in executing VQAs effectively. This paper makes significant strides in tackling these challenges by proposing advanced measurement partitioning techniques.

Key Contributions

The authors explore three primary strategies to enhance measurement efficiency:

1. Grouping of Commuting Operators: They employ a greedy heuristic to group commuting multi-qubit Pauli products, aiming to minimize the number of measurement fragments. The greedy approach, by optimizing the grouping based on fragment variances, outperforms traditional graph-coloring techniques, which minimize the number of fragments.
2. Non-Local Unitary Transformations: The paper innovatively uses non-local unitary (entangling) transformations for measuring groups of fully commuting Pauli operators. This approach involves selective entangling operations from the Clifford group, which, despite the additional circuit depth required, provides flexibility in reducing the total number of measurements by more effectively minimizing variances across measurement sets.
3. Covariance-Aware Grouping: The authors introduce a method whereby certain Pauli products are measured as part of multiple compatible groups, a concept they term as overlapping grouping. This strategy leverages the non-transitivity of operator commutativity to further reduce measurement redundancy. By employing classical computations to approximate operator covariances or utilizing empirical estimates from VQAs, the paper proposes two optimization techniques: iterative coefficient splitting (ICS) and iterative measurement allocation (IMA).

Numerical and Theoretical Insights

The numerical evaluations presented showcase a dramatic reduction in the required measurements. For example, employing fully commuting group strategies often resulted in variances up to ten times lower than traditional methods. Furthermore, the greedy-based overlap grouping methods (IMA and ICS) consistently demonstrated superior performance against both non-overlapping measurement strategies and contemporary shadow tomography techniques.

Implications and Future Directions

These deterministic improvements provide a robust, scalable solution to the measurement efficiency problem, paving the way for more practical and effective utilization of VQAs, especially in quantum chemistry. As prioritized by the authors, non-local transformations based on Clifford circuits offer a viable path for future advancements, potentially complementing error mitigation strategies in noisy intermediate-scale quantum (NISQ) devices.

Further work might delve into optimizing measurement combinations explicitly through insights gained from constrained optimization, given the considerable number of variables introduced in overlapping groupings. Another promising direction could involve adaptive iteration methodologies, where empirical data from initial VQA runs refine future measurement allocations dynamically.

In conclusion, this research enriches the quantum computing field with measurable improvements not only pertinent to variational techniques but potentially applicable across quantum algorithms where measuring complex observables is a bottleneck.