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

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 estimation 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.

Fidelity overhead for non-local measurements in variational quantum algorithms

“…Among various heuristics for grouping commuting Pauli products 7,10,16 we used the Sorted Insertion (SI) method 3 because it generally outperforms other grouping techniques in the number of measurements. 4,5 Since measurement circuits for Ûi of the FC and QWC groups can be constructed using only Clifford group transformations, according to the Gottesman-Knill theorem these circuits can be implemented efficiently on both classical and quantum computers. 17,18 For the FC grouping an asymptotically optimal scaling of O(N 2 q / log(N q )) in the number of 2qubit entangling gates and single qubit rotations can be achieved.…”

“…Recent comparisons of state-of-theart techniques in the two categories shown that one can achieve a lower number of the measurements using techniques within the qubit operator algebra. [4][5][6] These qubit techniques are based on partitioning of the Hamiltonian to sets of commuting Pauli operators. [6][7][8][9][10] Due to differences in accuracy of performing one-and two-qubit transformations in quantum computers, it is convenient to distinguish two types of commutativity, general or full commutativity (FC) and a more restrictive qubit-wise commutativity (QWC).…”

“…[11][12][13][14][15] Using the electronic structure Hamiltonians as examples of quantum observables, it was shown that for a single observable, CST schemes are generally inferior than grouping techniques in lowering the number of measurements required to estimate the expectation value. 5,6 Therefore, here we consider the grouping methods to measurements but our analysis involving Clifford transformations can be extended to the CST schemes as well.…”

Hardware-Tailored Diagonalization Circuits

“…As of today, the best method to experimentally estimate an expectation value O is unclear. Besides Pauli grouping [4][5][6][7][8][9][10][11][12][13], there is active research in addressing this problem with classical shadows [14][15][16][17], unitary partitioning [18,19], low-rank factorization [20], adaptive estimators [21][22][23], and decision diagrams [24].…”

“…(see SM [29,IV]). In this letter, we focus on grouped Pauli measurements (GPMs) which are more efficient than individual Pauli measurements (IPMs) [4][5][6][7][8][9][10][11][12][13]. To quantify this advantage, Crawford et al [9] introduce a figure of merit, called R, which approximates the difficult to compute quantity R = N shots IPM /N shots GPM , where N shots IPM is the number of shots that IPMs would require to estimate O to a desired accuracy and similarly for N shots GPM .…”