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Physical Review A

Optimization of variational-quantum-eigensolver measurement by partitioning Pauli operators using multiqubit Clifford gates on noisy intermediate-scale quantum hardware

Andrew Jena, Scott N. Genin, Michele Mosca

Published Oct 27, 2022• 12 citations
Publisher

Abstract

Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of Pauli operators allows for fewer measurements and an overall speedup of the measurement process. Using results from coloring random graphs, we give evidence for an upper bound on the expected minimum number of simultaneously measurable parts necessary to partition a set of Pauli operators. By expanding our gate set to allow multiqubit Clifford gates before measurement, we were able to cut down the number of measurements considerably compared to previous approaches. We conjecture that this improvement is, in general, linear with respect to the lengths of the Pauli operators. We conclude by demonstrating the feasibility of this approach by implementing the variational quantum eigensolver on the Aspen-4-4Q-A quantum processor unit. In doing so, we calculate the ground-state energies of ${\mathrm{H}}_{2}$ using fewer measurements than is possible with single-qubit gates with no statistically significant difference in accuracy.

DOI
10.1103/physreva.106.042443

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Physical Review Applied•Sep 20, 2023•7 citations
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of Pauli operators allows for fewer measurements and an overall speedup of the measurement process. Using results from coloring random graphs, we give evidence for an upper bound on the expected minimum number of simultaneously measurable parts necessary to partition a set of Pauli operators. By expanding our gate set to allow multiqubit Clifford gates before measurement, we were able to cut down the number of measurements considerably compared to previous approaches. We conjecture that this improvement is, in general, linear with respect to the lengths of the Pauli operators. We conclude by demonstrating the feasibility of this approach by implementing the variational quantum eigensolver on the Aspen-4-4Q-A quantum processor unit. In doing so, we calculate the ground-state energies of H2{\mathrm{H}}_{2}H2​ using fewer measurements than is possible with single-qubit gates with no statistically significant difference in accuracy.
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-qubit operations, leads to a decrease in the number of parts which is linear with respect to the lengths of the operators. We give evidence to confirm this conjecture and comment on the importance of this result for a specific near-term application: speeding up the measurement process of the variational quantum eigensolver.
Measuring the expectation value of Pauli operators on prepared quantum states is a \nfundamental task in the variational quantum eigensolver. Simultaneously measuring sets \nof operators allows for fewer measurements and an overall speedup of the measurement \nprocess. In this thesis, we look both at the task of partitioning all Pauli operators of a \n xed length and of partitioning a random subset of these Pauli operators. We rst show \nhow Singer cycles can be used to optimally partition the set of all Pauli operators, giving \nsome insight to the structure underlying many constructions of mutually unbiased bases. \nThereafter, we show how graph coloring algorithms promise to provide speedups linear \nwith respect to the lengths of the operators over currently-implemented techniques in the \nmeasurement step of the variational quantum eigensolver.
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule, this increases the uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: 1) grouping terms into parts whose eigenstates have a single-qubit product structure, and 2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in several-fold reduction of separately measured terms are illustrated on the H2 and LiH problems.
Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as N4N^4N4 for typical problems of interest--a daunting growth rate that poses a serious limitation for emerging applications such as quantum computational chemistry. We introduce a systematic technique for minimizing requisite state preparations by exploiting the simultaneous measurability of partitions of commuting Pauli strings. Our work encompasses algorithms for efficiently approximating a MIN-COMMUTING-PARTITION, as well as a synthesis tool for compiling simultaneous measurement circuits. For representative problems, we achieve 8-30x reductions in state preparations, with minimal overhead in measurement circuit cost. We demonstrate experimental validation of our techniques by estimating the ground state energy of deuteron on an IBM Q 20-qubit machine. We also investigate the underlying statistics of simultaneous measurement and devise an adaptive strategy for mitigating harmful covariance terms.
Variational algorithms have received significant attention in recent years due to their potential to solve practical problems using noisy intermediate-scale quantum (NISQ) devices. A fundamental step of these algorithms is the evaluation of the expected value of Hamiltonians, and hence efficient schemes to perform this task are required. The standard approach employs local measurements of Pauli operators and requires a large number of circuits. An alternative is to make use of entangled measurements, which might introduce additional gates between physically disconnected qubits that harm the performance. As a solution to this problem, we propose hardware-efficient entangled measurements (HEEM), that is, measurements that permit only entanglement between physically connected qubits. We show that this strategy enhances the evaluation of molecular Hamiltonians in NISQ devices by reducing the number of circuits required without increasing their depth. We provide quantitative metrics of how this approach offers better results than local measurements and arbitrarily entangled measurements. We estimate the ground-state energy of the H2_22​O molecule with classical simulators and quantum hardware using the variational quantum eigensolver with HEEM.
Variational algorithms have received significant attention in recent years due to their potential to solve practical problems using noisy intermediate-scale quantum (NISQ) devices. A fundamental step of these algorithms is the evaluation of the expected value of Hamiltonians, and hence efficient schemes to perform this task are required. The standard approach employs local measurements of Pauli operators and requires a large number of circuits. An alternative is to use entangled measurements, which might introduce additional gates between physically disconnected qubits that harm the performance. To solve this problem, we propose hardware-efficient entangled measurements (HEEMs), that is, measurements that permit only entanglement between physically connected qubits. We show that this strategy enhances the evaluation of molecular Hamiltonians in NISQ devices by reducing the required circuits without increasing their depth. We provide quantitative metrics of how this approach offers better results than local measurements and arbitrarily entangled measurements. We estimate the ground-state energy of the H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecule with classical simulators and quantum hardware using the variational quantum eigensolver with HEEM.