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arXiv:2105.13091 · Quantum

Overlapped grouping measurement: A unified framework for measuring quantum states

Bujiao Wu, Jinzhao Sun, Qi Jun Huang, Xiao Yuan

Published Jan 13, 2023• 54 citations
arXivHTMLPDF

Abstract

Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as importance sampling, observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme – overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.

DOI
10.22331/q-2023-01-13-896

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Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as importance sampling, observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme – overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.
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Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.
The measurement cost for estimating expectation values of Hamiltonians is a central bottleneck in variational quantum algorithms. Grouping strategies significantly reduce this cost, with overlapping techniques being the state of the art in the field. Overlapping grouping methods require i) a non-overlapping grouping of the Hamiltonian, typically obtained from the Sorted Insertion (SI) algorithm as initialization, and ii) the construction of covariance dictionaries from approximate wavefunctions to guide the optimization. It was recently shown that different initializations can potentially reduce measurement costs for overlapping methods. Motivated by these findings, we introduce variance-aware SI (VarSI), a family of covariance-informed non-overlapping Pauli grouping heuristics to reduce measurement counts. VarSI grouping leverages the covariance dictionaries, already required by overlapping methods, to construct better non-overlapping groups. We propose three variants: a global greedy grouping insertion rule, a variance-informed SI analog, and a local refinement step initialized from SI or our variance-informed variant. We showcase the use of groupings generated by our VarSI heuristic algorithms to initialize overlapping methods using the iterative coefficient-splitting (ICS) algorithm. Molecular benchmarks with 130 Hamiltonians demonstrate consistent, non-overlapping measurement improvements over SI of 38\% and enhanced downstream ICS results when initialized from VarSI groups. We find that the initializations considered here achieve up to 70\% measurement reductions for ICS, compared to the standard SI initialization with mean reductions of 9--15.3\% depending on qubit mappings and covariance dictionaries used. These results show that non-overlapping grouping remains a consequential design step even when the final estimator uses overlapping fragments.
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, one of the strategies 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 three previously studied directions that minimize the number of measurements: (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 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.
We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig [], which exhibits 103N2 scaling in the leading order. We develop a method based on a finite projective plane to construct sets of simultaneously measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth- O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. We perform numerical simulation of our method for molecular Hamiltonians of hydrogen chains. We count the number of sets of simultaneously measurable operators generated by our method and estimate the number of measurement shots required to achieve the small standard deviation of the energy expectation value for precise quantum chemistry calculations. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications. Published by the American Physical Society 2024
Quantum chemistry is among the most promising applications of quantum computing, offering the potential to solve complex electronic structure problems more efficiently than classical approaches. A critical component of any quantum algorithm is the measurement step, where the desired properties are extracted from a quantum computer. This review focuses on recent advancements in quantum measurement techniques tailored for quantum chemistry, particularly within the second quantized framework suitable for current and near-term quantum hardware. We provide a comprehensive overview of measurement strategies developed primarily for the Variational Quantum Eigensolver (VQE) and its derivatives. These strategies address the inherent challenges posed by complexity of the electronic Hamiltonian operator. Additionally, we examine methods for estimating excited states and one- and two-electron properties, extending the applicability of quantum algorithms to broader chemical phenomena. Key aspects of the review include approaches for constructing measurement operators with reduced classical preprocessing and quantum implementation costs, techniques to minimize the number of measurements required for a given accuracy, and error mitigation strategies that leverage symmetries and other properties of the measurement operators. Furthermore, we explore measurement schemes rooted in Quantum Phase Estimation (QPE), which are expected to become viable with the advent of fault-tolerant quantum computing. This review emphasizes foundational concepts and methodologies rather than numerical benchmarks, serving as a resource for researchers aiming to enhance the efficiency and accuracy of quantum measurements in quantum chemistry.