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arXiv:2408.11898

Non-Clifford diagonalization for measurement shot reduction in quantum expectation value estimation

Nicolas P. D. Sawaya, Daan Camps, DalFavero, Ben, Norm M. Tubman, Grant M. Rotskoff, Ryan LaRose

Published Aug 21, 2024• 2 citations
arXivHTMLPDF

Abstract

Estimating expectation values on near-term quantum computers often requires a prohibitively large number of measurements. One widely-used strategy to mitigate this problem has been to partition an operator's Pauli terms into sets of mutually commuting operators. Here, we introduce a method that relaxes this constraint of commutativity, instead allowing for entirely arbitrary terms to be grouped together, save a locality constraint. The key idea is that we decompose the operator into arbitrary tensor products with bounded tensor size, ignoring Pauli commuting relations. This method -- named $k$-NoCliD ($k$-local non-Clifford diagonalization) -- allows one to measure in far fewer bases in most cases, often (though not always) at the cost of increasing the circuit depth. We introduce several partitioning algorithms tailored to different Hamiltonian classes. For electronic structure, we numerically demonstrate the existence of threshold values of $k$ for which $k$-NoCliD leads to the lowest shot counts, though we leave improved partitioning algorithms to future work. We focus primarily on three Hamiltonian classes -- molecular vibrational structure, Fermi-Hubbard, and Bose-Hubbard -- and show that $k$-NoCliD reduces the number of circuit shots, often by a very large margin, and often even for $k$ as small as 2.

DOI
10.48550/arxiv.2408.11898

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Estimating expectation values on near-term quantum computers often requires a prohibitively large number of measurements. One widely-used strategy to mitigate this problem has been to partition an operator's Pauli terms into sets of mutually commuting operators. Here, we introduce a method that relaxes this constraint of commutativity, instead allowing for entirely arbitrary terms to be grouped together, save a locality constraint. The key idea is that we decompose the operator into arbitrary tensor products with bounded tensor size, ignoring Pauli commuting relations. This method -- named kkk-NoCliD (kkk-local non-Clifford diagonalization) -- allows one to measure in far fewer bases in most cases, often (though not always) at the cost of increasing the circuit depth. We introduce several partitioning algorithms tailored to different Hamiltonian classes. For electronic structure, we numerically demonstrate the existence of threshold values of kkk for which kkk-NoCliD leads to the lowest shot counts, though we leave improved partitioning algorithms to future work. We focus primarily on three Hamiltonian classes -- molecular vibrational structure, Fermi-Hubbard, and Bose-Hubbard -- and show that kkk-NoCliD reduces the number of circuit shots, often by a very large margin, and often even for kkk as small as 2.
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A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable SWAP gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and - as a proof-of-principle application - we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers.
A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable Swap gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and - as a proof-of-principle application - we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers.
Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.
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.
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