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arXiv:2311.08551 · International Symposium on Lattice Field Theory (LATTICE)

Fast Partitioning of Pauli Strings into Commuting Families for Expectation Value Measurements of Dense Operators

Andrew Lytle, Nouman Butt, Ben Reggio, Patrick Draper

Published May 3, 2024• 0 citations
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Abstract

The cost of measuring quantum expectation values of an operator can be reduced by grouping the Pauli string (SU(2) tensor product) decomposition of the operator into maximally commuting sets. We detail an algorithm, presented in [1], to partition the full set of m-qubit Pauli strings into the minimal number of commuting families, and benchmark the performance with dense Hamiltonians on IBM hardware. Here we also compare how our method scales compared to graph-theoretic techniques for the generally commuting case.

DOI
10.22323/1.453.0130

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The cost of measuring quantum expectation values of an operator can be reduced by grouping the Pauli string (SU(2) tensor product) decomposition of the operator into maximally commuting sets. We detail an algorithm, presented in [1], to partition the full set of m-qubit Pauli strings into the minimal number of commuting families, and benchmark the performance with dense Hamiltonians on IBM hardware. Here we also compare how our method scales compared to graph-theoretic techniques for the generally commuting case.
The Pauli strings appearing in the decomposition of an operator can be can be grouped into commuting families, reducing the number of quantum circuits needed to measure the expectation value of the operator. We detail an algorithm to completely partition the full set of Pauli strings acting on any number of qubits into the minimal number of sets of commuting families, and we provide python code to perform the partitioning. The partitioning method scales linearly with the size of the set of Pauli strings and it naturally provides a fast method of diagonalizing the commuting families with quantum gates. We provide a package that integrates the partitioning into qiskit, and use this to benchmark the algorithm with dense Hamiltonians, such as those that arise in matrix quantum mechanics models, on IBM hardware. We demonstrate computational speedups close to the theoretical limit of (3/2)m{(3/2)}^{m}(3/2)m relative to qubit-wise commuting groupings, for m=2,...,6m=2,...\phantom{\rule{0.16em}{0ex}},6m=2,...,6 qubits.
Simultaneous measurement of multiple Pauli strings (tensor products of Pauli matrices) is the basis for efficient measurement of observables on quantum computers by partitioning the observable into commuting sets of Pauli strings. We present the implementation and optimisation of the Baranyai grouping method for second quantised Hamiltonian partitioning in molecules up to CH4_44​ (cc-pVDZ, 68 qubits) and efficient construction of the diagonalisation circuit in O(N)O(N)O(N) quantum gates, compared to O(N2)O(N^2)O(N2), where NNN is the number of qubits. We show that this method naturally handles sparsity in the Hamiltonian and produces a O(1)O(1)O(1) number of groups for linearly scaling Hamiltonians, such as those formed by molecules in a line; rising to O(N3)O(N^3)O(N3) for fully connected two-body Hamiltonians. While this is more measurements than some other schemes it allows for the flexibility to move Pauli strings and optimise the variance. We also present an explicit optimisation for spin-symmetry which reduces the number of groups by a factor of 888, without extra computational effort.
We introduce a notion of commutativity between operators on a tensor product space, nominally Pauli strings on qubits, that interpolates between qubitwise commutativity and (full) commutativity. We apply this notion, which we call kkk-commutativity, to measuring expectation values of observables in quantum circuits and show a reduction in the number measurements at the cost of increased circuit depth. We discuss the asymptotic measurement complexity of kkk-commutativity for several families of nnn-qubit Hamiltonians, showing examples with O(1)O(1)O(1), O(n)O(\sqrt{n})O(n​), and O(n)O(n)O(n) scaling.
We propose an efficient algorithm for partitioning Pauli strings into subgroups, which can be simultaneously measured in a single quantum circuit. Our partitioning algorithm drastically reduces the total number of measurements in a variational quantum eigensolver for a quantum chemistry, one of the most promising applications of quantum computing. The algorithm is based on the Ising model optimization problem, which can be quickly solved using an Ising machine. We develop an algorithm that is applicable to problems with sizes larger than the maximum number of variables that an Ising machine can handle (nbitn_\text{bit}nbit​) through its iterative use. The algorithm has much better time complexity and solution optimality than other algorithms such as Boppana--Halldórsson algorithm and Bron--Kerbosch algorithm, making it useful for the quick and effective reduction of the number of quantum circuits required for measuring the expectation values of multiple Pauli strings. We investigate the performance of the algorithm using the second-generation Digital Annealer, a high-performance Ising hardware, for up to 65,53565,53565,535 Pauli strings using Hamiltonians of molecules and the full tomography of quantum states. We demonstrate that partitioning problems for quantum chemical calculations can be solved with a time complexity of O(N)O(N)O(N) for N≤nbitN\leq n_\text{bit}N≤nbit​ and O(N2)O(N^2)O(N2) for N>nbitN>n_\text{bit}N>nbit​ for the worst case, where NNN denotes the number of candidate Pauli strings and nbit=8,192n_\text{bit}=8,192nbit​=8,192 for the second-generation Digital Annealer used in this study. The reduction factor, which is the number of Pauli strings divided by the number of obtained partitions, can be 200200200 at maximum.
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.