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Variance reduction methods in the estimation of Pauli sums

Søren Fuglede Jørgensen, Rafael Emilio Barfknecht, Patrick Ettenhuber, Nikolaj Thomas Zinner

Abstract

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.

Variance reduction methods in the estimation of Pauli sums

Abstract

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.
Paper Structure (45 sections, 2 theorems, 64 equations, 9 figures, 19 tables)

This paper contains 45 sections, 2 theorems, 64 equations, 9 figures, 19 tables.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Structure of the paper
  4. Observables as Pauli sums
  5. Estimation through measurement
  6. Measurement circuits
  7. Benchmark: Electronic structure problem
  8. Variance reduction
  9. Stratified sampling and partitioning algorithms
  10. Pauli commutation graphs
  11. Algorithms for finding clique covers
  12. Integer programming
  13. Greedy covering algorithms
  14. Column generation
  15. Maximum weight clique solver
  16. ...and 30 more sections

Key Result

Proposition 5.1

Let $P = P_1 \cdots P_n$ and $Q = Q_1 \cdots Q_n$, where $P_i, Q_i \in \{I, X, Y, Z\}$. Let and let $\delta = \sum_{i=1}^n \delta_i$. Then $(P, Q) \in E^{\mathrm{QWC}}$ if and only if $\delta = 0$, and $(P, Q) \in E^{\mathrm{FC}}$ if and only if $\delta$ is even.

Figures (9)

  • Figure 1: Overview of the strategies we consider: The observable $O$ of interest is used as input for a measurement group generator to generate a collection of groups of Pauli strings. These groups may then be post-processed before an allocator decides how many shots to associate to each group, or which probabilities to associate to each group in a randomized setting. Given a circuit $U_\psi$ preparing the state of interest, for each group $G$, one synthesizes a measurement circuit $U_G$, and by measuring the combined circuit, measurements for each Pauli string are collected and combined into an estimate $\widehat{\langle O \rangle}$ of $\langle O \rangle$ in a way that depends on a choice of estimator type. The boxes highlighted in green indicate the subproblems for which we will consider several different strategies.
  • Figure 2: The average number of CNOT gates used in Clifford measurement circuits for the 6 benchmark molecules, together with the fit $n \mapsto 0.449 \tfrac{n^2}{\log n}$.
  • Figure 3: Variances of the estimators from Example \ref{['ex:min-var-example']}.
  • Figure 4: The relation between $c_P^2$ and $c_P^2\mathop{\mathrm{Var}}\nolimits(X_P)$ for the 6 benchmark Hamiltonians. Each plot shows the points $(c_P^2, c_P^2\mathop{\mathrm{Var}}\nolimits(X_P))$ for all Pauli strings $P$ for that Hamiltonian. The line is a linear least-squares fit.
  • Figure 5: Bias, standard deviation, and RMSE of $\widehat{\langle O \rangle}^{\mathrm{det},M}$ as obtained from ShadowGrouping for NH3, for different values of $M$. The horizontal gray dashed lines are $\pm E_{\mathrm{HF} - \langle O \rangle}$.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 3.1
  • Example 4.1
  • Example 4.2
  • Proposition 5.1
  • Remark 5.2
  • ...and 3 more