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Expectation value estimation with parametrized quantum circuits

Bujiao Wu, Lingyu Kong, Yuxuan Yan, Fuchuan Wei, Zhenhuan Liu

TL;DR

The paper introduces a framework for estimating linear properties Tr(ρH) using shallow parameterized quantum circuits by decomposing H into a sum of terms each diagonalizable by such circuits, via greedy projection (GPD) or tensor network (TND) methods, and then applying importance-sampling-based estimation. It provides a formal analysis of sample complexity, including a general bound T = O(||\bm{Λ}||_1^2 log(1/δ)/ε^2) and a fundamental lower bound Ω( Tr(H_0^2)^2/(ε^2 δ(H_0) 4^n) ) for shallow circuits, and validates the approach numerically on sparse/dense Hamiltonians and Slater-determinant inner products. The results show that GPD and TN decompositions can achieve lower estimation errors than traditional Pauli-group and classical-shadow methods, especially when circuit depth and decomposition terms are properly chosen. This framework is particularly relevant for near-term quantum devices by optimizing sample complexity under realistic hardware constraints and guiding efficient observable estimation in quantum chemistry and many-body physics.

Abstract

Estimating properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. Due to the limitation of practical quantum devices, including limited circuit depth and connectivity, estimating even linear properties encounters high sample complexity. To address this inefficiency, we propose a framework that optimizes sample complexity for estimating the expectation value of any observable using a shallow parameterized quantum circuit. Within this framework, we introduce two decomposition algorithms, a tensor network approach and a greedy projection approach that decompose the target observable into a linear combination of multiple observables, each of which can be diagonalized with the shallow circuit. Using this decomposition, we then apply an importance sampling algorithm to estimate the expectation value of the target observable. We numerically demonstrate the performance of our algorithm by estimating the expectation values of some specific Hamiltonians and inner product of a Slater determinant with a pure state, highlighting advantages compared to some conventional methods. Additionally, we derive the fundamental lower bound for the sample complexity required to estimate a target observable using a given shallow quantum circuit, thereby enhancing our understanding of the capabilities of shallow circuits in quantum learning tasks.

Expectation value estimation with parametrized quantum circuits

TL;DR

The paper introduces a framework for estimating linear properties Tr(ρH) using shallow parameterized quantum circuits by decomposing H into a sum of terms each diagonalizable by such circuits, via greedy projection (GPD) or tensor network (TND) methods, and then applying importance-sampling-based estimation. It provides a formal analysis of sample complexity, including a general bound T = O(||\bm{Λ}||_1^2 log(1/δ)/ε^2) and a fundamental lower bound Ω( Tr(H_0^2)^2/(ε^2 δ(H_0) 4^n) ) for shallow circuits, and validates the approach numerically on sparse/dense Hamiltonians and Slater-determinant inner products. The results show that GPD and TN decompositions can achieve lower estimation errors than traditional Pauli-group and classical-shadow methods, especially when circuit depth and decomposition terms are properly chosen. This framework is particularly relevant for near-term quantum devices by optimizing sample complexity under realistic hardware constraints and guiding efficient observable estimation in quantum chemistry and many-body physics.

Abstract

Estimating properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. Due to the limitation of practical quantum devices, including limited circuit depth and connectivity, estimating even linear properties encounters high sample complexity. To address this inefficiency, we propose a framework that optimizes sample complexity for estimating the expectation value of any observable using a shallow parameterized quantum circuit. Within this framework, we introduce two decomposition algorithms, a tensor network approach and a greedy projection approach that decompose the target observable into a linear combination of multiple observables, each of which can be diagonalized with the shallow circuit. Using this decomposition, we then apply an importance sampling algorithm to estimate the expectation value of the target observable. We numerically demonstrate the performance of our algorithm by estimating the expectation values of some specific Hamiltonians and inner product of a Slater determinant with a pure state, highlighting advantages compared to some conventional methods. Additionally, we derive the fundamental lower bound for the sample complexity required to estimate a target observable using a given shallow quantum circuit, thereby enhancing our understanding of the capabilities of shallow circuits in quantum learning tasks.
Paper Structure (16 sections, 3 theorems, 16 equations, 6 figures, 4 tables, 2 algorithms)

This paper contains 16 sections, 3 theorems, 16 equations, 6 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Expectation value estimation framework
  3. Greedy projection decomposition of the Hamiltonian
  4. Tensor network decomposition of the Hamiltonian
  5. Numerical results
  6. Ground energy estimation
  7. Expectation value estimation for a dense Hamiltonian
  8. Expectation value calculation with TND
  9. Lower bound for sample complexity
  10. Conclusion and discussion
  11. Review of Qubit-Commuting Random Measurements
  12. Qubit-commuting decomposition
  13. Review of Commuting Measurements
  14. Proof of Proposition \ref{['proposition:SampleComplexity_LB']}
  15. Preliminary to inner product calculation
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\hat{H}=\sum_{k=1}^K U_L\left( \theta^{(k)} \right)^\dagger \Lambda_k U_L\left( \theta^{(k)} \right)$ be an $\epsilon_1$ approximation of $H$ in spectral norm. Then $T=\mathcal{O}\left(\frac{\left\| \bm{\Lambda} \right\|_1^2\log(1/\delta)}{\epsilon_2^2}\right)$ copies of $\rho$ are required to

Figures (6)

  • Figure 1: Overview of the expectation value estimation with parameterized circuit. We utilize a classical algorithm (greedy projection or tensor network method) to generate a decomposition of $H \approx \sum_{k=1}^K U_L(\theta^{(k)})^\dagger \Lambda_k U_L(\theta^{(k)})$, followed by iteratively repeating $T$ rounds: sampling $k$ with probability $p_k\propto \left\| \Lambda_k \right\|_2$, performing $U_L(\theta^{(k)})$ on the prepared quantum state $\rho$, as illustrated by the $L$-depth circuit consisting of two-qubit parametrized gates. Afterward, we measure in the computational basis to obtain a result $b$, which is then used to compute the estimator $v$ by applying the Median-of-Means method. Here, $\Lambda_{k}(b)$ denotes the $b$-th diagonal entry of the diagonal matrix $\Lambda_k$ and $b$ ranges from $0$ to $2^n-1$.
  • Figure 2: Illustration of tensor network decomposition for a given Hamiltonian. (a) Tensor network representation of the parametrized unitary $U(\theta_k)$, the diagonal matrix $\Lambda_k$, and the Hamiltonian $H$. (b) Tensor network representation of $A_{kk'}$ and $B_k$, which are associated with the loss function calculations.
  • Figure 3: The variations in Frobenius distance as the number of decomposition terms increases for (a) a sparse Hermitian $H$ and (b) a random generated observable, and (c) a non-Hermitian operator $O$ associated with a Slater determinant.
  • Figure 4: Comparison of the TN algorithm with Derandom Huang2021Efficient, SG gresch2025guaranteed, RandomPauli huang2020predicting, and AdaptivePaulis Shlosberg2023Adaptive in terms of the average estimation error as a function of the number of measurements. The evaluation is performed over five experiments, each using a different randomly generated quantum input state. The system has 8 qubits. The TN algorithm uses a quantum parameterized circuit with depth $L=1$ and employs $K = 3$ decomposition terms.
  • Figure 5: Comparison of the GPD algorithm with Derandom Huang2021Efficient, SG gresch2025guaranteed, RandomPauli huang2020predicting, and AdaptivePaulis Shlosberg2023Adaptive in terms of average estimation error versus the number of measurements, evaluated over five randomly generated datasets. (a) Random sparse Hamiltonians and (b) Random dense (general) Hamiltonians. The system has 4 qubits. The GPD algorithm uses a quantum parameterized circuit with depth $L = 4$ and employs $K = 20$ decomposition terms.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3: Sample Complexity Lower Bound
  • proof : Proof of Proposition 3 in the main text