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Distributed Quantum Inner Product Estimation with Structured Random Circuits

Congcong Zheng, Kun Wang, Xutao Yu, Ping Xu, Zaichen Zhang

TL;DR

This work analyzes cross-platform verification via distributed inner product estimation (DIPE) using structured random circuits. It demonstrates that DIPE with a unitary $2$-design ensemble achieves an average sample complexity of $\mathcal{O}(\sqrt{2^n})$, while brickwork, local Clifford, and global Clifford ensembles yield $\mathcal{O}(\sqrt{2.18^n})$, $\mathcal{O}(\sqrt{2.5^n})$, and $\Theta(\sqrt{2^n})$ respectively, with state-dependent refinements showing deep connections to nonstabilizerness and tensor-network representations. The paper also analyzes $\varepsilon$-approximate $4$-designs, constructing a biased estimator whose bias decays exponentially with circuit depth and whose variance matches the exact design for $m = \mathcal{O}(\sqrt{2^n})$ shots. Numerical simulations up to $26$ qubits corroborate the theory and illustrate practical benefits of nonstabilizerness and depth scaling. Overall, the results provide rigorously guaranteed, experimentally feasible DIPE strategies for near-term quantum devices and cross-platform verification tasks.

Abstract

Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. Existing rigorous sample complexity analyses are limited to unitary $4$-designs, which pose significant practical challenges for near-term quantum devices. This work addresses this challenge by exploring DIPE with structured random circuits. We first establish that DIPE with an arbitrary unitary $2$-design ensemble achieves an average sample complexity of $\mathcal{O}(\sqrt{2^n})$, where $n$ is the number of qubits. We then analyze ensembles below unitary $2$-designs -- specifically, the brickwork and local unitary $2$-design ensembles -- demonstrating average sample complexities of $\mathcal{O}(\sqrt{2.18^n})$ and $\mathcal{O}(\sqrt{2.5^n})$, respectively. Furthermore, we analyze the state-dependent sample complexity. For brickwork ensembles, we develop a tensor network approach to compute the asymptotic state-dependent sample complexity, showing that it converges to $\mathcal{O}(\sqrt{2.18^n})$ as the circuit depth increases. Remarkably, we find that DIPE with the global Clifford ensemble requires $Θ(\sqrt{2^n})$ copies, matching the performance of unitary $4$-designs. For both local and global Clifford ensembles, we find that the efficiency can be further enhanced by the nonstabilizerness of states. Additionally, for approximate unitary $4$-designs, the performance exponentially approaches that of exact unitary $4$-designs as the circuit depth increases. Our results provide theoretically guaranteed methods for implementing DIPE with experimentally feasible unitary ensembles.

Distributed Quantum Inner Product Estimation with Structured Random Circuits

TL;DR

This work analyzes cross-platform verification via distributed inner product estimation (DIPE) using structured random circuits. It demonstrates that DIPE with a unitary -design ensemble achieves an average sample complexity of , while brickwork, local Clifford, and global Clifford ensembles yield , , and respectively, with state-dependent refinements showing deep connections to nonstabilizerness and tensor-network representations. The paper also analyzes -approximate -designs, constructing a biased estimator whose bias decays exponentially with circuit depth and whose variance matches the exact design for shots. Numerical simulations up to qubits corroborate the theory and illustrate practical benefits of nonstabilizerness and depth scaling. Overall, the results provide rigorously guaranteed, experimentally feasible DIPE strategies for near-term quantum devices and cross-platform verification tasks.

Abstract

Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. Existing rigorous sample complexity analyses are limited to unitary -designs, which pose significant practical challenges for near-term quantum devices. This work addresses this challenge by exploring DIPE with structured random circuits. We first establish that DIPE with an arbitrary unitary -design ensemble achieves an average sample complexity of , where is the number of qubits. We then analyze ensembles below unitary -designs -- specifically, the brickwork and local unitary -design ensembles -- demonstrating average sample complexities of and , respectively. Furthermore, we analyze the state-dependent sample complexity. For brickwork ensembles, we develop a tensor network approach to compute the asymptotic state-dependent sample complexity, showing that it converges to as the circuit depth increases. Remarkably, we find that DIPE with the global Clifford ensemble requires copies, matching the performance of unitary -designs. For both local and global Clifford ensembles, we find that the efficiency can be further enhanced by the nonstabilizerness of states. Additionally, for approximate unitary -designs, the performance exponentially approaches that of exact unitary -designs as the circuit depth increases. Our results provide theoretically guaranteed methods for implementing DIPE with experimentally feasible unitary ensembles.

Paper Structure

This paper contains 42 sections, 24 theorems, 161 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. General Framework for DIPE
  3. Protocol
  4. Classical Function
  5. Sample Complexity
  6. Average Sample Complexity
  7. DIPE with Brickwork Ensembles
  8. Classical Function and Average Variance
  9. Asymptotic State-dependent Variance
  10. DIPE with Clifford Ensembles
  11. Global Clifford Ensemble
  12. Local Clifford Ensemble
  13. DIPE with Approximate Unitary 4-design Ensembles
  14. Numerical Simulation
  15. Conclusions
  16. ...and 27 more sections

Key Result

Lemma 1

To guarantee that $\hat{\omega}$ defined in Eq. eq:estimator is an unbiased estimator, the classical function $f_{\mathcal{E}}$ should satisfy for all $P,P'\in\mathcal{P}_n$, where $\mathcal{P}_n := \{I,X,Y,Z\}^{\otimes n}$ is the $n$-qubit Pauli set, and $O := \sum_{\bm{a}, \bm{b}} f_{\mathcal{E}}(\bm{a},\bm{b}) \vert{\bm{a}\bm{b}}\rangle\!\langle{\bm{a}\bm{b}}\vert$.

Figures (6)

  • Figure 1: The general framework for distributed inner product estimation (DIPE). Here, $\rho$ and $\sigma$ are two quantum states independently prepared on two distant quantum platforms. The DIPE begins by applying randomized measurements on each platform using a unitary ensemble $\mathcal{E}$. The resulting measurement outcomes are then processed classically using a function $f_{\mathcal{E}}$, which depends on $\mathcal{E}$, to obtain an unbiased estimator of the inner product $\mathop{\mathrm{Tr}}\nolimits[\rho\sigma]$. In this work, we focus on the following experimentally feasible unitary ensembles: (i) the $n$-qubit global Clifford ensemble ${\rm Cl}_n$, (ii) the $n$-qubit unitary $2$-design ensemble $\mathcal{T}_n$, (iii) the brickwork ensemble $\mathcal{B}_d$, where $d$ denotes the depth, and (iv) the local Clifford ensemble ${\rm Cl}_1^{\otimes n}$. The average sample complexities for each ensemble are shown above, where the worst-case sample complexity for ${\rm Cl}_n$ is $\Theta(\sqrt{2^n})$.
  • Figure 2: Numerical results of DIPE with different unitary ensembles: the local Clifford ensemble ${\rm Cl}_1^{\otimes n}$, the brickwork ensemble $\mathcal{B}_d$ ($d=1,3,5$), and the global Clifford ensemble ${\rm Cl}_n$. The states are set as $\rho=\sigma$, (a) GHZ state, and (b)$\vert{S_{8,k}(\theta)}\rangle$ defined in Eq. \ref{['eq:definition of S state']}. Each data point is obtained with $10^2$ unitaries, $10^2$ state pairs, and $m$ shots. The green line is from Theorem \ref{['the:global clifford variance']}, the yellow line tracks the behavior of the stabilizer $2$-Rényi entropy ($2$-SRE), and the red line is from Theorem \ref{['the:local clifford variance']}, where $\xi(\theta)$ is defined in Eq. \ref{['eq:single-qubit 4-moment']}.
  • Figure 3: The MPO representation of $h(\bm{a}, P)$. For each $M_d$, the physical dimension of input leg is $2$, the physical dimension of output leg is $4$, and the bound dimension is $2^{d-1}$. Therefore, each $M_d$ has $2^{d-1}\times 2^{d-1}$ matrices in $\mathbb{R}^{4\times 2}$.
  • Figure 4: Numerical result of Haar random states.
  • Figure 5: Numerical result of $\mathbb{V}_{\mathcal{E}}^{(4)}(\rho,\sigma)-1$.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 4: Average Variances
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 41 more