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A Lightweight Protocol for Matchgate Fidelity Estimation

Jędrzej Burkat, Sergii Strelchuk

TL;DR

The paper addresses the challenge of benchmarking and calibrating noisy quantum devices implementing matchgate circuits by introducing a low-depth randomized protocol to estimate the entanglement fidelity $F_e(\mathcal{E},\mathcal{U})$ between an $n$-qubit MG circuit and its noisy implementation. It leverages a modified Pauli-Liouville representation with a Clifford basis, which makes MG superoperators block-diagonal and enables direct fidelity estimation via random sampling of Pauli-based observables, yielding a $1/\sqrt{n}$ speedup over prior methods. The approach accommodates Clifford interleavings, subgroups such as the $XY$ group and Givens rotations, and extends naturally to Clifford+MG+Clifford circuits, with explicit analyses of decay parameters $\lambda_k$ and the role of well-conditioning $\alpha$. It also introduces MG tomography by reconstructing the $R\in SO(2n)$ matrix and discusses practical illustrations through the fSim gate and numerical simulations, demonstrating favorable shot-count scaling and applicability to scalable MG-based quantum architectures. Overall, the method offers a simpler, faster route to MG fidelity estimation and direct gate benchmarking, with potential impact on routinely calibrating MG-based quantum processors.

Abstract

We present a low-depth randomised algorithm for the estimation of entanglement fidelity between an $n$-qubit matchgate circuit $\mathcal{U}$ and its noisy implementation $\mathcal{E}$. Our procedure makes use of a modified Pauli-Liouville representation of quantum channels, with Clifford algebra elements as a basis. We show that this choice of representation leads to a block-diagonal compound matrix structure of matchgate superoperators which enables construction of efficient protocols for estimating the fidelity, achieving a $1/\sqrt{n}$ speedup over protocols of Flammia & Liu [PRL 106, 230501]. Finally, we offer simple extensions of our protocol which (without additional overhead) benchmark matchgate circuits intertwined by Clifford circuits, and circuits composed of exclusively nearest-neighbour $XY(θ)$ gates or Givens rotations - forming the first known method for direct benchmarking of matchgate subgroups.

A Lightweight Protocol for Matchgate Fidelity Estimation

TL;DR

The paper addresses the challenge of benchmarking and calibrating noisy quantum devices implementing matchgate circuits by introducing a low-depth randomized protocol to estimate the entanglement fidelity between an -qubit MG circuit and its noisy implementation. It leverages a modified Pauli-Liouville representation with a Clifford basis, which makes MG superoperators block-diagonal and enables direct fidelity estimation via random sampling of Pauli-based observables, yielding a speedup over prior methods. The approach accommodates Clifford interleavings, subgroups such as the group and Givens rotations, and extends naturally to Clifford+MG+Clifford circuits, with explicit analyses of decay parameters and the role of well-conditioning . It also introduces MG tomography by reconstructing the matrix and discusses practical illustrations through the fSim gate and numerical simulations, demonstrating favorable shot-count scaling and applicability to scalable MG-based quantum architectures. Overall, the method offers a simpler, faster route to MG fidelity estimation and direct gate benchmarking, with potential impact on routinely calibrating MG-based quantum processors.

Abstract

We present a low-depth randomised algorithm for the estimation of entanglement fidelity between an -qubit matchgate circuit and its noisy implementation . Our procedure makes use of a modified Pauli-Liouville representation of quantum channels, with Clifford algebra elements as a basis. We show that this choice of representation leads to a block-diagonal compound matrix structure of matchgate superoperators which enables construction of efficient protocols for estimating the fidelity, achieving a speedup over protocols of Flammia & Liu [PRL 106, 230501]. Finally, we offer simple extensions of our protocol which (without additional overhead) benchmark matchgate circuits intertwined by Clifford circuits, and circuits composed of exclusively nearest-neighbour gates or Givens rotations - forming the first known method for direct benchmarking of matchgate subgroups.
Paper Structure (16 sections, 4 theorems, 59 equations, 3 figures, 1 algorithm)

This paper contains 16 sections, 4 theorems, 59 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matchgate Fidelity Estimation
  4. Comparison to existing methods
  5. Decay parameters lambda and the fidelity
  6. The XY Group
  7. Beyond Matchgates: Benchmarking Clifford + MG + Clifford circuits
  8. Efficient Matchgate Tomography
  9. Examples
  10. Google's fSim(theta, phi) gate
  11. Numerical Simulations
  12. Properties of the Clifford Basis
  13. Runtime Parameters
  14. General Case
  15. Well-conditioned Case
  16. ...and 1 more sections

Key Result

Theorem 1

For any matchgate circuit $U = e^{-iH}$ with an associated $SO(2n)$ matrix $R$, in the Clifford algebra generator basis the superoperator matrix elements $\chi_\mathcal{U}(I,J)$ are given by: where $R_{I, J}$ is a submatrix of $R$ for which each $i$th row $i \in I$ is preserved, and each $j$th column $j \in J$ is preserved. Consequently, $\hat{\mathcal{U}}$ is a block diagonal matrix with $2n+1$

Figures (3)

  • Figure 1: Comparison of Matchgate Benchmarking protocols.$F_e(\mathcal{E, U})$ and $F_e(\mathcal{E}) = F_e(\mathcal{E}, \mathbb{1} )$ are the entanglement fidelities, related to channel fidelity via Equation \ref{['eq:fidelity_def']}
  • Figure 2: Simulation of Algorithm \ref{['box:algorithm']} for $n=3$ Qubits. Each point is the output of Algorithm \ref{['box:algorithm']} applied to a Haar-randomly sampled circuit followed by an $n$-qubit depolarising channel. Colour maps indicate shot numbers in each run. Restricting to nearest-neighbour Givens rotations is seen to decrease the shot number by about $1/2$, in agreement with Section \ref{['sec:xygroup']}. Code used to simulate our algorithm is provided in mgrepo.
  • Figure 3: The Superoperator Matrix $\hat{\mathcal{U}}$ for the 2-qubit ${\text{fSim}(\theta, \phi)}$ Gate. If $\phi = 0$, this is an $XY(\theta)$ gate with a block-diagonal compound PL matrix. If $\theta = 0, \phi = \pi$ it is monomial, with entries from $\{ \pm 1, \pm i\}$. If $\theta \neq 0, \phi = \pi$, the gate is of the form $\hat{\mathcal{W}} = \hat{\mathcal{U}} \circ \hat{\mathcal{V}}$ discussed in Section \ref{['sec:cliffordmg']}.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof