A Lightweight Protocol for Matchgate Fidelity Estimation
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 $XY(\theta)$ gates or Givens rotations - forming the first known method for direct benchmarking of matchgate subgroups.
- R. Jozsa and A. Miyake, Matchgates and classical simulation of quantum circuits, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, 3089 (2008).
- L. G. Valiant, Quantum circuits that can be simulated classically in polynomial time, SIAM Journal on Computing 31, 1229 (2002), https://doi.org/10.1137/S0097539700377025 .
- E. Knill, Fermionic linear optics and matchgates (2001), arXiv:quant-ph/0108033 [quant-ph] .
- B. M. Terhal and D. P. DiVincenzo, Classical simulation of noninteracting-fermion quantum circuits, Physical Review A 65, 10.1103/physreva.65.032325 (2002).
- D. P. DiVincenzo and B. M. Terhal, Fermionic linear optics revisited, Foundations of Physics 35, 1967 (2005).
- S. Bravyi, Contraction of matchgate tensor networks on non-planar graphs (2008), arXiv:0801.2989 [quant-ph] .
- A. Sørensen and K. Mølmer, Entanglement and quantum computation with ions in thermal motion, Physical Review A 62, 10.1103/physreva.62.022311 (2000).
- S. M. Young, N. T. Jacobson, and J. R. Petta, Optimal control of a cavity-mediated iSWAPiSWAP\mathrm{i}\mathrm{SWAP}roman_iSWAP gate between silicon spin qubits, Phys. Rev. Appl. 18, 064082 (2022).
- S. T. Flammia and Y.-K. Liu, Direct fidelity estimation from few pauli measurements, Physical Review Letters 106, 10.1103/physrevlett.106.230501 (2011).
- M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A 303, 249 (2002).
- J. Burkat, Matchgate benchmarking - github repository (2024).
- E. S. Meckes, The Random Matrix Theory of the Classical Compact Groups, Cambridge Tracts in Mathematics (Cambridge University Press, 2019).
- D. K. Hoffman, R. C. Raffenetti, and K. Ruedenberg, Generalization of Euler Angles to N-Dimensional Orthogonal Matrices, Journal of Mathematical Physics 13, 528 (1972).
- C. Kravvaritis and M. Mitrouli, Compound matrices: properties, numerical issues and analytical computations, Numerical Algorithms 50, 155 (2009).
- K. Nambiar and S. Sreevalsan, Compound matrices and three celebrated theorems, Mathematical and Computer Modelling 34, 251 (2001).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.