Matchgate benchmarking: Scalable benchmarking of a continuous family of many-qubit gates

Jonas Helsen1,2, Sepehr Nezami3, Matthew Reagor4, and Michael Walter1,5,6

1QuSoft & Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 123, 1098 XG Amsterdam, The Netherlands
2Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The Netherlands
3Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA
4Rigetti Computing, 775 Heinz Ave, Berkeley, CA 94710, USA
5Institute for Theoretical Physics & ILLC, University of Amsterdam, Science Park 123, 1098 XG Amsterdam, The Netherlands
6Faculty of Computer Science, Ruhr University Bochum, Universitätsstraße 150, 44801 Bochum, Germany

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Abstract

We propose a method to reliably and efficiently extract the fidelity of many-qubit quantum circuits composed of continuously parametrized two-qubit gates called matchgates. This method, which we call $\textit{matchgate benchmarking}$, relies on advanced techniques from randomized benchmarking as well as insights from the representation theory of matchgate circuits. We argue the formal correctness and scalability of the protocol, and moreover deploy it to estimate the performance of matchgate circuits generated by two-qubit XY spin interactions on a quantum processor.

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► References

[1] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman. Real randomized benchmarking. Quantum, 2: 85, 2018. 10.22331/​q-2018-08-22-85.
https:/​/​doi.org/​10.22331/​q-2018-08-22-85

[2] J. Helsen, X. Xue, L. M. K. Vandersypen, and S. Wehner. A new class of efficient randomized benchmarking protocols. npj Quant. Inf., 5: 1–9, 2019. 10.1038/​s41534-019-0182-7.
https:/​/​doi.org/​10.1038/​s41534-019-0182-7

[3] A. W. Cross, E. Magesan, L. S. Bishop, J. A. Smolin, and J. M. Gambetta. Scalable randomised benchmarking of non-Clifford gates. npj Quant. Inf., 2: 16012, 2016. 10.1038/​npjqi.2016.12.
https:/​/​doi.org/​10.1038/​npjqi.2016.12

[4] A. Carignan-Dugas, J. J. Wallman, and J. Emerson. Characterizing universal gate sets via dihedral benchmarking. Phys. Rev. A, 92: 060302, 2015. 10.1103/​PhysRevA.92.060302.
https:/​/​doi.org/​10.1103/​PhysRevA.92.060302

[5] J. J. Wallman, M. Barnhill, and J. Emerson. Robust characterization of loss rates. Phys. Rev. Lett., 115: 060501, 2015. 10.1103/​PhysRevLett.115.060501.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.060501

[6] R. Barends, J. Kelly, A. Veitia, A. Megrant, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, E. Jeffrey, C. Neill, P. J. J. O'Malley, J. Mutus, C. Quintana, P. Roushan, D. Sank, J. Wenner, T. C. White, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Rolling quantum dice with a superconducting qubit. Phys. Rev. A, 90: 030303, 2014. 10.1103/​PhysRevA.90.030303.
https:/​/​doi.org/​10.1103/​PhysRevA.90.030303

[7] J. M. Gambetta, A. D. Córcoles, S. T. Merkel, B. R. Johnson, J. A. Smolin, J. M. Chow, C. A. Ryan, C. Rigetti, S. Poletto, T. A. Ohki, M. B. Ketchen, and M. Steffen. Characterization of addressability by simultaneous randomized benchmarking. Phys. Rev. Lett., 109: 240504, 2012. 10.1103/​PhysRevLett.109.240504.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.240504

[8] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland. Randomized benchmarking of quantum gates. Phys. Rev. A, 77: 012307, 2008. 10.1103/​PhysRevA.77.012307.
https:/​/​doi.org/​10.1103/​PhysRevA.77.012307

[9] Daniel Stilck França, Sergii Strelchuk, and Michał Studziński. Efficient classical simulation and benchmarking of quantum processes in the Weyl basis. Physical Review Letters, 126 (21): 210502, 2021. 10.1103/​PhysRevLett.126.210502.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.210502

[10] Jonas Helsen, Ingo Roth, Emilio Onorati, Albert Werner, and Jens Eisert. A general framework for randomized benchmarking. arXiv:2010.07974, 2020.
arXiv:2010.07974

[11] X Xue, TF Watson, J Helsen, Daniel R Ward, Donald E Savage, Max G Lagally, Susan N Coppersmith, MA Eriksson, S Wehner, and LMK Vandersypen. Benchmarking gate fidelities in a Si/​SiGe two-qubit device. Phys. Rev. X, 9 (2): 021011, 2019. 10.1103/​PhysRevX.9.021011.
https:/​/​doi.org/​10.1103/​PhysRevX.9.021011

[12] A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt. Characterizing large-scale quantum computers via cycle benchmarking. Nature Comm., 10, 2019. 10.1038/​s41467-019-13068-7.
https:/​/​doi.org/​10.1038/​s41467-019-13068-7

[13] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New J. Phys., 18 (2): 023023, 2016. 10.1088/​1367-2630/​18/​2/​023023.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[14] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv:1411.4028, 2014.
arXiv:1411.4028

[15] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. 10.1038/​s41586-019-1666-5.
https:/​/​doi.org/​10.1038/​s41586-019-1666-5

[16] Leslie G Valiant. Expressiveness of matchgates. Theor. Comput. Sci., 289 (1): 457–471, 2002. 10.1016/​S0304-3975(01)00325-5.
https:/​/​doi.org/​10.1016/​S0304-3975(01)00325-5

[17] Emanuel Knill. Fermionic linear optics and matchgates. arXiv:quant-ph/​0108033, 2001.
arXiv:quant-ph/0108033

[18] Barbara M Terhal and David P DiVincenzo. Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A, 65 (3): 032325, 2002. 10.1103/​PhysRevA.65.032325.
https:/​/​doi.org/​10.1103/​PhysRevA.65.032325

[19] D. P. DiVincenzo and B. M. Terhal. Fermionic linear optics revisited. Found. Phys., 35 (12): 1967–1984, 2005. 10.1007/​s10701-005-8657-0.
https:/​/​doi.org/​10.1007/​s10701-005-8657-0

[20] Sergey Bravyi. Lagrangian representation for fermionic linear optics. Quantum Inf. Comput., 5: 216–238, 2005. 10.26421/​qic5.3-3.
https:/​/​doi.org/​10.26421/​qic5.3-3

[21] Richard Jozsa and Akimasa Miyake. Matchgates and classical simulation of quantum circuits. Proc. Royal Soc. A, 464 (2100): 3089–3106, 2008. 10.1098/​rspa.2008.0189.
https:/​/​doi.org/​10.1098/​rspa.2008.0189

[22] Norbert Schuch and Jens Siewert. Natural two-qubit gate for quantum computation using the XY interaction. Phys. Rev. A, 67 (3): 032301, 2003. 10.1103/​PhysRevA.67.032301.
https:/​/​doi.org/​10.1103/​PhysRevA.67.032301

[23] Deanna M. Abrams, Nicolas Didier, Blake R. Johnson, Marcus P. da Silva, and Colm A. Ryan. Implementation of XY entangling gates with a single calibrated pulse. Nature Electronics, 2020. 10.1038/​s41928-020-00498-1.
https:/​/​doi.org/​10.1038/​s41928-020-00498-1

[24] Masao Ishikawa and Masato Wakayama. Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities. J. Combin. Theory Ser. A, 113 (1): 113–155, 2006. 10.1016/​j.jcta.2005.05.008.
https:/​/​doi.org/​10.1016/​j.jcta.2005.05.008

[25] Persi Diaconis and Laurent Saloff-Coste. Bounds for Kac's master equation. Commun. Math. Phys., 209: 729–755, 2000. 10.1007/​s002200050036.
https:/​/​doi.org/​10.1007/​s002200050036

[26] David C McKay, Christopher J Wood, Sarah Sheldon, Jerry M Chow, and Jay M Gambetta. Efficient Z gates for quantum computing. Phys. Rev. A, 96 (2): 022330, 2017. 10.1103/​PhysRevA.96.022330.
https:/​/​doi.org/​10.1103/​PhysRevA.96.022330

[27] Robert Koenig and John A Smolin. How to efficiently select an arbitrary Clifford group element. J. Math. Phys., 55 (12): 122202, 2014. 10.1063/​1.4903507.
https:/​/​doi.org/​10.1063/​1.4903507

[28] Yunjiang Jiang. Kac's random walk on the special orthogonal group mixes in polynomial time. Proc. Amer. Math. Soc., 145 (10): 4533–4541, 2017. 10.1090/​proc/​13598.
https:/​/​doi.org/​10.1090/​proc/​13598

[29] Data & code supplementary to the paper "Matchgate benchmarking: Scalable benchmarking of a continuous family of many-qubit gates". 10.5281/​zenodo.5833362.
https:/​/​doi.org/​10.5281/​zenodo.5833362

[30] Richard Jozsa, Akimasa Miyake, and Sergii Strelchuk. Jordan-Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation. Quant. Inform. Comp., 15: 0541–0556, 2015. 10.26421/​qic15.7-8-1.
https:/​/​doi.org/​10.26421/​qic15.7-8-1

[31] Easwar Magesan and Jay M Gambetta. Effective Hamiltonian models of the cross-resonance gate. Phys. Rev. A, 101 (5): 052308, 2020. 10.1103/​PhysRevA.101.052308.
https:/​/​doi.org/​10.1103/​PhysRevA.101.052308

[32] Sarah Sheldon, Easwar Magesan, Jerry M Chow, and Jay M Gambetta. Procedure for systematically tuning up cross-talk in the cross-resonance gate. Phys. Rev. A, 93 (6): 060302, 2016. 10.1103/​PhysRevA.93.060302.
https:/​/​doi.org/​10.1103/​PhysRevA.93.060302

[33] Daniel J Brod and Andrew M Childs. The computational power of matchgates and the XY interaction on arbitrary graphs. Quantum Inf. Comput., 14: 901–916, 2014. 10.26421/​qic14.11-12-1.
https:/​/​doi.org/​10.26421/​qic14.11-12-1

[34] Leonardo DiCarlo, Jerry M Chow, Jay M Gambetta, Lev S Bishop, Blake R Johnson, DI Schuster, J Majer, Alexandre Blais, Luigi Frunzio, SM Girvin, et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature, 460 (7252): 240–244, 2009. 10.1038/​nature08121.
https:/​/​doi.org/​10.1038/​nature08121

[35] J. Long, T. Zhao, M. Bal, R. Zhao, G. S. Barron, H.-S. Ku, J. A. Howard, X. Wu, C. R. H. McRae, X.-H. Deng, et al. A universal quantum gate set for transmon qubits with strong ZZ interactions. arXiv:2103.12305, 2021.
arXiv:2103.12305

[36] J. Claes, E. Rieffel, and Z. Wang. Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking. 2020. 10.1103/​PRXQuantum.2.010351.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010351

[37] Roe Goodman and Nolan R Wallach. Symmetry, representations, and invariants, volume 255. Springer, 2009.

[38] William Fulton and Joe Harris. Representation theory: a first course, volume 129. Springer, 2013.

[39] Linghang Kong. A framework for randomized benchmarking over compact groups. arXiv:2111.10357, 2021.
arXiv:2111.10357

[40] Sergey Bravyi and David Gosset. Complexity of quantum impurity problems. Commun. Math. Phys., 356 (2): 451–500, 2017. 10.1007/​s00220-017-2976-9.
https:/​/​doi.org/​10.1007/​s00220-017-2976-9

Cited by

[1] Martin Kliesch, "Randomized benchmarking with a tractable continuously generated group", Quantum Views 6, 64 (2022).

[2] Michał Oszmaniec, Ninnat Dangniam, Mauro E. S. Morales, and Zoltán Zimborás, "Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states", arXiv:2012.15825, PRX Quantum 3 2, 020328 (2020).

[3] Ashley Montanaro and Stasja Stanisic, "Error mitigation by training with fermionic linear optics", arXiv:2102.02120.

[4] Yunchao Liu, Matthew Otten, Roozbeh Bassirianjahromi, Liang Jiang, and Bill Fefferman, "Benchmarking near-term quantum computers via random circuit sampling", arXiv:2105.05232.

[5] Jonas Helsen, Ingo Roth, Emilio Onorati, Albert H. Werner, and Jens Eisert, "A general framework for randomized benchmarking", arXiv:2010.07974.

[6] Ryotaro Suzuki, Kosuke Mitarai, and Keisuke Fujii, "Computational power of one- and two-dimensional dual-unitary quantum circuits", arXiv:2103.09211.

[7] Jonas Helsen, Marios Ioannou, Ingo Roth, Jonas Kitzinger, Emilio Onorati, Albert H. Werner, and Jens Eisert, "Estimating gate-set properties from random sequences", arXiv:2110.13178.

[8] Timothy Proctor, Stefan Seritan, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young, "Scalable randomized benchmarking of quantum computers using mirror circuits", arXiv:2112.09853.

[9] Cupjin Huang, Dawei Ding, Qi Ye, Feng Wu, Linghang Kong, Fang Zhang, Xiaotong Ni, Yaoyun Shi, Hui-Hai Zhao, and Jianxin Chen, "Towards ultra-high fidelity quantum operations: SQiSW gate as a native two-qubit gate", arXiv:2105.06074.

[10] Jahan Claes, Eleanor Rieffel, and Zhihui Wang, "Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking", arXiv:2011.00007.

[11] Peter W. Evans, Dominik Hangleiter, and Karim P. Y. Thébault, "How to engineer a quantum wavefunction", arXiv:2112.01105.

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-28 19:38:53) and SAO/NASA ADS (last updated successfully 2022-05-28 19:38:54). The list may be incomplete as not all publishers provide suitable and complete citation data.

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