Randomized benchmarking with gate-dependent noise

Joel J. Wallman

Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

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We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.

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[1] Isaac L. Chuang and Michael A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics, 44, 2455 (1997).
https://doi.org/10.1080/09500349708231894

[2] J. F. Poyatos, J. Ignacioi Cirac, and P. Zoller, Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate, Physical Review Letters 78, 390 (1997).
https://doi.org/10.1103/PhysRevLett.78.390

[3] Marcus P. da Silva, Olivier Landon-Cardinal, and David Poulin, Practical Characterization of Quantum Devices without Tomography, Physical Review Letters 107, 210404 (2011).
https://doi.org/10.1103/PhysRevLett.107.210404

[4] Steven T. Flammia and Yi-Kai Liu, Direct Fidelity Estimation from Few Pauli Measurements, Physical Review Letters 106, 230501 (2011).
https://doi.org/10.1103/PhysRevLett.106.230501

[5] Steven T. Flammia, David Gross, Yi-Kai Liu, and Jens Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators, New Journal of Physics 14, 095022 (2012).
https://doi.org/10.1088/1367-2630/14/9/095022

[6] Daniel M. Reich, Giulia Gualdi, and Christiane P. Koch, Optimal Strategies for Estimating the Average Fidelity of Quantum Gates, Physical Review Letters 111, 200401 (2013).
https://doi.org/10.1103/PhysRevLett.111.200401

[7] Martin Kliesch, Richard Kueng, Jens Eisert, and David Gross, Guaranteed recovery of quantum processes from few measurements, arXiv:1701.03135 [quant-ph].
arXiv:1701.03135

[8] Joseph Emerson, Robert Alicki, and Karol Życzkowski, Scalable noise estimation with random unitary operators, Journal of Optics B 7, S347 (2005).
https://doi.org/10.1088/1464-4266/7/10/021

[9] Benjamin Lévi, Cecilia C López, Joseph Emerson, and David G. Cory, Efficient error characterization in quantum information processing, Physical Review A 75, 022314 (2007).
https://doi.org/10.1103/PhysRevA.75.022314

[10] Emanuel Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and David J. Wineland, Randomized benchmarking of quantum gates, Physical Review A 77, 012307 (2008).
https://doi.org/10.1103/PhysRevA.77.012307

[11] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Physical Review A 80, 012304 (2009).
https://doi.org/10.1103/PhysRevA.80.012304

[12] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson, Scalable and Robust Randomized Benchmarking of Quantum Processes, Physical Review Letters 106, 180504 (2011).
https://doi.org/10.1103/PhysRevLett.106.180504

[13] Joseph Emerson, Marcus P. da Silva, Osama Moussa, Colm A. Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme, Symmetrized characterization of noisy quantum processes. Science 317, 1893 (2007).
https://doi.org/10.1126/science.1145699

[14] Easwar Magesan, Jay M. Gambetta, Blake R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and Matthias Steffen, Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking, Physical Review Letters 109, 080505 (2012).
https://doi.org/10.1103/PhysRevLett.109.080505

[15] Joel J. Wallman, Christopher Granade, Robin Harper, and Steven T. Flammia, Estimating the Coherence of Noise, New Journal of Physics 17, 113020 (2015).
https://doi.org/10.1088/1367-2630/17/11/113020

[16] Joel J. Wallman, Marie Barnhill, and Joseph Emerson, Robust Characterization of Loss Rates, Physical Review Letters 115, 060501 (2015).
https://doi.org/10.1103/PhysRevLett.115.060501

[17] Joel J. Wallman, Marie Barnhill, and Joseph Emerson, Robust characterization of leakage errors, New Journal of Physics 18, 043021 (2016).
https://doi.org/10.1088/1367-2630/18/4/043021

[18] Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson, Characterizing universal gate sets via dihedral benchmarking, Physical Review A 92, 060302(R) (2015).
https://doi.org/10.1103/PhysRevA.92.060302

[19] Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin, and Jay M. Gambetta, Scalable randomised benchmarking of non-Clifford gates, npj Quantum Information 2, 16012 (2016).
https://doi.org/10.1038/npjqi.2016.12

[20] Antonio D. Córcoles, Jay M. Gambetta, Jerry M. Chow, John A. Smolin, Matthew Ware, Joel Strand, B. L. T. Plourde, and Matthias Steffen, Process verification of two-qubit quantum gates by randomized benchmarking, Physical Review A 87, 030301(R) (2013).
https://doi.org/10.1103/PhysRevA.87.030301

[21] R. Barends, Julian Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, Austin G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. J. J. O`Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis, Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500 (2014).
https://doi.org/10.1038/nature13171

[22] Julian Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, Austin G. Fowler, I.-C. Hoi, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Cleland, and John M. Martinis, Optimal Quantum Control Using Randomized Benchmarking, Physical Review Letters 112, 240504 (2014).
https://doi.org/10.1103/PhysRevLett.112.240504

[23] Jeffrey M. Epstein, Andrew W. Cross, Easwar Magesan, and Jay M. Gambetta, Investigating the limits of randomized benchmarking protocols, Physical Review A 89, 062321 (2014).
https://doi.org/10.1103/PhysRevA.89.062321

[24] Tobias Chasseur and Frank K. Wilhelm, Complete randomized benchmarking protocol accounting for leakage errors, Physical Review A 92, 042333 (2015).
https://doi.org/10.1103/PhysRevA.92.042333

[25] Harrison Ball, Thomas M. Stace, Steven T. Flammia, and Michael J. Biercuk, Effect of noise correlations on randomized benchmarking, Physical Review A 93, 022303 (2016).
https://doi.org/10.1103/PhysRevA.93.022303

[26] Timothy Proctor, Kenneth Rudinger, Kevin Young, Mohan Sarovar, and Robin Blume-kohout, What randomized benchmarking actually measures, Physical Review Letters 119, 130502 (2017).
https://doi.org/10.1103/PhysRevLett.119.130502

[27] Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T. Flammia, Logical Randomized Benchmarking, arXiv:1702.03688 [quant-ph].
arXiv:1702.03688

[28] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson, Characterizing quantum gates via randomized benchmarking, Physical Review A 85, 042311 (2012).
https://doi.org/10.1103/PhysRevA.85.042311

[29] Yuval R. Sanders, Joel J. Wallman, and Barry C. Sanders, Bounding quantum gate error rate based on reported average fidelity, New Journal of Physics 18, 012002 (2016).
https://doi.org/10.1088/1367-2630/18/1/012002

[30] Joel J. Wallman, and Steven T. Flammia, Randomized benchmarking with confidence, New Journal of Physics 16, 103032 (2014).
https://doi.org/10.1088/1367-2630/16/10/103032

[31] Christopher Granade, Christopher Ferrie, and David G. Cory, Accelerated randomized benchmarking, New Journal of Physics 17, 013042 (2015).
https://doi.org/10.1088/1367-2630/17/1/013042

[32] Michael A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A 303, 249 (2002).
https://doi.org/10.1016/S0375-9601(02)01272-0

[33] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numerische Mathematik 2, 137 (1960).
https://doi.org/10.1007/BF01386217

[34] David Pérez-García, Michael M. Wolf, Denes Petz, and Mary Beth Ruskai, Contractivity of positive and trace-preserving maps under $L_p$ norms, Journal of Mathematical Physics 47, 083506 (2006).
https://doi.org/10.1063/1.2218675

[35] Robin Blume-Kohout, John King Gamble, Erik Nielsen, Kenneth Rudinger, Jonathan Mizrahi, Kevin Fortier, and Peter Maunz, Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nature Communications 8, 14485 (2017).
https://doi.org/10.1038/ncomms14485

[36] Mark D. Bowdrey, Daniel K. L. Oi, Anthony J. Short, Konrad Banaszek, and Jonathan A. Jones, Fidelity of single qubit maps, Physics Letters A 294, 258 (2002).
https://doi.org/10.1016/S0375-9601(02)00069-5

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