The rapid progress in the development of quantum devices is in large part due to the availability of a wide range of characterization techniques allowing to probe, test and adjust them. Nevertheless, these methods often make use of approximations that hold in rather simplistic circumstances. In particular, assuming that error mechanisms stay constant in time and have no dependence in the past, is something that will be impossible to do as quantum processors continue scaling up in depth and size. We establish a theoretical framework for the Randomized Benchmarking protocol encompassing temporally-correlated, so-called non-Markovian noise, at the gate level, for any gate set belonging to a wide class of finite groups. We obtain a general expression for the Average Sequence Fidelity (ASF) and propose a way to obtain average gate fidelities of full non-Markovian noise processes. Moreover, we obtain conditions that are fulfilled when an ASF displays authentic non-Markovian deviations. Finally, we show that even though gate-dependence does not translate into a perturbative term within the ASF, as in the Markovian case, the non-Markovian sequence fidelity nevertheless remains stable under small gate-dependent perturbations.
 B. Lévi, C. C. López, J. Emerson, and D. G. Cory, ``Efficient error characterization in quantum information processing,'' Phys. Rev. A 75, 022314 (2007).
 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).
 E. Magesan, J. M. Gambetta, and J. Emerson, ``Scalable and robust randomized benchmarking of quantum processes,'' Phys. Rev. Lett. 106, 180504 (2011).
 I. L. Chuang and M. A. Nielsen, ``Prescription for experimental determination of the dynamics of a quantum black box,'' J. Mod. Optic 44, 2455–2467 (1997).
 S. J. van Enk and R. Blume-Kohout, ``When quantum tomography goes wrong: drift of quantum sources and other errors,'' New J. Phys. 15, 025024 (2013).
 M. A. Fogarty, M. Veldhorst, R. Harper, C. H. Yang, S. D. Bartlett, S. T. Flammia, and A. S. Dzurak, ``Nonexponential fidelity decay in randomized benchmarking with low-frequency noise,'' Phys. Rev. A 92, 022326 (2015).
 T. Proctor, M. Revelle, E. Nielsen, K. Rudinger, D. Lobser, P. Maunz, R. Blume-Kohout, and K. Young, ``Detecting and tracking drift in quantum information processors,'' Nat. Commun. 11, 5396 (2020).
 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).
 M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, ``Detecting crosstalk errors in quantum information processors,'' Quantum 4, 321 (2020).
 P. Parrado-Rodríguez, C. Ryan-Anderson, A. Bermudez, and M. Müller, ``Crosstalk Suppression for Fault-tolerant Quantum Error Correction with Trapped Ions,'' Quantum 5, 487 (2021).
 J. J. Wallman, M. Barnhill, and J. Emerson, ``Robust characterization of leakage errors,'' New J. Phys. 18, 043021 (2016).
 K. Young, S. Bartlett, R. J. Blume-Kohout, J. K. Gamble, D. Lobser, P. Maunz, E. Nielsen, T. J. Proctor, M. Revelle, and K. M. Rudinger, Diagnosing and Destroying Non-Markovian Noise, Tech. Rep. (U.S. Department of Energy, Office of Scientific and Technical Information, 2020).
 C. A. Ryan, M. Laforest, and R. Laflamme, ``Randomized benchmarking of single- and multi-qubit control in liquid-state NMR quantum information processing,'' New J. Phys. 11, 013034 (2009).
 J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, ``Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,'' Nat. Phys. 7, 565 (2011).
 C. Müller, J. Lisenfeld, A. Shnirman, and S. Poletto, ``Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits,'' Phys. Rev. B 92, 035442 (2015).
 K. W. Chan, W. Huang, C. H. Yang, J. C. C. Hwang, B. Hensen, T. Tanttu, F. E. Hudson, K. M. Itoh, A. Laucht, A. Morello, and A. S. Dzurak, ``Assessment of a silicon quantum dot spin qubit environment via noise spectroscopy,'' Phys. Rev. Applied 10, 044017 (2018).
 S. M. Meißner, A. Seiler, J. Lisenfeld, A. V. Ustinov, and G. Weiss, ``Probing individual tunneling fluctuators with coherently controlled tunneling systems,'' Phys. Rev. B 97, 180505 (2018).
 J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, ``Decoherence benchmarking of superconducting qubits,'' npj Quantum Inf. 5 (2019), 10.1038/s41534-019-0168-5.
 S. Mavadia, C. L. Edmunds, C. Hempel, H. Ball, F. Roy, T. M. Stace, and M. J. Biercuk, ``Experimental quantum verification in the presence of temporally correlated noise,'' npj Quantum Inf. 4, 7 (2018).
 H. Ball, T. M. Stace, S. T. Flammia, and M. J. Biercuk, ``Effect of noise correlations on randomized benchmarking,'' Phys. Rev. A 93, 022303 (2016).
 P. Figueroa-Romero, K. Modi, R. J. Harris, T. M. Stace, and M.-H. Hsieh, ``Randomized benchmarking for non-Markovian noise,'' PRX Quantum 2, 040351 (2021).
 J. Helsen, X. Xue, L. M. K. Vandersypen, and S. Wehner, ``A new class of efficient randomized benchmarking protocols,'' npj Quantum Inf. 5, 71 (2019).
 T. Proctor, K. Rudinger, K. Young, M. Sarovar, and R. Blume-Kohout, ``What randomized benchmarking actually measures,'' Phys. Rev. Lett. 119, 130502 (2017).
 A. W. Cross, E. Magesan, L. S. Bishop, J. A. Smolin, and J. M. Gambetta, ``Scalable randomised benchmarking of non-Clifford gates,'' npj Quantum Inf. 2 (2016).
 J. J. Wallman and S. T. Flammia, ``Randomized benchmarking with confidence,'' New J. Phys. 16, 103032 (2014).
 S. T. Merkel, E. J. Pritchett, and B. H. Fong, ``Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors,'' Quantum 5, 581 (2021).
 A. Carignan-Dugas, K. Boone, J. J. Wallman, and J. Emerson, ``From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters,'' New J. Phys. 20, 092001 (2018).
 G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum circuit architecture,'' Phys. Rev. Lett. 101, 060401 (2008).
 C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, ``Causal boxes: Quantum information-processing systems closed under composition,'' IEEE Trans. Inf. Theory , 1–1 (2017).
 H. I. Nurdin and J. Gough, ``From the heisenberg to the schrödinger picture: Quantum stochastic processes and process tensors,'' 2021 60th IEEE Conference on Decision and Control (CDC) (2021), 10.1109/cdc45484.2021.9683765.
 F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys. 18, 063032 (2016).
 F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Non-Markovian quantum processes: Complete framework and efficient characterization,'' Phys. Rev. A 97, 012127 (2018a).
 F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, ``Operational Markov condition for quantum processes,'' Phys. Rev. Lett. 120, 040405 (2018b).
 S. Milz, F. Sakuldee, F. A. Pollock, and K. Modi, ``Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories,'' Quantum 4, 255 (2020a).
 P. Taranto, F. A. Pollock, and K. Modi, ``Non-Markovian memory strength bounds quantum process recoverability,'' npj Quantum Inf. 7 (2021), 10.1038/s41534-021-00481-4.
 S. Milz, M. S. Kim, F. A. Pollock, and K. Modi, ``Completely positive divisibility does not mean Markovianity,'' Phys. Rev. Lett. 123, 040401 (2019).
 P. Figueroa-Romero, K. Modi, and F. A. Pollock, ``Equilibration on average in quantum processes with finite temporal resolution,'' Phys. Rev. E 102, 032144 (2020).
 S. Milz, D. Egloff, P. Taranto, T. Theurer, M. B. Plenio, A. Smirne, and S. F. Huelga, ``When is a non-Markovian quantum process classical?'' Phys. Rev. X 10, 041049 (2020b).
 G. A. L. White, C. D. Hill, F. A. Pollock, L. C. L. Hollenberg, and K. Modi, ``Demonstration of non-Markovian process characterisation and control on a quantum processor,'' Nat. Commun. 11 (2020), 10.1038/s41467-020-20113-3.
 G. D. Berk, S. Milz, F. A. Pollock, and K. Modi, ``Extracting quantum dynamical resources: Consumption of non-Markovianity for noise reduction,'' (2021), arXiv:2110.02613 [quant-ph].
 J. Claes, E. Rieffel, and Z. Wang, ``Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking,'' PRX Quantum 2, 010351 (2021).
 P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, ``Quantum Markov order,'' Phys. Rev. Lett. 122, 140401 (2019a).
 P. Taranto, S. Milz, F. A. Pollock, and K. Modi, ``Structure of quantum stochastic processes with finite Markov order,'' Phys. Rev. A 99, 042108 (2019b).
 J. Helsen, M. Ioannou, I. Roth, J. Kitzinger, E. Onorati, A. H. Werner, and J. Eisert, ``Estimating gate-set properties from random sequences,'' (2021), arXiv:2110.13178 [quant-ph].
 D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, ``Contractivity of positive and trace-preserving maps under lp norms,'' J. Math. Phys. 47, 083506 (2006).
 H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, ``Colloquium: Non-markovian dynamics in open quantum systems,'' Rev. Mod. Phys. 88, 021002 (2016).
 M. Tinkham, Group Theory and Quantum Mechanics, Dover Books on Chemistry and Earth Sciences (Dover Publications, 2003).
 W. Harris, W. Fulton, and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics (Springer New York, 1991).
 M. Horodecki and P. Horodecki, ``Reduction criterion of separability and limits for a class of protocols of entanglement distillation,'' (1997), arXiv:quant-ph/9708015 [quant-ph].
 J. Helsen, M. Ioannou, J. Kitzinger, E. Onorati, A. H. Werner, J. Eisert, and I. Roth, "Shadow estimation of gate-set properties from random sequences", Nature Communications 14 1, 5039 (2023).
 Philip Taranto, Thomas J. Elliott, and Simon Milz, "Hidden Quantum Memory: Is Memory There When Somebody Looks?", Quantum 7, 991 (2023).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-22 14:50:51) and SAO/NASA ADS (last updated successfully 2023-09-22 14:50:52). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.