Abstract
We estimate the success probability of quantum protocols composed of Clifford operations in the presence of Pauli errors. Our method is derived from the fault-point formalism previously used to determine the success rate of low-distance error correction codes. Here we apply it to a wider range of quantum protocols and identify circuit structures that allow for efficient calculation of the exact success probability and even the final distribution of output states. As examples, we apply our method to the Bernstein–Vazirani algorithm and the Steane [[7,1,3]] quantum error correction code and compare the results to Monte Carlo simulations.
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Chuang, I.L., Nielsen, M.A.: Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 44, 2455 (1997)
Emerson, J., Silva, M., Moussa, O., Ryan, C., Laforest, M., Baugh, J., Cory, D.G., Laflamme, R.: Symmetrized characterization of noisy quantum processes. Science 317, 1893 (2007)
Knill, E., Leibfried, D., Reichle, R., Britton, J., Blakestad, R.B., Jost, J.D., Langer, C., Ozeri, R., Seidelin, S., Wineland, D.J.: Randomized benchmarking of quantum gates. Phys. Rev. A 77, 012307 (2008)
Flammia, S.T., Liu, Y.K.: Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011)
Flammia, S.T., Gross, D., Liu, Y.K., Eisert, J.: Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14, 095022 (2012)
da Silva, M.P., Landon-Cardinal, O., Poulin, D.: Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011)
Blume-Kohout, R., Gamble, J.K., Nielsen, E., Mizrahi, J., Sterk, J.D., Maunz, P.: Robust, Self-consistent, Closed-form Tomography of Quantum Logic Gates on a Trapped Ion Qubit. arXiv:1310.4492 (2013)
Merkel, S.T., Gambetta, J.M., Smolin, J.A., Poletto, S., Córcoles, A.D., Johnson, B.R., Ryan, C.A., Steffen, M.: Self-consistent quantum process tomography. Phys. Rev. A 87, 062119 (2013)
Wallman, J., Granade, C., Harper, R., Flammia, S.T.: Estimating the coherence of noise. New J. Phys. 17(11), 113020 (2015)
Shabani, A., Kosut, R.L., Mohseni, M., Rabitz, H., Broome, M.A., Almeida, M.P., Fedrizzi, A., White, A.G.: Efficient measurement of quantum dynamics via compressive sensing. Phys. Rev. Lett. 106, 100401 (2011)
Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009)
Clader, B.D., Jacobs, B.C., Sprouse, C.R.: Preconditioned quantum linear system algorithm. Phys. Rev. Lett. 110, 250504 (2013)
Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004)
Anders, S., Briegel, H.J.: Fast simulation of stabilizer circuits using a graph-state representation. Phys. Rev. A 73, 022334 (2006)
Magesan, E., Puzzuoli, D., Granade, C.E., Cory, D.G.: Modeling quantum noise for efficient testing of fault-tolerant circuits. Phys. Rev. A 87, 012324 (2013)
Gutiérrez, M., Svec, L., Vargo, A., Brown, K.R.: Approximation of realistic errors by Clifford channels and Pauli measurements. Phys. Rev. A 87, 030302 (2013)
Gutiérrez, M., Brown, K.R.: Comparison of a quantum error-correction threshold for exact and approximate errors. Phys. Rev. A 91, 022335 (2015)
Aliferis, P., Gottesman, D., Preskill, J.: Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97 (2006)
Bernstein, E., Vazirani, U.: Quantum complexity theory, In: Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing (ACM, New York, NY, USA, 1993), STOC ’93, pp. 11–20
Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996)
Tomita, Y., Gutiérrez, M., Kabytayev, C., Brown, K.R., Hutsel, M.R., Morris, A.P., Stevens, K.E., Mohler, G.: Comparison of ancilla preparation and measurement procedures for the Steane [[7,1,3]] code on a model ion-trap quantum computer. Phys. Rev. A 88, 042336 (2013)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th edn. Cambridge University Press, Cambridge (2010)
Tomita, Y.: Numerical and Analytical Studies of Quantum Error Correction. Ph.D. thesis, Georgia Institute of Technology (2014)
DiVincenzo, D.P., Aliferis, P.: Effective fault-tolerant quantum computation with slow measurements. Phys. Rev. Lett. 98, 220501 (2007)
Aliferis, P., Cross, A.W.: Subsystem fault tolerance with the Bacon-Shor code. Phys. Rev. Lett. 98, 220502 (2007)
Walt, S.v d, Colbert, S.C., Varoquaux, G.: The NumPy array: a structure for efficient numerical computation. CiSE 13, 22–30 (2011)
Orús, R.: A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014)
Acknowledgments
We would like to thank Andrew Cross, Ryan Sheehan, Alonzo Hernandez and Silas Fradley for useful discussions. This project was supported by Office of the Director of National Intelligence (ODNI)—Intelligence Advanced Research Project Activity (IARPA) through Army Research Office Grant No. W911NF-10-1-0231 and Department of Interior Contract No. D11PC20167. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US Government.
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Janardan, S., Tomita, Y., Gutiérrez, M. et al. Analytical error analysis of Clifford gates by the fault-path tracer method. Quantum Inf Process 15, 3065–3079 (2016). https://doi.org/10.1007/s11128-016-1330-z
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DOI: https://doi.org/10.1007/s11128-016-1330-z