# Optimal length of decomposition sequences composed of imperfect gates

- 142 Downloads
- 1 Citations

## Abstract

Quantum error correcting circuitry is both a resource for correcting errors and a source for generating errors. A balance has to be struck between these two aspects. Perfect quantum gates do not exist in nature. Therefore, it is important to investigate how flaws in the quantum hardware affect quantum computing performance. We do this in two steps. First, in the presence of realistic, faulty quantum hardware, we establish how quantum error correction circuitry achieves reduction in the extent of quantum information corruption. Then, we investigate fault-tolerant gate sequence techniques that result in an approximate phase rotation gate, and establish the existence of an optimal length \(L_{\text {opt}}\) of the length *L* of the decomposition sequence. The existence of \(L_{\text {opt}}\) is due to the competition between the increase in gate accuracy with increasing *L*, but the decrease in gate performance due to the diffusive proliferation of gate errors due to faulty basis gates. We present an analytical formula for the gate fidelity as a function of *L* that is in satisfactory agreement with the results of our simulations and allows the determination of \(L_{\text {opt}}\) via the solution of a transcendental equation. Our result is universally applicable since gate sequence approximations also play an important role, e.g., in atomic and molecular physics and in nuclear magnetic resonance.

## Keywords

Quantum computing Quantum information processing Qubit Quantum gates Decomposition sequences Quantum error correction Fault tolerance## References

- 1.Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Goldwasser, S. (ed.) Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, pp. 124–134. IEEE, Santa Fe (1994)Google Scholar
- 2.Gaitan, F.: Quantum Error Correction and Fault Tolerant Quantum Computing. CRC Press, Boca Raton (2008)CrossRefMATHGoogle Scholar
- 3.Fowler, A.G., Mariantoni, M., Martinis, J.M., Cleland, A.N.: Surface codes: towards practical large-scale quantum computation. Phys. Rev. A
**86**, 032324 (2012)ADSCrossRefGoogle Scholar - 4.Barends, R., Kelly, J., Megrant, A., Veitia, A., Sank, D., Jeffrey, E., Chen, Y., Chiaro, B., Mutus, J., Neill, C., et al.: Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature
**508**, 500 (2014)ADSCrossRefGoogle Scholar - 5.Muhonen, J.T., Dehollain, J.P., Laucht, A., Hudson, F.E., Karia, R., Sekiguchi, T., Itoh, K.M., Jamieson, D.N., McCallum, J.C., Dzurak, A.S., Morello, A.: Storing quantum information for 30 s in a nanoelectronic device. Nat. Nanotechnol.
**9**, 986 (2014)ADSCrossRefGoogle Scholar - 6.Coppersmith, D.: An approximate Fourier transform useful in quantum factoring. arXiv:quant-ph/0201067 (1994)
- 7.Cirac, J.I., Zoller, P.: Quantum computations with cold trapped ions. Phys. Rev. Lett.
**74**, 4091 (1995)ADSCrossRefGoogle Scholar - 8.Miquel, C., Paz, J.P., Zurek, W.H.: Quantum computation with phase drift errors. Phys. Rev. Lett.
**78**, 3971 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar - 9.Fowler, A.G., Hollenberg, L.C.L.: Scalability of Shor’s algorithm with a limited set of rotation gates. Phys. Rev. A
**70**, 032329 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar - 10.García-Mata, I., Frahm, K.M., Shepelyansky, D.L.: Shor’s factorization algorithm with a single control qubit and imperfections. Phys. Rev. A
**78**, 062323 (2008)ADSCrossRefGoogle Scholar - 11.Nam, Y.S., Blümel, R.: Scaling laws for Shor’s algorithm with a banded quantum Fourier transform. Phys. Rev. A
**87**, 032333 (2013)ADSCrossRefGoogle Scholar - 12.Nam, Y.S., Blümel, R.: Robustness of the quantum Fourier transform with respect to static gate defects. Phys. Rev. A
**89**, 042337 (2014)ADSCrossRefGoogle Scholar - 13.Nam, Y.S., Blümel, R.: Structural stability of the quantum Fourier transform. Quantum Inf. Process.
**14**, 1179 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar - 14.Nam, Y.S., Blümel, R.: Performance scaling of the quantum Fourier transform with defective rotation gates. Quantum Inf. Comput.
**15**, 721 (2015)MathSciNetGoogle Scholar - 15.Nam, Y.S., Blümel, R.: Analytical formulas for the performance scaling of quantum processors with a large number of defective gates. Phys. Rev. A
**92**, 042301 (2015)ADSCrossRefGoogle Scholar - 16.Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
- 17.Jin, C.Y., Johne, R., Swinkels, M.Y., Hoang, T.B., Midolo, L., van Veldhoven, P.J.: Ultrafast non-local control of spontaneous emission. Nat. Nanotechnol.
**9**, 886 (2014)ADSCrossRefGoogle Scholar - 18.Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phys.
**2**, 754 (2006)CrossRefGoogle Scholar - 19.Dyakonov, M.I.: Revisiting the hopes for scalable quantum computation. JETP Lett.
**98**, 514 (2013)ADSCrossRefGoogle Scholar - 20.DiVincenzo, D.P., Shor, P.W.: Fault-tolerant error correction with efficient quantum codes. Phys. Rev. Lett.
**77**, 3260 (1996)ADSCrossRefGoogle Scholar - 21.Plenio, M.B., Vedral, V., Knight, P.L.: Conditional generation of error syndromes in fault-tolerant error correction. Phys. Rev. A
**55**, 4593 (1997)ADSCrossRefGoogle Scholar - 22.Contact the author at
*ynam@wesleyan.edu*Google Scholar - 23.Hogben, L.: Handbook of Linear Algebra. Chapman and Hall/CRC, Boca Raton (2014)MATHGoogle Scholar
- 24.Selinger, P.: Newsynth: exact and approximate synthesis of quantum circuits. http://www.mathstat.dal.ca/~selinger/newsynth/ (2013)
- 25.Gottesman, D.: An introduction to quantum error correction and fault-tolerant quantum computation. In: Lomonaco, S.J. (ed.) Proceedings of Symposia in Applied Mathematics, Volume 68, Quantum Information Science and its Contributions to Mathematics. American Mathematical Society Short Course Lecture Notes (2010)Google Scholar
- 26.Reichardt, B.W., Grover, L.K.: Quantum error correction of systematic errors using a quantum search framework. Phys. Rev. A
**72**, 042326 (2005)ADSMathSciNetCrossRefGoogle Scholar - 27.Kliuchnikov, V., Maslov, D., Mosca, M.: Asymptotically optimal approximation of single qubit unitaries by Clifford and \(T\) circuits using a constant number of ancillary qubits. Phys. Rev. Lett.
**110**, 190502 (2013)ADSCrossRefGoogle Scholar - 28.Ross, N.J., Selinger, P.: Optimal ancilla-free Clifford+\(T\) approximation of \(z\)-rotations. arXiv:1403.2975v1 [quant-ph] (2014)
- 29.Bocharov, A., Roetteler, M., Svore, K.M.: Efficient synthesis of universal repeat-until-success quantum circuits. Phys. Rev. Lett.
**114**, 080502 (2015)ADSCrossRefGoogle Scholar - 30.Giles, B., Selinger, P.: Exact synthesis of multiqubit Clifford+\(T\) circuits. Phys. Rev. A
**87**, 032332 (2013)ADSCrossRefGoogle Scholar - 31.Kliuchnikov, V., Maslov, D., Mosca, M.: Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and \(T\) gates. Quantum Inf. Comput.
**13**, 607 (2013)MathSciNetGoogle Scholar - 32.Selinger, P.: Quantum circuits of \(T\)-depth one. Phys. Rev. A
**87**, 042302 (2013)ADSCrossRefGoogle Scholar - 33.Buhrman, H., Cleve, R., Laurent, M., Linden, N., Schrijver, A., Unger, F.: New limits on fault-tolerant quantum computation. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 411–419, IEEE Computer Society, Los Alamitos (2006)Google Scholar
- 34.Bravyi, S., Kitaev, A.: Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A
**71**, 022316 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar - 35.Reichardt, B.W.: Quantum universality from magic states distillation applied to CSS codes. Quantum Inf. Process.
**4**, 251 (2005)MathSciNetCrossRefMATHGoogle Scholar - 36.Bravyi, S., Haah, J.: Magic-state distillation with low overhead. Phys. Rev. A
**86**, 052329 (2012)ADSCrossRefGoogle Scholar - 37.Raussendorf, R., Harrington, J., Goyal, K.: Topological fault-tolerance in cluster state quantum computation. N. J. Phys.
**9**, 199 (2007)MathSciNetCrossRefGoogle Scholar - 38.Fowler, A.G., Stephens, A.M., Groszkowski, P.: High-threshold universal quantum computation on the surface code. Phys. Rev. A
**80**, 052312 (2009)ADSCrossRefGoogle Scholar - 39.Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A
**52**, R2493 (1995)ADSCrossRefGoogle Scholar - 40.Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran 77. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
- 41.Papoulis, A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1965)MATHGoogle Scholar
- 42.Beckmann, P.: Statistical distribution of the amplitude and phase of a multiply scattered field. J. Res. Natl. Bur. Stand.
**66D**, 231 (1962)MathSciNetGoogle Scholar