Abstract
While it is important to investigate the negative effects of decoherence on the performance of quantum information processors, Landauer was one of the first to point out that an equally basic problem, i.e., the effects of unavoidable hardware flaws in the real-world implementations of quantum gates, needs to be investigated as well. Following Landauer’s suggestion, we investigated the structural stability of the quantum Fourier transform (QFT) via significantly changing the analytical form of its controlled rotation gates, thus modeling structural flaws in the Hamiltonian of the QFT. Three types of modified rotation gates were investigated, numerically and analytically, changing the exact QFT rotation angles \(\pi /2^j\) to (1) \(\pi /\alpha ^j\), (2) \(\pi /2 j^{\beta }\), and (3) \(\pi /\log _{\gamma }(j+1)\), where \(\alpha \), \(\beta \), and \(\gamma \) are constants and \(j\) is the integer distance between QFT qubits. Surprisingly good performance is observed in all the three cases for a wide range of \(\alpha \), \(\beta \), and \(\gamma \). This demonstrates the structural stability of the QFT Hamiltonian. Our results also demonstrate that the precise implementation of QFT rotation angles is not critical as long as the angles (roughly) observe a monotonic decrease in \(j\) (hierarchy). This result is important since it indicates that stringent tolerances do not need to be imposed in the actual manufacturing process of quantum information hardware components.
Similar content being viewed by others
References
Landauer, R.: Information is physical, but slippery. In: Brooks, M. (ed.) Quantum Computing and Communication, pp. 59–62. Springer, London (1999)
Cirac, J.I., Zoller, P.: Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091 (1995)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Mermin, N.D.: Quantum Computer Science. Cambridge University Press, Cambridge (2007)
Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 (1995)
Steane, A.M.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551 (1996)
Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996)
Steane, A.M.: Efficient fault-tolerant quantum computing. Nature 399, 124 (1999)
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, NM (1994)
Barenco, A., Ekert, A., Suominen, K.-A., Törmä, P.: Approximate quantum Fourier transform and decoherence. Phys. Rev. A 54, 139 (1996)
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)
Niwa, J., Matsumoto, K., Imai, H.: General-purpose parallel simulator for quantum computing. Phys. Rev. A 66, 062317 (2002)
Nam, Y.S., Blümel, R.: Performance scaling of Shor’s algorithm with a banded quantum Fourier transform. Phys. Rev. A 86, 044303 (2012)
Nam, Y.S., Blümel, R.: Scaling laws for Shor’s algorithm with a banded quantum Fourier transform. Phys. Rev. A 87, 032333 (2013)
Nam, Y.S., Blümel, R.: Robustness of the quantum Fourier transform with respect to static gate defects. Phys. Rev. A 89, 042337 (2014)
Papoulis, A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1965)
Chiaverini, J., Britton, J., Leibfried, D., Knill, E., Barrett, M.D., Blakestad, R.B., Itano, W.M., Jost, J.D., Langer, C., Ozeri, R., Schaetz, T., Wineland, D.J.: Implementation of the semiclassical quantum Fourier transform in a scalable system. Science 308, 997 (2005)
Leibfried, D., Wineland, D.J., Blakestad, R.B., Bollinger, J.J., Britton, J., Chiaverini, J., Epstein, R.J., Itano, W.M., Jost, J.D., Knill, E., Langer, C., Ozeri, R., Reichle, R., Seidelin, S., Shiga, N., Wesenberg, J.H.: Towards scaling up trapped ion quantum information processing. Hyperfine Interact 174, 1 (2007)
Nam, Y.S., Blümel, R.: In preparation
Kitaev, A.Y.: Quantum computations: algorithms and error correction. Russ. Math. Surv. 52, 1191 (1997)
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)
Ross, N.J., Selinger, P.: Optimal Ancilla-free Clifford+ T Approximation of z-rotations arXiv:1403.2975v1 [quant-ph] (2014)
Selinger, P.: Efficient Clifford+ T Approximation of Single-qubit Operators. arXiv:1212.6253v2 [quant-ph] (2012)
Bocharov, A., Roetteler, M., Svore, K.M.: Efficient Synthesis of Universal Repeat-Until-Success Circuits. arXiv:1404.5320v2 [quant-ph] (2014)
Bocharov, A., Roetteler, M., Svore, K.M.: Efficient Synthesis of Probabilistic Quantum Circuits with Fallback. arXiv:1409.3552v2 [quant-ph] (2014)
Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nam, Y.S., Blümel, R. Structural stability of the quantum Fourier transform. Quantum Inf Process 14, 1179–1192 (2015). https://doi.org/10.1007/s11128-015-0923-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-015-0923-2