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Selecting efficient phase estimation with constant-precision phase shift operators

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Abstract

We investigate the cost of three phase estimation procedures that require only constant-precision phase shift operators. The cost is in terms of the number of elementary gates, not just the number of measurements. Faster phase estimation requires the minimal number of measurements with a logarithmic factor of reduction when the required precision \(n\) is large. The arbitrary constant-precision approach (ACPA) requires the minimal number of elementary gates with a minimal factor of 14 of reduction in comparison with Kitaev’s approach. The reduction factor increases as the precision gets higher in ACPA. Kitaev’s approach is with a reduction factor of 14 in comparison with the faster phase estimation in terms of elementary gate counts.

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Notes

  1. In quantum computing, the quantum Fourier transform is a linear transformation on quantum bits and is the quantum analog of the discrete Fourier transform.

  2. The iterated logarithm. It is defined as follows:

    $$\begin{aligned} \log ^* n := \left\{ \begin{array}{ll} 0 &{} \quad \text {if } n \le 1 \\ 1 + \log ^*(\log n) &{} \quad \text {if } n > 1 \end{array} \right. . \end{aligned}$$

  3. A similar analysis can be applied to \(x_{j+1}\) = 1.

  4. No matter how many rounds there are, the very last round must have \(S\) of size \(\log n\).

  5. As \(k\) gets larger, the figure would be identical to Fig. 4.

References

  1. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In 45th Annual IEEE Symposium on Foundation, pp. 32–41 (2004)

  3. Shor, P.W.: Algorithms for quantum computation: Discrete logarithms and factoring. In 35th Annual IEEE Symposium on Foundation (Santa Fe, NM), pp. 124–134 (1994)

  4. Brassard, G., Høyer, P., Tapp, A.: Lecture Notes in Computer Science, vol. 1443, pp. 820–831. Springer (1998)

  5. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  6. Kitaev, A., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. American Mathematical Society, Providence, RI (2002)

    Book  MATH  Google Scholar 

  7. Kitaev, A.: Quantum Measurements and the Abelian stabilizer problem, Technical report (1996)

  8. Ahmedi, H., Chiang, C.: Quantum phase estimation with arbitrary constant-precision phase shift operators. Quantum Inf. Comput. 12(9 &10), 0864–0875 (2012)

    MathSciNet  Google Scholar 

  9. Cheung, D.: Improved bounds for the approximate QFT. In Proceedings of the Winter International Symposium on Information and Communication Technologies (WISICT), pp. 1–6. Trinity College Dublin (2004)

  10. Kaye, P., Laflamme, R., Mosca, M.: An Introduction to Quantum Computing. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  11. Svore, K., Hastings, M., Freedman, M.: Faster Phase Estimation. Quantum Inf. Comput. 14(3 &4), 306–328 (2013)

    MathSciNet  Google Scholar 

  12. Duclos-Cianci, G., Svore, K.: A state distillation protocol to implement arbitrary single-qubit rotations, arXiv:1210.1980v1

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Acknowledgments

We would like to thank an anonymous referee for many insightful comments and questions. We also thank J. Anderson, P. Iyer, and D. Poulin for their suggestions. C. C gratefully acknowledges the support of Lockheed Martin Corporation.

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  1. Département de Physique, Université de Sherbrooke Sherbrooke, Québec, J1K 2R1S, Canada

    Chen-Fu Chiang

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  1. Chen-Fu Chiang
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Correspondence to Chen-Fu Chiang.

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Chiang, CF. Selecting efficient phase estimation with constant-precision phase shift operators. Quantum Inf Process 13, 415–428 (2014). https://doi.org/10.1007/s11128-013-0659-9

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