Degradation of Grover’s search under collective phase flips in queries to the oracle

Abstract

We address the case in which querying the oracle in Grover’s algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposing party that tries to prevent reception of correct data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study this problem, we introduce a simple model of collective phase distortions on the basis of a phase-damping channel. In the model, the probability of success is not altered via the oracle-box wires per se. Phase distortions of the considered type can hardly be detected via any one-time query to the oracle. However, the probability of success is significantly changed when such errors are introduced as an intermediate step in the Grover iteration. We investigate the probability of success with respect to variations of the parameter that characterizes the amount of phase errors. It turns out that the probability of success decreases significantly even if the error is not very high. Moreover, this probability quickly reduces to the value of one half, which corresponds to the completely mixed state. We also study trade-off relations between quantum coherence and the probability of success in the presence of noise of the considered type.

Keywords

Grover’s algorithm / phase noise / relative entropy of coherence

References

[1] A. Galindo and M. A. Martin-Delgado, Information and computation: Classical and quantum aspects, Rev. Mod. Phys. 74(2), 347 (2002)

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

[3] D. Haase and H. Maier, Quantum algorithms for number fields, Fortschr. Phys. 54(8–10), 866 (2006)

[4] S. Hallgren, Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem, J. Assoc. Comput. Mach. 54(1), 1 (2007)

[5] A. M. Childs and W. van Dam, Quantum algorithms for algebraic problems, Rev. Mod. Phys. 82(1), 1 (2010)

[6] L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79(2), 325 (1997)

[7] L. K. Grover, Quantum computers can search arbitrarily large databases by a single query, Phys. Rev. Lett. 79(23), 4709 (1997)

[8] L. K. Grover, Quantum computers can search rapidly by using almost any transformation, Phys. Rev. Lett. 80(19), 4329 (1998)

[9] A. D. Patel and L. K. Grover, Quantum search, in: M.- Y. Kao (Ed.), Encyclopedia of Algorithms, New York: Springer, 2016, pp 1707–1716

[10] S. J. Jr Lomonaco and L. H. Kauffman, Is Grover’s algorithm a quantum hidden subgroup algorithm? Quantum Inform. Process. 6(6), 461 (2007)

[11] C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput. 26(5), 1510 (1997)

[12] C. Zalka, Grover’s quantum searching algorithm is optimal, Phys. Rev. A 60(4), 2746 (1999)

[13] E. Biham, O. Biham, D. Biron, M. Grassl, and D. A. Lidar, Grover’s quantum search algorithm for an arbitrary initial amplitude distribution, Phys. Rev. A 60(4), 2742 (1999)

[14] A. Galindo and M. A. Martin-Delgado, Family of Grover’s quantum-searching algorithms, Phys. Rev. A 62(6), 062303 (2000)

[15] E. Biham, O. Biham, D. Biron, M. Grassl, D. A. Lidar, and D. Shapira, Analysis of generalized Grover quantum search algorithms using recursion equations, Phys. Rev. A 63(1), 012310 (2000)

[16] E. Biham and D. Kenigsberg, Grover’s quantum search algorithm for an arbitrary initial mixed state, Phys. Rev. A 66(6), 062301 (2002)

[17] J. Watrous, The Theory of Quantum Information, Cambridge: Cambridge University Press, 2018

[18] D. Deutsch, Quantum theory, the Church–Turing principle and the universal quantum computer, Proc. R. Soc. Lond. A 400(1818), 97 (1985)

[19] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000

[20] S. L. Braunstein and A. K. Pati, Speed-up and entanglement in quantum searching, Quantum Inf. Comput. 2, 399 (2002)

[21] R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proc. R. Soc. Lond. A 459(2036), 2011 (2003)

[22] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113(14), 140401 (2014)

[23] A. Streltsov, G. Adesso, and M. B. Plenio, Quantum coherence as a resource, Rev. Mod. Phys. 89(4), 041003 (2017)

[24] G. Adesso, T. R. Bromley, and M. Cianciaruso, Measures and applications of quantum correlations, J. Phys. A Math. Theor. 49(47), 473001 (2016)

[25] M. L. Hu and H. Fan, Relative quantum coherence, incompatibility, and quantum correlations of states, Phys. Rev. A 95(5), 052106 (2017)

[26] M. L. Hu, X. Hu, Y. Peng, Y. R. Zhang, and H. Fan, Quantum coherence and quantum correlations, arXiv: 1703.01852 [quant-ph] (2017)

[27] V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74(1), 197 (2002)

[28] A. E. Rastegin, Quantum-coherence quantifiers based on the Tsallis relative a entropies, Phys. Rev. A 93(3), 032136 (2016)

[29] E. Chitambar and G. Gour, Comparison of incoherent operations and measures of coherence, Phys. Rev. A 94(5), 052336 (2016)

[30] L. H. Shao, Y. Li, Y. Luo, and Z. Xi, Quantum coherence quantifiers based on the Rényi a-relative entropy, Commum. Theor. Phys. 67(6), 631 (2017)

[31] A. Streltsov, H. Kampermann, S. Wölk, M. Gessner, and D. Bruß, Maximal coherence and the resource theory of purity, New J. Phys. 20(5), 053058 (2018)

[32] L. H. Shao, Z. Xi, H. Fan, and Y. Li, Fidelity and tracenorm distances for quantifying coherence, Phys. Rev. A 91(4), 042120 (2015)

[33] S. Rana, P. Parashar, and M. Lewenstein, Tracedistance measure of coherence, Phys. Rev. A 93(1), 012110 (2016)

[34] H. J. Zhang, B. Chen, M. Li, S. M. Fei, and G. L. Long, Estimation on geometric measure of quantum coherence, Commum. Theor. Phys. 67(2), 166 (2017)

[35] S. Cheng and M. J. W. Hall, Complementarity relations for quantum coherence, Phys. Rev. A 92(4), 042101 (2015)

[36] U. Singh, A. K. Pati, and M. N. Bera, Uncertainty relations for quantum coherence, Mathematics 4(3), 47 (2016)

[37] Y. Peng, Y. R. Zhang, Z.Y. Fan, S. Liu, and H. Fan, Complementary relation of quantum coherence and quantum correlations in multiple measurements, arXiv: 1608.07950 [quant-ph] (2016)

[38] X. Yuan, G. Bai, T. Peng, and X. Ma, Quantum uncertainty relation using coherence, Phys. Rev. A 96(3), 032313 (2017)

[39] A. E. Rastegin, Uncertainty relations for quantum coherence with respect to mutually unbiased bases, Front. Phys. 13(1), 130304 (2018)

[40] M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, Duality of quantum coherence and path distinguishability, Phys. Rev. A 92(1), 012118 (2015)

[41] E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Relations between coherence and path information, Phys. Rev. Lett. 116(16), 160406 (2016)

[42] T. Qureshi and M. A. Siddiqui, Wave-particle duality in N-path interference, Ann. Phys. 385, 598 (2017)

[43] M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation, Phys. Rev. A 93(1), 012111 (2016)

[44] H. L. Shi, S. Y. Liu, X. H. Wang, W. L. Yang, Z. Y. Yang, and H. Fan, Coherence depletion in the Grover quantum search algorithm, Phys. Rev. A 95(3), 032307 (2017)

[45] N. Anand and A. K. Pati, Coherence and entanglement monogamy in the discrete analogue of analog Grover search, arXiv: 1611.04542 [quant-ph] (2016)

[46] A. E. Rastegin, On the role of dealing with quantum coherence in amplitude amplification, Quantum Inf. Progress 17(7), 179 (2018)

[47] D. Reitzner and M. Hillery, Grover search under localized dephasing, arXiv: 1712.06558 [quant-ph] (2017)

[48] M. Hillery, J. Bergou, and E. Feldman, Quantum walks based on an interferometric analogy, Phys. Rev. A 68(3), 032314 (2003)

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