A Universal Adiabatic Quantum Query Algorithm
Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.
Quantum Algorithms
Query Complexity
Adiabatic Quantum Computation
Adversary Method
163-179
Regular Paper
Mathieu
Brandeho
Mathieu Brandeho
Jérémie
Roland
Jérémie Roland
10.4230/LIPIcs.TQC.2015.163
Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, and Seth Lloyd. Adiabatic quantum computation is equivalent to standard quantum computation. In Proceedings of the 45th Annual Symposium on the Foundations of Computer Science, pages 42-51, New York, 2004. IEEE Computer Society Press.
Andris Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System Sciences, 64(4):750-767, 2002.
J. E. Avron, R. Seiler, and L. G. Yaffe. Adiabatic theorems and applications to the quantum Hall effect. Communications in Mathematical Physics, 110(1):33-49, March 1987.
Joseph E. Avron and Alexander Elgart. Adiabatic Theorem without a Gap Condition. Communications in Mathematical Physics, 203(2):445-463, June 1999.
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48:778-797, 2001.
M. Born and V. Fock. Beweis des adiabatensatzes. Zeitschrift für Physik, 51(3-4):165-180, 1928.
Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21-43, 2002.
Andrew Childs and Jeffrey Goldstone. Spatial search by quantum walk. Physical Review A, 70(2):022314, August 2004.
Andrew M. Childs. On the Relationship Between Continuous- and Discrete-Time Quantum Walk. Communications in Mathematical Physics, 294(2):581-603, October 2009.
Andrew M. Childs and Jeffrey Goldstone. Spatial search and the Dirac equation. Physical Review A, 70(4):042312, October 2004.
Richard Cleve, Daniel Gottesman, Michele Mosca, Rolando D. Somma, and David Yonge-Mallo. Efficient discrete-time simulations of continuous-time quantum query algorithms. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 409-416. ACM, 2009.
P. Ehrenfest. Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift fur Physik, 45:455-457, 1927.
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum algorithm for the Hamiltonian NAND tree. Theory of Computing, 4:169-190, 2008.
Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Quantum computation by adiabatic evolution. arXiv:0001106, 2000.
Edward Farhi and Sam Gutmann. An analog analogue of a digital quantum computation. arXiv:9612026, 1996.
Iain Foulger, Sven Gnutzmann, and Gregor Tanner. Quantum Search on Graphene Lattices. Physical Review Letters, 112(7):070504, February 2014.
Christopher A. Fuchs and Jeroen van De Graaf. Cryptographic distinguishability measures for quantum-mechanical states. In IEEE Transactions on Information Theory, volume 45, pages 1216-1227, 1999.
Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 526-535. ACM, 2007.
S. Jansen, M.-B. Ruskai, and R. Seiler. Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys., 48(10):102111, 2007.
Tosio Kato. On the Adiabatic Theorem of Quantum Mechanics. Journal of the Physical Society of Japan, 5(6):435-439, November 1950.
Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pages 344-353. IEEE Computer Society, 2011.
Troy Lee and Jérémie Roland. A strong direct product theorem for quantum query complexity. Computational Complexity, 22(2):429-462, 2013.
Carlos Mochon. Hamiltonian oracles. Physical Review A - Atomic, Molecular, and Optical Physics, 75(4), 2007.
M. Reed and B. Simon. Methods of modern mathematical physics. 2. Fourier analysis, self-adjointness. Fourier Analysis, Self-adjointness. Academic Press, 1975.
Ben Reichardt and Robert Špalek. Span-program-based quantum algorithm for evaluating formulas. Theory of Computing, 8(13):291-319, 2012.
Ben W. Reichardt. Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pages 544-551. IEEE Computer Society, 2009.
Ben W. Reichardt. Reflections for quantum query algorithms. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms, pages 560-569, 2011.
Jérémie Roland and Nicolas J. Cerf. Quantum search by local adiabatic evolution. Physical Review A, 65:042308, 2002.
W. van Dam, M. Mosca, and U. Vazirani. How powerful is adiabatic quantum computation? Proceedings 2001 IEEE International Conference on Cluster Computing, pages 279-287, 2002.
David Yonge-Mallo. Adversary lower bounds in the Hamiltonian oracle model. arXiv:1108.2479, 2011.
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