Highlighting the Mechanism of the Quantum Speedup by Time-Symmetric and Relational Quantum Mechanics

Bob hides a ball in one of four drawers. Alice is to locate it. Classically she has to open up to three drawers, quantally just one. The fundamental reason for this quantum speedup is not known. The usual representation of the quantum algorithm is limited to the process of solving the problem. We extend it to the process of setting the problem. The number of the drawer with the ball becomes a unitary transformation of the random outcome of the preparation measurement. This extended, time-symmetric, representation brings in relational quantum mechanics. It is with respect to Bob and any external observer and cannot be with respect to Alice. It would tell her the number of the drawer with the ball before she opens any drawer. To Alice, the projection of the quantum state due to the preparation measurement should be retarded at the end of her search; in the input state of the search, the drawer number is determined to Bob and undetermined to Alice. We show that, mathematically, one can ascribe any part of the selection of the random outcome of the preparation measurement to the final Alice’s measurement. Ascribing half of it explains the speedup of the present algorithm. This leaves the input state to Bob unaltered and projects that to Alice on a state of lower entropy where she knows half of the number of the drawer with the ball in advance. The quantum algorithm turns out to be a sum over histories in each of which Alice knows in advance that the ball is in a pair of drawers and locates it by opening one of the two. In the sample of quantum algorithms examined, the part of the random outcome of the initial measurement selected by the final measurement is one half or slightly above it. Conversely, given an oracle problem, the assumption it is one half always corresponds to an existing quantum algorithm and gives the order of magnitude of the number of oracle queries required by the optimal one.

[1]  F. M. Toyama,et al.  Quantum search with certainty based on modified Grover algorithms: optimum choice of parameters , 2013, Quantum Inf. Process..

[2]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[3]  Fumiaki Morikoshi.  M ay 2 00 6 Information-theoretic temporal Bell inequality and quantum computation , 2006 .

[4]  Satosi Watanabe,et al.  Symmetry of Physical Laws. Part III. Prediction and Retrodiction , 1955 .

[5]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[6]  G. Long.  Grover algorithm with zero theoretical failure rate , 2001, quant-ph/0106071.

[7]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[8]  S Bose,et al.  Communication capacity of quantum computation. , 2000, Physical review letters.

[9]  F. Morikoshi.  Problem-Solution Symmetry in Grover’s Quantum Search Algorithm , 2011 .

[10]  Space-time structure in high energy interactions , 1969 .

[11]  Stephen W. Hawking,et al.  On The Shoulders Of Giants , 2016 .

[12]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[13]  Garg,et al.  Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? , 1985, Physical review letters.

[14]  Sandu Popescu,et al.  A time-symmetric formulation of quantum mechanics , 2010 .

[15]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[16]  Michele Mosca,et al.  The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer , 1998, QCQC.

[17]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[18]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[19]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[20]  D. Pines.  A Collective Description of-Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas , 1953 .

[21]  S. Braunstein,et al.  Information-theoretic Bell inequalities. , 1988, Physical review letters.

[22]  Giuseppe Castagnoli,et al.  Theory of the quantum speed-up , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[23]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[24]  J. Lebowitz,et al.  TIME SYMMETRY IN THE QUANTUM PROCESS OF MEASUREMENT , 1964 .

[25]  G. Castagnoli.  The mechanism of the quantum speed up , 2010, 1101.4359.

[26]  R. Feynman.  Simulating physics with computers , 1999 .

[27]  A. Elitzur,et al.  Non-Sequential Behavior of the Wave Function , 2001, quant-ph/0102109.

[28]  Vaidman,et al.  How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. , 1988, Physical review letters.

[29]  A. Zeilinger,et al.  A snapshot of foundational attitudes toward quantum mechanics , 2013, 1301.1069.

[30]  G. Castagnoli.  The Mechanism of Quantum Computation , 2008 .

[31]  C. Rovelli,et al.  Relational Quantum Mechanics , 2006 .