Quantum Computation and Quantum Information

Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering. They will likely lead to a new wave of technological innovations in communication, computation and cryptography. As the theory of quantum physics is fundamentally stochastic, randomness and uncertainty are deeply rooted in quantum computation, quantum simulation and quantum information. Consequently quantum algorithms are random in nature, and quantum simulation utilizes Monte Carlo techniques extensively. Thus statistics can play an important role in quantum computation and quantum simulation, which in turn offer great potential to revolutionize computational statistics. While only pseudo-random numbers can be generated by classical computers, quantum computers are able to produce genuine random numbers; quantum computers can exponentially or quadratically speed up median evaluation, Monte Carlo integration and Markov chain simulation. This paper gives a brief review on quantum computation, quantum simulation and quantum information. We introduce the basic concepts of quantum computation and quantum simulation and present quantum algorithms that are known to be much faster than the available classic algorithms. We provide a statistical framework for the analysis of quantum algorithms and quantum simulation.

[1]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[2]  P. Grangier,et al.  Experimental Tests of Realistic Local Theories via Bell's Theorem , 1981 .

[3]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[4]  Felix Wu,et al.  The quantum query complexity of approximating the median and related statistics , 1998, STOC '99.

[5]  C. Butucea,et al.  Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data , 2005, math/0504058.

[6]  L. Artiles,et al.  An invitation to quantum tomography , 2003, quant-ph/0303020.

[7]  D. A. Edwards.  The mathematical foundations of quantum mechanics , 1979, Synthese.

[8]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[9]  Mazyar Mirrahimi,et al.  Real-time quantum feedback prepares and stabilizes photon number states , 2011, Nature.

[10]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[11]  Christof Zalka.  Simulating quantum systems on a quantum computer , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Yazhen Wang.  Quantum gaussian processes , 1994 .

[13]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[14]  Charles H. Bennett,et al.  Optimal Simulation of Two-Qubit Hamiltonians Using General Local Operations , 2001, quant-ph/0107035.

[15]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[16]  Stefan Heinrich.  From Monte Carlo to quantum computation , 2003, Math. Comput. Simul..

[17]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  M. Freedman,et al.  Simulation of Topological Field Theories¶by Quantum Computers , 2000, quant-ph/0001071.

[19]  B. Boghosian,et al.  Simulating quantum mechanics on a quantum computer , 1997, quant-ph/9701019.

[20]  M. Head‐Gordon,et al.  Simulated Quantum Computation of Molecular Energies , 2005, Science.

[21]  R. Cleve,et al.  Efficient Quantum Algorithms for Simulating Sparse Hamiltonians , 2005, quant-ph/0508139.

[22]  M. Nussbaum,et al.  THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING , 2006, quant-ph/0607216.

[23]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[24]  Elham Kashefi,et al.  Demonstration of Blind Quantum Computing , 2011, Science.

[25]  Amnon Ta-Shma,et al.  Adiabatic quantum state generation and statistical zero knowledge , 2003, STOC '03.

[26]  Erik Lucero,et al.  Implementing the Quantum von Neumann Architecture with Superconducting Circuits , 2011, Science.

[27]  P. Wocjan,et al.  Speedup via quantum sampling , 2008, 0804.4259.

[28]  Michel Devoret,et al.  Superconducting quantum bits , 2005 .

[29]  Kuldeep Kumar,et al.  Robust Statistics, 2nd edn , 2011 .

[30]  Shuntaro Takeda,et al.  Teleportation of Nonclassical Wave Packets of Light , 2011, Science.

[31]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[32]  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.

[33]  Stochastic Green function algorithm. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  J. Ignacio Cirac,et al.  Simulation of quantum dynamics with quantum optical systems , 2003, Quantum Inf. Comput..

[35]  Frédéric Magniez,et al.  Search via quantum walk , 2006, STOC '07.

[36]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[37]  Lov K. Grover,et al.  Quantum computation , 1999, Proceedings Twelfth International Conference on VLSI Design. (Cat. No.PR00013).

[38]  J. Wrachtrup,et al.  Multipartite Entanglement Among Single Spins in Diamond , 2008, Science.

[39]  Jiangfeng Du,et al.  Search via Quantum Walk , 2011 .

[40]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[41]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[42]  E. Davies,et al.  PROBABILISTIC AND STATISTICAL ASPECTS OF QUANTUM THEORY (North‐Holland Series in Statistics and Probability, 1) , 1984 .

[43]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[44]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[45]  林 正人.  Quantum information : an introduction , 2006 .

[46]  Avatar Tulsi,et al.  Faster quantum-walk algorithm for the two-dimensional spatial search , 2008, 0801.0497.

[47]  Andrew M. Childs.  On the Relationship Between Continuous- and Discrete-Time Quantum Walk , 2008, 0810.0312.

[48]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[49]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[50]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[51]  C. Umrigar,et al.  Quantum Monte Carlo methods in physics and chemistry , 1999 .

[52]  L. DiCarlo,et al.  Demonstration of two-qubit algorithms with a superconducting quantum processor , 2009, Nature.

[53]  Yazhen Wang.  Quantum Monte Carlo simulation , 2011 .

[54]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[55]  Peter C. Richter.  Quantum speedup of classical mixing processes , 2006, quant-ph/0609204.

[56]  M. W. Johnson,et al.  Quantum annealing with manufactured spins , 2011, Nature.

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

[58]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

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

[60]  O. Barndorff-Nielsen,et al.  On quantum statistical inference , 2003, quant-ph/0307189.

[61]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[62]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[63]  Principles of Quantum Mechanics: Index , 1999 .

[64]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[65]  Claus Kiefer.  On the Interpretation of Quantum Theory — from Copenhagen to the Present Day , 2002 .

[66]  D. Griffiths,et al.  Introduction to Quantum Mechanics , 1960 .

[67]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[68]  M. Kashiwara,et al.  The Campbell-Hausdorff Formula , 1978 .

[69]  Timothy F. Havel,et al.  EXPERIMENTAL QUANTUM ERROR CORRECTION , 1998, quant-ph/9802018.

[70]  Robin L. Hudson.  An introduction to quantum stochastic calculus , 2014 .

[71]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[72]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[73]  P. Grangier,et al.  Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell's Inequalities , 1982 .

[74]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[75]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[76]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

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