Multiple-particle interference and quantum error correction

The concept of multiple-particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K /n can be greater than 1 - 2H(2x/n), but must be less than 1 - 2H(2x/n) This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).

[1]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .

[2]  E. Gilbert.  A comparison of signalling alphabets , 1952 .

[3]  B. D'espagnat.  Conceptual Foundations Of Quantum Mechanics , 1971 .

[4]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[5]  Richard W. Hamming,et al.  Coding and Information Theory , 1980 .

[6]  W. Zurek.  Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse? , 1981 .

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

[8]  N. Gisin.  Quantum measurements and stochastic processes , 1984 .

[9]  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.

[10]  Richard Wesley Hamming,et al.  Coding and information theory (2. ed.) , 1986 .

[11]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[12]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[13]  M. Kafatos.  Bell's theorem, quantum theory and conceptions of the universe , 1989 .

[14]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[15]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[16]  W. Zurek.  The Environment, Decoherence and the Transition from Quantum to Classical , 1991 .

[17]  Zoller,et al.  Monte Carlo simulation of the atomic master equation for spontaneous emission. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[18]  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.

[19]  K. Mølmer,et al.  Wave-function approach to dissipative processes in quantum optics. , 1992, Physical review letters.

[20]  W. Zurek.  Preferred States, Predictability, Classicality and the Environment-Induced Decoherence , 1993 .

[21]  H. Carmichael.  An open systems approach to quantum optics , 1993 .

[22]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[23]  Klaus Mølmer,et al.  A Monte Carlo wave function method in quantum optics , 1993, Optical Society of America Annual Meeting.

[24]  Daniel R. Simon.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[25]  D. Deutsch,et al.  The stabilisation of quantum computations , 1994, Proceedings Workshop on Physics and Computation. PhysComp '94.

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

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

[28]  Simon J. D. Phoenix,et al.  Quantum cryptography: How to beat the code breakers using quantum mechanics , 1995 .

[29]  A. Ekert,et al.  Universality in quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[30]  A. Barenco.  A universal two-bit gate for quantum computation , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[31]  J. Cirac,et al.  Quantum Computations with Cold Trapped Ions. , 1995, Physical review letters.

[32]  Unruh.  Maintaining coherence in quantum computers. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[33]  DiVincenzo.  Two-bit gates are universal for quantum computation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

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

[35]  R. Jozsa,et al.  Quantum Computation and Shor's Factoring Algorithm , 1996 .

[36]  Deutsch,et al.  Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels. , 1996, Physical review letters.

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

[38]  Charles H. Bennett,et al.  Purification of noisy entanglement and faithful teleportation via noisy channels. , 1995, Physical review letters.

[39]  Artur Ekert,et al.  Quantum computers and dissipation , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[40]  N. Mermin.  Quantum theory: Concepts and methods , 1997 .

[41]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .