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  • Cited by 13
  • Print publication year: 2009
  • Online publication date: July 2014

7 - Extended Measurement Calculus

Summary

Abstract

Measurement-based quantum computation (MBQC) has emerged as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that takes unitary operations as fundamental. Among measurement-based quantum computation methods the recently introduced one-way quantum computer stands out as basic and fundamental. The key idea is to start from an entangled state and then use measurements and one-qubit unitaries, which may be dependent on the outcomes of measurements, to guide the computation. The main point is that one never has to perform unitaries on more than one qubit at a time after the initial preparation of an entangled state. The “programs” that one writes in this model are traditionally called “patterns.”

In this chapter, we develop a rigorous mathematical model underlying measurement-based quantum computation. We give syntax, operational semantics, denotational semantics, and an algebra of programs derived from the denotational semantics. We also present a rewrite theory and prove a general standardization theorem that allows all programs to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements, and expressiveness theorems.

We use our general measurement calculus not just to formalize the one-way model but also several other measurement-based models, e.g., Teleportation, Phase, and Pauli models, and present compositional embeddings of them into and from the one-way model.

Abramsky, S., and Coecke, B. (2004) A categorical semantics of quantum protocols. In Press, I. C. S., editor, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS). Quant-ph/0402130.
Aliferis, P., and Leung, D. W. (2004) Computation by measurements: a unifying picture. Physical Review A 70. Quant-ph/0404082.
Barendregt, H. P. (1984) The Lambda Calculus, Its Syntax and Semantics. Studies in Logic. North-Holland.
Benjamin, S., Eisert, J., and Stace, T. M. (2005) Optical generation of matter qubit graph states. New Journal of Physics 7. Quant-ph/0506110.
Benjamin, S. C., Browne, D. E., Fitzsimons, J., and Morton, J. J. L. (2006) Brokered graph state quantum computing. New Journal of Physics 8. Quant-ph/0509209.
Bennett, C., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., and Wootters, W. (1993) Teleporting an unknown quantum state via dual classical and EPR channels. Physical Review Letters.
Bernstein, E., and Vazirani, U. (1997) Quantum complexity theory. SIAM Journal of Computing 5(26).
Bravyi, S., and Kitaev, A. (2005) Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A 71. Quant-ph/0403025.
Broadbent, A., and Kashefi, E. (2009) On parallelizing quantum circuits. Theoretical Computer Science.
Browne, D., Kashefi, E., Mhalla, M., and Perdrix, S. (2007) Generalized flow and determinism in measurement-based quantum computation. New Journal of Physics 9.
Browne, D. E., and Rudolph, T. (2005) Resource-efficient linear optical quantum computation. Physical Review Letters 95. Quant-ph/0405157.
Chen, Q., Cheng, J., Wang, K. L., and Du, J. (2006) Efficient construction of two-dimensional cluster states with probabilistic quantum gates. Physical Review A 73. Quant-ph/0507066.
Childs, A. M., Leung, D. W., and Nielsen, M. A. (2005) Unified derivations of measurement-based schemes for quantum computation. Physical Review A 71. Quant-ph/0404132.
Clark, S. R., Alves, C. M., and Jaksch, D. (2005) Efficient generation of graph states for quantum computation. New Journal of Physics 7. Quant-ph/0406150.
Danos, V, and Kashefi, E. (2005) Pauli measurements are universal. In Selinger (2005b).
Danos, V, and Kashefi, E. (2006) Determinism in the one-way model. Physical Review A.
Danos, V, Kashefi, E., Olivier, H., and Silva, M. (2006) A direct approach to fault-tolerance in measurement-based quantum computation via teleportation. New Journal of Physics. Quant-ph/0611273.
Danos, V, Kashefi, E., and Panangaden, P. (2005) Parsimonious and robust realizations of unitary maps in the one-way model. Physical Review A 72.
Danos, V, Kashefi, E., and Panangaden, P. (2007) The measurement calculus. Journal of ACM.
Dawson, C. M., Haselgrove, H. L., and Nielsen, M. A. (2006) Noise thresholds for optical cluster-state quantum computation. Physical Review A 73. Quant-ph/0601066.
de Beaudrap, N. (2008) Finding flows in the one-way measurement model. Physical Review A 77.
de Beaudrap, N., Danos, V, and Kashefi, E. (2006) Phase map decomposition for unitaries. Quant-ph/0603266.
de Beaudrap, N., Danos, V, Kashefi, E., and Roetteler, M. (2008) Quadratic form expansions for unitaries. In Theory ofQuantum Computation, Communication, and Cryptography Third Workshop, TQC 2008 Tokyo, Japan, number 5106 in Lecture Notes in Computer Science.
den Nest, M. V., Dehaene, J., and Moor, B. D. (2004a) An efficient algorithm to recognize local clifford equivalence of graph states. Physical Review A 70. Quant-ph/0405023.
den Nest, M. V., Dehaene, J., and Moor, B. D. (2004b) Graphical description of the action of local clifford transformations on graph states. Physical Review A 69. Quant-ph/0308151.
Deutsch, D. (1985) Quantum theory, the Church-Turing principle and the universal quantum computer. In Proceedings of the Royal Society of London, volume A 400.
Deutsch, D. (1989) Quantum computational networks. Proceedings of the Royal Society of London A 425.
Diestel, R. (2005) Graph Theory. Springer-Verlag.
Duncan, R. (2005) An abstract approach to entanglement. Mathematical Structures In Quantum Informatics, QDay II.
Dür, W., Aschauer, H., and Briegel, H. J. (2003) Multiparticle entanglement purification for graph state. Physical Review Letters 91. Quant-ph/0303087.
Dürr, C., and Santha, M. (1996) A decision procedure for unitary linear quantum cellular automata. In Proceedings of FOCS'96 – Symposium on Foundations of Computer Science. LNCS. Quant-ph/9604007.
Gilbert, G., Hamrick, M., and Weinstein, Y. S. (2005) Efficient construction of photonic quantum computational clusters. Quant-ph/0512110.
Gottesman, D., and Chuang, I. L. (1999) Quantum teleportation is a universal computational primitive. Nature 402.
Gottesman, D. (1997) Stabilizer codes and quantum error correction. Ph.D. thesis, California Institute of Technology.
Griffiths, R., and Niu, C. (1996) Semiclassical Fourier transform for quantum computation. Physical Review Letters 76:3228–3231.
Hartmann, L., Dur, W., and Briegel, H. J. (2005) Steady state entanglement in open and noisy quantum systems at high temperature. Quant-ph/0512219.
Hein, M., Eisert, J., and Briegel, H. J. (2004) Multi-party entanglement in graph states. Physical Review A 69. Quant-ph/0307130.
Jorrand, P., and Perdrix, S. (2005) Unifying quantum computation with projective measurements only and one-way quantum computation. In Ozhigov, Y. I., editor, Quantum Informatics 2004, volume 5833 of SPIE Proceedings. Quant-ph/0404125.
Jozsa, R. (2005) An introduction to measurement based quantum computation. Quant-ph/0508124.
Kay, A., Pachos, J. K., and Adams, C. S. (2006) Graph-state preparation and quantum computation with global addressing of optical lattices. Physical Review A 73.
Leung, D. W. (2004) Quantum computation by measurements. International Journal of Quantum Information 2(1). Quant-ph/0310189.
Mhalla, M., and Perdrix, S. (2004) Complexity of graph state preparation. Quant-ph/0412071.
Mhalla, M., and Perdrix, S. (2008) Finding optimal flows efficiently. In Proceedings of 35th International Colloquium on Automata, Languages and Programming.
Moore, C., and Nilsson, M. (2002) Parallel quantum computation and quantum codes. SIAM Journal on Computing 31.
Nielsen, M. A. (2003) Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state. Physical Review A 308.
Nielsen, M. A. (2004) Optical quantum computation using cluster states. Physical Review Letters 93. Quant-ph/0402005.
Nielsen, M. A., and Chuang, I. L. (2000) Quantum Computation and Quantum Information. Cambridge University Press.
Perdrix, S. (2003) State transfer instead of teleportation in measurement-based quantum computation. International Journal of Quantum Information 3(1). Quant-ph/0402204.
Perdrix, S. (2006) Formal models of quantum computation: resources, abstract machines and measurement-based quantum computation. Ph.D. thesis, Institut National Polytechnique de Grenoble, Laboratoire Leibniz.
Perdrix, S. (2007) Towards minimal resources of measurement-based quantum computation. New Journal of Physics 9.
Perdrix, S., and Jorrand, P. (2004) Measurement-based quantum turing machines and their universality. Quant-ph/0404146.
Raussendorf, R., Anders, S., and Briegel, H. J. (2004) Fault-tolerant quantum computation using graph states. Communication to the Quantum Information and Quantum Control Conference, Fields Institute, Toronto. http://atlas-conferences.com/c/a/n/n/80.htm.
Raussendorf, R., and Briegel, H. J. (2001) A one-way quantum computer. Physical Review Letters 86.
Raussendorf, R., and Briegel, H. J. (2002) Computational model underlying the one-way quantum computer. Quantum Information & Computation 2. Quant-ph/0108067.
Raussendorf, R., Browne, D. E., and Briegel, H. J. (2003) Measurement-based quantum computation on cluster states. Physical Review A 68.
Schlingemann, D. (2003) Cluster states, algorithms and graphs. Quant-ph/0305170.
Schumacher, B., and Werner, R. F. (2004) Reversible quantum cellular automata. Quant-ph/0405174.
Selinger, P. (2004) Towards a quantum programming language. Mathematical Structures in Computer Science 14(4).
Selinger, P. (2005a) Dagger compact closed categories and completely positive maps. In Selinger (2005b).
Selinger, P., editor (2005b) Proceedings of the 3nd International Workshop on Quantum Programming Languages, Electronic Notes in Theoretical Computer Science.
Tame, M. S., Paternostro, M., Kim, M. S., and Vedral, V. (2004) Toward a more economical cluster state quantum computation. Quant-ph/0412156.
Tame, M. S., Paternostro, M., Kim, M. S., and Vedral, V. (2006) Natural three-qubit interactions in one-way quantum computing. Physical Review A 73. Quant-ph/0507173.
Unruh, D. (2005) Quantum programs with classical output streams. In Selinger (2005b).
van Dam, W. (1996) Quantum cellular automata. Master's thesis, Computer Science, Nijmegen.
Walther, P., k. J., Resch, Rudolph, T., Schenck, E., Weinfurter, H., Vedral, V., Aspelmeyer, M., and Zeilinger, A. (2005) Experimental one-way quantum computing. Nature 434. Quant-ph/0503126.
Watrous, J. (1995) On one-dimensional quantum cellular automata. In Proceedings of FOCS'95 – Symposium on Foundations of Computer Science. LNCS.