Finding flows in the one-way measurement model

The one-way measurement model is a framework for universal quantum computation in which algorithms are partially described by a graph G of entanglement relations on a collection of qubits. A sufficient condition for an algorithm to perform a unitary embedding between two Hilbert spaces is for the graph G, together with input and output I, O vertices I,O is contained in V(G), to have a flow in the sense introduced by Danos and Kashefi [Phys. Rev. A 74, 052310 (2006)]. For the special case of |I|=|O|, using a graph-theoretic characterization, I show that such flows are unique when they exist. This leads to an efficient algorithm for finding flows by a reduction to solved problems in graph theory.

[1]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  Elham Kashefi,et al.  The measurement calculus , 2004, JACM.

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

[4]  Michael A. Nielsen,et al.  Fault-tolerant quantum computation with cluster states , 2005 .

[5]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[6]  E. Kashefi,et al.  Generalized flow and determinism in measurement-based quantum computation , 2007, quant-ph/0702212.

[7]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[8]  Esko Nuutila,et al.  Efficient transitive closure computation in large digraphs , 1995 .

[9]  Niel de Beaudrap.  A complete algorithm to find flows in the one-way measurement model , 2006 .

[10]  John Clark,et al.  A First Look at Graph Theory , 1991 .

[11]  Niel de Beaudrap,et al.  Characterizing & constructing flows in the one-way measurement model in terms of disjoint I--O paths , 2006 .

[12]  J. Niel de Beaudrap,et al.  An extremal result for geometries in the one-way measurement model , 2008, Quantum Inf. Comput..

[13]  Elham Kashefi,et al.  Phase map decompositions for unitaries , 2006 .

[14]  C. Moura Alves,et al.  Efficient generation of graph states for quantum computation , 2005 .

[15]  S. C. Benjamin,et al.  Optical generation of matter qubit graph states , 2005, quant-ph/0506110.

[16]  M. Horodecki,et al.  Properties of quantum nonsignaling boxes , 2006 .

[17]  P. Panangaden,et al.  Parsimonious and robust realizations of unitary maps in the one-way model , 2005 .

[18]  Hans-J. Briegel,et al.  Computational model underlying the one-way quantum computer , 2002, Quantum Inf. Comput..

[19]  K.J.Resch,et al.  Experimental One-Way Quantum Computing , 2005, quant-ph/0503126.

[20]  Elham Kashefi,et al.  Parallelizing quantum circuits , 2007, Theor. Comput. Sci..

[21]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[22]  M. Nielsen.  Optical quantum computation using cluster States. , 2004, Physical review letters.

[23]  Pieter Kok,et al.  Efficient high-fidelity quantum computation using matter qubits and linear optics , 2005 .

[24]  Frank Harary,et al.  Graph Theory , 2016 .

[25]  E. Kashefi,et al.  Determinism in the one-way model , 2005, quant-ph/0506062.

[26]  Dan E. Browne,et al.  Efficient linear optical quantum computation , 2004 .