Graph States, Pivot Minor, and Universality of (X, Z)-Measurements

The graph state formalism offers strong connections between quantum information processing and graph theory. Exploring these connections, first we show that any graph is a pivot-minor of a planar graph, and even a pivot minor of a triangular grid. Then, we prove that the application of measurements in the (X,Z) plane (i.e. one-qubit measurement according to the basis {cos(�)|0i+ sin(�)|1i,sin(�)|0i −cos(�)|1i} for some �) over graph states represented by triangular grids is a universal measurementbased model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.

[1]  Salman Beigi,et al.  Graph concatenation for quantum codes , 2009, 0910.4129.

[2]  Michael A. Nielsen,et al.  Quantum computation by measurement and quantum memory , 2003 .

[3]  Simon Perdrix.  STATE TRANSFER INSTEAD OF TELEPORTATION IN MEASUREMENT-BASED QUANTUM COMPUTATION , 2005 .

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

[5]  Ronald de Wolf,et al.  Quantum Proofs for Classical Theorems , 2009, Theory Comput..

[6]  Simon Perdrix.  Towards minimal resources of measurement-based quantum computation , 2007 .

[7]  Bart De Moor,et al.  Edge-local equivalence of graphs , 2005 .

[8]  W Dür,et al.  Multiparticle entanglement purification for graph states. , 2003, Physical review letters.

[9]  A. Bouchet.  Connectivity of Isotropic Systems , 1989 .

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

[11]  Mehdi Mhalla,et al.  Which Graph States are Useful for Quantum Information Processing? , 2010, TQC.

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

[13]  Michael A. Nielsen,et al.  Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state , 2001 .

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

[15]  A. Zeilinger,et al.  Experimental one-way quantum computing , 2005, Nature.

[16]  André Bouchet,et al.  Graphic presentations of isotropic systems , 1987, J. Comb. Theory, Ser. B.

[17]  Elham Kashefi,et al.  Computational Depth Complexity of Measurement-Based Quantum Computation , 2009, TQC.

[18]  R. Prevedel,et al.  High-speed linear optics quantum computing using active feed-forward , 2007, Nature.

[19]  J. Eisert,et al.  Entanglement in Graph States and its Applications , 2006, quant-ph/0602096.

[20]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

[21]  Debbie W. Leung,et al.  Quantum computation by measurements , 2003 .

[22]  Mehdi Mhalla,et al.  Finding Optimal Flows Efficiently , 2007, ICALP.

[23]  Wolfgang Dür,et al.  Universal resources for measurement-based quantum computation. , 2006, Physical review letters.

[24]  Hubert de Fraysseix,et al.  Local complementation and interlacement graphs , 1981, Discret. Math..

[25]  Elham Kashefi,et al.  Extended Measurement Calculus , 2009 .

[26]  Sang-il Oum,et al.  Excluding a bipartite circle graph from line graphs , 2009, J. Graph Theory.

[27]  D. Markham,et al.  Graph states for quantum secret sharing , 2008, 0808.1532.

[28]  SIMPLE SETS OF MEASUREMENTS FOR UNIVERSAL QUANTUM COMPUTATION AND GRAPH STATE PREPARATION , 2010 .

[29]  Elham Kashefi,et al.  Universal Blind Quantum Computation , 2008, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[30]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

[31]  Sang-il Oum,et al.  Rank-width and vertex-minors , 2005, J. Comb. Theory, Ser. B.

[32]  Mehdi Mhalla,et al.  New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States , 2011, TQC.

[33]  Damian Markham,et al.  Information Flow in Secret Sharing Protocols , 2009, DCM.

[34]  James F. Geelen,et al.  A Generalization of Tutte's Characterization of Totally Unimodular Matrices , 1997, J. Comb. Theory, Ser. B.

[35]  Mehdi Mhalla,et al.  Resources Required for Preparing Graph States , 2006, ISAAC.

[36]  Sylvain Gravier,et al.  Optimal accessing and non-accessing structures for graph protocols , 2011, ArXiv.

[37]  André Bouchet,et al.  Circle Graph Obstructions , 1994, J. Comb. Theory, Ser. B.

[38]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory, Ser. B.

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