A Quantum Approach to the Unique Sink Orientation Problem

We consider quantum algorithms for the unique sink orientation problem on cubes. This problem is widely considered to be of intermediate computational complexity. This is because there no known polynomial algorithm (classical or quantum) from the problem and yet it arrises as part of a series of problems for which it being intractable would imply complexity theoretic collapses. We give a reduction which proves that if one can efficiently evaluate the kth power of the unique sink orientation outmap, then there exists a polynomial time quantum algorithm for the unique sink orientation problem on cubes.

[1]  Kathy Williamson Hoke,et al.  Completely unimodal numberings of a simple polytope , 1988, Discret. Appl. Math..

[2]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[3]  N. Megiddo.  A Note on the Complexity of P � Matrix LCP and Computing an Equilibrium , 1988 .

[4]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

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

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

[7]  Bernd Gärtner.  Combinatorial Linear Programming: Geometry Can Help , 1998, RANDOM.

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

[9]  Tibor Szabó,et al.  Unique sink orientations of cubes , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[10]  Tibor Szabó,et al.  Finding the Sink Takes Some Time , 2002, ESA.

[11]  Bernd Gärtner.  The Random-Facet simplex algorithm on combinatorial cubes , 2002, Random Struct. Algorithms.

[12]  Jirí Matousek,et al.  Random edge can be exponential on abstract cubes , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[13]  Bernd Gärtner,et al.  The smallest enclosing ball of balls: combinatorial structure and algorithms , 2004, Int. J. Comput. Geom. Appl..

[14]  Bernd Gärtner,et al.  Simple Stochastic Games and P-Matrix Generalized Linear Complementarity Problems , 2005, FCT.

[15]  O. Svensson.  Mean Payoff Games and Linear Complementarity , 2005 .

[16]  Bernd Gärtner,et al.  Linear programming and unique sink orientations , 2006, SODA '06.

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

[18]  Sergei G. Vorobyov,et al.  Cyclic games and linear programming , 2008, Discret. Appl. Math..

[19]  Sebastian Dörn.  Quantum Algorithms for Algebraic Problems ∗ , 2008 .

[20]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[21]  Bernd Gärtner,et al.  Counting unique-sink orientations , 2014, Discret. Appl. Math..

[22]  Vitor Bosshard,et al.  Classical and Quantum Algorithms for USO Recognition , 2015 .

[23]  Bernd Gärtner,et al.  The Complexity of Recognizing Unique Sink Orientations , 2015, STACS.

[24]  Mehdi Soleimanifar,et al.  No-go theorem for iterations of unknown quantum gates , 2016 .

[25]  D. Aharonov,et al.  Fast-forwarding of Hamiltonians and exponentially precise measurements , 2016, Nature Communications.