Fast amplification of QMA

Given a verifier circuit for a problem in QMA, we show how to exponentially amplifythe gap between its acceptance probabilities in the 'yes' and 'no' cases, with a methodthat is quadratically faster than the procedure given by Marriott and Watrous [1]. Ourconstruction is natively quantum, based on the analogy of a product of two reflections anda quantum walk. Second, in some special cases we show how to amplify the acceptanceprobability for good witnesses to 1, making a step towards the proof that QMA withone-sided error (QMA1) is equal to QMA. Finally, we simplify the filter-state method tosearch for QMA witnesses by Poulin and Wocjan [2].

[1]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

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

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Pawel Wocjan,et al.  The Jones polynomial: quantum algorithms and applications in quantum complexity theory , 2008, Quantum Inf. Comput..

[5]  Isaac L. Chuang,et al.  Quantum Information And Computation , 1996 .

[6]  John Watrous,et al.  Succinct quantum proofs for properties of finite groups , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[7]  Oded Goldreich,et al.  Another proof that bpp?ph (and more) , 1997 .

[8]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[9]  Scott Aaronson,et al.  On perfect completeness for QMA , 2008, Quantum Inf. Comput..

[10]  Stathis Zachos,et al.  Probabalistic Quantifiers vs. Distrustful Adversaries , 1987, FSTTCS.

[11]  D. Poulin,et al.  Preparing ground States of quantum many-body systems on a quantum computer. , 2008, Physical review letters.

[12]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[13]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

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

[15]  W. Browder,et al.  Annals of Mathematics , 1889 .

[16]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[17]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

[18]  Oded Goldreich,et al.  On Promise Problems (a survey in memory of Shimon Even [1935-2004]) , 2005, Electron. Colloquium Comput. Complex..

[19]  Chris Marriott,et al.  Quantum Arthur–Merlin games , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[20]  Pawel Wocjan,et al.  A Simple PromiseBQP-complete Matrix Problem , 2007, Theory Comput..

[21]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[22]  Sergey Bravyi,et al.  Efficient algorithm for a quantum analogue of 2-SAT , 2006, quant-ph/0602108.