Achieving perfect completeness in classical-witness quantum merlin-arthur proof systems

This paper proves that classical-witness quantum Merlin-Arthur proof systems can achieve perfect completeness. That is, QCMA = QCMA1. This holds under any gate set with which the Hadamard and arbitrary classical reversible transformations can be exactly implemented, e.g., {Hadamard, Toffoli, NOT}. The proof is quantumly nonrelativizing, and uses a simple but novel quantum technique that additively adjusts the success probability, which may be of independent interest.

[1]  Pawel Wocjan,et al.  Two QCMA-complete problems , 2003, Quantum Inf. Comput..

[2]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[3]  Oded Goldreich,et al.  Interactive proof systems: Provers that never fail and random selection , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

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

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

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

[7]  E. Knill.  Quantum Randomness and Nondeterminism , 1996 .

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

[9]  Keiji Matsumoto,et al.  Using Entanglement in Quantum Multi-prover Interactive Proofs , 2008, Computational Complexity Conference.

[10]  Jin-Soo Kim,et al.  Quantum Database Search by a Single Query , 1998, QCQC.

[11]  Yong Zhang,et al.  Fast amplification of QMA , 2009, Quantum Inf. Comput..

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

[13]  Noam Nisan,et al.  Quantum circuits with mixed states , 1998, STOC '98.

[14]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[15]  Dong Pyo Chi,et al.  Quantum Database Searching by a Single Query , 1997 .

[16]  John Watrous,et al.  Quantum Computational Complexity , 2008, Encyclopedia of Complexity and Systems Science.

[17]  Yacov Yacobi,et al.  The Complexity of Promise Problems with Applications to Public-Key Cryptography , 1984, Inf. Control..

[18]  Ben Reichardt,et al.  Fault-Tolerant Quantum Computation , 2016, Encyclopedia of Algorithms.

[19]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[20]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[21]  Oded Goldreich,et al.  On Completeness and Soundness in Interactive Proof Systems , 1989, Adv. Comput. Res..

[22]  Jamie Sikora,et al.  QMA variants with polynomially many provers , 2011, Quantum Inf. Comput..

[23]  D. Aharonov.  A Simple Proof that Toffoli and Hadamard are Quantum Universal , 2003, quant-ph/0301040.

[24]  John Watrous,et al.  Zero-knowledge against quantum attacks , 2005, STOC '06.

[25]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

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

[27]  Dorit Aharonov,et al.  Quantum NP - A Survey , 2002, quant-ph/0210077.

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

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

[30]  Alexei Y. Kitaev,et al.  Parallelization, amplification, and exponential time simulation of quantum interactive proof systems , 2000, STOC '00.