Characterising the intersection of QMA and coQMA

We show that the functional analogue of QMA ∩ coQMA, denoted F(QMA ∩ coQMA), equals the complexity class Total Functional QMA (TFQMA). To prove this we need to introduce an alternative definition of QMA ∩ coQMA in terms of a single quantum verification procedure. We show that if TFQMA equals the functional analogue of BQP (FBQP), then QMA ∩ coQMA = BQP. We show that if there is a QMA complete problem that (robustly) reduces to a problem in TFQMA, then QMA ∩ coQMA = QMA. Our results thus imply that if some of the inclusions between functional classes FBQP ⊆ TFQMA ⊆ FQMA are in fact equalities, then the corresponding inclusions in BQP ⊆ QMA ∩ coQMA ⊆ QMA are also equalities.

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

[2]  Mihalis Yannakakis,et al.  On the complexity of local search , 1990, STOC '90.

[3]  Mark W. Krentel,et al.  Structure in locally optimal solutions , 1989, 30th Annual Symposium on Foundations of Computer Science.

[4]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

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

[6]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[7]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

[8]  Dorit Aharonov,et al.  The Pursuit For Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings preliminary version , 2008 .

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

[10]  Pawel Wocjan,et al.  Cooling and Low Energy State Preparation for 3-local Hamiltonians are FQMA-complete , 2003 .

[11]  Miklos Santha,et al.  Total functions in QMA , 2018, Quantum Information Processing.

[12]  Adam D. Bookatz.  QMA-complete problems , 2012, Quantum Inf. Comput..

[13]  Christos H. Papadimitriou,et al.  On Total Functions, Existence Theorems and Computational Complexity , 1991, Theor. Comput. Sci..

[14]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[15]  Paul W. Goldberg,et al.  Towards a Unified Complexity Theory of Total Functions , 2017, ITCS.

[16]  Bill Fefferman,et al.  The importance of the spectral gap in estimating ground-state energies , 2020, ArXiv.

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