The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings

Dorit Aharonov1, Michael Ben-Or1, Fernando G.S.L. Brandão2,3, and Or Sattath4

1School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel.
2AWS Center for Quantum Computing, Pasadena, CA 91125, USA
3IQIM, California Institute of Technology, Pasadena, CA 91125, USA.
4Computer Science Department, Ben-Gurion University of the Negev, Beer-Sheva, Israel.

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Valiant-Vazirani showed in 1985 [45] that solving NP with the promise that "yes" instances have only one witness is powerful enough to solve the entire NP class (under randomized reductions).
We are interested in extending this result to the quantum setting. We prove extensions to the classes Merlin-Arthur MA and Quantum-Classical-Merlin-Arthur QCMA [7]. Our results have implications for the complexity of approximating the ground state energy of a quantum local Hamiltonian with a unique ground state and an $\textit{inverse polynomial}$ spectral gap. We show that the estimation (to within polynomial accuracy) of the ground state energy of poly-gapped 1-D local Hamiltonians is QCMA-hard, under randomized reductions. This is in stark contrast to the case of constant gapped 1-D Hamiltonians, which is in NP [24]. Moreover, it shows that unless QCMA can be reduced to NP by randomized reductions, there is no classical description of the ground state of every poly-gapped local Hamiltonian that allows efficient calculation of expectation values.
Finally, we discuss a few of the obstacles to the establishment of an analogous result to the class Quantum-Merlin-Arthur (QMA). In particular, we show that random projections fail to provide a polynomial gap between two witnesses.

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[2] Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka, "Quantum generalizations of the polynomial hierarchy with applications to QMA(2)", computational complexity 31 2, 13 (2022).

[3] Sergey Bravyi, Anirban Chowdhury, David Gosset, and Pawel Wocjan, "Quantum Hamiltonian complexity in thermal equilibrium", Nature Physics 18 11, 1367 (2022).

[4] Tamara Kohler, Stephen Piddock, Johannes Bausch, and Toby Cubitt, "General Conditions for Universality of Quantum Hamiltonians", PRX Quantum 3 1, 010307 (2022).

[5] Dorit Aharonov, Itai Arad, and Thomas Vidick, "The Quantum PCP Conjecture", arXiv:1309.7495, (2013).

[6] Jordi Weggemans, Marten Folkertsma, and Chris Cade, "Guidable Local Hamiltonian Problems with Implications to Heuristic Ansätze State Preparation and the Quantum PCP Conjecture", arXiv:2302.11578, (2023).

[7] Brielin Brown, Steven T. Flammia, and Norbert Schuch, "Computational Difficulty of Computing the Density of States", Physical Review Letters 107 4, 040501 (2011).

[8] Thomas Vidick and John Watrous, "Quantum Proofs", arXiv:1610.01664, (2016).

[9] Abhinav Deshpande, Alexey V. Gorshkov, and Bill Fefferman, "Importance of the spectral gap in estimating ground-state energies", arXiv:2007.11582, (2020).

[10] Sevag Gharibian, "Approximation, Proof Systems, and Correlations in a Quantum World", Ph.D. Thesis (2013).

[11] Andris Ambainis, "On physical problems that are slightly more difficult than QMA", arXiv:1312.4758, (2013).

[12] Sevag Gharibian, Jamie Sikora, and Sarvagya Upadhyay, "QMA variants with polynomially many provers", arXiv:1108.0617, (2011).

[13] Serge Massar and Miklos Santha, "Characterising the intersection of QMA and coQMA", Quantum Information Processing 20 12, 396 (2021).

[14] Sevag Gharibian and Justin Yirka, "The complexity of simulating local measurements on quantum systems", Quantum 3, 189 (2019).

[15] Rahul Jain, Iordanis Kerenidis, Greg Kuperberg, Miklos Santha, Or Sattath, and Shengyu Zhang, "On the power of a unique quantum witness", arXiv:0906.4425, (2009).

[16] Sevag Gharibian and Justin Yirka, "The complexity of simulating local measurements on quantum systems", arXiv:1606.05626, (2016).

[17] Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka, "Quantum generalizations of the polynomial hierarchy with applications to QMA(2)", arXiv:1805.11139, (2018).

[18] Serge Massar and Miklos Santha, "Total functions in QMA", Quantum Information Processing 20 1, 35 (2021).

[19] Dorit Aharonov, Michael Ben-Or, Fernando G. S. L. Brandão, and Or Sattath, "The Pursuit of Uniqueness: Extending Valiant-Vazirani Theorem to the Probabilistic and Quantum Settings", Quantum 6, 668 (2022).

[20] Tamara Kohler, Stephen Piddock, Johannes Bausch, and Toby Cubitt, "General Conditions for Universality of Quantum Hamiltonians", PRX Quantum 3 1, 010308 (2022).

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