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

Dorit Aharonov, Michael Ben-Or, Fernando G.S.L. Brandão, Or Sattath

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4 Scopus citations

Abstract

Valiant-Vazirani showed in 1985 [VV85] 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) [AN02]. 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 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 [Has07]. 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.

Original languageAmerican English
Number of pages27
JournalQuantum
Volume6
DOIs
StatePublished - 1 Jan 2022

Bibliographical note

Funding Information:
This work is part of the QIP-IRC supported by EPSRC (GR/S82176/0) as well as the Integrated Project Qubit Applications (QAP) supported by the IST directorate as Contract Number 015848' and was supported by an EPSRC Postdoctoral Fellowship for Theoretical Physics. Most of this work was done while O.S. was at the Hebrew University, and F.G.S.L.B. was at the Imperial College London.

Funding Information:
This work is part of the QIP-IRC supported by EPSRC (GR/S82176/0) as well as the Integrated Project Qubit Applications (QAP) supported by the IST directorate as Contract Number 015848’ and was supported by an EPSRC Postdoctoral Fellowship for Theoretical Physics. Most of this work was done while O.S. was at the Hebrew University, and F.G.S.L.B. was at the Imperial College London.

Publisher Copyright:
© The Author(s), 2021.

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