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

TL;DR

This paper extends the Valiant-Vazirani theorem to quantum complexity classes, demonstrating the hardness of approximating ground state energies of certain quantum Hamiltonians and discussing challenges in quantum Merlin-Arthur classes.

Contribution

It generalizes the Valiant-Vazirani theorem to MA and QCMA, and explores implications for quantum Hamiltonian ground state energy approximation.

Findings

Estimating ground state energy of poly-gapped 1-D Hamiltonians is QCMA-hard.

Constant gapped 1-D Hamiltonians are in NP.

Random projections do not create polynomial gaps between quantum witnesses.

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. 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 [AN02], 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.