[QIP-Sem] QIP seminar, Tue 1/20, 4:15, 36-428, Sattath,Or

[Peter Shor]

MIT Quantum Information Processing seminar
Tuesday 1/20 at 4:15 in 36-428
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 Sattath,Or ()

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

Abstract:

In 1985, Valiant-Vazirani showed [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  on 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, within  polynomial accuracy, of the ground state energy of poly-gapped 1-D  local Hamiltonians is QCMA-hard, under randomized reductions. This  is in strong 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 loc
 al  Hamiltonian which allows the efficient calculation of expectation  values.Finally, we conjecture that an analogous result to the class Quantum  Merlin-Arthur(QMA) is impossible. This is formulated by separating  between the two relavant classes, QMA and its "unique" version UQMA,  using the definition by Aaronson and Kuperberg [AK06] of quantum  oracles.

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