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TR19-131 | 11th September 2019 20:50

A Simple Proof of Vyalyi's Theorem and some Generalizations


Authors: Lieuwe Vinkhuijzen, André Deutz
Publication: 30th September 2019 17:57
Downloads: 1007


In quantum computational complexity theory, the class QMA models the set of problems efficiently verifiable by a quantum computer the same way that NP models this for classical computation. Vyalyi proved that if $\text{QMA}=\text{PP}$ then $\text{PH}\subseteq \text{QMA}$. In this note, we give a simple, self-contained proof of the theorem, using only the closure properties of the complexity classes in the theorem statement. We then extend the theorem in two directions: (i) we strengthen the consequent, proving that if $\text{QMA}=\text{PP}$ then $\text{QMA}=\text{PH}^{\text{PP}}$, and (ii) we weaken the hypothesis, proving that if $\text{QMA}=\text{coQMA}$ then $\text{PH}\subseteq \text{QMA}$. Lastly, we show that all the above results hold, without loss of generality, for the class QAM instead of QMA. We also formulate a ``Quantum Toda's Conjecture''.

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