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arXiv:1806.09732

QMA(2) with postselection equals to NEXP

Kinoshita, Yusuke

Published Jun 25, 2018• 0 citations
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Abstract

We study the power of QMA(2) with postselection and show that the power is equal to NEXP. Our method for showing this equality can be also used to prove that other classes with exponentially small completeness-soundness gap equals to the corresponding postselection versions.
DOI
10.48550/arxiv.1806.09732

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•Jan 31, 2015•1 citation
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We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆\subseteq⊆ NEXP.
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We consider restricted versions of postBQP{\rm postBQP}postBQP where postselection probabilities can be efficiently calculated by classical or quantum computers. We show that such restricted postBQP{\rm postBQP}postBQP classes are in AWPP(⊆PP=postBQP){\rm AWPP}(\subseteq{\rm PP}={\rm postBQP})AWPP(⊆PP=postBQP). This result suggests that postselecting an event with an exponentially small probability does not necessarily boost BQP{\rm BQP}BQP to PP. The best upperbound of BQP{\rm BQP}BQP is AWPP{\rm AWPP}AWPP, and therefore restricted postBQP{\rm postBQP}postBQP classes are other examples of complexity classes slightly above BQP{\rm BQP}BQP. In [C. M. Lee and J. Barrett, arXiv:1412.8671], it was shown that the computational capacity of a general probabilistic theory which satisfies the tomographic locality is in AWPP{\rm AWPP}AWPP. Our result therefore implies that quantum physics with restricted postselections is an example of a super quantum theory which seems to be outside of their general probabilistic theory but its computational capacity is also in AWPP{\rm AWPP}AWPP. Although it is physically natural to expect that such restricted postBQP{\rm postBQP}postBQP classes are not equivalent to BQP, we show that UP∩coUP{\rm UP}\cap{\rm coUP}UP∩coUP, which is unlikely to be in BQP, is contained in the restricted postBQP{\rm postBQP}postBQP classes. Finally, we also consider another restricted class of postBQP{\rm postBQP}postBQP, where the postselected probability depends only on the size of inputs. We show that this restricted version is contained in APP{\rm APP}APP, where AWPP⊆APP⊆PP=postBQP{\rm AWPP}\subseteq{\rm APP}\subseteq{\rm PP}={\rm postBQP}AWPP⊆APP⊆PP=postBQP.