dorsal/arxiv
View SchemaQuantum Versus Classical Proofs and Advice
| Authors | Scott Aaronson, Greg Kuperberg |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0604056 |
| URL | https://arxiv.org/abs/quant-ph/0604056 |
| Journal | Theory Comput. 3 (2007), 129-157 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCMA. More concretely, we show that any quantum algorithm needs $\Omega(\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit "marked state" $\lvert\psi\rangle$, even if given an $m$-bit classical description of $\lvert\psi\rangle$ together with a quantum black box that recognizes $\lvert\psi\rangle$. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.
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"abstract": "This paper studies whether quantum proofs are more powerful than classical\nproofs, or in complexity terms, whether QMA=QCMA. We prove three results about\nthis question. First, we give a \"quantum oracle separation\" between QMA and\nQCMA. More concretely, we show that any quantum algorithm needs\n$\\Omega(\\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit \"marked state\"\n$\\lvert\\psi\\rangle$, even if given an $m$-bit classical description of\n$\\lvert\\psi\\rangle$ together with a quantum black box that recognizes\n$\\lvert\\psi\\rangle$. Second, we give an explicit QCMA protocol that nearly\nachieves this lower bound. Third, we show that, in the one previously-known\ncase where quantum proofs seemed to provide an exponential advantage, classical\nproofs are basically just as powerful. In particular, Watrous gave a QMA\nprotocol for verifying non-membership in finite groups. Under plausible\ngroup-theoretic assumptions, we give a QCMA protocol for the same problem. Even\nwith no assumptions, our protocol makes only polynomially many queries to the\ngroup oracle. We end with some conjectures about quantum versus classical\noracles, and about the possibility of a classical oracle separation between QMA\nand QCMA.",
"arxiv_id": "quant-ph/0604056",
"authors": [
"Scott Aaronson",
"Greg Kuperberg"
],
"categories": [
"quant-ph",
"cs.CC"
],
"journal_ref": "Theory Comput. 3 (2007), 129-157",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Quantum Versus Classical Proofs and Advice",
"url": "https://arxiv.org/abs/quant-ph/0604056"
},
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