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BQP/qpoly EXP/poly Scott Aaronson UC Berkeley

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BQP/qpoly Class of languages recognized by a bounded-error polytime quantum algorithm, with a polysize quantum advice state | n that depends only on the input size Buhrman: Is BQP/qpoly anything/poly?

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Our Result BQP/qpoly EXP/poly Means we shouldnt hope for an unrelativized separation between BQP/poly and BQP/qpolysince it would imply P/poly EXP/poly, which is equivalent to EXP P/poly

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Proof Sketch Given a BQP/qpoly algorithm, make error prob. exponentially small by taking | n p(n) as advice On input x {0,1} n, loop through all y x in lexicographic order For i {0,1}, let S i be set of advice states that cause algorithm to output i with prob. 1-c -n. Then there exist orthogonal subspaces H 0,H 1 s.t. all states in S i are exponentially close to H i To see this: acceptance probability on advice | can be written | x |, for some Hermitian p.s.d. x with eigenvalues in [0,1]. Let H 0,H 1 be subspaces spanned by eigenvectors of x corresponding to eigenvalues in [0,1/3], [2/3,1] respectively

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The Subspaces Let T y be subspace of | s compatible with inputs 1,…,y (initially T 0 = whole Hilbert space) Let T y = whichever has larger dimension: projection of T y-1 onto H 0, or projection of T y-1 onto H 1 Unless classical advice says to pick the subspace of smaller dimension! Each time we pick smaller subspace, dim(T y ) is at least halved. So advice needs to intervene only polynomially many times H1H1 H0H0

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The Subspaces Can do everything in EXP (diagonalize exponentially large matrix y, loop over all inputs, etc.) Main technical fact: Error (distance from T y to | n p(n) ) stays bounded over all iterations

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Open Problems Oracle separation between BQP/poly and BQP/qpoly Is BQP/qpoly PSPACE/poly? Is BQP/qpoly PP/poly relative to an oracle? Any natural problems in BQP/qpoly (besides cousins of QMA problems)?

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