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BQP/qpoly  EXP/poly

BQP/qpoly  EXP/poly. Scott Aaronson UC Berkeley. 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?. Our Result.

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BQP/qpoly  EXP/poly

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  1. BQP/qpoly  EXP/poly Scott Aaronson UC Berkeley

  2. 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?

  3. Our Result • BQP/qpoly  EXP/poly • Means we shouldn’t hope for an unrelativized separation between BQP/poly and BQP/qpoly—since it would imply P/poly  EXP/poly, which is equivalent to EXP  P/poly

  4. 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 Si be set of advice states that cause algorithm to output i with prob. 1-c-n. Then there exist orthogonal subspaces H0,H1 s.t. all states in Si are exponentially close to Hi • To see this: acceptance probability on advice | can be written |x|, for some Hermitian p.s.d. x with eigenvalues in [0,1]. Let H0,H1 be subspaces spanned by eigenvectors of x corresponding to eigenvalues in [0,1/3], [2/3,1] respectively

  5. H1 The Subspaces H0 • Let Ty be subspace of |’s compatible with inputs 1,…,y (initially T0 = whole Hilbert space) • Let Ty = whichever has larger dimension: projection of Ty-1 onto H0, or projection of Ty-1 onto H1 • Unless classical advice says to pick the subspace of smaller dimension! • Each time we pick smaller subspace, dim(Ty) is at least halved. So advice needs to intervene only polynomially many times

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

  7. 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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