339 research outputs found
A universal adiabatic quantum query algorithm
Quantum query complexity is known to be characterized by the so-called
quantum adversary bound. While this result has been proved in the standard
discrete-time model of quantum computation, it also holds for continuous-time
(or Hamiltonian-based) quantum computation, due to a known equivalence between
these two query complexity models. In this work, we revisit this result by
providing a direct proof in the continuous-time model. One originality of our
proof is that it draws new connections between the adversary bound, a modern
technique of theoretical computer science, and early theorems of quantum
mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's
theorem, while the upper bound relies on the adiabatic theorem, as it goes by
constructing a universal adiabatic quantum query algorithm. Another originality
is that we use for the first time in the context of quantum computation a
version of the adiabatic theorem that does not require a spectral gap.Comment: 22 pages, compared to v1, includes a rigorous proof of the
correctness of the algorithm based on a version of the adiabatic theorem that
does not require a spectral ga
Quantum SDP-Solvers: Better upper and lower bounds
Brand\~ao and Svore very recently gave quantum algorithms for approximately
solving semidefinite programs, which in some regimes are faster than the
best-possible classical algorithms in terms of the dimension of the problem
and the number of constraints, but worse in terms of various other
parameters. In this paper we improve their algorithms in several ways, getting
better dependence on those other parameters. To this end we develop new
techniques for quantum algorithms, for instance a general way to efficiently
implement smooth functions of sparse Hamiltonians, and a generalized
minimum-finding procedure.
We also show limits on this approach to quantum SDP-solvers, for instance for
combinatorial optimizations problems that have a lot of symmetry. Finally, we
prove some general lower bounds showing that in the worst case, the complexity
of every quantum LP-solver (and hence also SDP-solver) has to scale linearly
with when , which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will
appear in Quantu
Quantum rejection sampling
Rejection sampling is a well-known method to sample from a target
distribution, given the ability to sample from a given distribution. The method
has been first formalized by von Neumann (1951) and has many applications in
classical computing. We define a quantum analogue of rejection sampling: given
a black box producing a coherent superposition of (possibly unknown) quantum
states with some amplitudes, the problem is to prepare a coherent superposition
of the same states, albeit with different target amplitudes. The main result of
this paper is a tight characterization of the query complexity of this quantum
state generation problem. We exhibit an algorithm, which we call quantum
rejection sampling, and analyze its cost using semidefinite programming. Our
proof of a matching lower bound is based on the automorphism principle which
allows to symmetrize any algorithm over the automorphism group of the problem.
Our main technical innovation is an extension of the automorphism principle to
continuous groups that arise for quantum state generation problems where the
oracle encodes unknown quantum states, instead of just classical data.
Furthermore, we illustrate how quantum rejection sampling may be used as a
primitive in designing quantum algorithms, by providing three different
applications. We first show that it was implicitly used in the quantum
algorithm for linear systems of equations by Harrow, Hassidim and Lloyd.
Secondly, we show that it can be used to speed up the main step in the quantum
Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum
algorithm for the hidden shift problem of an arbitrary Boolean function and
relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to
appear in proceedings of ITCS 2012
Quantum vs Classical Proofs and Subset Verification
We study the ability of efficient quantum verifiers to decide properties of
exponentially large subsets given either a classical or quantum witness. We
develop a general framework that can be used to prove that QCMA machines, with
only classical witnesses, cannot verify certain properties of subsets given
implicitly via an oracle. We use this framework to prove an oracle separation
between QCMA and QMA using an "in-place" permutation oracle, making the first
progress on this question since Aaronson and Kuperberg in 2007. We also use the
framework to prove a particularly simple standard oracle separation between
QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe
Hamiltonian Oracles
Hamiltonian oracles are the continuum limit of the standard unitary quantum
oracles. In this limit, the problem of finding the optimal query algorithm can
be mapped into the problem of finding shortest paths on a manifold. The study
of these shortest paths leads to lower bounds of the original unitary oracle
problem. A number of example Hamiltonian oracles are studied in this paper,
including oracle interrogation and the problem of computing the XOR of the
hidden bits. Both of these problems are related to the study of geodesics on
spheres with non-round metrics. For the case of two hidden bits a complete
description of the geodesics is given. For n hidden bits a simple lower bound
is proven that shows the problems require a query time proportional to n, even
in the continuum limit. Finally, the problem of continuous Grover search is
reexamined leading to a modest improvement to the protocol of Farhi and
Gutmann.Comment: 16 pages, REVTeX 4 (minor corrections in v2
Quantum query complexity of state conversion
State conversion generalizes query complexity to the problem of converting
between two input-dependent quantum states by making queries to the input. We
characterize the complexity of this problem by introducing a natural
information-theoretic norm that extends the Schur product operator norm. The
complexity of converting between two systems of states is given by the distance
between them, as measured by this norm.
In the special case of function evaluation, the norm is closely related to
the general adversary bound, a semi-definite program that lower-bounds the
number of input queries needed by a quantum algorithm to evaluate a function.
We thus obtain that the general adversary bound characterizes the quantum query
complexity of any function whatsoever. This generalizes and simplifies the
proof of the same result in the case of boolean input and output. Also in the
case of function evaluation, we show that our norm satisfies a remarkable
composition property, implying that the quantum query complexity of the
composition of two functions is at most the product of the query complexities
of the functions, up to a constant. Finally, our result implies that discrete
and continuous-time query models are equivalent in the bounded-error setting,
even for the general state-conversion problem.Comment: 19 pages, 2 figures; heavily revised with new results and simpler
proof
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