6,487 research outputs found
Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function
The general adversary bound is a semi-definite program (SDP) that
lower-bounds the quantum query complexity of a function. We turn this lower
bound into an upper bound, by giving a quantum walk algorithm based on the dual
SDP that has query complexity at most the general adversary bound, up to a
logarithmic factor.
In more detail, the proof has two steps, each based on "span programs," a
certain linear-algebraic model of computation. First, we give an SDP that
outputs for any boolean function a span program computing it that has optimal
"witness size." The optimal witness size is shown to coincide with the general
adversary lower bound. Second, we give a quantum algorithm for evaluating span
programs with only a logarithmic query overhead on the witness size.
The first result is motivated by a quantum algorithm for evaluating composed
span programs. The algorithm is known to be optimal for evaluating a large
class of formulas. The allowed gates include all constant-size functions for
which there is an optimal span program. So far, good span programs have been
found in an ad hoc manner, and the SDP automates this procedure. Surprisingly,
the SDP's value equals the general adversary bound. A corollary is an optimal
quantum algorithm for evaluating "balanced" formulas over any finite boolean
gate set. The second result extends span programs' applicability beyond the
formula evaluation problem.
A strong universality result for span programs follows. A good quantum query
algorithm for a problem implies a good span program, and vice versa. Although
nearly tight, this equivalence is nontrivial. Span programs are a promising
model for developing more quantum algorithms.Comment: 70 pages, 2 figure
New Developments in Quantum Algorithms
In this survey, we describe two recent developments in quantum algorithms.
The first new development is a quantum algorithm for evaluating a Boolean
formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This
provides quantum speedups for any problem that can be expressed via Boolean
formulas. This result can be also extended to span problems, a generalization
of Boolean formulas. This provides an optimal quantum algorithm for any Boolean
function in the black-box query model.
The second new development is a quantum algorithm for solving systems of
linear equations. In contrast with traditional algorithms that run in time
O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in
time O(\log^c N). It outputs a quantum state describing the solution of the
system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201
Span programs and quantum algorithms for st-connectivity and claw detection
We introduce a span program that decides st-connectivity, and generalize the
span program to develop quantum algorithms for several graph problems. First,
we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries
to the n x n adjacency matrix to decide if vertices s and t are connected,
under the promise that they either are connected by a path of length at most d,
or are disconnected. We also show that if T is a path, a star with two
subdivided legs, or a subdivision of a claw, its presence as a subgraph in the
input graph G can be detected with O(n) quantum queries to the adjacency
matrix. Under the promise that G either contains T as a subgraph or does not
contain T as a minor, we give O(n)-query quantum algorithms for detecting T
either a triangle or a subdivision of a star. All these algorithms can be
implemented time efficiently and, except for the triangle-detection algorithm,
in logarithmic space. One of the main techniques is to modify the
st-connectivity span program to drop along the way "breadcrumbs," which must be
retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure
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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