109 research outputs found
An optimal quantum algorithm for the oracle identification problem
In the oracle identification problem, we are given oracle access to an
unknown N-bit string x promised to belong to a known set C of size M and our
task is to identify x. We present a quantum algorithm for the problem that is
optimal in its dependence on N and M. Our algorithm considerably simplifies and
improves the previous best algorithm due to Ambainis et al. Our algorithm also
has applications in quantum learning theory, where it improves the complexity
of exact learning with membership queries, resolving a conjecture of Hunziker
et al.
The algorithm is based on ideas from classical learning theory and a new
composition theorem for solutions of the filtered -norm semidefinite
program, which characterizes quantum query complexity. Our composition theorem
is quite general and allows us to compose quantum algorithms with
input-dependent query complexities without incurring a logarithmic overhead for
error reduction. As an application of the composition theorem, we remove all
log factors from the best known quantum algorithm for Boolean matrix
multiplication.Comment: 16 pages; v2: minor change
Simulating sparse Hamiltonians with star decompositions
We present an efficient algorithm for simulating the time evolution due to a
sparse Hamiltonian. In terms of the maximum degree d and dimension N of the
space on which the Hamiltonian H acts for time t, this algorithm uses
(d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the
sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders,
which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a
general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of
non-zero entries have the property that every connected component is a star,
and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
Dequantizing read-once quantum formulas
Quantum formulas, defined by Yao [FOCS '93], are the quantum analogs of
classical formulas, i.e., classical circuits in which all gates have fanout
one. We show that any read-once quantum formula over a gate set that contains
all single-qubit gates is equivalent to a read-once classical formula of the
same size and depth over an analogous classical gate set. For example, any
read-once quantum formula over Toffoli and single-qubit gates is equivalent to
a read-once classical formula over Toffoli and NOT gates. We then show that the
equivalence does not hold if the read-once restriction is removed. To show the
power of quantum formulas without the read-once restriction, we define a new
model of computation called the one-qubit model and show that it can compute
all boolean functions. This model may also be of independent interest.Comment: 14 pages, 8 figures, to appear in proceedings of TQC 201
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