573 research outputs found
Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002],
we introduce a new query complexity model, which we call bomb query complexity
. We investigate its relationship with the usual quantum query complexity
, and show that .
This result gives a new method to upper bound the quantum query complexity:
we give a method of finding bomb query algorithms from classical algorithms,
which then provide nonconstructive upper bounds on .
We subsequently were able to give explicit quantum algorithms matching our
upper bound method. We apply this method on the single-source shortest paths
problem on unweighted graphs, obtaining an algorithm with quantum
query complexity, improving the best known algorithm of [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite
matching problem gives an algorithm, improving the best known
trivial upper bound.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof
that P(OR) = \Omega(N) remove
Oracles with Costs
While powerful tools have been developed to analyze quantum query complexity, there are still many natural problems that do not fit neatly into the black box model of oracles. We create a new model that allows multiple oracles with differing costs. This model captures more of the difficulty of certain natural problems. We test this model on a simple problem, Search with Two Oracles, for which we create a quantum algorithm that we prove is asymptotically optimal. We further give some evidence, using a geometric picture of Grover\u27s algorithm, that our algorithm is exactly optimal
On relating one-way classical and quantum communication complexities
Communication complexity is the amount of communication needed to compute a
function when the function inputs are distributed over multiple parties. In its
simplest form, one-way communication complexity, Alice and Bob compute a
function , where is given to Alice and is given to Bob, and
only one message from Alice to Bob is allowed. A fundamental question in
quantum information is the relationship between one-way quantum and classical
communication complexities, i.e., how much shorter the message can be if Alice
is sending a quantum state instead of bit strings? We make some progress
towards this question with the following results.
Let
be a partial function and be a distribution with support contained in
. Denote . Let
be the classical one-way communication
complexity of ; be the quantum one-way
communication complexity of and be the
entanglement-assisted quantum one-way communication complexity of , each
with distributional error (average error over ) at most . We
show:
1) If is a product distribution, and , then,
2)If is a non-product distribution and , then
such that ,
where
\[\mathsf{CS}(f) = \max_{y} \min_{z\in\{0,1\}} \vert \{x~|~f(x,y)=z\} \vert
\enspace.\
Efficient learning of -doped stabilizer states with single-copy measurements
One of the primary objectives in the field of quantum state learning is to
develop algorithms that are time-efficient for learning states generated from
quantum circuits. Earlier investigations have demonstrated time-efficient
algorithms for states generated from Clifford circuits with at most
non-Clifford gates. However, these algorithms necessitate multi-copy
measurements, posing implementation challenges in the near term due to the
requisite quantum memory. On the contrary, using solely single-qubit
measurements in the computational basis is insufficient in learning even the
output distribution of a Clifford circuit with one additional gate under
reasonable post-quantum cryptographic assumptions. In this work, we introduce
an efficient quantum algorithm that employs only nonadaptive single-copy
measurement to learn states produced by Clifford circuits with a maximum of
non-Clifford gates, filling a gap between the previous positive and
negative results.Comment: 6 page
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Semidefinite programming (SDP) is a central topic in mathematical
optimization with extensive studies on its efficient solvers. In this paper, we
present a proof-of-principle sublinear-time algorithm for solving SDPs with
low-rank constraints; specifically, given an SDP with constraint matrices,
each of dimension and rank , our algorithm can compute any entry and
efficient descriptions of the spectral decomposition of the solution matrix.
The algorithm runs in time
given access to a sampling-based low-overhead data structure for the constraint
matrices, where is the precision of the solution. In addition, we
apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix
multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2)
sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to
compute the matrix exponential required in the MMW framework, we introduce two
new techniques that may be of independent interest:
Weighted sampling: assuming sampling access to each individual
constraint matrix , we propose a procedure that gives a
good approximation of .
Symmetric approximation: we propose a sampling procedure that gives
the \emph{spectral decomposition} of a low-rank Hermitian matrix . To the
best of our knowledge, this is the first sampling-based algorithm for spectral
decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th
International Symposium on Mathematical Foundations of Computer Science (MFCS
2020
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