7,836 research outputs found
Quantum lower bound for inverting a permutation with advice
Given a random permutation as a black box and ,
we want to output . Supplementary to our input, we are given
classical advice in the form of a pre-computed data structure; this advice can
depend on the permutation but \emph{not} on the input . Classically, there
is a data structure of size and an algorithm that with the help
of the data structure, given , can invert in time , for
every choice of parameters , , such that . We prove a
quantum lower bound of for quantum
algorithms that invert a random permutation on an fraction of
inputs, where is the number of queries to and is the amount of
advice. This answers an open question of De et al.
We also give a quantum lower bound for the simpler but
related Yao's box problem, which is the problem of recovering a bit ,
given the ability to query an -bit string at any index except the
-th, and also given bits of advice that depend on but not on .Comment: To appear in Quantum Information & Computation. Revised version based
on referee comment
An Introduction to Quantum Computing for Non-Physicists
Richard Feynman's observation that quantum mechanical effects could not be
simulated efficiently on a computer led to speculation that computation in
general could be done more efficiently if it used quantum effects. This
speculation appeared justified when Peter Shor described a polynomial time
quantum algorithm for factoring integers.
In quantum systems, the computational space increases exponentially with the
size of the system which enables exponential parallelism. This parallelism
could lead to exponentially faster quantum algorithms than possible
classically. The catch is that accessing the results, which requires
measurement, proves tricky and requires new non-traditional programming
techniques.
The aim of this paper is to guide computer scientists and other
non-physicists through the conceptual and notational barriers that separate
quantum computing from conventional computing. We introduce basic principles of
quantum mechanics to explain where the power of quantum computers comes from
and why it is difficult to harness. We describe quantum cryptography,
teleportation, and dense coding. Various approaches to harnessing the power of
quantum parallelism are explained, including Shor's algorithm, Grover's
algorithm, and Hogg's algorithms. We conclude with a discussion of quantum
error correction.Comment: 45 pages. To appear in ACM Computing Surveys. LATEX file. Exposition
improved throughout thanks to reviewers' comment
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
Rank penalized estimation of a quantum system
We introduce a new method to reconstruct the density matrix of a
system of -qubits and estimate its rank from data obtained by quantum
state tomography measurements repeated times. The procedure consists in
minimizing the risk of a linear estimator of penalized by
given rank (from 1 to ), where is previously obtained by the
moment method. We obtain simultaneously an estimator of the rank and the
resulting density matrix associated to this rank. We establish an upper bound
for the error of penalized estimator, evaluated with the Frobenius norm, which
is of order and consistency for the estimator of the rank. The
proposed methodology is computationaly efficient and is illustrated with some
example states and real experimental data sets
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