41,710 research outputs found
On the volume of the set of mixed entangled states II
The problem of of how many entangled or, respectively, separable states there
are in the set of all quantum states is investigated. We study to what extent
the choice of a measure in the space of density matrices describing
N--dimensional quantum systems affects the results obtained. We demonstrate
that the link between the purity of the mixed states and the probability of
entanglement is not sensitive to the measure chosen. Since the criterion of
partial transposition is not sufficient to distinguish all separable states for
N > 6, we develop an efficient algorithm to calculate numerically the
entanglement of formation of a given mixed quantum state, which allows us to
compute the volume of separable states for N=8 and to estimate the volume of
the bound entangled states in this case.Comment: 14 pages in Latex, Revtex + epsf; 7 figures in .ps included (one new
figure in the revised version, several minor changes
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
On the relationship between continuous- and discrete-time quantum walk
Quantum walk is one of the main tools for quantum algorithms. Defined by
analogy to classical random walk, a quantum walk is a time-homogeneous quantum
process on a graph. Both random and quantum walks can be defined either in
continuous or discrete time. But whereas a continuous-time random walk can be
obtained as the limit of a sequence of discrete-time random walks, the two
types of quantum walk appear fundamentally different, owing to the need for
extra degrees of freedom in the discrete-time case.
In this article, I describe a precise correspondence between continuous- and
discrete-time quantum walks on arbitrary graphs. Using this correspondence, I
show that continuous-time quantum walk can be obtained as an appropriate limit
of discrete-time quantum walks. The correspondence also leads to a new
technique for simulating Hamiltonian dynamics, giving efficient simulations
even in cases where the Hamiltonian is not sparse. The complexity of the
simulation is linear in the total evolution time, an improvement over
simulations based on high-order approximations of the Lie product formula. As
applications, I describe a continuous-time quantum walk algorithm for element
distinctness and show how to optimally simulate continuous-time query
algorithms of a certain form in the conventional quantum query model. Finally,
I discuss limitations of the method for simulating Hamiltonians with negative
matrix elements, and present two problems that motivate attempting to
circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian
oracles; v3: published version, with improved analysis of phase estimatio
Revisiting Shor's quantum algorithm for computing general discrete logarithms
We heuristically demonstrate that Shor's algorithm for computing general
discrete logarithms, modified to allow the semi-classical Fourier transform to
be used with control qubit recycling, achieves a success probability of
approximately 60% to 82% in a single run. By slightly increasing the number of
group operations that are evaluated quantumly, and by performing a limited
search in the classical post-processing, we furthermore show how the algorithm
can be modified to achieve a success probability exceeding 99% in a single run.
We provide concrete heuristic estimates of the success probability of the
modified algorithm, as a function of the group order, the size of the search
space in the classical post-processing, and the additional number of group
operations evaluated quantumly. In analogy with our earlier works, we show how
the modified quantum algorithm may be simulated classically when the logarithm
and group order are both known. Furthermore, we show how slightly better
tradeoffs may be achieved, compared to our earlier works, if the group order is
known when computing the logarithm.Comment: The pre-print has been extended to show how slightly better tradeoffs
may be achieved, compared to our earlier works, if the group order is known.
A minor issue with an integration limit, that lead us to give a rough success
probability estimate of 60% to 70%, as opposed to 60% to 82%, has been
corrected. The heuristic and results reported in the original pre-print are
otherwise unaffecte
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum normalizer circuits were recently introduced as generalizations of
Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian
group is composed of the quantum Fourier transform (QFT) over G, together
with gates which compute quadratic functions and automorphisms. In
[arXiv:1201.4867] it was shown that every normalizer circuit can be simulated
efficiently classically. This result provides a nontrivial example of a family
of quantum circuits that cannot yield exponential speed-ups in spite of usage
of the QFT, the latter being a central quantum algorithmic primitive. Here we
extend the aforementioned result in several ways. Most importantly, we show
that normalizer circuits supplemented with intermediate measurements can also
be simulated efficiently classically, even when the computation proceeds
adaptively. This yields a generalization of the Gottesman-Knill theorem (valid
for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum
circuits described by arbitrary finite Abelian groups. Moreover, our
simulations are twofold: we present efficient classical algorithms to sample
the measurement probability distribution of any adaptive-normalizer
computation, as well as to compute the amplitudes of the state vector in every
step of it. Finally we develop a generalization of the stabilizer formalism
[quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian
groups: for example we characterize how to update stabilizers under generalized
Pauli measurements and provide a normal form of the amplitudes of generalized
stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To
appear in Quantum Information and Computation, Vol.14 No.3&4, 201
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