37,922 research outputs found
The quantum adversary method and classical formula size lower bounds
We introduce two new complexity measures for Boolean functions, or more
generally for functions of the form f:S->T. We call these measures sumPI and
maxPI. The quantity sumPI has been emerging through a line of research on
quantum query complexity lower bounds via the so-called quantum adversary
method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the
realization that these many different formulations are in fact equivalent.
Given that sumPI turns out to be such a robust invariant of a function, we
begin to investigate this quantity in its own right and see that it also has
applications to classical complexity theory.
As a surprising application we show that sumPI^2(f) is a lower bound on the
formula size, and even, up to a constant multiplicative factor, the
probabilistic formula size of f. We show that several formula size lower bounds
in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93],
including a key lemma of [Has98], are in fact special cases of our method.
The second quantity we introduce, maxPI(f), is always at least as large as
sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a
lower bound on formula size. While sumPI(f) is always a lower bound on the
quantum query complexity of f, this is not the case in general for maxPI(f). A
strong advantage of sumPI(f) is that it has both primal and dual
characterizations, and thus it is relatively easy to give both upper and lower
bounds on the sumPI complexity of functions. To demonstrate this, we look at a
few concrete examples, for three functions: recursive majority of three, a
function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200
Quantum Entanglement Capacity with Classical Feedback
For any quantum discrete memoryless channel, we define a quantity called
quantum entanglement capacity with classical feedback (), and we show that
this quantity lies between two other well-studied quantities. These two
quantities - namely the quantum capacity assisted by two-way classical
communication () and the quantum capacity with classical feedback ()
- are widely conjectured to be different: there exists quantum discrete
memoryless channel for which . We then present a general scheme to
convert any quantum error-correcting codes into adaptive protocols for this
newly-defined quantity of the quantum depolarizing channel, and illustrate with
Cat (repetition) code and Shor code. We contrast the present notion with
entanglement purification protocols by showing that whilst the Leung-Shor
protocol can be applied directly, recurrence methods need to be supplemented
with other techniques but at the same time offer a way to improve the
aforementioned Cat code. For the quantum depolarizing channel, we prove a
formula that gives lower bounds on the quantum capacity with classical feedback
from any protocols. We then apply this formula to the protocols
that we discuss to obtain new lower bounds on the quantum capacity with
classical feedback of the quantum depolarizing channel
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
Further developments for the auxiliary field method
The auxiliary field method is a technique to obtain approximate closed
formulae for the solutions of both nonrelativistic and semirelativistic
eigenequations in quantum mechanics. For a many-body Hamiltonian describing
identical particles, it is shown that the approximate eigenvalues can be
written as the sum of the kinetic operator evaluated at a mean momentum
and of the potential energy computed at a mean distance . The quantities
and are linked by a simple relation depending on the quantum
numbers of the state considered and are determined by an equation which is
linked to the generalized virial theorem. The (anti)variational character of
the method is discussed, as well as its connection with the perturbation
theory. For a nonrelativistic kinematics, general results are obtained for the
structure of critical coupling constants for potentials with a finite number of
bound states.Comment: New improved presentatio
On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson
bound for the isotropic XY chain with Fibonacci external magnetic field at
arbitrary coupling. It is anomalous in that the usual exponential decay in
is replaced by exponential decay in with . In
fact, we can characterize the values of for which such a bound holds
as those exceeding , the upper transport exponent of the one-body
Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate
Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian
corresponding to the XY chain via the Jordan-Wigner transformation; in our case
the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by
adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes.
We also explain why our method does not extend to yield anomalous
Lieb-Robinson bounds of power-law type for the random dimer model.Comment: 21 pages, Final version to appear in J. Spectr. Theor
Detector decoy quantum key distribution
Photon number resolving detectors can enhance the performance of many
practical quantum cryptographic setups. In this paper, we employ a simple
method to estimate the statistics provided by such a photon number resolving
detector using only a threshold detector together with a variable attenuator.
This idea is similar in spirit to that of the decoy state technique, and is
specially suited for those scenarios where only a few parameters of the photon
number statistics of the incoming signals have to be estimated. As an
illustration of the potential applicability of the method in quantum
communication protocols, we use it to prove security of an entanglement based
quantum key distribution scheme with an untrusted source without the need of a
squash model and by solely using this extra idea. In this sense, this detector
decoy method can be seen as a different conceptual approach to adapt a single
photon security proof to its physical, full optical implementation. We show
that in this scenario the legitimate users can now even discard the double
click events from the raw key data without compromising the security of the
scheme, and we present simulations on the performance of the BB84 and the
6-state quantum key distribution protocols.Comment: 27 pages, 7 figure
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