13,197 research outputs found
Adversary Lower Bound for Element Distinctness with Small Range
The Element Distinctness problem is to decide whether each character of an
input string is unique. The quantum query complexity of Element Distinctness is
known to be ; the polynomial method gives a tight lower bound
for any input alphabet, while a tight adversary construction was only known for
alphabets of size .
We construct a tight adversary lower bound for Element
Distinctness with minimal non-trivial alphabet size, which equals the length of
the input. This result may help to improve lower bounds for other related query
problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos
fixed. v3: minor typos fixe
On the Power of Non-Adaptive Learning Graphs
We introduce a notion of the quantum query complexity of a certificate
structure. This is a formalisation of a well-known observation that many
quantum query algorithms only require the knowledge of the disposition of
possible certificates in the input string, not the precise values therein.
Next, we derive a dual formulation of the complexity of a non-adaptive
learning graph, and use it to show that non-adaptive learning graphs are tight
for all certificate structures. By this, we mean that there exists a function
possessing the certificate structure and such that a learning graph gives an
optimal quantum query algorithm for it.
For a special case of certificate structures generated by certificates of
bounded size, we construct a relatively general class of functions having this
property. The construction is based on orthogonal arrays, and generalizes the
quantum query lower bound for the -sum problem derived recently in
arXiv:1206.6528.
Finally, we use these results to show that the learning graph for the
triangle problem from arXiv:1210.1014 is almost optimal in these settings. This
also gives a quantum query lower bound for the triangle-sum problem.Comment: 16 pages, 1.5 figures v2: the main result generalised for all
certificate structures, a bug in the proof of Proposition 17 fixe
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page
Towards Rigorous Derivation of Quantum Kinetic Equations
We develop a rigorous formalism for the description of the evolution of
states of quantum many-particle systems in terms of a one-particle density
operator. For initial states which are specified in terms of a one-particle
density operator the equivalence of the description of the evolution of quantum
many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and
by the Cauchy problem of the generalized quantum kinetic equation together with
a sequence of explicitly defined functionals of a solution of stated kinetic
equation is established in the space of trace class operators. The links of the
specific quantum kinetic equations with the generalized quantum kinetic
equation are discussed.Comment: 25 page
Boundary Conditions on Internal Three-Body Wave Functions
For a three-body system, a quantum wave function with definite
and quantum numbers may be expressed in terms of an internal wave
function which is a function of three internal coordinates. This
article provides necessary and sufficient constraints on to
ensure that the external wave function is analytic. These
constraints effectively amount to boundary conditions on and its
derivatives at the boundary of the internal space. Such conditions find
similarities in the (planar) two-body problem where the wave function (to
lowest order) has the form at the origin. We expect the boundary
conditions to prove useful for constructing singularity free three-body basis
sets for the case of nonvanishing angular momentum.Comment: 41 pages, submitted to Phys. Rev.
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