78 research outputs found
Fault-ignorant Quantum Search
We investigate the problem of quantum searching on a noisy quantum computer.
Taking a 'fault-ignorant' approach, we analyze quantum algorithms that solve
the task for various different noise strengths, which are possibly unknown
beforehand. We prove lower bounds on the runtime of such algorithms and thereby
find that the quadratic speedup is necessarily lost (in our noise models).
However, for low but constant noise levels the algorithms we provide (based on
Grover's algorithm) still outperform the best noiseless classical search
algorithm.Comment: v1: 15+8 pages, 4 figures; v2: 19+8 pages, 4 figures, published
version (Introduction section significantly expanded, presentation clarified,
results and order unchanged
Quantum incompatibility in collective measurements
We study the compatibility (or joint measurability) of quantum observables in
a setting where the experimenter has access to multiple copies of a given
quantum system, rather than performing the experiments on each individual copy
separately. We introduce the index of incompatibility as a quantifier of
incompatibility in this multi-copy setting, as well as the notion of
compatibility stack representing the various compatibility relations present in
a given set of observables. We then prove a general structure theorem for
multi-copy joint observables, and use it to prove that all abstract
compatibility stacks with three vertices have realizations in terms of quantum
observables.Comment: 22 pages, 13 figure
Noise robustness of the incompatibility of quantum measurements
The existence of incompatible measurements is a fundamental phenomenon having
no explanation in classical physics. Intuitively, one considers given
measurements to be incompatible within a framework of a physical theory, if
their simultaneous implementation on a single physical device is prohibited by
the theory itself. In the mathematical language of quantum theory, measurements
are described by POVMs (positive operator valued measures), and given POVMs are
by definition incompatible if they cannot be obtained via coarse-graining from
a single common POVM; this notion generalizes noncommutativity of projective
measurements. In quantum theory, incompatibility can be regarded as a resource
necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH)
Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do
not have classical explanation. We define operational ways of quantifying this
resource via the amount of added classical noise needed to render the
measurements compatible, i.e., useless as a resource. In analogy to
entanglement measures, we generalize this idea by introducing the concept of
incompatibility measure, which is monotone in local operations. In this paper,
we restrict our consideration to binary measurements, which are already
sufficient to explicitly demonstrate nontrivial features of the theory. In
particular, we construct a family of incompatibility monotones operationally
quantifying violations of certain scaled versions of the CHSH Bell inequality,
prove that they can be computed via a semidefinite program, and show how the
noise-based quantities arise as special cases. We also determine maximal
violations of the new inequalities, demonstrating how Tsirelson's bound appears
as a special case. The resource aspect is further motivated by simple quantum
protocols where our incompatibility monotones appear as relevant figures of
merit.Comment: 13 pages, 4 figures; small changes to v0
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