546 research outputs found
Testing non-isometry is QMA-complete
Determining the worst-case uncertainty added by a quantum circuit is shown to
be computationally intractable. This is the problem of detecting when a quantum
channel implemented as a circuit is close to a linear isometry, and it is shown
to be complete for the complexity class QMA of verifiable quantum computation.
This is done by relating the problem of detecting when a channel is close to an
isometry to the problem of determining how mixed the output of the channel can
be when the input is a pure state. How mixed the output of the channel is can
be detected by a protocol making use of the swap test: this follows from the
fact that an isometry applied twice in parallel does not affect the symmetry of
the input state under the swap operation.Comment: 12 pages, 3 figures. Presentation improved, results unchange
Computational Distinguishability of Quantum Channels
The computational problem of distinguishing two quantum channels is central
to quantum computing. It is a generalization of the well-known satisfiability
problem from classical to quantum computation. This problem is shown to be
surprisingly hard: it is complete for the class QIP of problems that have
quantum interactive proof systems, which implies that it is hard for the class
PSPACE of problems solvable by a classical computation in polynomial space.
Several restrictions of distinguishability are also shown to be hard. It is
no easier when restricted to quantum computations of logarithmic depth, to
mixed-unitary channels, to degradable channels, or to antidegradable channels.
These hardness results are demonstrated by finding reductions between these
classes of quantum channels. These techniques have applications outside the
distinguishability problem, as the construction for mixed-unitary channels is
used to prove that the additivity problem for the classical capacity of quantum
channels can be equivalently restricted to the mixed unitary channels.Comment: Ph.D. Thesis, 178 pages, 35 figure
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
All entangled states are useful for channel discrimination
We prove that every entangled state is useful as a resource for the problem
of minimum-error channel discrimination. More specifically, given a single copy
of an arbitrary bipartite entangled state, it holds that there is an instance
of a quantum channel discrimination task for which this state allows for a
correct discrimination with strictly higher probability than every separable
state.Comment: 5 pages, more similar to the published versio
Floodplains : the forgotten and abused component of the fluvial system
River restoration is strongly focussed on in-channel initiatives driven by fisheries interests and a continued
desire for river stability. This contrasts greatly with the inherently mobile nature of watercourses. What is often
overlooked is the fact that many rivers have developed floodplain units that would naturally operate as integrated
functional systems, moderating the effects of extreme floods by distributing flow energy and sediment transport
capacity through out of bank flooding. Floodplain utilisation for farming activities and landowner intransigence when
it comes to acknowledging that the floodplain is part of the river system, has resulted in floodplains being the most
degraded fluvial morphologic unit, both in terms of loss of form and function and sheer levels of spatial impact. The
degradation has been facilitated by the failure of regulatory mechanisms to adequately acknowledge floodplain form
and function. This is testament to the ‘inward looking’ thinking behind national assessment strategies. This paper
reviews the state of floodplain systems drawing on quantitative data from England and Wales to argue for greater
consideration of the floodplain in relation to river management. The database is poor and must be improved, however
it does reveal significant loss of watercourse-floodplain connectivity linked to direct flood alleviation measures and
also to altered flood frequency as a result of river downcutting following river engineering. These latter effects have
persisted along many watercourses despite the historic nature of the engineering interventions and will continue to
exacerbate the risk of flooding to downstream communities. We also present several examples of the local and wider
values of reinstating floodplain form and function, demonstrating major ecological gains, improvement to
downstream flood reduction, elevation of water quality status and reductions in overall fine sediment loss from
farmland. A re-think is required regarding our approach to managing floodplains and funding floodplain restoration,
arguing for greater recognition of the natural role of the floodplain as a resource for upstream flood management and
as an agent for overall biotic improvement in line with restoration objectives
Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
The set of doubly-stochastic quantum channels and its subset of mixtures of
unitaries are investigated. We provide a detailed analysis of their structure
together with computable criteria for the separation of the two sets. When
applied to O(d)-covariant channels this leads to a complete characterization
and reveals a remarkable feature: instances of channels which are not in the
convex hull of unitaries can return to it when either taking finitely many
copies of them or supplementing with a completely depolarizing channel. In
these scenarios this implies that a channel whose noise initially resists any
environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page
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