4,918 research outputs found
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
Quantum simulation of partially distinguishable boson sampling
Boson Sampling is the problem of sampling from the same output probability
distribution as a collection of indistinguishable single photons input into a
linear interferometer. It has been shown that, subject to certain computational
complexity conjectures, in general the problem is difficult to solve
classically, motivating optical experiments aimed at demonstrating quantum
computational "supremacy". There are a number of challenges faced by such
experiments, including the generation of indistinguishable single photons. We
provide a quantum circuit that simulates bosonic sampling with arbitrarily
distinguishable particles. This makes clear how distinguishabililty leads to
decoherence in the standard quantum circuit model, allowing insight to be
gained. At the heart of the circuit is the quantum Schur transform, which
follows from a representation theoretic approach to the physics of
distinguishable particles in first quantisation. The techniques are quite
general and have application beyond boson sampling.Comment: 25 pages, 4 figures, 2 algorithms, comments welcom
Mirror Symmetry And Loop Operators
Wilson loops in gauge theories pose a fundamental challenge for dualities.
Wilson loops are labeled by a representation of the gauge group and should map
under duality to loop operators labeled by the same data, yet generically, dual
theories have completely different gauge groups. In this paper we resolve this
conundrum for three dimensional mirror symmetry. We show that Wilson loops are
exchanged under mirror symmetry with Vortex loop operators, whose microscopic
definition in terms of a supersymmetric quantum mechanics coupled to the theory
encode in a non-trivial way a representation of the original gauge group,
despite that the gauge groups of mirror theories can be radically different.
Our predictions for the mirror map, which we derive guided by branes in string
theory, are confirmed by the computation of the exact expectation value of
Wilson and Vortex loop operators on the three-sphere.Comment: 92 pages, v2: minor clarifications in the introduction, to be
published in JHE
The wave-particle duality of the qudit-based quantum space demonstrated by the wave-like quantum functionals
The wave-particle duality of the qudit-based quantum space is demonstrated by
the wave-like quantum functionals and the particle-like quantum states. The
quantum functionals are quantum objects generated by the basis qudit
functionals, which are the duals of the basis qudit states. The relation
between the quantum states and quantum functionals is analogous to the relation
between the position and momentum in fundamental quantum physics. In
particular, the quantum states and quantum functionals are related by a Fourier
transform, and the quantum functionals have wave-like interpretations. The
quantum functionals are not just mathematical constructs but have clear
physical meanings and quantum circuit realizations. Any arbitrary qudit-based
quantum state has dual characters of wave and particle, and its wave character
can be evaluated by an observable and realized by a quantum circuit
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