12 research outputs found
Almost Everything About the Unitary Almost Mathieu Operator
We introduce a unitary almost-Mathieu operator, which is obtained from a
two-dimensional quantum walk in a uniform magnetic field. We exhibit a version
of Aubry--Andr\'{e} duality for this model, which partitions the parameter
space into three regions: a supercritical region and a subcritical region that
are dual to one another, and a critical regime that is self-dual. In each
parameter region, we characterize the cocycle dynamics of the transfer matrix
cocycle generated by the associated generalized eigenvalue equation. In
particular, we show that supercritical, critical, and subcritical behavior all
occur in this model. Using Avila's global theory of one-frequency cocycles, we
exactly compute the Lyapunov exponent on the spectrum in terms of the given
parameters. We also characterize the spectral type for each value of the
coupling constant, almost every frequency, and almost every phase. Namely, we
show that for almost every frequency and every phase the spectral type is
purely absolutely continuous in the subcritical region, pure point in the
supercritical region, and purely singular continuous in the critical region. In
some parameter regions, we refine the almost-sure results. In the critical case
for instance, we show that the spectrum is a Cantor set of zero Lebesgue
measure for arbitrary irrational frequency and that the spectrum is purely
singular continuous for all but countably many phases.Comment: 41 pages, 5 figure
Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations
Quantum walks subject to decoherence generically suffer the loss of their
genuine quantum feature, a quadratically faster spreading compared to classical
random walks. This intuitive statement has been verified analytically for
certain models and is also supported by numerical studies of a variety of
examples. In this paper we analyze the long-time behavior of a particular class
of decoherent quantum walks, which, to the best of our knowledge, was only
studied at the level of numerical simulations before. We consider a local coin
operation which is randomly and independently chosen for each time step and
each lattice site and prove that, under rather mild conditions, this leads to
classical behavior: With the same scaling as needed for a classical diffusion
the position distribution converges to a Gaussian, which is independent of the
initial state. Our method is based on non-degenerate perturbation theory and
yields an explicit expression for the covariance matrix of the asymptotic
Gaussian in terms of the randomness parameters
A single-particle framework for unitary lattice gauge theory in discrete time
International audienceWe construct a real-time lattice-gauge-theory (LGT)-type action for a spin-1/2 matter field of a single particle on a ( 1 + 1 ) -dimensional spacetime lattice. The framework is based on a discrete-time quantum walk, and is hence inherently unitary and strictly local, i.e. transition amplitudes exactly vanish outside of a lightcone on the lattice. We then provide a lattice Noether’s theorem for internal symmetries of this action. We further couple this action to an electromagnetic field by a minimal substitution on the lattice. Finally, we suggest a real-time LGT-type action for the electromagnetic field in arbitrary spacetime dimensions, and derive its classical equations of motion, which are lattice versions of Maxwell’s equations
Addressable quantum gates
38 pages, 15 figuresWe extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quantum switch and the polarizing beam splitter within our model. One difficulty is that the names used as addresses should not matter beyond the wiring they describe, i.e. the evolution should commute with "renamings". Yet, the evolution may act nontrivially on these names. Our main technical contribution is a full characterization of such "nameblind" matrices
Exact mobility edges for almost-periodic CMV matrices via gauge symmetries
We investigate the symmetries of so-called generalized extended CMV matrices.
It is well-documented that problems involving reflection symmetries of standard
extended CMV matrices can be subtle. We show how to deal with this in an
elegant fashion by passing to the class of generalized extended CMV matrices
via explicit diagonal unitaries in the spirit of
Cantero-Gr\"unbaum-Moral-Vel\'azquez. As an application of these ideas, we
construct an explicit family of almost-periodic CMV matrices, which we call the
mosaic unitary almost-Mathieu operator, and prove the occurrence of exact
mobility edges. That is, we show the existence of energies that separate
spectral regions with absolutely continuous and pure point spectrum and exactly
calculate them.Comment: 35 pages, 3 figure
Addressable quantum gates
38 pages, 15 figuresWe extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quantum switch and the polarizing beam splitter within our model. One difficulty is that the names used as addresses should not matter beyond the wiring they describe, i.e. the evolution should commute with "renamings". Yet, the evolution may act nontrivially on these names. Our main technical contribution is a full characterization of such "nameblind" matrices
Addressable quantum gates
38 pages, 15 figuresWe extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quantum switch and the polarizing beam splitter within our model. One difficulty is that the names used as addresses should not matter beyond the wiring they describe, i.e. the evolution should commute with "renamings". Yet, the evolution may act nontrivially on these names. Our main technical contribution is a full characterization of such "nameblind" matrices
Addressable quantum gates
38 pages, 15 figuresWe extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quantum switch and the polarizing beam splitter within our model. One difficulty is that the names used as addresses should not matter beyond the wiring they describe, i.e. the evolution should commute with "renamings". Yet, the evolution may act nontrivially on these names. Our main technical contribution is a full characterization of such "nameblind" matrices
Addressable quantum gates
38 pages, 15 figuresWe extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quantum switch and the polarizing beam splitter within our model. One difficulty is that the names used as addresses should not matter beyond the wiring they describe, i.e. the evolution should commute with "renamings". Yet, the evolution may act nontrivially on these names. Our main technical contribution is a full characterization of such "nameblind" matrices