5,139 research outputs found
Quantum proof systems for iterated exponential time, and beyond
© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. We show that any language solvable in nondeterministic time exp(exp(· · · exp(n))), where the number of iterated exponentials is an arbitrary function R(n), can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−C exp(· · · exp(n))), where the number of iterated exponentials is R(n) − 1 and C > 0 is a universal constant. The result was previously known for R = 1 and R = 2; we obtain it for any time-constructible function R. The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC’17). As a separate consequence of this technique we obtain a different proof of Slofstra’s recent result on the uncomputability of the entangled value of multiprover games (Forum of Mathematics, Pi 2019). Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP∗ contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson’s problem on the relation between the commuting operator and tensor product models for quantum correlations
Coherent States Measurement Entropy
Coherent states (CS) quantum entropy can be split into two components. The
dynamical entropy is linked with the dynamical properties of a quantum system.
The measurement entropy, which tends to zero in the semiclassical limit,
describes the unpredictability induced by the process of a quantum approximate
measurement. We study the CS--measurement entropy for spin coherent states
defined on the sphere discussing different methods dealing with the time limit
. In particular we propose an effective technique of computing
the entropy by iterated function systems. The dependence of CS--measurement
entropy on the character of the partition of the phase space is analysed.Comment: revtex, 22 pages, 14 figures available upon request (e-mail:
[email protected]). Submitted to J.Phys.
Exact Results on Dynamical Decoupling by -Pulses in Quantum Information Processes
The aim of dynamical decoupling consists in the suppression of decoherence by
appropriate coherent control of a quantum register. Effectively, the
interaction with the environment is reduced. In particular, a sequence of
pulses is considered. Here we present exact results on the suppression of the
coupling of a quantum bit to its environment by optimized sequences of
pulses. The effect of various cutoffs of the spectral density of the
environment is investigated. As a result we show that the harder the cutoff is
the better an optimized pulse sequence can deal with it. For cutoffs which are
neither completely hard nor very soft we advocate iterated optimized sequences.Comment: 12 pages and 3 figure
Hyperoctahedral Chen calculus for effective Hamiltonians
The algebraic structure of iterated integrals has been encoded by Chen.
Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and
Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie
algebras. Here, we tackle the problem of unraveling the algebraic structure of
computations of effective Hamiltonians. This is an important subject in view of
applications to chemistry, solid state physics, quantum field theory or
engineering. We show, among others, that the correct framework for these
computations is provided by the hyperoctahedral group algebras. We define
several structures on these algebras and give various applications. For
example, we show that the adiabatic evolution operator (in the time-dependent
interaction representation of an effective Hamiltonian) can be written
naturally as a Picard-type series and has a natural exponential expansion.Comment: Minor corrections. Some misleading notations and typos in the first
version have been fixe
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties
Hamiltonian tridiagonal matrices characterized by multi-fractal spectral
measures in the family of Iterated Function Systems can be constructed by a
recursive technique here described. We prove that these Hamiltonians are
almost-periodic. They are suited to describe quantum lattice systems with
nearest neighbours coupling, as well as chains of linear classical oscillators,
and electrical transmission lines.
We investigate numerically and theoretically the time dynamics of the systems
so constructed. We derive a relation linking the long-time, power-law behaviour
of the moments of the position operator, expressed by a scaling function
of the moment order , and spectral multi-fractal dimensions,
D_q, via . We show cases in which this relation
is exact, and cases where it is only approximate, unveiling the reasons for the
discrepancies.Comment: 13 pages, Latex, 6 postscript figures. Accepted for publication in
Physica
Quantized recurrence time in iterated open quantum dynamics
The expected return time to the original state is a key concept
characterizing systems obeying both classical or quantum dynamics. We consider
iterated open quantum dynamical systems in finite dimensional Hilbert spaces, a
broad class of systems that includes classical Markov chains and unitary
discrete time quantum walks on networks. Starting from a pure state, the time
evolution is induced by repeated applications of a general quantum channel, in
each timestep followed by a measurement to detect whether the system has
returned to the original state. We prove that if the superoperator is unital in
the relevant Hilbert space (the part of the Hilbert space explored by the
system), then the expectation value of the return time is an integer, equal to
the dimension of this relevant Hilbert space. We illustrate our results on
partially coherent quantum walks on finite graphs. Our work connects the
previously known quantization of the expected return time for bistochastic
Markov chains and for unitary quantum walks, and shows that these are special
cases of a more general statement. The expected return time is thus a
quantitative measure of the size of the part of the Hilbert space available to
the system when the dynamics is started from a certain state
Computation in Finitary Stochastic and Quantum Processes
We introduce stochastic and quantum finite-state transducers as
computation-theoretic models of classical stochastic and quantum finitary
processes. Formal process languages, representing the distribution over a
process's behaviors, are recognized and generated by suitable specializations.
We characterize and compare deterministic and nondeterministic versions,
summarizing their relative computational power in a hierarchy of finitary
process languages. Quantum finite-state transducers and generators are a first
step toward a computation-theoretic analysis of individual, repeatedly measured
quantum dynamical systems. They are explored via several physical systems,
including an iterated beam splitter, an atom in a magnetic field, and atoms in
an ion trap--a special case of which implements the Deutsch quantum algorithm.
We show that these systems' behaviors, and so their information processing
capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous
corrections and update
Digital-analog quantum simulation of generalized Dicke models with superconducting circuits
We propose a digital-analog quantum simulation of generalized Dicke models
with superconducting circuits, including Fermi-Bose condensates, biased and
pulsed Dicke models, for all regimes of light-matter coupling. We encode these
classes of problems in a set of superconducting qubits coupled with a bosonic
mode implemented by a transmission line resonator. Via digital-analog
techniques, an efficient quantum simulation can be performed in
state-of-the-art circuit quantum electrodynamics platforms, by suitable
decomposition into analog qubit-bosonic blocks and collective single-qubit
pulses through digital steps. Moreover, just a single global analog block would
be needed during the whole protocol in most of the cases, superimposed with
fast periodic pulses to rotate and detune the qubits. Therefore, a large number
of digital steps may be attained with this approach, providing a reduced
digital error. Additionally, the number of gates per digital step does not grow
with the number of qubits, rendering the simulation efficient. This strategy
paves the way for the scalable digital-analog quantum simulation of many-body
dynamics involving bosonic modes and spin degrees of freedom with
superconducting circuits.Comment: Published version, with added reference
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