23 research outputs found
Positive Wigner functions render classical simulation of quantum computation efficient
We show that quantum circuits where the initial state and all the following
quantum operations can be represented by positive Wigner functions can be
classically efficiently simulated. This is true both for continuous-variable as
well as discrete variable systems in odd prime dimensions, two cases which will
be treated on entirely the same footing. Noting the fact that Clifford and
Gaussian operations preserve the positivity of the Wigner function, our result
generalizes the Gottesman-Knill theorem. Our algorithm provides a way of
sampling from the output distribution of a computation or a simulation,
including the efficient sampling from an approximate output distribution in
case of sampling imperfections for initial states, gates, or measurements. In
this sense, this work highlights the role of the positive Wigner function as
separating classically efficiently simulatable systems from those that are
potentially universal for quantum computing and simulation, and it emphasizes
the role of negativity of the Wigner function as a computational resource.Comment: 7 pages, minor change
Directly estimating non-classicality
We establish a method of directly measuring and estimating non-classicality -
operationally defined in terms of the distinguishability of a given state from
one with a positive Wigner function. It allows to certify non-classicality,
based on possibly much fewer measurement settings than necessary for obtaining
complete tomographic knowledge, and is at the same time equipped with a full
certificate. We find that even from measuring two conjugate variables alone,
one may infer the non-classicality of quantum mechanical modes. This method
also provides a practical tool to eventually certify such features in
mechanical degrees of freedom in opto-mechanics. The proof of the result is
based on Bochner's theorem characterizing classical and quantum characteristic
functions and on semi-definite programming. In this joint
theoretical-experimental work we present data from experimental optical Fock
state preparation, demonstrating the functioning of the approach.Comment: 4+1 pages, 2 figures, minor change
Wick's theorem for matrix product states
Matrix-product states and their continuous analogues are variational classes
of states that capture quantum many-body systems or quantum fields with low
entanglement; they are at the basis of the density-matrix renormalization group
method and continuous variants thereof. In this work we show that, generically,
N-point functions of arbitrary operators in discrete and continuous
translationally invariant matrix product states are completely characterized by
the corresponding two- and three-point functions. Aside from having important
consequences for the structure of correlations in quantum states with low
entanglement, this result provides a new way of reconstructing unknown states
from correlation measurements, e.g., for one-dimensional continuous systems of
cold atoms. We argue that such a relation of correlation functions may help in
devising perturbative approaches to interacting theories.Comment: 6 pages, final versio
Negative Quasi-Probability as a Resource for Quantum Computation
A central problem in quantum information is to determine the minimal physical
resources that are required for quantum computational speedup and, in
particular, for fault-tolerant quantum computation. We establish a remarkable
connection between the potential for quantum speed-up and the onset of negative
values in a distinguished quasi-probability representation, a discrete analog
of the Wigner function for quantum systems of odd dimension. This connection
allows us to resolve an open question on the existence of bound states for
magic-state distillation: we prove that there exist mixed states outside the
convex hull of stabilizer states that cannot be distilled to non-stabilizer
target states using stabilizer operations. We also provide an efficient
simulation protocol for Clifford circuits that extends to a large class of
mixed states, including bound universal states.Comment: 15 pages v4: This is a major revision. In particular, we have added a
new section detailing an explicit extension of the Gottesman-Knill simulation
protocol to deal with positively represented states and measurement (even
when these are non-stabilizer). This paper also includes significant
elaboration on the two main results of the previous versio
Quasi-probability representations of quantum theory with applications to quantum information science
This article comprises a review of both the quasi-probability representations
of infinite-dimensional quantum theory (including the Wigner function) and the
more recently defined quasi-probability representations of finite-dimensional
quantum theory. We focus on both the characteristics and applications of these
representations with an emphasis toward quantum information theory. We discuss
the recently proposed unification of the set of possible quasi-probability
representations via frame theory and then discuss the practical relevance of
negativity in such representations as a criteria for quantumness.Comment: v3: typos fixed, references adde
Gently modulating opto-mechanical systems
We introduce a framework of opto-mechanical systems that are driven with a
mildly amplitude modulated light field, but that are not subject to classical
feedback or squeezed input light. We find that in such a system one can achieve
large degrees of squeezing of a mechanical micromirror signifying quantum
properties of opto-mechanical systems - without the need of any feedback and
control, and within parameters reasonable in experimental settings.
Entanglement dynamics is shown of states following classical quasi-periodic
orbits in their first moments. We discuss the complex time dependence of the
modes of a cavity-light field and a mechanical mode in phase space. Such
settings give rise to certifiable quantum properties within experimental
conditions feasible with present technology.Comment: 4+3 pages, 2 figures, additional brief appendix compared to version
in press in Phys. Rev. Lett