20 research outputs found
Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates
We study the quantum query complexity of constant-sized subgraph containment.
Such problems include determining whether an -vertex graph contains a
triangle, clique or star of some size. For a general subgraph with
vertices, we show that containment can be solved with quantum query
complexity , with a strictly positive
function of . This is better than \tilde{O}\s{n^{2-2/k}} by Magniez et
al. These results are obtained in the learning graph model of Belovs.Comment: 14 pages, 1 figure, published under title:"Quantum Query Complexity
of Constant-sized Subgraph Containment
Holographic Trace Anomaly and Local Renormalization Group
The Hamilton-Jacobi method in holography has produced important results both
at a renormalization group (RG) fixed point and away from it. In this paper we
use the Hamilton-Jacobi method to compute the holographic trace anomaly for
four- and six-dimensional boundary conformal field theories (CFTs), assuming
higher-derivative gravity and interactions of scalar fields in the bulk. The
scalar field contributions to the anomaly appear in CFTs with exactly marginal
operators. Moving away from the fixed point, we show that the Hamilton-Jacobi
formalism provides a deep connection between the holographic and the local RG.
We derive the local RG equation holographically, and verify explicitly that it
satisfies Weyl consistency conditions stemming from the commutativity of Weyl
scalings. We also consider massive scalar fields in the bulk corresponding to
boundary relevant operators, and comment on their effects to the local RG
equation.Comment: 27 pages. v3: References adde
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
Superadditivity of the Classical Capacity with Limited Entanglement Assistance
Finding the optimal encoding strategies can be challenging for communication
using quantum channels, as classical and quantum capacities may be
superadditive. Entanglement assistance can often simplify this task, as the
entanglement-assisted classical capacity for any channel is additive, making
entanglement across channel uses unnecessary. If the entanglement assistance is
limited, the picture is much more unclear. Suppose the classical capacity is
superadditive, then the classical capacity with limited entanglement assistance
could retain superadditivity by continuity arguments. If the classical capacity
is additive, it is unknown if superadditivity can still be developed with
limited entanglement assistance. We show this is possible, by providing an
example. We construct a channel for which, the classical capacity is additive,
but that with limited entanglement assistance can be superadditive. This shows
entanglement plays a weird role in communication and we still understand very
little about it.Comment: 13 page
Superadditivity in trade-off capacities of quantum channels
In this article, we investigate the additivity phenomenon in the dynamic
capacity of a quantum channel for trading classical communication, quantum
communication and entanglement. Understanding such additivity property is
important if we want to optimally use a quantum channel for general
communication purpose. However, in a lot of cases, the channel one will be
using only has an additive single or double resource capacity, and it is
largely unknown if this could lead to an superadditive double or triple
resource capacity. For example, if a channel has an additive classical and
quantum capacity, can the classical-quantum capacity be superadditive? In this
work, we answer such questions affirmatively.
We give proof-of-principle requirements for these channels to exist. In most
cases, we can provide an explicit construction of these quantum channels. The
existence of these superadditive phenomena is surprising in contrast to the
result that the additivity of both classical-entanglement and classical-quantum
capacity regions imply the additivity of the triple capacity region.Comment: 15 pages. v2: typo correcte
Doubly infinite separation of quantum information and communication
We prove the existence of (one-way) communication tasks with a subconstant
versus superconstant asymptotic gap, which we call "doubly infinite," between
their quantum information and communication complexities. We do so by studying
the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for
which there exist instances where the quantum information complexity tends to
zero as the size of the input increases. By showing that the quantum
communication complexity of these games scales at least logarithmically in ,
we obtain our result. We further show that the established lower bounds and
gaps still hold even if we allow a small probability of error. However in this
case, the -qubit quantum message of the zero-error strategy can be
compressed polynomially.Comment: 16 pages, 2 figures. v4: minor errors fixed; close to published
version; v5: financial support info adde
Hyperfine interaction induced decoherence and deterministic teleportation of electrons in a quantum dot nanostructure
Recently, de Visser and Blaauboer [Phys. Rev. Lett. {\bf 96}, 246801 (2006)]
proposed the most efficient deterministic teleportation protocol for
electron spins in a semiconductor nanostructure consisting of a single and a
double quantum dot. However, it is as yet unknown if can be completed
before decoherence sets in. In this paper we analyze the detrimental effect of
nuclear spin baths, the main source of decoherence, on . We show that
nonclassical teleportation fidelity can be achieved with provided
certain conditions are met. Our study indicates that realization of quantum
computation with quantum dots is indeed promising.Comment: 10 page
Iteration Complexity of Variational Quantum Algorithms
There has been much recent interest in near-term applications of quantum
computers. Variational quantum algorithms (VQA), wherein an optimization
algorithm implemented on a classical computer evaluates a parametrized quantum
circuit as an objective function, are a leading framework in this space.
In this paper, we analyze the iteration complexity of VQA, that is, the
number of steps VQA required until the iterates satisfy a surrogate measure of
optimality. We argue that although VQA procedures incorporate algorithms that
can, in the idealized case, be modeled as classic procedures in the
optimization literature, the particular nature of noise in near-term devices
invalidates the claim of applicability of off-the-shelf analyses of these
algorithms. Specifically, the form of the noise makes the evaluations of the
objective function via circuits biased, necessitating the perspective of
convergence analysis of variants of these classical optimization procedures,
wherein the evaluations exhibit systematic bias. We apply our reasoning to the
most often used procedures, including SPSA the parameter shift rule, which can
be seen as zeroth-order, or derivative-free, optimization algorithms with
biased function evaluations. We show that the asymptotic rate of convergence is
unaffected by the bias, but the level of bias contributes unfavorably to both
the constant therein, and the asymptotic distance to stationarity.Comment: 39 pages, 11 figure
Generative Quantum Learning of Joint Probability Distribution Functions
Modeling joint probability distributions is an important task in a wide
variety of fields. One popular technique for this employs a family of
multivariate distributions with uniform marginals called copulas. While the
theory of modeling joint distributions via copulas is well understood, it gets
practically challenging to accurately model real data with many variables. In
this work, we design quantum machine learning algorithms to model copulas. We
show that any copula can be naturally mapped to a multipartite maximally
entangled state. A variational ansatz we christen as a `qopula' creates
arbitrary correlations between variables while maintaining the copula structure
starting from a set of Bell pairs for two variables, or GHZ states for multiple
variables. As an application, we train a Quantum Generative Adversarial Network
(QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz
to generate samples from joint distributions of two variables for historical
data from the stock market. We demonstrate our generative learning algorithms
on trapped ion quantum computers from IonQ for up to 8 qubits and show that our
results outperform those obtained through equivalent classical generative
learning. Further, we present theoretical arguments for exponential advantage
in our model's expressivity over classical models based on communication and
computational complexity arguments.Comment: 19 pages, 11 figures. v2: published versio