14 research outputs found
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
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
Explainable AI using expressive Boolean formulas
We propose and implement an interpretable machine learning classification
model for Explainable AI (XAI) based on expressive Boolean formulas. Potential
applications include credit scoring and diagnosis of medical conditions. The
Boolean formula defines a rule with tunable complexity (or interpretability),
according to which input data are classified. Such a formula can include any
operator that can be applied to one or more Boolean variables, thus providing
higher expressivity compared to more rigid rule-based and tree-based
approaches. The classifier is trained using native local optimization
techniques, efficiently searching the space of feasible formulas. Shallow rules
can be determined by fast Integer Linear Programming (ILP) or Quadratic
Unconstrained Binary Optimization (QUBO) solvers, potentially powered by
special purpose hardware or quantum devices. We combine the expressivity and
efficiency of the native local optimizer with the fast operation of these
devices by executing non-local moves that optimize over subtrees of the full
Boolean formula. We provide extensive numerical benchmarking results featuring
several baselines on well-known public datasets. Based on the results, we find
that the native local rule classifier is generally competitive with the other
classifiers. The addition of non-local moves achieves similar results with
fewer iterations, and therefore using specialized or quantum hardware could
lead to a speedup by fast proposal of non-local moves.Comment: 28 pages, 16 figures, 4 table
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
Information-theoretic aspects of quantum channels
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2019Cataloged from PDF version of thesis.Includes bibliographical references (pages 165-172).Quantum information theory is an important element of quantum computing and quantum communication systems. Whenever a quantum computer needs to send an output state to another party, or two parties need to establish quantum entanglement or secure keys via quantum communication, a quantum channel is inevitably involved. Hence it is absolutely important to understand the properties of quantum channels for the purpose of communication. Here, quantum entanglement plays a huge role. Pre-shared entanglement could enhance the capacity, whereas entanglement across inputs could render the capacity formulae impossible to compute. The first part of this thesis seeks to address this issue, by studying the additivity properties in the communication of classical and quantum information, with or without entanglement assistance. I also study the reverse problem that, given a channel capacity, what can be said about the quantum channel itself. Quantum information theory also serves as an important tool in understanding other systems, for example, black holes. In this thesis, I model a closed random system by a unitary channel, and study how typical unitary channels process information. This provides huge insight into the strength of generalized entanglement measures, and the hierarchies in the complexity of information scrambling.by Elton Yechao Zhu.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Physic
Quantum state preparation of normal distributions using matrix product states
Abstract State preparation is a necessary component of many quantum algorithms. In this work, we combine a method for efficiently representing smooth differentiable probability distributions using matrix product states with recently discovered techniques for initializing quantum states to approximate matrix product states. Using this, we generate quantum states encoding a class of normal probability distributions in a trapped ion quantum computer for up to 20 qubits. We provide an in depth analysis of the different sources of error which contribute to the overall fidelity of this state preparation procedure. Our work provides a study in quantum hardware for scalable distribution loading, which is the basis of a wide range of algorithms that provide quantum advantage
Superadditivity in Trade-Off Capacities of Quantum Channels
© 2018 IEEE. We investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes. 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, respectively. 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 resource capacity region
Generalized Entanglement Entropies of Quantum Designs
The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This Letter investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Renyi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Renyi entanglement entropies and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Renyi entanglement entropies to the complexity of scrambling and quantum chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture