[...] In this thesis, we are interested in generalizing factor graphs and the relevant methods toward describing quantum systems. Two generalizations of classical graphical models are investigated, namely double-edge factor graphs (DeFGs) and quantum factor graphs (QFGs). Conventionally, a factor in a factor graph represents a nonnegative real-valued local functions. Two different approaches to generalize factors in classical factor graphs yield DeFGs and QFGs, respectively. We proposed/re-proposed and analyzed generalized versions of belief-propagation algorithms for DeFGs/QFGs. As a particular application of the DeFGs, we investigate the information rate and their upper/lower bounds of classical communications over quantum channels with memory. In this study, we also propose a data-driven method for optimizing the upper/lower bounds on information rate.
[...] In this thesis, we are interested in generalizing factor graphs and the relevant methods toward describing quantum systems. Two generalizations of classical graphical models are investigated, namely double-edge factor graphs (DeFGs) and quantum factor graphs (QFGs). Conventionally, a factor in a factor graph represents a nonnegative real-valued local functions. Two different approaches to generalize factors in classical factor graphs yield DeFGs and QFGs, respectively. We proposed/re-proposed and analyzed generalized versions of belief-propagation algorithms for DeFGs/QFGs. As a particular application of the DeFGs, we investigate the information rate and their upper/lower bounds of classical communications over quantum channels with memory. In this study, we also propose a data-driven method for optimizing the upper/lower bounds on information rate.
We are interested in classical communication over a quantum channel with memory, in particular, we are interested in quantum-auxiliary-channel-based upper and lower bounds on the information rate. Toward improving these bounds, we propose efficient algorithms for optimizing the auxiliary channel. From a practical perspective, the lower bounds are of significant interest because they represent mismatched information rates, i.e., information rates that are achievable by a communication system where the decoder is matched to the auxiliary channel, instead of the original channel. Moreover, the auxiliary channels are learned in a data-driven approach, i.e., only input/output sequences of the true channel are needed, but not the channel model of the true channel.
Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.
Optimization is a fundamental area in mathematics and computer science, with many real-world applications. In this thesis we study the efficiency with which we can solve certain optimization problems from two different perspectives. Firstly, we study it from the perspective of matrix factorization ranks, which comes with connections to quantum information theory. Secondly, we study it from the perspective of quantum computing. This thesis is accordingly divided into two parts, where we take these perspectives. In the first part of this thesis we study several matrix factorization ranks and their connections to quantum information theory. In Chapter 5, we first give a unified approach to lower bounding matrix factorization ranks, using polynomial optimization techniques. In Chapter 6, we exploit the connection between one particular factorization rank, the completely positive semidefinite rank, and quantum correlations to provide an explicit family of matrices with a large completely positive semidefinite factorization rank. In Chapter 8, we use the same connection to study quantum versions of classical graph parameters from the perspective of polynomial optimization (albeit in noncommutative variables). In Chapter 7 we propose and study a new measure for the amount of entanglement needed to realize quantum correlations. We can approximate this new measure using our approach to matrix factorization ranks, i.e., through polynomial optimization in noncommutative variables. In the second part of this thesis we turn our attention to the question: Can we solve optimization problems faster on a quantum computer? First, in Chapter 10, we look at the problem of evaluating a Boolean function. We give a new semidefinite programming characterization of the minimum number of quantum queries to the input that are needed to determine the corresponding function value. In Chapter 11 we revisit the semidefinite programming problem that plays a crucial role in the rest of the thesis; we provide a quantum algorithm for solving semidefinite programs. Finally, in Chapter 12, we study the general problem of solving convex optimization problems in the oracle model. We provide both upper and lower bounds on the efficiency of various reductions in the quantum setting. In particular, we show that quantum computers are more efficient than classical computers for the task of answering separation queries when access to the convex problem is given through membership queries.
The belief propagation (BP) algorithm is a powerful tool in a wide range of disciplines from statistical physics to machine learning to computational biology, and is ubiquitous in decoding classical error-correcting codes. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding of the channel, in some cases even up to the Shannon capacity. Here we construct the first BP algorithm which passes quantum messages on the factor graph and is capable of decoding the classical–quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the code length. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.
Some of the most interesting quantities associated with a factor graph are its marginals and its partition sum. For factor graphs without cycles and moderate message-update complexities, the sum-product algorithm (SPA) can be used to efficiently compute these quantities exactly. Moreover, for various classes of factor graphs with cycles, the SPA has been successfully applied to efficiently compute good approximations to these quantities. Note that in the case of factor graphs with cycles, the local functions are usually non-negative real-valued functions. In this paper we introduce a class of factor graphs, called double-edge factor graphs (DE-FGs), which allow local functions to be complex-valued and only require them, in some suitable sense, to be positive semi-definite kernel functions. We discuss various properties of the SPA when running it on DE-FGs and we show promising numerical results for various example DE-FGs, some of which have connections to quantum information processing.
Belief propagation (BP) is a classical algorithm that approximates the marginal distribution associated with a factor graph by passing messages between adjacent nodes in the graph. It gained popularity in the 1990’s as a powerful decoding algorithm for LDPC codes. In 2016, Renes introduced a belief propagation with quantum messages (BPQM) and described how it could be used to decode classical codes defined by tree factor graphs that are sent over the classical-quantum pure-state channel. In this work, we propose an extension of BPQM to general binary-input symmetric classical-quantum (BSCQ) channels based on the implementation of a symmetric "paired measurement". While this new paired-measurement BPQM (PMBPQM) approach is suboptimal in general, it provides a concrete BPQM decoder that can be implemented with local operations. Finally, we demonstrate that density evolution can be used to analyze the performance of PMBPQM on tree factor graphs. As an application, we compute noise thresholds of some LDPC codes with BPQM decoding for a class of BSCQ channels.