arXiv subjects

Jacob Miller

Publications and source records attributed to Jacob Miller.

At least 37 records · Page 2Linked to original sources

arXiv2023

Synergy Between Quantum Circuits and Tensor Networks: Short-cutting the Race to Practical Quantum Advantage

While recent breakthroughs have proven the ability of noisy intermediate-scale quantum (NISQ) devices to achieve quantum advantage in classically-intractable sampling tasks, the use of these devices for solving more practically relevant computational problems remains a challenge. Proposals for attaining practical quantum advantage typically involve parametrized quantum circuits (PQCs), whose parameters can be optimized to find solutions to diverse problems throughout quantum simulation and machine learning. However, training PQCs for real-world problems remains a significant practical challenge, largely due to the phenomenon of barren plateaus in the optimization landscapes of randomly-initialized quantum circuits. In this work, we introduce a scalable procedure for harnessing classical computing resources to provide pre-optimized initializations for PQCs, which we show significantly improves the trainability and performance of PQCs on a variety of problems. Given a specific optimization task, this method first utilizes tensor network (TN) simulations to identify a promising quantum state, which is then converted into gate parameters of a PQC by means of a high-performance decomposition procedure. We show that this learned initialization avoids barren plateaus, and effectively translates increases in classical resources to enhanced performance and speed in training quantum circuits. By demonstrating a means of boosting limited quantum resources using classical computers, our approach illustrates the promise of this synergy between quantum and quantum-inspired models in quantum computing, and opens up new avenues to harness the power of modern quantum hardware for realizing practical quantum advantage.

quant-ph↗

arXiv2023

On the Perception of Small Sub-graphs

Interpreting a node-link graph is enhanced if similar subgraphs (or motifs) are depicted in a similar manner; that is, they have the same visual form. Small motifs within graphs may be perceived to be identical when they are structurally dissimilar, or may be perceived to be dissimilar when they are identical. This issue primarily relates to the Gestalt principle of similarity, but may also include an element of quick, low-level pattern-matching. We believe that if motifs are identical, they should be depicted identically; if they are nearly-identical, they should be depicted nearly-identically. This principle is particularly important in domains where motifs hold meaning and where their identification is important. We identified five small motifs: bi-cliques, cliques, cycles, double-cycles, and stars. For each, we defined visual variations on two dimensions: same or different structure, same or different shape. We conducted a crowd-sourced empirical study to test the perception of similarity of these varied motifs, and found that determining whether motifs are identical or similar is affected by both shape and structure.

cs.HC↗

arXiv2022

Qubit seriation: Improving data-model alignment using spectral ordering

With the advent of quantum and quantum-inspired machine learning, adapting the structure of learning models to match the structure of target datasets has been shown to be crucial for obtaining high performance. Probabilistic models based on tensor networks (TNs) are prime candidates to benefit from data-dependent design considerations, owing to their bias towards correlations which are local with respect to the topology of the model. In this work, we use methods from spectral graph theory to search for optimal permutations of model sites which are adapted to the structure of an input dataset. Our method uses pairwise mutual information estimates from the target dataset to ensure that strongly correlated bits are placed closer to each other relative to the model's topology. We demonstrate the effectiveness of such preprocessing for probabilistic modeling tasks, finding substantial improvements in the performance of generative models based on matrix product states (MPS) across a variety of datasets. We also show how spectral embedding, a dimensionality reduction technique from spectral graph theory, can be used to gain further insights into the structure of datasets of interest.

quant-ph↗

arXiv2022

Spherical Graph Drawing by Multi-dimensional Scaling

We describe an efficient and scalable spherical graph embedding method. The method uses a generalization of the Euclidean stress function for Multi-Dimensional Scaling adapted to spherical space, where geodesic pairwise distances are employed instead of Euclidean distances. The resulting spherical stress function is optimized by means of stochastic gradient descent. Quantitative and qualitative evaluations demonstrate the scalability and effectiveness of the proposed method. We also show that some graph families can be embedded with lower distortion on the sphere, than in Euclidean and hyperbolic spaces.

cs.CG↗

arXiv2022

Decomposition of Matrix Product States into Shallow Quantum Circuits

The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed tensor network (TN) algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The "quantum-inspired" nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum-classical training frameworks which benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, something which remains an unresolved question. In this work, we compare a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimensions into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition scheme with constrained optimization of circuit unitaries, and all possess efficient classical runtimes. Our experimental results reveal one particular protocol, involving sequential growth and optimization of the quantum circuit, to outperform all other methods, with even greater benefits seen in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, in turn further unlocking the rich and complementary benefits of classical and quantum computation.

quant-ph↗

arXiv2022

Browser-based Hyperbolic Visualization of Graphs

Hyperbolic geometry offers a natural focus + context for data visualization and has been shown to underlie real-world complex networks. However, current hyperbolic network visualization approaches are limited to special types of networks and do not scale to large datasets. With this in mind, we designed, implemented, and analyzed three methods for hyperbolic visualization of networks in the browser based on inverse projections, generalized force-directed algorithms, and hyperbolic multi-dimensional scaling (H-MDS). A comparison with Euclidean MDS shows that H-MDS produces embeddings with lower distortion for several types of networks. All three methods can handle node-link representations and are available in fully functional web-based systems.

cs.GR↗

arXiv2021

Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework

We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturing structure in discrete probability distributions, this behavior also renders the direct application of standard PGM tools impossible. We aim to bridge this gap by introducing a hybrid PGM-TN formalism that integrates quantum-like correlations into PGM models in a principled manner, using the physically-motivated concept of decoherence. We first prove that applying decoherence to the entirety of a BM model converts it into a discrete UGM, and conversely, that any subgraph of a discrete UGM can be represented as a decohered BM. This method allows a broad family of probabilistic TN models to be encoded as partially decohered BMs, a fact we leverage to combine the representational strengths of both model families. We experimentally verify the performance of such hybrid models in a sequential modeling task, and identify promising uses of our method within the context of existing applications of graphical models.

stat.ML↗

arXiv2021

Adaptive Learning of Tensor Network Structures

Tensor Networks (TN) offer a powerful framework to efficiently represent very high-dimensional objects. TN have recently shown their potential for machine learning applications and offer a unifying view of common tensor decomposition models such as Tucker, tensor train (TT) and tensor ring (TR). However, identifying the best tensor network structure from data for a given task is challenging. In this work, we leverage the TN formalism to develop a generic and efficient adaptive algorithm to jointly learn the structure and the parameters of a TN from data. Our method is based on a simple greedy approach starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify TN structures with small number of parameters that effectively optimize any differentiable objective function. Experiments on tensor decomposition, tensor completion and model compression tasks demonstrate the effectiveness of the proposed algorithm. In particular, our method outperforms the state-of-the-art evolutionary topology search [Li and Sun, 2020] for tensor decomposition of images (while being orders of magnitude faster) and finds efficient tensor network structures to compress neural networks outperforming popular TT based approaches [Novikov et al., 2015].

cs.LG↗

arXiv2021

Adaptive Epidemic Forecasting and Community Risk Evaluation of COVID-19

Pandemic control measures like lock-down, restrictions on restaurants and gatherings, social-distancing have shown to be effective in curtailing the spread of COVID-19. However, their sustained enforcement has negative economic effects. To craft strategies and policies that reduce the hardship on the people and the economy while being effective against the pandemic, authorities need to understand the disease dynamics at the right geo-spatial granularity. Considering factors like the hospitals' ability to handle the fluctuating demands, evaluating various reopening scenarios, and accurate forecasting of cases are vital to decision making. Towards this end, we present a flexible end-to-end solution that seamlessly integrates public health data with tertiary client data to accurately estimate the risk of reopening a community. At its core lies a state-of-the-art prediction model that auto-captures changing trends in transmission and mobility. Benchmarking against various published baselines confirm the superiority of our forecasting algorithm. Combined with the ability to extend to multiple client-specific requirements and perform deductive reasoning through counter-factual analysis, this solution provides actionable insights to multiple client domains ranging from government to educational institutions, hospitals, and commercial establishments.

cs.CY↗

arXiv2021

Tensor Networks for Probabilistic Sequence Modeling

Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We first show that u-MPS enable sequence-level parallelism, with length-n sequences able to be evaluated in depth O(log n). We then introduce a novel generative algorithm giving trained u-MPS the ability to efficiently sample from a wide variety of conditional distributions, each one defined by a regular expression. Special cases of this algorithm correspond to autoregressive and fill-in-the-blank sampling, but more complex regular expressions permit the generation of richly structured data in a manner that has no direct analogue in neural generative models. Experiments on sequence modeling with synthetic and real text data show u-MPS outperforming a variety of baselines and effectively generalizing their predictions in the presence of limited data.

cs.LG↗

arXiv2020

Quantum Tensor Networks, Stochastic Processes, and Weighted Automata

Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix product states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.

cs.LG↗

arXiv2018

Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry

An emerging insight is that ground states of symmetry-protected topological orders (SPTO's) possess latent computational complexity in terms of their many-body entanglement. By introducing a fractional symmetry of SPTO, which requires the invariance under 3-colorable symmetries of a lattice, we prove that every renormalization fixed-point state of 2D $(\mathbb{Z}_2)^m$ SPTO with fractional symmetry can be utilized for universal quantum computation using only Pauli measurements, as long as it belongs to a nontrivial 2D SPTO phase. Our infinite family of fixed-point states may serve as a base model to demonstrate the idea of a "quantum computational phase" of matter, whose states share universal computational complexity ubiquitously.

quant-ph↗

arXiv2017

Quantum supremacy in constant-time measurement-based computation: A unified architecture for sampling and verification

While quantum speed-up in solving certain decision problems by a fault-tolerant universal quantum computer has been promised, a timely research interest includes how far one can reduce the resource requirement to demonstrate a provable advantage in quantum devices without demanding quantum error correction, which is crucial for prolonging the coherence time of qubits. We propose a model device made of locally-interacting multiple qubits, designed such that simultaneous single-qubit measurements on it can output probability distributions whose average-case sampling is classically intractable, under similar assumptions as the sampling of non-interacting bosons and instantaneous quantum circuits. Notably, in contrast to these previous unitary-based realizations, our measurement-based implementation has two distinctive features. (i) Our implementation involves no adaptation of measurement bases, leading output probability distributions to be generated in constant time, independent of the system size. Thus, it could be implemented in principle without quantum error correction. (ii) Verifying the classical intractability of our sampling is done by changing the Pauli measurement bases only at certain output qubits. Our usage of random commuting quantum circuits in place of computationally universal circuits allows a unique unification of sampling and verification, so that they require the same physical resource requirements in contrast to the more demanding verification protocols seen elsewhere in the literature.

quant-ph↗

arXiv2016

Hierarchy of universal entanglement in 2D measurement-based quantum computation

Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. While MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here, we utilize recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behavior differs fundamentally from the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X, Y, and Z operators. This novel computational feature is related to the "genuine" 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity.

quant-ph↗

arXiv2016

Estimating Propensity Parameters using Google PageRank and Genetic Algorithms

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) is a modeling framework that considers two propensity parameters for updating each node and uses one when the update has a positive impact on the variable, that is, when the update causes the variable to increase its value, and uses the other when the update has a negative impact, that is, when the update causes it to decrease its value. This framework offers additional features for simulations but also adds a complexity in parameter estimation of the propensities. This paper presents a method for estimating the propensity parameters for SDDS. The method is based on adding noise to the system using the Google PageRank approach to make the system ergodic and thus guaranteeing the existence of a stationary distribution. Then with the use of a genetic algorithm, the propensity parameters are estimated. Approximation techniques that make the search algorithms efficient are also presented and Matlab/Octave code to test the algorithms are available at http://www.ms.uky.edu/~dmu228/GeneticAlg/Code.html.

q-bio.MN↗

arXiv2015

Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data

Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed (see the seminal paper [1]). However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measures with finite support. In this paper, we develop theoretical results for Wasserstein barycenters in this discrete setting. Our results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals. Our results closely mirror those in the continuous case with a few exceptions. In this discrete setting we establish that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse. Moreover, for each Wasserstein barycenter there exists a non-mass-splitting optimal transport to each of the discrete marginals. Such non-mass-splitting transports do not generally exist between two discrete measures unless special mass balance conditions hold. This makes Wasserstein barycenters in this discrete setting special in this regard. We illustrate the results of our discrete barycenter theory with a proof-of-concept computation for a hypothetical transportation problem with multiple marginals: distributing a fixed set of goods when the demand can take on different distributional shapes characterized by the discrete marginal distributions. A Wasserstein barycenter, in this case, represents an optimal distribution of inventory facilities which minimize the squared distance/transportation cost totaled over all demands.

math.OC↗

arXiv2015

The Hierarchy of Circuit Diameters and Transportation Polytopes

The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch conjecture is true for general $m{\times}n$--transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that this hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarchy of diameters for the $m{\times}n$--transportation polytopes. We show that the Hirsch conjecture bound of $m+n-1$ is actually valid in most of these diameter notions. Second, we prove that for $3{\times}n$--transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for $2{\times}n$--transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter.

math.CO↗

arXiv2015

Resource quality of a symmetry-protected topologically ordered phase for quantum computation

We investigate entanglement naturally present in the 1D topologically ordered phase protected with the on-site symmetry group of an octahedron as a potential resource for teleportation-based quantum computation. We show that, as long as certain characteristic lengths are finite, all its ground states have the capability to implement any unit-fidelity one-qubit gate operation asymptotically as a key computational building block. This feature is intrinsic to the entire phase, in that perfect gate fidelity coincides with perfect string order parameters under a state-insensitive renormalization procedure. Our approach may pave the way toward a novel program to classify quantum many-body systems based on their operational use for quantum information processing.

quant-ph↗