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arXiv:1803.08220 · Physical Review Letters

Matrix Product States for Quantum Stochastic Modeling

Chengran Yang, Felix C. Binder, Varun Narasimhachar, Mile Gu

Published Dec 27, 2018• 26 citations
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Abstract

In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.

DOI
10.1103/physrevlett.121.260602

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In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.
Predicting a stochastic process' future lies at the heart of many scientific areas.A predictive model extracts information from a stochastic process' past and uses it to generate future statistics.There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information.By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits.Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems.In this thesis, we explore the connection between tensor networks and quantum predictive models.A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models.First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models.MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory.Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models.Third, we propose a family of divergence measures to quantify the distance between two stochastic processes.This family of divergence measures overcomes certain weaknesses of the existing measures.Moreover, we propose an efficient means of computing the divergence measure using our MPS methods.Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process.We use machine learning methods to propose a systematic algorithm for learning the quantum model.This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.
We propose an entanglement classification for symmetric quantum states based on their diagonal matrix-product-state (MPS) representation. The proposed classification, which preserves the stochastic local operation assisted with classical communication (SLOCC) criterion, relates entanglement families to the interaction length of Hamiltonians. In this manner, we establish a connection between entanglement classification and condensed matter models from a quantum information perspective. Moreover, we introduce a scalable nesting property for the proposed entanglement classification, in which the families for N parties carry over to the N + 1 case. Finally, using techniques from algebraic geometry, we prove that the minimal nontrivial interaction length n for any symmetric state is bounded by .
Matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an NNN-qubit MPS with large NNN and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.
The matrix product state (MPS) belongs to the most important models in, for example, quantum information sciences and condensed-matter physics. However, realizing an NNN-qubit MPS with large NNN and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or nnn-qubit gates with $n\ensuremath{\gg}2$ to carry or produce the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. Essentially different from the existing compiling methods, our idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPSs of the strongly correlated spin models show that the constructed quantum circuits can simulate the MPSs with much fewer qubits than the sizes of the MPSs themselves. This method paves a feasible and efficient path to realizing useful and/or exotic quantum states and MPS-based models as quantum circuits on near-term quantum platforms.
Isolating past information relevant for future prediction is central to quantitative science. Quantum models offer a promising approach, enabling statistically faithful modeling while using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of simulating a stochastic process unitarily in causal order. The resulting constructions surpass previous state-of-art methods---both in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information. Moreover, these two features are generally competing factors in optimization---leading to an ambiguity in optimal modeling that is unique to the quantum regime. Our results simultaneously offer quantum advantages for stochastic simulation and illustrate further qualitative differences between classical and quantum notions of complexity.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.