Matrix Product States for Quantum Stochastic Modeling

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.The quest for simple representations and models of the physical world, often phrased as Occam's famous razor, underlies most scientific pursuits. In this spirit, computational mechanics seeks the most memory-efficient predictive models for stochastic processes-models which track relevant past information about a process, in order to generate statistically faithful future predictions [1-5]. The classically minimal models, ε-machines, have been used in diverse contexts from neuroscience to nonequilibrium contextuality [6][7][8][9][10][11][12][13][14][15]. Recently, it was shown that quantum extensions of ε-machines can further reduce their memory [16], leading recent studies to find memoryefficient quantum means of predictive modeling [17][18][19][20][21][22][23][24][25][26][27].In condensed matter, on the other hand, simplicity is sought after for the description of quantum many-body systems. Tensor networks, such as matrix product states (MPS), for instance, provide an efficient and useful description of one-dimensional quantum systems-i.e., spin chains [28][29][30]. This has led to reliable and powerful numerical methods for probing and simulating properties of multi-partite systems, whose study would be otherwise intractable [31][32][33][34].In this Letter, we develop a connection between εmachines and the MPS representation, shown in Fig. 1. We associate each stochastic process with a suitable quantum state of a spin chain called q-sample, measurement of which generates the corresponding stochastic process. We then show that the classical ε-machine of the process leads to a systematic MPS representation of the q-sample. Conversely, applying MPS methods to this state allows us to construct a q-simulator for the associated stochastic process -the best known quantum model [17,21,22]. Lastly, we show that the entanglement across the past-future bipartition of the q-sample coincides exactly with the quantum memory requirements * Yangchengran92@gmail.com † quantum@felix-binder.net ‡ mgu@quantumcomplexity.org Fig. 1. This Letter connects the complexity of stochastic processes and the representational complexity of spin chains. These (i) lin...

Cavitylike strong coupling in macroscopic waveguide QED using three coupled qubits in the deep non-Markovian regime

“…).-To represent the time evolution of the probe qubit in the non-Markovian regime, and to confirm our semiclassical results in the frequency domain, we next solve the same system using MPSs [37][38][39]. To do this, we first consider the Hamiltonian for three qubits [29], dependent boson operator.…”

Non-Hermitian Physics

“…( 296) may not be upper bounded by cµ l ∀l with c being a constant. We may readily construct such a state from a classical Markov chain specified by a transition-probability matrix [T ] ba = p a→b , which satisfies p a→b ≥ 0 and b p a→b = 1 ∀a [884,885]. The MPS is generated by M a→b = √ p a→b |a)(b| 22 and is normal whenever the classical steady state π determined by T π = π is unique and satisfies π a > 0 ∀a.…”

Modelling quantum light-matter interactions in waveguide-QED with retardation and a time-delayed feedback: matrix product states versus a space-discretized waveguide model

“…The aim of this paper is to present and compare two powerful, but quite different, approaches for solving several classes of non-Markovian feedback and waveguide QED, which can be applied to study both vacuum dynamics and nonlinear (i.e., multi-photon) excitation regimes. Specifically, these methods are based on: (a) matrix product states (MPSs) [15,[38][39][40] and (b) a new space-discretized waveguide (SDW) model, using a collision approach for the waveguide environment [41][42][43][44]. The collision model is solved explicitly by allowing for 1 or 2 photons in the waveguide, which helps to show when several photons need to be included in the model, while the MPS model is not restricted in the number of photons.…”