Temporal Matrix Product States

• Temporal Matrix Product States are matrix-product constructions that treat time slices as sites, efficiently compressing temporal correlations and memory effects.
• They employ tools like transfer matrices and entanglement entropy to analyze stochastic processes, quantum impurity dynamics, Euclidean-time evolution, and deterministic dynamics in models such as Rule 54.
• The framework underpins practical simulation algorithms, enabling predictive modeling and efficient memory truncation in time-dependent, many-body and non-equilibrium systems.

Searching arXiv for relevant papers on temporal matrix product states and closely related formulations. Temporal Matrix Product States are matrix-product constructions in which the ordered one-dimensional tensor-network direction is temporal rather than spatial, or in which a past–future stochastic axis is recast as an MPS chain. In the literature, the term covers several distinct but structurally related objects: q-samples of stationary stochastic processes, Feynman–Vernon influence functionals represented as MPS in the time domain, continuous MPS obtained from Euclidean-time evolution, and time states encoding multi-time probabilities of local observables (Yang et al., 2018, Thoenniss et al., 2022, Tirrito et al., 2018, Klobas et al., 2019). Across these settings, the central idea is that temporal correlations can be compressed into bond degrees of freedom, so that entanglement spectra, transfer matrices, and canonical forms become tools for analyzing memory, predictability, and simulation cost.

1. Conceptual scope and formal setting

A temporal MPS is not a single universal ansatz but a family of constructions in which time slices, time bins, or past–future partitions play the role ordinarily occupied by spatial lattice sites. In stochastic modeling, the basic object is a pure state over time steps,

$$|\Psi\rangle=\sum_{x_1\cdots x_N\in\mathcal A}\sqrt{P(x_1,\ldots,x_N)}\,|x_1\rangle\otimes\cdots\otimes|x_N\rangle,$$

whose local measurements reproduce the classical joint distribution exactly (Yang et al., 2018). In non-equilibrium impurity problems, the temporal object is instead the discrete-time influence functional on a Keldysh contour, which can be interpreted as a fictitious Gaussian wavefunction on a one-dimensional temporal lattice of length $T$ (Thoenniss et al., 2022). In Euclidean formulations of quantum critical systems, repeated imaginary-time evolution generates an MPS along the time direction, whose continuum limit is a cMPS characterized by matrices $Q$ and $R$ (Tirrito et al., 2018). In deterministic lattice dynamics such as Rule 54, the time state is a probability distribution over bit configurations observed at a fixed spatial point over multiple times, and this distribution itself admits an exact MPS form (Klobas et al., 2019).

The common structure is an ordered chain of local temporal indices with auxiliary bonds carrying compressed information about temporal dependence. The meaning of the bond variables, however, is model-dependent. In the q-sample construction they encode causal states of an $\varepsilon$-machine; in influence-functional methods they encode memory inherited from integrating out a reservoir; in Euclidean-time tMPS they encode the fixed-point structure of imaginary-time evolution; and in Rule 54 they encode a finite temporal memory of admissible soliton configurations. A plausible implication is that “temporal MPS” is best understood as a tensor-network perspective on memory-bearing dynamics rather than as a single domain-specific formalism.

2. Stochastic processes, q-samples, and predictive memory

For a stationary stochastic process, Yang, Binder, Narasimhachar, and Gu associate the process with a q-sample state and show that the optimal predictive model leads directly to an MPS representation of that state (Yang et al., 2018). An open-boundary MPS of bond dimension $D$ has the form

$$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$

where $A^x$ is a $D\times D$ matrix for each symbol $x$, and $T$0 controls the maximum Schmidt rank across any bipartition. For a classical $T$1-machine with causal states $T$2 and transition probabilities

$T$3

the site matrices are defined by

$T$4

With bond dimension $T$5, the resulting boundary-free MPS reproduces precisely $T$6.

This construction gives the temporal bond index a direct predictive interpretation. The left bond index represents the causal state “at the left,” the physical index samples the emitted symbol, and the right bond index becomes the new causal state. The past–future cut of the infinite MPS then yields a Schmidt decomposition

$T$7

with entanglement entropy

$T$8

Yang et al. prove that this $T$9 coincides with the quantum Shannon entropy of the q-simulator memory state,

$Q$0

so that

$Q$1

They further identify the rank entropy $Q$2 with the Schmidt rank $Q$3 (Yang et al., 2018).

The transfer matrix

$Q$4

governs ergodicity and canonicalization. Ergodicity of the original stochastic process is equivalent to $Q$5 having a unique leading eigenvalue. From the corresponding left and right eigenmatrices $Q$6, one constructs a canonical form in which the diagonal matrix $Q$7 explicitly displays the Schmidt coefficients $Q$8. In this gauge, the minimal exact bond dimension is the Schmidt rank, while the entanglement spectrum is read directly from $Q$9. The same framework also motivates approximate predictive quantum models via MPS truncation, for example by cutting small Schmidt values at the cost of small errors in reproducing process statistics (Yang et al., 2018).

3. Influence functionals and temporal entanglement in impurity dynamics

In non-equilibrium quantum impurity problems, the temporal MPS is built not from a state of physical spins along time, but from the Feynman–Vernon influence functional obtained after integrating out a non-interacting reservoir. After Trotter discretization into $R$0 time steps of size $R$1, the influence functional becomes a tensor with forward and backward Keldysh legs. In a fermionic coherent-state representation it has Gaussian form,

$R$2

in $R$3 Grassmann components, and can be interpreted as the wavefunction of $R$4 fictitious fermions on a one-dimensional temporal lattice (Thoenniss et al., 2022). Thoenniss et al. write the corresponding fictitious state in a BCS-like form and define temporal entanglement by bipartitioning the temporal chain at some time $R$5.

For a bipartition $R$6, $R$7, the temporal entanglement entropy is

$R$8

The maximum typically occurs near $R$9. The scaling law depends on reservoir initial conditions. For short-range correlated or finite-temperature initial states with finite correlation length, the temporal entanglement obeys a temporal area law,

$\varepsilon$0

For a one-dimensional zero-temperature Fermi sea, it shows a logarithmic violation,

$\varepsilon$1

with $\varepsilon$2 for spinless fermions and $\varepsilon$3 for Majorana modes (Thoenniss et al., 2022).

The exact MPS conversion is based on an extension of the Fishman–White algorithm. One computes the two-point correlation matrix of the Gaussian state, iteratively peels off nearly localized natural orbitals, maps them onto physical sites using Givens and Bogoliubov rotations, inverts the resulting circuit, and contracts it onto the vacuum to obtain an exact MPS. At a cut $\varepsilon$4, the bond dimension is bounded by $\varepsilon$5, where $\varepsilon$6 is the subsystem size used to localize the $\varepsilon$7-th mode. Numerically, $\varepsilon$8 remains $\varepsilon$9, or grows only logarithmically, for reservoirs with finite correlation time, so $D$0 grows at most polynomially in $D$1. Truncation is implemented by stopping the mode extraction once all residual eigenvalues satisfy $D$2, producing an approximate MPS $D$3 with infidelity bounded by $D$4 (Thoenniss et al., 2022).

Once the influence matrix is encoded as an MPS of bond dimension $D$5, impurity observables are obtained by contracting local impurity super-operators of dimension $D$6 with the reservoir tensors along the Keldysh contour. The overall complexity scales as

$D$7

All time-ordered and out-of-time-ordered impurity correlators can be inserted by modifying the local impurity tensor at the appropriate times (Thoenniss et al., 2022).

A distinct but related advance exploits time-translational invariance of the influence functional. Guo and Chen construct a translationally invariant MPO for the exponent

$D$8

after fitting the memory kernel as

$D$9

The generator is represented by a single site tensor of bond dimension $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$0, and the full influence functional is obtained from

$$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$1

by only $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$2 successive squarings, with compression after each multiplication (Guo et al., 2024). The total cost scales as

$$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$3

while the number of costly MPS multiplications becomes independent of $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$4. In the Toulouse model, for $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$5, $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$6, and $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$7, both partial-IF and TTI-IF reach maximum Green’s-function error $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$8, while CPU time scales as $$|\Psi\rangle=\sum_{x_1\cdots x_N}\langle b_l|A^{x_1}A^{x_2}\cdots A^{x_N}|b_r\rangle\,|x_1\cdots x_N\rangle,$$9 s for partial-IF and $A^x$0 s for TTI-IF; in that setting $A^x$1 squarings gave negligible gain when further increased (Guo et al., 2024).

4. Euclidean-time tMPS and the quantum field theory vacuum

A different usage of temporal MPS appears in Euclidean-time tensor networks for ground states of one-dimensional quantum systems. Starting from a Hamiltonian

$A^x$2

the ground state is projected by imaginary-time evolution,

$A^x$3

Trotterizing $A^x$4 into $A^x$5 steps builds a two-dimensional tensor network of height $A^x$6 in the imaginary-time direction and width $A^x$7 in physical space. Contracting this network from left to right, rather than downward, yields an MPS in the time direction with identical time-site tensors $A^x$8; the fixed point of this procedure is the temporal MPS, or tMPS (Tirrito et al., 2018).

As $A^x$9 and $D\times D$0, the temporal lattice becomes continuous and one obtains the Verstraete–Cirac cMPS ansatz with $D\times D$1-independent $D\times D$2 matrices $D\times D$3 and $D\times D$4. The discrete tensors satisfy

$D\times D$5

and the continuum limit becomes

$D\times D$6

The matrices $D\times D$7 and $D\times D$8 are extracted numerically by extrapolating finite-$D\times D$9 data for $x$0 and $x$1 (Tirrito et al., 2018).

At the critical point of a translationally invariant lattice Hamiltonian, low-energy excitations have linear dispersion $x$2, and the continuum limit acquires emergent Lorentz symmetry. This makes it natural to exchange space and Euclidean time by rotation. The temporal and spatial correlation lengths are then related by

$x$3

where the temporal transfer matrix is

$x$4

Numerically, $x$5 was found for the Ising example discussed in the paper (Tirrito et al., 2018).

The same work uses the Bisognano–Wichmann theorem to connect the half-chain entanglement spectrum of the tMPS to the spectrum of the QFT Hamiltonian on a strip with free boundary conditions. For half a temporal or spatial chain, the entropy follows the Calabrese–Cardy form

$x$6

with measured central charge $x$7 and finite-entanglement scaling exponent $x$8 for both tMPS and spatial MPS (Tirrito et al., 2018). The low-lying entanglement levels converge to the conformal field theory values for the Ising vacuum on a strip, with a tower of the identity starting at $x$9 and a tower of the spin field $T$00 at $T$01.

Away from criticality, the same temporal framework is applied to the weakly perturbed Ising field theory with Hamiltonian density

$T$02

where the physics depends on $T$03. The authors attempted to match tMPS transfer-matrix gaps to the meson masses $T$04 satisfying Zamolodchikov’s semiclassical quantization condition, but found systematic discrepancies. Their explicit conclusion is that the tMPS transfer-matrix spectrum does not trivially coincide with the full QFT mass spectrum without a proper zero-momentum projection (Tirrito et al., 2018). This is one of the clearest examples in the literature of a nontrivial interpretive caveat in temporal-MPS spectroscopy.

5. Time states and multi-time correlations in Rule 54

In the reversible cellular automaton Rule 54, temporal MPS appear as exact matrix-product representations of multi-time probability distributions at a fixed spatial point. The sampled observable is the occupation bit at alternating sites over times $T$05, producing a time state

$T$06

where $T$07 is a space–time-translation-invariant equilibrium measure and $T$08 for even $T$09, $T$10 for odd $T$11 (Klobas et al., 2019). This is an exact probability distribution over time configurations, not a wavefunction in the ordinary quantum sense.

The equilibrium reference state itself is a two-parameter MPS ansatz with $T$12 matrices $T$13 and $T$14, stabilized by cubic cancellation relations involving the local Rule-54 permutation $T$15 and an auxiliary swap $T$16. In the thermodynamic limit, finite blocks are governed by the leading eigenvalue $T$17 of

$T$18

and the left- and right-moving soliton densities are fixed functions of $T$19, $T$20, and $T$21 (Klobas et al., 2019).

The key simplification is that, in this equilibrium measure, solitons at different times are statistically independent subject only to exclusion, so the conditional probability

$T$22

depends only on the last four bits and alternates between two functions related by $T$23. This allows an exact embedding into a three-dimensional auxiliary space with local matrices

$T$24

and odd-time counterparts $T$25. With boundary vectors

$T$26

the full time state is

$T$27

(Klobas et al., 2019).

This exact temporal MPS gives direct access to equal-space multi-time observables. For example, the density–density correlator

$T$28

is expressed in terms of the temporal transfer matrix

$T$29

whose subleading eigenvalues $T$30 determine a sum of two exponentials. At the maximum-entropy point $T$31, corresponding to $T$32, the result reproduces the previously known exact formula. The same construction yields closed-form exchange-time and persistence-time distributions and leads to the inequality

$T$33

which is used to prove the absence of decoupling of timescales in Rule 54 (Klobas et al., 2019).

6. Entanglement, transfer matrices, and interpretive boundaries

Across these formulations, three structural motifs recur: a transfer matrix or transfer operator, an entanglement or entropy measure associated with a temporal cut, and an auxiliary bond space that stores compressed temporal information. In the stochastic q-sample setting, the transfer matrix $T$34 determines ergodicity and canonical form, while the past–future entanglement entropy equals the optimal quantum memory cost $T$35 (Yang et al., 2018). In impurity influence-function methods, the temporal entanglement of the fictitious time-domain wavefunction controls the feasibility of MPS simulation, producing temporal area-law behavior for finite-correlation-time initial states and logarithmic growth for one-dimensional critical Fermi seas (Thoenniss et al., 2022). In Euclidean-time tMPS, the temporal transfer matrix $T$36 supplies correlation lengths, entanglement spectra, and critical data of the emergent continuum theory (Tirrito et al., 2018). In Rule 54, the temporal transfer matrix $T$37 generates exact multi-time probabilities and waiting-time statistics (Klobas et al., 2019).

These parallels should not obscure important differences. First, “temporal entanglement” is not uniformly a physical entanglement in laboratory time. In the impurity formulation it is the entanglement of a fictitious wavefunction representing an influence functional on a temporal lattice (Thoenniss et al., 2022). Second, a temporal MPS need not represent a quantum state at all; it may encode a classical probability distribution, as in q-samples and Rule-54 time states (Yang et al., 2018, Klobas et al., 2019). Third, temporal transfer-matrix spectra do not automatically coincide with spectra of physical excitations; the perturbed-Ising analysis explicitly shows that such identifications can fail without further projection conditions (Tirrito et al., 2018).

The algorithmic message is similarly nuanced. Several papers show that temporal ordering can make otherwise intractable dynamics compressible: q-samples reduce predictive modeling to MPS canonicalization, Gaussian influence functionals admit exact or controllably approximate MPS conversion, and time-translational invariance can reduce the number of required MPS multiplications from $T$38 to $T$39 (Yang et al., 2018, Thoenniss et al., 2022, Guo et al., 2024). At the same time, the Rule-54 analysis states explicitly that its fixed auxiliary dimension $T$40 relies on special integrable and probabilistic simplifications, and that for more general interacting systems one expects the time-MPS auxiliary dimension to grow, typically exponentially, with time (Klobas et al., 2019). This suggests that the usefulness of temporal MPS is governed less by the mere presence of time ordering than by whether the underlying temporal correlations obey an effective area law, finite-memory condition, or other compressibility criterion.