Matrix Product States

• Matrix Product States are a structured ansatz that efficiently represent 1D quantum many-body systems using low-dimensional tensor networks.
• They employ gauge freedom and canonical forms to optimize state representations, underpinning techniques like DMRG and time-dependent variational methods.
• Their rich algebraic and geometric structure links quantum physics with invariant theory, extending to cMPS and Gaussian MPS for broader applications.

A matrix product state (MPS) is a structured, low-parameter ansatz for representing elements of tensor product spaces, most notably wavefunctions of quantum many-body systems on one-dimensional lattices. The formalism provides the foundation of modern tensor network methods in both physics and computer science, including the density matrix renormalization group, and admits extensions to stochastic processes, quantum information, algebraic geometry, and operator algebraic frameworks.

1. Definitions, Algebraic Structure, and Parameterization

For a chain of $N$ sites, each with local Hilbert space $\mathbb{C}^d$, an MPS of bond (auxiliary) dimension $D$ is the state

$$|\psi_N\rangle = \sum_{s_1,\dots,s_N=1}^d \operatorname{Tr}\left[ A^{s_1} A^{s_2} \cdots A^{s_N} \right]\, |s_1 s_2 \cdots s_N\rangle,$$

where the $A^s$ are $D \times D$ complex matrices, and the trace is over the auxiliary indices. With open boundary conditions, the trace is replaced by boundary vectors or matrices. The bond dimension $D$ controls the representational power, as the maximal Schmidt rank across any bipartition is at most $D$ (Souissi et al., 5 Nov 2024).

In the language of tensor networks, the MPS is a 1D tensor contraction: each $A^{s_n}$ corresponds to a physical index $s_n$ and two virtual (bond) indices contracted along the chain. MPS with site-dependent tensors $A^{[n]s_n}$ remain within the class, with translation-invariant MPS corresponding to $A^{[n]} \equiv A$ for all $n$ (Seynnaeve, 2022, Critch et al., 2012).

Algebraically, the image of the MPS parameterization forms a low-dimensional (typically nonlinear) algebraic variety within the exponentially large tensor space, governed by classical trace identities and invariant theory (see Section 6) (Seynnaeve, 2022, Critch et al., 2012).

2. Canonical Form, Gauge Redundancy, and Geometry

MPS parameterizations are not unique. There exists a gauge freedom: for any invertible $X$, $A^s \mapsto X^{-1}A^s X$ leaves $|\psi_N\rangle$ invariant, up to possible changes in boundary conditions (Haegeman et al., 2012, Haegeman et al., 2012). This redundancy underlies a principal fiber bundle structure for the manifold of MPS: the parameter space is the total space, the physical states form the base manifold, and the structure group is $\mathrm{PGL}(D)$ (Haegeman et al., 2012).

Gauge fixing (canonical form) is implemented by imposing local orthogonality constraints:

• Left-canonical: $\sum_{s}(A^s)^\dagger A^s = I$.
• Right-canonical: $\sum_{s} A^s (A^s)^\dagger = I$.

The geometry of the MPS manifold is Kählerian, with a well-defined induced metric (the pullback of the Fubini–Study metric) and explicit formulas for tangent spaces, connections, and curvature tensors (Haegeman et al., 2012). This structure is crucial for variational optimization (DMRG), time-dependent variational principle (TDVP), and systematic error control.

3. Physical Significance: Entanglement, Expressiveness, and Representational Power

The bond dimension $D$ quantifies the amount of entanglement an MPS can express: the entanglement entropy across any bipartition is at most $\log D$ (Yang et al., 2018, Souissi et al., 5 Nov 2024). MPS naturally obey an area law, efficiently approximating ground states of gapped 1D local Hamiltonians and some critical systems (modulo logarithmic corrections). For generic states, exact representation requires $D$ scaling exponentially with system size, but most physical states of interest are efficiently approximable, as shown in lattice gauge theory applications (Bañuls et al., 2013).

The expressiveness of MPS is characterized algebraically: for translation-invariant $D=2$ MPS on $N$ qubits, the MPS variety saturates at dimension 5 once $N \gtrsim 5$, indicating that most quantum states cannot be represented as low-bond MPS (Critch et al., 2012, Seynnaeve, 2022). Explicit polynomial equations characterize which states admit an exact MPS representation.

4. Computational Methods and Applications

4.1 DMRG and Variational Algorithms

The density matrix renormalization group algorithm is the variational optimization of the MPS ansatz. The energy expectation value is minimized over the tensor parameters, typically via alternating least-squares (sweeping) (Bañuls et al., 2013). The computational cost per sweep scales as $O(dD^3)$, where $d$ is the physical dimension.

Excited states and dynamics can also be addressed within the MPS manifold using tangent-space formalism, which allows systematic construction of time-evolved or excited wavefunctions (Haegeman et al., 2012, Haegeman et al., 2012).

4.2 Infinite-Volume and Operator-Algebraic Framework

Consistency and projectivity conditions ensure the extension of finite MPS to states on the infinite quasi-local $C^*$-algebra. Specifically, the matrices must satisfy

$$\sum_{s} (A^s)^\dagger A^s = I_D, \quad \sum_{s} (A^s)^\dagger \otimes A^s = \sum_{s} A^s \otimes (A^s)^\dagger,$$

ensuring the existence and uniqueness of the limiting (infinite-volume) state and correspond to fixed-point equations of quantum channels (Souissi et al., 5 Nov 2024).

4.3 Physical Models and Numerical Benchmarks

Applications include precise simulation of the Schwinger model (lattice QED) (Bañuls et al., 2013), efficient encoding of ground states, and computation of order parameters and excitation spectra. The approach extends to systems with chemical potential, out-of-equilibrium situations, and systems with non-Abelian or dynamically adapted symmetries (Guo et al., 2019).

5. Extensions: cMPS and Gaussian MPS

5.1 Continuous Matrix Product States (cMPS)

The cMPS formalism extends the tensor network paradigm to quantum field theories in one spatial dimension. A cMPS of bond dimension $D$ for bosonic or fermionic fields $\hat\psi(x)$ is

$$|\chi\rangle = \operatorname{Tr}_{\mathrm{aux}} \left[ \mathcal{P}\exp\left( \int_0^L dx\, \big( Q(x) \otimes I + R(x) \otimes \hat\psi^\dagger(x) \big) \right) \right] | \Omega \rangle,$$

where $Q(x), R(x)$ are $D\times D$ auxiliary matrices, and $\mathcal{P}$ denotes path-ordering (Verstraete et al., 2010, Haegeman et al., 2012).

cMPS arise as exact continuum limits of discrete MPS under appropriate scaling, and the formalism enables direct variational optimization for continuum models (e.g., Lieb–Liniger gas), producing results indistinguishable from Bethe ansatz solutions for modest $D$ (Verstraete et al., 2010). The area law remains valid for cMPS, and the bond dimension controls an effective intrinsic ultraviolet cutoff.

5.2 Gaussian MPS (GMPS)

Gaussian MPS generalize MPS to lattices of bosonic modes. Bonds correspond to projected infinitely squeezed EPR states, and onsite maps are Gaussian channels. Any pure translationally invariant Gaussian state can be approximated arbitrarily well by a GMPS of finite bond dimension (Schuch et al., 2012). GMPS arise as unique ground states of local quadratic Hamiltonians whose range is determined by the bond structure.

Correlation functions can be computed efficiently via Fourier-space techniques, and exponential decay of correlations is guaranteed in 1D gapped systems.

6. Algebraic Geometry, Invariant Theory, and Connections to Stochastic Models

The MPS varieties are low-dimensional algebraic subvarieties indexed by physical and bond dimensions and chain length. The defining ideals are generated by polynomial identities of matrices (e.g., Cayley–Hamilton, trace identities), and for $D=2$, $n=2$, explicit Hilbert series and dimension formulas for spans are known (Seynnaeve, 2022).

Parameter identifiability: For $D=d=2$ MPS, global state reconstruction up to phase is possible from 3- or 4-site reduced density matrices, depending on boundary conditions, and the corresponding rational maps are generically $N$-to-one (Critch et al., 2012).

There exists a direct connection between MPS and hidden Markov models (HMM): classical Markov processes map to MPS with positive tensors, and their algebraic parametrizations align, facilitating algorithms and rational parameterizations (Critch et al., 2012, Souissi, 18 Feb 2025). Every MPS can be viewed as an "observed" process from a quantum hidden Markov chain (EHMM), with a dual construction connecting any MPS to a corresponding EHMM and quantifying the relative entropy divergence between observed and ideal amplitudes (Souissi, 18 Feb 2025).

In classical probability, stochastic MPS (sMPS) represent steady states of non-equilibrium processes and encode "Schmidt-like" coefficients whose Shannon entropy bounds the necessary bond dimension (Temme et al., 2010). The MPS formalism accurately captures steady-state statistics (e.g., asymmetric exclusion process) with manageable bond dimension except at criticality.

7. Generalizations, Limitations, and Current Directions

MPS efficiently represent a broad, yet fundamentally restricted, class of quantum states: those satisfying area laws in 1D and possessing only moderate entanglement or correlation length. The approach extends to infinite chains, stochastic and quantum processes, non-exclusion models with unbounded occupation numbers (Chatterjee et al., 2017), and quantum systems with adaptive or emergent symmetries (Guo et al., 2019).

However, for highly entangled (volume-law) states or systems with long-range correlations, the required bond dimension becomes prohibitive. Algebraic geometry methods rigorously identify the locus of tensor states outside the MPS variety, and property-testing protocols reveal sharp sample complexity separations between product states and nontrivial MPS in quantum information tasks (Soleimanifar et al., 2022).

Recent work aims to systematize infinite-volume limits via $C^*$-algebraic projectivity conditions (Souissi et al., 5 Nov 2024), to connect all MPS with quantum Markov models (Souissi, 18 Feb 2025), and to clarify the role of invariants, parameter spaces, and identifiability for practical state and process tomography.

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References:

• "Matrix Product States in Quantum Spin Chains" (Souissi et al., 5 Nov 2024)
• "Calculus of continuous matrix product states" (Haegeman et al., 2012)
• "Continuous Matrix Product States for Quantum Fields" (Verstraete et al., 2010)
• "Algebraic Geometry of Matrix Product States" (Critch et al., 2012)
• "Matrix product states, geometry, and invariant theory" (Seynnaeve, 2022)
• "Matrix Product States as Observations of Entangled Hidden Markov Models" (Souissi, 18 Feb 2025)
• "Stochastic Matrix Product States" (Temme et al., 2010)
• "Matrix Product States for Interacting Particles without Hardcore Constraints" (Chatterjee et al., 2017)
• "Matrix Product States for Lattice Field Theories" (Bañuls et al., 2013)
• "Gaussian Matrix Product States" (Schuch et al., 2012)
• "Matrix Product States: Entanglement, symmetries, and state transformations" (Sauerwein et al., 2019)
• "Testing matrix product states" (Soleimanifar et al., 2022)
• "Matrix Product States for Quantum Stochastic Modelling" (Yang et al., 2018)
• "Geometry of Matrix Product States: metric, parallel transport and curvature" (Haegeman et al., 2012)
• "Matrix Product States with adaptive global symmetries" (Guo et al., 2019)
• "Implementing Entangled States on a Quantum Computer" (Bhatia et al., 2018)
• "Matrix Product States for Trial Quantum Hall States" (Estienne et al., 2012)