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arXiv:2511.04545 · Physical Review Research

Continuous matrix product operators for quantum fields

Erickson Tjoa, J. Ignacio Cirac

quant-phcond-mat.str-elhep-lathep-thSubmitted Nov 6, 2025• Announced Nov 7, 2025• 0 citations• Revised Nov 7, 2025 (v2)
v1 · Nov 6, 2025v2 · Nov 7, 2025v3 · Apr 18, 2026
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Abstract

In this work we introduce an ansatz for continuous matrix product operators for quantum field theory. We show that (i) they admit a closed-form expression in terms of finite number of matrix-valued functions without reference to any lattice parameter; (ii) they are obtained as a suitable continuum limit of matrix product operators; (iii) they preserve the entanglement area law directly in the continuum, and in particular they map continuous matrix product states (cMPS) to another cMPS. As an application, we use this ansatz to construct several families of continuous matrix product unitaries beyond quantum cellular automata.

Journal reference
Phys. Rev. Research 8, L022013 (2026)
DOI
10.1103/d61y-cxx3
Comments
4+1+10 pages, no figures; v2: minor fixes/typos

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In this work we introduce an ansatz for continuous matrix product operators for quantum field theory. We show that (i) they admit a closed-form expression in terms of finite number of matrix-valued functions without reference to any lattice parameter; (ii) they are obtained as a suitable continuum limit of matrix product operators; (iii) they preserve the entanglement area law directly in the continuum, and in particular they map continuous matrix product states (cMPS) to another cMPS. As an application, we use this ansatz to construct several families of continuous matrix product unitaries beyond quantum cellular automata.
We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states (cMPS) to spatial dimensions d≥2d\geq 2d≥2. By construction, they are Euclidean invariant, and are genuine continuum limits of discrete tensor network states. Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the NNN-point functions of local observables. While we discuss mostly the continuous tensor network states extending Projected Entangled Pair States (PEPS), we propose a generalization bearing similarities with the continuum Multi-scale Entanglement Renormalization Ansatz (cMERA).
By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic quantum field theories. This construction works both for translation-invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian HHH, we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of HHH. Two key ingredients of our proposal are as follows: (i) a procedure to discretize HHH into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.
Continuous tensor networks give a variational ansatz for the ground state of the quantum field theories (QFTs). Notable examples are the continuous matrix product state (cMPS) and the continuous multiscale entanglement renormalization ansatz (cMERA). While cMPS is just adapted to the nonrelativistic QFTs, only the Gaussian cMERA is well understood, which we cannot use to approximate the ground state of the interacting relativistic QFTs. But, instead, cMERA also corresponds to a real-space renormalization group flow in the context of the wave functions. In this paper, we investigate the backward Gaussian cMERA renormalization group flow of the class of cMPS by putting the standard cMPS at the IR scale. At the UV scale, for the bosonic systems in the thermodynamic limit, we achieve the variational class of states that has been proposed recently, as the relativistic cMPS (RCMPS) is adapted to the relativistic QFTs without requiring one to introduce of any additional IR or UV cutoff. We also extend the RCMPS to fermionic systems and theories on a finite circle.
We present a way of constructing continuous matrix product states (cMPS) for coupled fields. The cMPS is a variational ansatz for the ground state of quantum field theories in one dimension. Our proposed scheme is based on a physical interpretation in which the cMPS class can be produced by means of a dissipative dynamic of a system interacting with a bath. We study the case of coupled bosonic fields. We test the method with previous DMRG results in coupled Lieb-Liniger models. Besides, we discuss a novel application for characterizing the Luttinger liquid theory emerging in the low-energy regime of these theories. Finally, we propose a circuit QED architecture as a quantum simulator for coupled fields.
The central topic of this thesis is on the fermionic continuous matrix product states (cMPS) variational ansatz for many-body quantum systems in a one-dimensional (1D) continuum.The matrix product states (MPS) is a variational ansatz which has been tremendously successful for nearly two decades in efficiently approximating groundstate properties of low-dimensional quantum lattice systems.The MPS is intrinsically designed for discrete lattice systems, however, and the cMPS has been introduced in 2010 as a continuum analogue of the MPS, enabling the possibility to directly tackle systems of quantum gases and liquids without resorting to discretization of space.
Continuous tensor networks are variational wavefunctions proposed in recent years to efficiently simulate quantum field theories (QFTs). Prominent examples include the continuous matrix product state (cMPS) and the continuous multi-scale entanglement renormalization ansatz (cMERA). While the cMPS can approximate ground states of a class of QFT Hamiltonians that are both local and interacting, cMERA is only well-understood for QFTs that are quasi-local and non-interacting. In this paper we propose the magic cMERA, a concrete realization of cMERA for a free boson QFT that simultaneously satisfies four remarkable properties: (i) it is the exact ground state of a strictly local Hamiltonian; (ii) in the massless case, its spectrum of scaling operators is exactly soluble in real space; (iii) it has the short-distance structure of a cMPS; (iv) it is generated by a quasi-local entangler that can be written as a continuous matrix product operator. None of these properties is fulfilled by previous cMERA proposals. Properties (iii)-(iv) establish a firm connection between cMERA and cMPS wavefunctionals, opening the path to applying powerful cMPS numerical techniques, valid for interacting QFTs, also to cMERA calculations.