Continuous matrix product operators for quantum fields
Erickson Tjoa, J. Ignacio Cirac
TL;DR
This work introduces continuous matrix product operators (cMPO) for bosonic quantum fields, defined as $O = \mathrm{Tr}_D\left(B\,\mathcal{P}e^{\int dx\,\mathfrak{L}_x}\right)\left(|\Omega\rangle\langle\Omega|\right)$ with $\mathfrak{L}_x = Q(x)\otimes \mathrm{Id} + L(x)\otimes l_x + R(x)\otimes r_x + T(x)\otimes \mathrm{Ad}_x$, providing a continuum limit of MPOs. They preserve entanglement area-law and map cMPS to cMPS, enabling a coherent continuum tensor-network framework for quantum fields. As an application, the authors construct several families of continuous matrix product unitaries (cMPU), including phase cMPU and displacement-based variants, such as $U_\theta = e^{-i\omega K}$ with $K = \int_I dx\, x\,(-1)^{\Pi(x)} n(x)$, illustrating nontrivial continuum dynamics beyond quantum cellular automata. This framework opens avenues for continuum non-Gaussian operations, cMPDOs, and extensions to fermionic theories and higher dimensions, with potential implications for continuum quantum information and field-theory tensor networks.
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.