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Beyond-QCA MPUs: generation, certification, and circuit synthesis

Marco Vassallo

Published Sep 24, 2026• 0 citations

Abstract

Matrix Product Unitaries (MPUs) provide compact tensor-network representations of many-body quantum operations. MPU theory has been developed in the translationally invariant setting with periodic boundaries, where MPUs correspond, up to blocking, to one-dimensional quantum cellular automata (QCA) and are therefore characterized by a strict causal cone. Only recently, Styliaris et al. identified a broader class of open-boundary beyond-QCA MPUs that can generate long-range correlations while retaining the area-law operator entanglement characteristic of MPUs. Building on these developments, this thesis presents a framework for the generation, certification, and circuit synthesis of non-QCA MPUs. First, we develop the analytical and numerical tools needed to certify the validity of an input MPU and optimize the exact circuit decomposition algorithm introduced by Styliaris et al. These methods reduce both the asymptotic and finite-size costs of the decomposition. We then implement the complete procedure in the open-source library \textsc{TreeMPU}. We also present a theoretically motivated variational ansatz for approximating non-QCA MPUs with quantum circuits and benchmark its performance against the standard brickwall structure used for MPUs. The second main contribution is a systematic framework for generating MPUs. The local constraints guarantee global unitarity at arbitrary system size and define parameterized families capable of generating long-range correlations and beyond-QCA behavior. They reproduce known examples of non-QCA MPUs as special cases while enabling the random sampling of broader classes of unitary tensor networks. Finally, we assess the performance of the optimized exact decomposition, the variational compilation scheme, and the certified MPU generators developed in this thesis. Together, the certified generators and \textsc{TreeMPU} provide a systematic computational platform for constructing, benchmarking, and implementing beyond-QCA MPUs.

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Matrix Product Unitaries (MPUs) provide compact tensor-network representations of many-body quantum operations. MPU theory has been developed in the translationally invariant setting with periodic boundaries, where MPUs correspond, up to blocking, to one-dimensional quantum cellular automata (QCA) and are therefore characterized by a strict causal cone. Only recently, Styliaris et al. identified a broader class of open-boundary beyond-QCA MPUs that can generate long-range correlations while retaining the area-law operator entanglement characteristic of MPUs. Building on these developments, this thesis presents a framework for the generation, certification, and circuit synthesis of non-QCA MPUs. First, we develop the analytical and numerical tools needed to certify the validity of an input MPU and optimize the exact circuit decomposition algorithm introduced by Styliaris et al. These methods reduce both the asymptotic and finite-size costs of the decomposition. We then implement the complete procedure in the open-source library \textsc{TreeMPU}. We also present a theoretically motivated variational ansatz for approximating non-QCA MPUs with quantum circuits and benchmark its performance against the standard brickwall structure used for MPUs. The second main contribution is a systematic framework for generating MPUs. The local constraints guarantee global unitarity at arbitrary system size and define parameterized families capable of generating long-range correlations and beyond-QCA behavior. They reproduce known examples of non-QCA MPUs as special cases while enabling the random sampling of broader classes of unitary tensor networks. Finally, we assess the performance of the optimized exact decomposition, the variational compilation scheme, and the certified MPU generators developed in this thesis. Together, the certified generators and \textsc{TreeMPU} provide a systematic computational platform for constructing, benchmarking, and implementing beyond-QCA MPUs.

Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems, while their action on states by construction preserves the entanglement area law. MPU which are formed by a single repeated tensor are known to coincide with 1D quantum cellular automata (QCA), i.e., unitaries with an exact light cone. However, this correspondence breaks down for MPU with open boundary conditions, even if the resulting operator is translation-invariant. Such unitaries can turn short- to long-range correlations and thus alter the underlying phase of matter. Here we make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary. In particular, we study the structure of a subclass with a direct-sum form which maximally violates the QCA property. We also consider the general case of MPU formed by site-dependent (nonuniform) tensors and show a correspondence between MPU and locally maximally entanglable states.

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an N-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth T=O(N^{α}) realizing the MPU, where the constant α depends only on the bulk and boundary tensor and not the system size N. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of C^{*}-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally varying MPUs and show that they can be implemented by a circuit of depth O(N^{β}polyD) where β≤1+log_{2}sqrt[D]/s_{min}, with D being the bond dimension and s_{min} the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations when applied on product states. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.

Abstract Matrix product vectors form the appropriate framework to study and classify one-dimensional quantum systems. In this work, we develop the structure theory of matrix product unitary operators (MPUs) which appear e.g. in the description of time evolutions of one-dimensional systems. We prove that all MPUs have a strict causal cone, making them quantum cellular automata (QCAs), and derive a canonical form for MPUs which relates different MPU representations of the same unitary through a local gauge. We use this canonical form to prove an index theorem for MPUs which gives the precise conditions under which two MPUs are adiabatically connected, providing an alternative derivation to that of (Gross et al 2012 Commun. Math. Phys . 310 419) for QCAs. We also discuss the effect of symmetries on the MPU classification. In particular, we characterize the tensors corresponding to MPU that are invariant under conjugation, time reversal, or transposition. In the first case, we give a full characterization of all equivalence classes. Finally, we give several examples of MPU possessing different symmetries.

Quantum circuit theory has long treated Matrix‑Product Unitaries (MPUs) as elegant but unreachable objects—beautiful on paper, impossible to extract, and as far as we know never once executed on real hardware. This work breaks that barrier. We introduce the first geometric MPU extraction pipeline capable of reconstructing full multi‑qubit unitaries with machine‑precision fidelity, validating them across uniform CZ layers, bidirectional sweeps, non‑uniform constructions, QFT‑derived operators, high‑bond entangling layers, and multi‑layer MPU compositions. Every MPU class tested—without exception—compiles cleanly through Qiskit into hardware‑native CX+U3 circuits, producing thousands‑gate decompositions that match the original MPU operators with Frobenius error near 10^{-11} and fidelity 1.0. For the first time, MPUs move from abstract tensor‑network theory into executable quantum computation. This establishes a practical bridge between MPU mathematics and real quantum devices, opening a new path for hardware‑compatible tensor‑network operators, circuit compression, and quantum algorithm design.

We develop a block-encoding compiler that treats matrix product operators as compressed, virtual-path linear combination of unitaries programs. The compiler constructs conditional PREP and local SELECT stages directly from a parent matrix product operator, establishing tensor networks as structured quantum intermediate representations that can be efficiently compiled to block-encoded circuits. We apply the construction to real-time evolution in the Heisenberg chain and two perturbed Heisenberg-family models. Across the regimes studied, the compressed, approximately unitary propagator MPOs retain mild bond dimension and LCU normalization. Relative to an LCU that explicitly lists the

\mathcal{O}(N^K)

Pauli-product branches of an order-K truncated Taylor polynomial, our virtual-transition implementation replaces combinatorial branch enumeration by a circuit complexity scaling as

\mathcal{O}(α_{\rm MPO}Nχ^2)

, approaching

\mathcal{O}(Nχ^2)

when

α_{\rm MPO}

remains mild. We numerically characterize how truncation order, bond-dimension budget, and system size affect approximation error, normalization, and compiler cost. These results demonstrate how classically compressed tensor-network representations can serve as quantum compiler intermediate representations for block encoding and opens new avenues to accelerate quantum algorithms.