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.
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.
This paper, “Matrix‑Product Unitaries on Quantum Hardware,” presents a set of experimental studies examining how different classes of Matrix‑Product Unitaries behave when executed on real superconducting quantum hardware. The work investigates several MPU constructions—including the CZ layer, non‑uniform MPUs, high‑bond MPUs, and multilayer MPUs—and evaluates their statistical, geometric, and correlation signatures under a variety of structured superposition inputs. As far as is known, this is the first time MPU operators have been systematically implemented and analyzed on physical quantum devices. The experiments use a geometric operator framework to extract, project, and reconstruct MPU operators before synthesizing them into native gate sets for hardware execution. Although the geometric method is used to generate the operators, the resulting circuits are valid MPU implementations independent of the extraction technique. The paper compares hardware‑executed MPU behavior against several baseline circuits, analyzes entropy, mutual information, collapse geometry, and classical statistical distances, and demonstrates that MPU operators form a distinctive operator class with reproducible hardware fingerprints.
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 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 O(NK) Pauli-product branches of an order-K truncated Taylor polynomial, our virtual-transition implementation replaces combinatorial branch enumeration by a circuit complexity scaling as O(αMPONχ2), approaching O(Nχ2) when αMPO 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.
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.
A geometric operator is a single mathematical object that contains its computation internally. Unlike a traditional compiler—which must parse external symbolic code, build abstract syntax trees, generate intermediate representations, and translate instructions into machine operations—a geometric operator requires no interpretation, no translation, and no external program structure. The computation “lives” inside the operator’s geometry itself. In a compiler-based system, meaning is assembled from many external components. In a geometric system, meaning is intrinsic: the operator already encodes the transformation, the structure, and the behavior. When executed, Python performs only numerical operations (matrix multiplication, SVD, QR factorization), but the semantic content is already present inside the operator. The processor is not executing a compiled program — it is revealing the operator’s internal geometry. This project demonstrates a result that, to our knowledge, has never been shown before: a geometric operator framework reproducing the full unitary behavior of deep quantum circuits (multi-layer, entangling, QFT‑mixed circuits) without gates, decomposition, transpilation, or circuit synthesis. The optimizer starts from identity and matches Qiskit-generated circuit unitaries with perfect operator fidelity and perfect functional fidelity, using only deviation geometry, extremal spectral steering, and polar re‑unitarization. Although the claim appears improbable, the mathematics and the code are fully transparent. Reviewers can inspect every module, every operator, and every update rule. The results are reproducible, and the behavior is consistent across structured manifolds, Haar targets, and deep circuit workloads. This repository provides the full implementation, tests, and validation suite for independent verification.
Quantum-inspired algorithms can deliver substantial speedups over classical state-of-the-art methods by executing quantum algorithms with tensor networks on conventional hardware. Unlike circuit models restricted to unitary gates, tensor networks naturally accommodate non-unitary maps. This flexibility lets us design quantum-inspired methods that start from a quantum algorithmic structure, yet go beyond unitarity to achieve speedups. Here we introduce a tensor-network approach to compute the discrete Laplace transform, a non-unitary, aperiodic transform (in contrast to the Fourier transform). We encode a length-N signal on two paired n-qubit registers and decompose the overall map into a non-unitary exponential Damping Transform followed by a Quantum Fourier Transform, both compressed in a single matrix-product operator. This decomposition admits strong MPO compression to low bond dimension resulting in significant acceleration. We demonstrate simulations up to N=230 input data points, with up to 260 output data points, and quantify how bond dimension controls runtime and accuracy, including precise and efficient pole identification.