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Direct Dense Unitary Optimization Without Circuits or Tensor Networks

David Mulnix

Published Aug 24, 2026• 0 citations
Publisher

Abstract

This work introduces a mathematically structured and numerically stable method for directly optimizing dense unitary operators at scales (1024–4096) where standard approaches fail. Conventional dense‑unitary optimization—gradient descent, coordinate perturbation, iterative refinement—breaks down at these dimensions because unconstrained updates leave the unitary manifold, accumulate floating‑point error, and cause eigenvalue drift. For this reason, the quantum‑computing ecosystem avoids dense operators entirely, relying instead on gate decompositions, tensor networks, sparse factorizations, or multi‑GPU distributed simulation. The method presented here is fundamentally different. It defines a multiscale dyadic block hierarchy, a block‑sum embedding Φ(U), and a coarse‑to‑fine energy E(U; U_ref) that acts as a Lyapunov‑like function. Structured blockwise perturbations, followed by QR re‑unitarization and monotone energy acceptance, produce a stable dynamical system on U(n) that preserves unitarity and descends predictably in feature space. This multiscale geometry makes dense optimization tractable at sizes where classical methods become numerically unstable or computationally prohibitive. The result is a capability not present in existing literature or industry practice: direct, repeated, stable optimization of fully dense 4096×4096 unitary matrices—over sixteen million complex parameters—without gate decompositions, tensor networks, special hardware, or distributed execution. The collapse flow demonstrates monotonic descent, preserved unitarity, and practical runtimes on a single GPU, revealing a new structural pathway for manipulating high‑dimensional quantum operators. In short: this paper presents a multiscale spectral algebra that enables dense unitary optimization at scales previously considered infeasible, providing a novel and structurally coherent alternative to the circuit‑based and tensor‑network methods that dominate current quantum‑computational practice.

DOI
10.5281/zenodo.22087829

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•May 1, 2026•0 citations
This work introduces a mathematically structured and numerically stable method for directly optimizing dense unitary operators at scales (1024–4096) where standard approaches fail. Conventional dense‑unitary optimization—gradient descent, coordinate perturbation, iterative refinement—breaks down at these dimensions because unconstrained updates leave the unitary manifold, accumulate floating‑point error, and cause eigenvalue drift. For this reason, the quantum‑computing ecosystem avoids dense operators entirely, relying instead on gate decompositions, tensor networks, sparse factorizations, or multi‑GPU distributed simulation. The method presented here is fundamentally different. It defines a multiscale dyadic block hierarchy, a block‑sum embedding Φ(U), and a coarse‑to‑fine energy E(U; U_ref) that acts as a Lyapunov‑like function. Structured blockwise perturbations, followed by QR re‑unitarization and monotone energy acceptance, produce a stable dynamical system on U(n) that preserves unitarity and descends predictably in feature space. This multiscale geometry makes dense optimization tractable at sizes where classical methods become numerically unstable or computationally prohibitive. The result is a capability not present in existing literature or industry practice: direct, repeated, stable optimization of fully dense 4096×4096 unitary matrices—over sixteen million complex parameters—without gate decompositions, tensor networks, special hardware, or distributed execution. The collapse flow demonstrates monotonic descent, preserved unitarity, and practical runtimes on a single GPU, revealing a new structural pathway for manipulating high‑dimensional quantum operators. In short: this paper presents a multiscale spectral algebra that enables dense unitary optimization at scales previously considered infeasible, providing a novel and structurally coherent alternative to the circuit‑based and tensor‑network methods that dominate current quantum‑computational practice.
In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an eigenvalue/eigenvector search, this algorithm features wide applicability. Operationally, power iterations consist of performing repeated matrix-to-vector multiplications (or MatVec) followed by a renormilization step in order to converge to the dominant eigenvalue/eigenvector. However, classical realizations, including novel tensor network based approaches, necessitate an exponential scaling for the algorithm's run-time. In this paper, we propose a quantum realiziation to circumvent this pitfall. Our methodology involves casting low-rank representations; Matrix Product Operators (MPO) for matrices and Matrix Product States (MPS) for vectors, into quantum circuits. Specifically, we recover a unitary approximation by variationally minimizing the Frobenius distance between a target MPO and an MPO ansatz wherein the tensor cores are constrained to unitaries. Such an unitary MPO can easily be implemented as a quantum circuit with the addition of ancillary qubits. Thereafter, with appropriate initialization and post-selection on the ancillary space, we realize a single iteration of the classical algorithm. With our proposed methodology, power iterations can be realized entirely on a quantum computer via repeated, static circuit blocks; therefore, a run-time advantage can indeed be guaranteed. Moreover, by exploiting Riemannian optimization and cross-approximation techniques, our methodology becomes instance agnostic and thus allows one to address black-box optimization within the framework of quantum computing.
Quantum operations on pure states can be fully represented by unitary matrices. Variational quantum circuits, also known as quantum neural networks, embed data and trainable parameters into gate-based operations and optimize the parameters via gradient descent. The high cost of training and low fidelity of current quantum devices, however, restricts much of quantum machine learning to classical simulation. For few-qubit problems with large datasets, training the matrix elements directly, as is done with weight matrices in classical neural networks, can be faster than decomposing data and parameters into gates. We propose a method that trains matrices directly while maintaining unitarity through a single regularization term added to the loss function. A second training step, circuit alignment, then recovers a gate-based architecture from the resulting soft-unitary. On a five-qubit supervised classification task with 1000 datapoints, this two-step process produces a trained variational circuit in under four minutes, compared to over two hours for direct circuit training, while achieving lower binary cross-entropy loss. In a second experiment, soft-unitaries are embedded in a hybrid quantum-classical network for a reinforcement learning cartpole task, where the hybrid agent outperforms a purely classical baseline of comparable size.
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-NNN signal on two paired nnn-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=230N=2^{30}N=230 input data points, with up to 2602^{60}260 output data points, and quantify how bond dimension controls runtime and accuracy, including precise and efficient pole identification.
An efficient representation of a quantum circuit is of great importance in achieving a quantum advantage on current noisy intermediate scale quantum (NISQ) devices and the classical simulation of quantum manybody systems.The quantum circuits are playing the key ingredient in the performance of variational quantum algorithms and quantum dynamics in problems of physics and chemistry.In this paper, we study the role of the network structure of a quantum circuit in its performance.We discuss the variational optimization of quantum circuit (a unitary tensor-network circuit) with different network structures.The ansatz is performed based on a generalization of well-developed multiscale entanglement renormalization algorithm and also the conjugategradient method with an effective line search.We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network QR decomposition.Our work can contribute to achieve the most out of NISQ hardware and to classically develop isometric tensor network states.
Jacobi diagonalization is a long-established numerical algorithm for the spectral decomposition of Hermitian and, more generally, normal matrices. In this work, we develop a qudit-native quantum realization of the Jacobi diagonalization algorithm for unknown unitary operators. The proposed framework avoids explicit reconstruction of the operator and controlled-unitary operations. Each elementary two-parameter Givens rotation is implemented through two sequential single-parameter quantum variational optimizations that are directly accessible experimentally. An interferometric protocol is further introduced for extracting the eigenvalues of the diagonalized unitary, up to an overall global phase. Numerical simulations on ensembles of Haar-random unitary matrices demonstrate that the proposed algorithm preserves the characteristic convergence behavior of the classical Jacobi method and exhibits the expected quadratic scaling in the number of elementary operations with the dimension ddd, comparable to its classical counterpart. The results establish the proposed algorithm as a structured quantum numerical linear algebra algorithm naturally suited to qudit architectures and provide a bridge between classical iterative matrix algorithms and their quantum realizations.
The rapid advancement of quantum computing has created new opportunities for solving computationally intensive problems across diverse fields such as optimization, physical system simulation, and machine learning. However, practical implementation on Noisy Intermediate-Scale Quantum (NISQ) devices remains challenging due to hardware limitations including noise, decoherence, and restricted qubit connectivity. A fundamental constraint further complicating this effort is that all quantum operations must be unitary, while many real-world problems are naturally expressed in non-unitary forms. This discrepancy necessitates novel techniques for transforming non-unitary mathematical structures into unitary or Hamiltonian representations compatible with quantum hardware.This thesis proposes a unified framework for converting structured, sparse, or otherwise non-unitary classical matrices into unitary forms that can be executed efficiently on NISQ devices. The methodology introduces general transformation strategies such as linear combinations of unitaries, structured embeddings, and resource-aware circuit construction, enabling the deployment of quantum algorithms without incurring prohibitive overhead. Each technique is developed with an emphasis on preserving problem structure, minimizing circuit depth, and ensuring compatibility with IBM’s NISQ hardware. The proposed frameworks are validated through theoretical analysis and quantum simulations, with performance benchmarks based on gate count, circuit fidelity, convergence behavior, and scalability. This work demonstrates how non-unitary formulations arising in various computational contexts can be systematically adapted for quantum execution, advancing the practical utility of NISQ-era devices.