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 $d$, 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.
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 d, 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.
Computing electronic structures of quantum systems is a key task underpinning many applications in photonics, solid-state physics, and quantum technologies. This task is typically performed through iterative algorithms to find the energy eigenstates of a Hamiltonian, which are usually computationally expensive and suffer from convergence issues. In this work, we develop and implement the Quantum Jacobi-Davidson (QJD) method and its quantum diagonalization variant, the Sample-Based Quantum Jacobi-Davidson (SBQJD) method, and demonstrate their fast convergence for ground state energy estimation. We assess the intrinsic algorithmic performance of our methods through exact numerical simulations on a variety of quantum systems, including 8-qubit diagonally dominant matrices, 12-qubit one-dimensional Ising models, and a 10-qubit water molecule (H2O) Hamiltonian. Our results show that both QJD and SBQJD achieve significantly faster convergence and require fewer Pauli measurements compared to the recently reported Quantum Davidson method, with SBQJD further benefiting from optimized reference state preparation. These findings establish the QJD framework as an efficient general-purpose subspace-based technique for solving quantum eigenvalue problems, providing a promising foundation for sparse Hamiltonian calculations on future fault-tolerant quantum hardware.
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in O(log2n) parallel time with a total cost of O(nω+1) Boolean operations. This Boolean complexity matches the best known rigorous O(log2n) parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases (mod(2π)) is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in O(log2n) parallel time and O(nω) Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.
The optimization of circuit parameters of variational quantum algorithms such as the variational quantum eigensolver (VQE) or the quantum approximate optimization algorithm (QAOA) is a key challenge for the practical deployment of near-term quantum computing algorithms. Here, we develop a hybrid quantum/classical optimization procedure inspired by the Jacobi diagonalization algorithm for classical eigendecomposition, and combined with Anderson acceleration. In the first stage, analytical tomography fittings are performed for a local cluster of circuit parameters via sampling of the observable objective function at quadrature points in the circuit angles. Classical optimization is used to determine the optimal circuit parameters within the cluster, with the other circuit parameters frozen. Different clusters of circuit parameters are then optimized in "sweeps,'' leading to a monotonically-convergent fixed-point procedure. In the second stage, the iterative history of the fixed-point Jacobi procedure is used to accelerate the convergence by applying Anderson acceleration/Pulay's direct inversion of the iterative subspace (DIIS). This Jacobi+Anderson method is numerically tested using a quantum circuit simulator (without noise) for a representative test case from the multistate, contracted variant of the variational quantum eigensolver (MC-VQE), and is found to be competitive with and often faster than Powell's method and L-BFGS.
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.
Currently available quantum hardware allows for small-scale implementations of quantum machine learning algorithms. Such experiments aid the search for applications of quantum computers by benchmarking the near-term feasibility of candidate algorithms. Here we demonstrate the quantum learning of a two-qubit unitary by a sequence of three parameterized quantum circuits containing a total of 21 variational parameters. Moreover, we variationally diagonalize the unitary to learn its spectral decomposition, i.e., its eigenvalues and eigenvectors. We illustrate how this can be used as a subroutine to compress the depth of dynamical quantum simulations. One can view our implementation as a demonstration of entanglement-enhanced machine learning, as only a single (entangled) training data pair is required to learn a $4\ifmmode\times\else\texttimes\fi{}4$ unitary matrix.
We describe criteria for implementation of quantum computation in qudits. A qudit is a -dimensional system whose Hilbert space is spanned by states \ensuremath{\mid}0⟩, \ensuremath{\mid}1⟩, \dots{}, . An important earlier work [] describes how to exactly simulate an arbitrary unitary on multiple qudits using a parameter family of single qudit and two qudit gates. That technique is based on the spectral decomposition of unitaries. Here we generalize this argument to show that exact universality follows given a discrete set of single qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to the -matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of and . A coupling graph results taking nodes 0, \dots{}, and edges are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian is also allowed. We discuss implementation in the eight dimensional ground electronic states of and construct an optimal gate sequence using Raman laser pulses.