Eigenanalysis-Based Method Reconstructs Quantum Gates Across Thousands of Dimensions

Quantum computers depend on quantum gates, and knowing how those gates actually behave is essential to building reliable machines. A newly published study presents a family of methods for reconstructing the mathematical description of such gates, even when they involve many qubits and dense, non-sparse process matrices. The work, published in Discover Quantum Science, introduces eigenanalysis-based quantum process tomography algorithms that were validated through numerical simulation on an ordinary desktop computer.

Quantum process tomography, or QPT, is the quantum counterpart of classical system identification. Its goal is to estimate the transformation that a quantum process, such as a gate, applies to its input states. Because gates are the building blocks of gate-based quantum computers, one of the most prominent quantum computing paradigms, QPT is a major tool for experimentally characterizing their actual behavior. The new research focuses on unitary processes, which correspond to systems isolated from their environment, and deliberately places no sparsity constraints on the process matrix, in contrast to approaches from the literature that request sparsity.

The core idea of the proposed methods rests on a specific mathematical property. When the input to the process is a mixed state whose density matrix is diagonal in the computational basis, the density matrix of the process output takes the form of an eigendecomposition: the input density matrix supplies the eigenvalues and the unknown unitary process matrix supplies the eigenvectors. In practice, the researchers start from an estimate of the output density matrix obtained through quantum state tomography, ensure it is Hermitian and has unit trace, and then perform an eigendecomposition of that estimate. The resulting eigenvalues and eigenvectors are related to the true input density matrix and process matrix, up to estimation errors and certain indeterminacies.

Those indeterminacies are handled in two steps. First, the user constrains the diagonal values of the input density matrix to be distinct and arranged in a known order, which allows the algorithm to resolve the arbitrary ordering of eigenvalues and eigenvectors. This part of the procedure requires no knowledge of the exact input state, making it unsupervised in the machine learning sense. Second, a known pure state is applied to the process, and comparing its measured output with the partially reconstructed process matrix resolves the remaining unknown phase factor in each column, up to a single global phase that a process matrix can only be defined up to. This second part is supervised, or semi-supervised if the input pure state is itself estimated by measurement.

The single-stage version of the method, called EQPT1, performs only one eigendecomposition. However, it faces a fundamental limitation: all eigenvalues of the input density matrix must be distinct and separated by a minimum gap to be distinguishable despite estimation errors, and because these eigenvalues all range from zero to one, this imposes an upper bound on the state space dimension that can be handled. To overcome this, the authors introduce two-stage methods, EQPT2 and EQPT3, which use two different diagonal input density matrices containing repeated eigenvalues. Each eigendecomposition identifies a set of subspaces spanned by groups of columns of the unknown process matrix, and the intersections of these subspaces, computed with canonical correlation analysis, isolate each individual column.

This two-stage construction raises the achievable dimension from a bound set by the number of distinguishable eigenvalues to its square. The authors then generalize further to multi-stage methods that perform several eigendecompositions, with the number of stages increasing with the state space dimension. A particularly attractive variant, called dichotomic multi-stage eigenanalysis-based QPT, or DEQPT, reduces the number of distinct eigenvalues to just two, which helps the algorithm assign each estimated eigenvalue to the correct value despite noise. Unlike the two-stage methods, this general approach potentially applies to any state space dimension, provided the number of stages and the associated computational complexity are acceptable.

The validation was carried out entirely through software simulation rather than on quantum hardware. The researchers modeled the estimation errors that quantum state tomography would produce by adding random, uniformly distributed fluctuations to the theoretical output states, with a parameter controlling the error magnitude. Each test used a randomly drawn unitary process matrix, and performance was measured with a normalized root mean square error that accounts for the unavoidable global phase. For each configuration, results were averaged over 100 independent tests, each with a different random process matrix.

The simulations showed that the two-stage methods consistently outperform the single-stage method, reducing the normalized mean square error by a factor of up to 36 depending on the conditions. For the lowest quantum state tomography error magnitude considered, the two-stage methods achieved a normalized mean square error of about 0.01 across all analyzed dimensions. The dichotomic multi-stage method performed even better at higher dimensions, reducing the error by a factor of up to more than 10 compared with the two-stage method and reaching a normalized mean square error down to about 0.001, though at a substantially higher computational cost.

A striking practical detail is the hardware on which these results were obtained. All tests ran on an eight-year-old PC with an Intel Core i5-7500 processor and only 16 gigabytes of RAM, using the Matlab environment, without high-performance computing or graphical processing units. The single-stage and two-stage methods remained efficient up to 13 qubits, corresponding to a state space dimension of about 8,000, with memory rather than computation being the limiting factor. Execution times ranged from a few milliseconds for 4 qubits to a few minutes for 12 qubits for the two-stage methods, while the dichotomic method needed roughly two hours for 11 qubits.

The authors note that their tests were restricted to a standard PC, and future work will include experiments on much more powerful platforms to reach significantly higher qubit counts. They also emphasize that the tested configurations, involving up to 11 to 13 qubits and state space dimensions of a few thousand, already exceed the scale of various earlier investigations, including a previous unitary-process method tested only for a Hilbert space dimension of five. Because the proposed methods belong to the class that first performs complete quantum state tomography and then exploits its results, their overall complexity depends on the chosen tomography algorithm, and the authors designed their error model so that readers can compare the simulated conditions with the accuracy of their preferred tomography approach.

Subject of Research: Eigenanalysis-based quantum process tomography methods for high-dimensional dense unitary processes in gate-based quantum computers.

Article Title: Multi-stage tomography based on eigenanalysis for high-dimensional dense unitary processes in gate-based quantum computers

Article References: Deville, Y., & Deville, A. (2026). Multi-stage tomography based on eigenanalysis for high-dimensional dense unitary processes in gate-based quantum computers. Discover Quantum Science, 2(1), Article 29. https://doi.org/10.1007/s44464-026-00030-y

Image Credits: AI Generated

DOI: 10.1007/s44464-026-00030-y

Keywords: quantum computing, quantum process tomography, quantum state tomography, eigendecomposition, unitary processes, quantum gates, density matrices, simulation, high-dimensional systems, machine learning, Multi-stage, tomography