Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues

Abstract: We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary $N\times N$ complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new values of optimization parameters for the next iteration of the loss function in the gradient optimization method. The depth and size of the circuit for this algorithm are, respectively, $O(N² \log N)$ and $O(\log N)$.

• The paper introduces a novel variational method that encodes non-Hermitian matrices into quantum state amplitudes to compute eigenvalues and generalized eigenvalues.
• The approach employs parameterized unitaries and a tailored loss function to achieve simultaneous triangularization, as confirmed by numerical simulations.
• The method circumvents traditional Hermitian embeddings and LCU techniques, offering advantages for applications in quantum chemistry, machine learning, and numerical simulations.

Matrix Encoding Methods for Variational Calculation of Eigenvalues and Generalized Eigenvalues

Introduction

The paper "Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues" (2605.06167) develops a variational quantum-classical hybrid approach for calculating eigenvalues and generalized eigenvalues (GEVs) for general $N \times N$ complex matrices. The method is characterized by direct encoding of non-Hermitian matrices into the amplitudes of a quantum superposition state, circumventing the need to construct Hermitian embeddings or deploy linear-combination-of-unitaries (LCU) techniques commonly found in existing VQA approaches for non-unitary or non-Hermitian operators. This essay provides an in-depth summary of the methodology, quantum circuit constructions, optimization routines, accuracy analysis, and empirical results demonstrated through numerical simulation, making explicit the algorithmic innovations and their implications for quantum numerical linear algebra.

Variational Quantum Algorithm with Matrix Encoding

The approach targets generalized eigenvalue problems (GEVP) of the form $A|\psi\rangle = \lambda B|\psi\rangle$ for arbitrary (not necessarily Hermitian or unitary) matrices $A, B \in \mathbb{C}^{N \times N}$, where $N=2^n$. The key technical novelty lies in representing $A$ and $B$ jointly in the amplitudes of a quantum register state, in particular via the encoding

$$|\Psi\rangle_{RCL} = \sum_{ij}(a_{ij}|0\rangle_L + b_{ij}|1\rangle_L)|i\rangle_R |j\rangle_C$$

where registers $R, C$ index rows and columns, and a label qubit $L$ differentiates between $A$ and $A|\psi\rangle = \lambda B|\psi\rangle$0. The matrix elements are normalized as needed to ensure quantum mechanical normalization.

Unitary parameters ($A|\psi\rangle = \lambda B|\psi\rangle$1, $A|\psi\rangle = \lambda B|\psi\rangle$2) are applied to bring $A|\psi\rangle = \lambda B|\psi\rangle$3 and $A|\psi\rangle = \lambda B|\psi\rangle$4 simultaneously to upper-triangular form as required by the generalized Schur decomposition. This is accomplished by a loss function

$A|\psi\rangle = \lambda B|\psi\rangle$5

which is minimized with respect to variational parameters $A|\psi\rangle = \lambda B|\psi\rangle$6 and $A|\psi\rangle = \lambda B|\psi\rangle$7. A vanishing loss corresponds to simultaneous triangularization. Diagonal entries then provide the GEVs via $A|\psi\rangle = \lambda B|\psi\rangle$8.

The quantum subroutine is tasked with efficient and accurate estimation of the loss function and its gradients, using quantum measurement to extract the relevant amplitude weightings, subject to probabilistic post-selection to remove "garbage" ancillary states.

Quantum Circuit Construction

The circuit introduces several logical quantum subsystems: register qubits ($A|\psi\rangle = \lambda B|\psi\rangle$9, $A, B \in \mathbb{C}^{N \times N}$0), auxiliary/ancillary indexes ($A, B \in \mathbb{C}^{N \times N}$1), label qubits, and selective ancilla ($A, B \in \mathbb{C}^{N \times N}$2) used to isolate computational paths via measurement.

The general procedure consists of:

• State preparation: Amplitude-encoding of the input matrices into a quantum register.
• Successive application of controlled and uncontrolled unitary blocks (realized by parametrized one- and two-qubit gates) to effect the variational ansatz.
• Controlled duplication, selection, and routing of relevant quantum states using projectors and CNOT networks.
• Post-processing by measurement, where the projection onto certain ancilla subspaces effectively filters the computational basis elements that correspond to non-vanishing off-diagonal components relevant for evaluating the loss.
• Probabilistic extraction of the loss value, whose amplitude is then used in a classical update (gradient descent or related optimizer).

Circuit complexity is carefully analyzed: the overall depth and size are $A, B \in \mathbb{C}^{N \times N}$3 (dominated by selecting vanishing elements) and $A, B \in \mathbb{C}^{N \times N}$4, respectively. The circuit construction supports efficient calculation of both the loss and its gradient with respect to all $A, B \in \mathbb{C}^{N \times N}$5 parameters for a layered ansatz, where $A, B \in \mathbb{C}^{N \times N}$6 is the layer count in the parameterized unitary block.

Optimization and Numerical Aspects

The optimization loop involves both the quantum estimation of the objective function (loss) and its gradients, and the classical updating of variational parameters. Gradients are computed via parameter-shift rules exploiting the exponential-Pauli structure of single-qubit rotations for all parameter indices. For the GEVP, both sets of circuit parameters ($A, B \in \mathbb{C}^{N \times N}$7) are optimized; for the standard EVP ($A, B \in \mathbb{C}^{N \times N}$8), only one set ($A, B \in \mathbb{C}^{N \times N}$9) is sufficient.

The paper delivers an explicit mapping between the accuracy in evaluating the loss function ($N=2^n$0) and the resulting accuracy in the computed (generalized) eigenvalues ($N=2^n$1), showing that $N=2^n$2 holds universally. The success probability due to post-selection steps is $N=2^n$3, which impacts the number of shots required in realistic implementations.

Parameter encoding accuracy and effects of digital/finite precision are quantitatively assessed; minimal achievable error for eigenvalues scales as $N=2^n$4, where $N=2^n$5 is the parameter encoding resolution.

Numerical Results

Empirical validation is performed using numerically simulated $N=2^n$6 complex matrix pairs ($N=2^n$7), covering 100 random instances per experiment. The simulation substantiates theoretical predictions on convergence rates, scaling of required iterations, and parameter relationships. Key findings:

• The number of optimization iterations scales linearly with $N=2^n$8 for the achieved accuracy.
• The quadratic relation $N=2^n$9 is robust across problem instances.
• When parameter precision is limited, the minimal eigenvalue error exhibits the theoretical scaling in terms of encoding accuracy.
• Algorithms for the standard EVP (including only $A$0) typically converge faster than GEVP.

All numerical observations corroborate the predicted asymptotic behavior.

Practical and Theoretical Implications

The direct amplitude encoding approach depends critically on efficient state preparation and high-probability post-selection. For small $A$1, as shown, the scheme is feasible and demonstrates clear structure-encoding advantages for arbitrary matrices—most markedly, avoiding the inefficiencies of Hermitianization or LCU for non-unitary systems. This confers potential for wider applicability in quantum chemistry, machine learning (e.g., in SVD, generalized PCA, kernel methods), and large-scale numerical simulation.

Scalability remains bottlenecked by the $A$2 probabilistic success, which is a challenge for near-term hardware. The authors suggest that improvements in controlled measurement and amplitude amplification techniques (citing, e.g., [FZQWarxive2025]) could potentially mitigate this issue.

Theoretical implications include the general validity of accuracy-scaling laws for variational quantum algorithms, the utility of amplitude encoding in a hybrid context, and robust error models for finite-precision parameterization in quantum optimization.

Conclusion

This work presents a comprehensive, mathematically rigorous variational quantum algorithmic framework for both eigenvalues and generalized eigenvalues using direct matrix amplitude encoding in quantum states. The approach dispenses with several restrictive assumptions common in prior quantum numeric work, supporting arbitrary complex matrices and providing detailed scaling laws for accuracy, resource cost, and convergence. Success on small systems in simulation validates the theoretical framework, while highlighting the prevailing challenge of success probabilities in garbage removal and the importance of precise low-level parameter control. Future research will likely target enhanced amplitude amplification mechanisms, adaptive state preparation, and integration within larger quantum-classical numerical simulation workflows.