Generalised Quantum Eigenvalue Estimation

• Generalised Quantum Eigenvalue Estimation (QEVE) is a framework that extends conventional quantum phase estimation to include non-Hermitian and generalized operators using advanced algorithmic techniques.
• It integrates hybrid quantum-classical routines, block encoding, polynomial transformations, and compressed sensing to efficiently extract eigenvalue spectra from complex matrices.
• QEVE offers improved resource efficiency and robust precision, making it valuable for applications in quantum chemistry, material science, and signal processing.

Generalised Quantum Eigenvalue Estimation (QEVE) encompasses a suite of quantum algorithms and frameworks extending eigenvalue estimation beyond standard Hermitian or unitary matrices. QEVE unifies, generalizes, and often improves upon traditional quantum phase estimation (QPE), enabling efficient estimation of spectra for generalized, non-Hermitian, non-normal, or ill-conditioned matrices ubiquitous in quantum chemistry, physics, engineering, and data science. This domain integrates variational, block-encoding, compressed sensing, and polynomial transformation approaches for eigenvalue extraction, often with robust performance against device or sampling noise, applicability to high-dimensional and dense spectra, and, in several cases, guaranteed resource or accuracy improvements over both classical and earlier quantum techniques.

1. Fundamental Concepts and Motivations

QEVE generalizes the quantum computing paradigm for eigenvalue estimation by relaxing constraints on the matrix class (from Hermitian/unitary to generalized, possibly non-normal or non-Hermitian matrices) and by allowing more versatile subroutines for eigenvalue extraction. Key concepts include block-encoded operator models, expectation estimation via repeated short coherent evolutions, variational minimization of generalized Rayleigh quotients, time-series analysis, compressed sensing on autocorrelation signals, and polynomial or rational function transformations applied to operator spectra.

Unlike the canonical QPE, which encodes the eigenphases of a unitary evolution into ancilla registers via deep, coherent circuits, QEVE-based techniques often improve near-term feasibility by (i) reducing circuit depth/coherence time, (ii) supporting matrices beyond Hermitian/unitary through similarity transforms or resolvent approaches, and (iii) providing robust or even Heisenberg-limited error scaling through advanced measurement and signal processing techniques [(1304.3061), 2019.07.26, (Low et al., 11 Jan 2024, Castaldo et al., 16 Jul 2025)].

2. Algorithmic Methodologies

2.1 Quantum–Classical Hybrid Algorithms

Variational and hybrid routines (e.g., VQE, variational GE solvers) replace extended coherence requirements of QPE with iterative parameter optimization. Here, the quantum processor is used for rapid, repeated measurement of operator expectation values (via Hamiltonian decomposition into Pauli terms), while a classical routine (e.g., Nelder–Mead simplex) adjusts circuit parameters to variationally minimize the energy (1304.3061).

The generalized Rayleigh quotient is the principle guiding the search for the smallest (or largest) generalized eigenvalue:

$$E_0 = \min_{|\psi\rangle} \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}$$

or, for GE problems,

$$\lambda = \frac{\langle\psi|\mathcal{A}|\psi\rangle}{\langle\psi|\mathcal{B}|\psi\rangle}$$

with quantum circuits encoding trial states controlled by parameterized ansätze, such as the unitary coupled cluster (UCC) for quantum chemistry. Extensions to orthogonality constraints and quantum gradient descent further generalize variational QE solvers for sequential nontrivial eigenstates (Liang et al., 2021, Sato et al., 2023).

2.2 Block-Encoding and Polynomial Transformations

The block-encoding framework systematically represents non-unitary (possibly non-Hermitian or generalized) matrices within a larger unitary, allowing quantum circuits to access spectra of arbitrary operators (Sünderhauf, 2023, Lin et al., 25 Feb 2025). Once an $(\alpha, a, \epsilon)$ block encoding is established,

$$U_A = \begin{bmatrix} A/\alpha & * \ * & * \end{bmatrix}$$

the application of polynomial transformations $f(A)$ (via Chebyshev polynomials or integer powers) becomes possible. The Hadamard test estimates the action of $f(A)$ by measuring expectation values, with the circuits designed so the measurement outcomes peak or filter at the desired eigenvalues (Lin et al., 25 Feb 2025).

Recent advances such as Generalized Quantum Eigenvalue Transformation (GQET) extend QSVT to allow complex coefficient and indefinite parity polynomials, leveraging block-encoded and "qubitized" operators for fast and expressive spectral transformation (Sünderhauf, 2023). Hermitianisation techniques further embed arbitrary matrices in symmetric blocks for polynomial manipulation.

2.3 Resolvent-Based and Parametric Eigenvalue Estimation

For non-normal or non-Hermitian matrices with spectra on general curves in the complex plane, resolvent-based methods invert the operator $M = Z \otimes \mathds{1}_n - \mathds{1}_a \otimes A$ (where $Z$ encodes candidate eigenvalues) and prepare quantum linear system states whose amplitudes encode proximity to an eigenvalue. Quantum Linear Systems Algorithms (QLSA) construct "resolvent states" which, when measured, yield additive-approximate eigenvalue estimates with query complexity depending on parameters such as the (restricted) Kreiss constant and the Jordan condition number (Alase et al., 7 Oct 2024). This approach naturally generalizes QPE to "parametric" eigenvalue estimation on arbitrary curves (e.g., real line, unit circle, ellipses).

2.4 Compressed Sensing and Time-Domain Approaches

Compressed sensing frameworks view the autocorrelation function $$z(t) = \langle\psi(0)|\psi(t)\rangle = \sum_{n}|c_n|^2 e^{-i E_n t}$$ as a sparse signal in the frequency domain. By taking only a small subset of quantum measurements (via the Hadamard test), the algorithm reconstructs the full spectrum using off-grid compressed sensing and atomic-norm minimization. Signal classification (e.g., MUSIC algorithm) then super-resolves closely spaced eigenvalues from reconstructed autocorrelation data (Castaldo et al., 16 Jul 2025). This approach is robust to state preparation errors, achieves Heisenberg-limited precision with short evolution times, and can reconstruct many eigenvalues simultaneously.

Time-series analysis protocols estimate Fourier components of $\mathrm{Tr}[\rho e^{-iHt}]$ at selected $t_k$, converting these into binned spectral weights after smoothing, with sample complexity polynomial in the inverse precision and no requirement for QFT or long coherent ancilla registers (Somma, 2019, Ding et al., 2023).

2.5 ODE and Collocation-Based Algorithms

For generalized eigenvalue problems $A x = \lambda B x$, ODE-based methods simulate the time evolution of $x(t)$ governed by an ODE $B \frac{d}{dt} x(t) = 2\pi i A x(t)$; by discretizing this evolution (e.g., via a Fourier-spectral method) and then inverting the coefficients with a QLSA, one recovers eigenvalue–eigenvector encodings efficiently. These methods extend to cases where $B^{-1}A$ is diagonalizable with real spectra (Shao et al., 2020).

Collocation-based eigenvalue solvers for the Schrödinger equation recast the eigenvalue problem as minimizing singular values of parameter-dependent residue matrices $M(\alpha) = M_0 - \alpha M_1$ without explicit inversion of ill-conditioned matrices, using a combination of QPE and amplitude amplification over a parameter grid (Rajchel-Mieldzioć et al., 16 Jun 2025). Landscape scanning across $\alpha$ identifies eigenvalues as singular value minima.

3. Classes of Generalized Problems and Applicability

3.1 Hermitian, Non-Hermitian, and Non-Normal Operators

QEVE encompasses spectral estimation for:

• Hermitian Hamiltonians (standard phase estimation, variational methods).
• Generalized eigenproblems $A v = \lambda B v$, converted to Hermitian form via $u = B^{1/2}v$ and transformation $B^{-1/2} A B^{-1/2}$ (Parker et al., 2020).
• Non-Hermitian (but diagonalizable, often with real spectrum) Hamiltonians—e.g., via transcorrelated similarity transforms in quantum chemistry (Feniou et al., 28 Jul 2025).
• Non-normal operators, with spectral estimation on curves in the complex plane (resolvent-based approaches) (Alase et al., 7 Oct 2024, Low et al., 11 Jan 2024).

3.2 Finite and Infinite-Dimensional Systems

Extensions to infinite-dimensional operators (e.g., differential operators; see ODE and Sturm–Liouville examples (Parker et al., 2020)), and flexibility to adapt to domains such as quantum chemistry, materials science, and signal processing, are core to QEVE's generality.

3.3 Subspace Methods and Krylov/Lanczos Extensions

QEVE frames are central to subspace-based quantum algorithms for excited states (Quantum Subspace Expansion QSE, quantum Equation-of-Motion qEOM, and quantum self-consistent EOM), which convert large eigenvalue problems into effective generalized eigenproblems on measured small matrices (Kwao et al., 12 Mar 2025). However, solving these reliably is sensitive to condition numbers in overlap matrices; basis orthonormalization (q-sc-EOM) can mitigate instability from sampling noise.

4. Resource Efficiency and Scaling

Substantial improvements in resource scaling distinguish QEVE approaches from both classical and earlier quantum methods:

• Hybrid variational algorithms require only short coherent evolution, with total resource cost scaling polynomially in precision (number of repetitions $\sim$ $1/p^2$ for achieving precision $p$) (1304.3061).
• Compressed sensing and time-series methods can achieve Heisenberg scaling (error $\sim 1/T_{max}^2$), exponential speedup in matrix size for certain block-encoded approaches, and circuit depths up to two orders of magnitude shallower than QPE (Castaldo et al., 16 Jul 2025, Ding et al., 2023, Lin et al., 25 Feb 2025).
• Block-encoded polynomial methods yield query complexity independent of the matrix size $n$—logarithmic in precision and failure probability, a strict improvement over previous methods (Lin et al., 25 Feb 2025).
• For generalized collocation algorithms, resource requirements scale as $O(\sqrt{N})$ in system size $N$ for certain problem classes, offering quartic improvement over the classical scaling for high-dimensional quantum chemistry tasks (Rajchel-Mieldzioć et al., 16 Jun 2025).

5. Error Analysis, Robustness, and Limitations

QEVE frameworks carefully address sources of error (sampling noise, discretization, circuit depth, condition numbers). In subspace expansion methods, high condition numbers in the overlap (Gram) matrix amplify statistical errors unless thresholding or orthonormalization is used—potentially at the cost of missing excited states (Kwao et al., 12 Mar 2025). Compressed sensing methods exhibit robustness to low-fidelity state preparation by leveraging sparsity and denoising from classical subroutines; residual errors are analyzed in terms of measurement noise and spectral density (Castaldo et al., 16 Jul 2025). In polynomial transformation-based QEVE, downscaling is sometimes required to ensure norm bounds, but this is often modest (Sünderhauf, 2023).

Limitations include increased classical postprocessing, the need for block encodings or matrix decompositions, possible circuit depth requirements for highly accurate solutions, and tailoring to specific spectral properties (e.g., diagonalizability). The area remains active, particularly in refining robustness, efficiency for non-diagonalizable operators, and resource performance on near-term hardware (Low et al., 11 Jan 2024).

6. Applications and Prospective Directions

QEVE methods have been applied to:

• Quantum chemistry: Accurate computation of ground and excited states (e.g., via VQE, QSE, transcorrelated extensions, adaptive real-space grids) (1304.3061, Rajchel-Mieldzioć et al., 16 Jun 2025, Feniou et al., 28 Jul 2025).
• Material science: Estimation of vibrational and electromagnetic eigenmodes in continuum and discrete models (Kerzner et al., 2023).
• Optimization, data analysis, and machine learning: Extraction of spectral information from kernel matrices and use in principal component analysis, classification, and robust signal estimation (Shao et al., 2020, Sato et al., 2023).
• Quantum dynamics and control, including simulation of open quantum systems and general non-normal operator dynamics (Alase et al., 7 Oct 2024).
• PDE solvers and time-evolution problems, using Faber and Chebyshev polynomial expansions (Low et al., 11 Jan 2024, Rajchel-Mieldzioć et al., 16 Jun 2025).

Prospective directions include integration of QEVE tools in hybrid quantum-classical workflows, extensions to unbounded or infinite-dimensional operator classes, improved analysis for non-diagonalizable matrices, and hardware-adapted implementations minimizing state overhead and measurement depth (Low et al., 11 Jan 2024).

7. Tabular Overview of Principal QEVE Methods and Features

Method/Class	Operator Types	Circuit/Resource Model
Hybrid VQE/Variational Solver	Hermitian, generalized	Shallow circuits, repeated expectation measurement (1304.3061, Liang et al., 2021, Sato et al., 2023)
Block-encoding + Poly. Transf.	Hermitian, non-Herm.	Chebyshev/polynomial, log-scaling (Sünderhauf, 2023, Lin et al., 25 Feb 2025)
Resolvent/Parametric QEVT	Non-normal, general	QLSA, curve-based state prep (Alase et al., 7 Oct 2024)
Compressed Sensing	Hermitian/unitary	Hadamard, few shots, MUSIC, off-grid (Castaldo et al., 16 Jul 2025)
Time Series/Fourier	Hermitian	One-ancilla time samples, binning (Somma, 2019, Ding et al., 2023)
Subspace QSE/qEOM/q-sc-EOM	Hermitian	Small matrix, sampling and classical postproc. (Kwao et al., 12 Mar 2025)

Each approach is characterized by its operator class (whether it allows non-Hermitian or non-normal matrices), resource efficiency (circuit depth, sample complexity, classical postprocessing), and robustness to errors or device limitations.

---

Generalised Quantum Eigenvalue Estimation thus represents a unifying and extensible framework in quantum algorithmics, subsuming and generalizing earlier eigenvalue extraction routines, and enabling robust and efficient spectral estimation for a wide variety of challenging operators in both quantum simulation and broader computational science.