Generalized Quantum Eigenvalue Transformation

• GQET is a framework that transforms Hermitian matrices using generalized quantum signal processing and Chebyshev polynomial mappings.
• It generalizes standard QET and QSVT by allowing complex coefficients, indefinite parity, and enforcing a unit-circle bounded condition.
• GQET enables efficient logarithmic-depth state preparation and phase synthesis, providing practical quantum functional calculus improvements.

Searching arXiv for papers on generalized quantum eigenvalue transformation and closely related frameworks. Generalized Quantum Eigenvalue Transformation (GQET) is a Hermitian-matrix polynomial transformation framework obtained by applying generalized quantum signal processing (GQSP) to a qubitized Hermitian projected unitary encoding of a matrix. In its standard form, GQET takes a Hermitian matrix $A$ encoded as a projected unitary and produces an encoding of $p(A/\alpha)$, where $p$ is specified in a Chebyshev basis and may have complex coefficients and indefinite parity. This distinguishes it from standard quantum eigenvalue transformation (QET) and quantum singular value transformation (QSVT), whose admissible polynomials are more restricted. Subsequent work has used the same mechanism as a concrete primitive for logarithmic-depth polynomial state preparation, while other papers have broadened the surrounding notion of “quantum eigenvalue transformation” toward Laplace-transform, contour-integral, and Weyl-calculus constructions for non-normal matrices (Sünderhauf, 2023, Claudon et al., 17 Mar 2026).

1. Formal definition and spectral mechanism

In the formulation introduced for Hermitian matrices, one starts from a Hermitian projected unitary encoding $(U,\Pi)$ of a Hermitian square matrix $A=A^\dagger$ with subnormalization $\alpha \ge \|A\|_2$, so that

$\Pi^\dagger U \Pi = A/\alpha.$

From this encoding one defines the reflection

$\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$

and qubitizes the encoded operator via $\mathcal R_\Pi U$. If $\lambda_i$ is an eigenvalue of $p(A/\alpha)$0, then the qubitized operator has eigenvalues

$p(A/\alpha)$1

A GQSP sequence acting on this unitary implements a polynomial

$p(A/\alpha)$2

and, after projection back to the encoding subspace, yields the Chebyshev polynomial

$p(A/\alpha)$3

evaluated on the encoded matrix. The output is therefore a projected unitary encoding of $p(A/\alpha)$4 (Sünderhauf, 2023).

The central algebraic identity is the Chebyshev–monomial correspondence

$p(A/\alpha)$5

together with

$p(A/\alpha)$6

GQET therefore acts on the qubitized eigenphases and reprojects the result as an eigenvalue transformation on the original Hermitian matrix. In the later SPUE/PUE formulation, the same mechanism is stated as: if $p(A/\alpha)$7 is a SPUE of $p(A/\alpha)$8, $p(A/\alpha)$9, and the associated signal-processing polynomial $p$0 satisfies $p$1, then the GQSP transformation of the qubitized walk operator yields a PUE of $p$2 (Claudon et al., 17 Mar 2026).

2. Relation to QET and QSVT

GQET is best understood as a strict generalization of standard QET for Hermitian matrices. The standard QSP-derived QET/QSVT formalism imposes parity constraints and real coefficients. GQET removes those two restrictions in the Hermitian setting, but replaces the usual interval boundedness condition with a unit-circle boundedness condition on the associated monomial polynomial $p$3 (Sünderhauf, 2023).

Framework	Target matrix class	Polynomial admissibility
QET	Hermitian	Real coefficients; definite parity
QSVT	Arbitrary matrices	Real coefficients; definite parity
GQET	Hermitian	Complex coefficients; indefinite parity; $p$4

This increase in expressivity is accompanied by a different normalization issue. A polynomial $p$5 that is bounded on $p$6 need not be directly realizable by GQET, because its associated $p$7 can violate the unit-circle bound. The relevant downscaling factor is

$p$8

The analysis in the generalized QSVT/GQET paper gives a worst-case upper bound $p$9. It also identifies parity subclasses for which $(U,\Pi)$0, and reports numerical matrix-inversion examples in which no scaling was needed in the tested cases and the scaling factors were below about $(U,\Pi)$1 (Sünderhauf, 2023).

GQET also inherits an important algorithmic advantage from GQSP: phase-factor synthesis runs in almost linear time $(U,\Pi)$2, whereas standard QSP/QET phase-finding is typically quadratic time $(U,\Pi)$3. This matters whenever classical phase synthesis is itself a bottleneck, not merely the quantum query count (Sünderhauf, 2023).

3. Encoding variants and normalization subtleties

The original Hermitian-block-encoding formulation and the later SPUE/PUE theorem describe the same basic transformation at slightly different levels of abstraction. In both cases, the operative object is not the raw matrix $(U,\Pi)$4, but an encoded unitary from which a qubitized walk operator can be constructed and then processed by GQSP (Sünderhauf, 2023, Claudon et al., 17 Mar 2026).

The SPUE/PUE formulation became operationally important in logarithmic-depth state preparation. There, the initial affine operator is block-encoded only with a constant success overlap $(U,\Pi)$5, rather than with normalization exactly $(U,\Pi)$6. The state-preparation paper notes that applying GQET directly to such an encoding would make the success probability scale like

$(U,\Pi)$7

which becomes too small for higher-degree polynomials. The construction therefore first performs amplitude amplification to obtain an encoding of approximately $(U,\Pi)$8 with normalization $(U,\Pi)$9, and only then applies GQET to implement the target polynomial of that operator (Claudon et al., 17 Mar 2026).

This normalization issue is conceptually central. Standard expositions of eigenvalue transformation often emphasize only the polynomial mapping $A=A^\dagger$0, but practical constructions depend just as strongly on the quality of the input encoding, the cost of controlled access to it, and the way postselection amplitudes compose under degree-$A=A^\dagger$1 signal-processing sequences. The state-preparation application makes those dependencies explicit (Claudon et al., 17 Mar 2026).

4. Polynomial state preparation as a canonical application

A direct application of GQET appears in logarithmic-depth quantum state preparation for polynomial amplitudes. The construction first block-encodes a simple affine diagonal operator $A=A^\dagger$2, described as a “linear position operator” on dyadic grid points. The affine operator $A=A^\dagger$3 has a Pauli decomposition involving only $A=A^\dagger$4 terms, which enables a modified PREPARE/SELECT implementation in logarithmic depth. GQET is then used to promote the block-encoded affine operator to a block-encoding of an arbitrary polynomial $A=A^\dagger$5 (Claudon et al., 17 Mar 2026).

The resulting transformed operator acts diagonally: $A=A^\dagger$6 and when applied to the uniform superposition

$A=A^\dagger$7

it yields a state proportional to $A=A^\dagger$8. In this application, GQET is explicitly the bridge from a logarithmic-depth affine construction to a logarithmic-depth polynomial state-preparation map (Claudon et al., 17 Mar 2026).

The surrounding circuit architecture is designed so that the low-depth property survives the transformation stage. PREPARE and SELECT each have depth $A=A^\dagger$9; the EXACT-one oracle used in PREPARE requires only two ancilla qubits and depth $\alpha \ge \|A\|_2$0; and the GQET stage adds a multiplicative factor proportional to the polynomial degree $\alpha \ge \|A\|_2$1. The full depth is

$\alpha \ge \|A\|_2$2

with size

$\alpha \ge \|A\|_2$3

ancilla count $\alpha \ge \|A\|_2$4, and asymptotically constant success probability in $\alpha \ge \|A\|_2$5 and $\alpha \ge \|A\|_2$6 (Claudon et al., 17 Mar 2026).

The same paper reports a proof-of-principle trapped-ion implementation using $\alpha \ge \|A\|_2$7 qubits and more than $\alpha \ge \|A\|_2$8 primitive quantum gates. In that setting, GQET is not a purely formal theorem: it is the explicit mechanism that lifts the block-encoding of a simple diagonal operator to the block-encoding of a polynomial amplitude profile without destroying the logarithmic dependence on the number of qubits (Claudon et al., 17 Mar 2026).

5. Extensions and neighboring eigenvalue-transformation frameworks

Although GQET is formally a Hermitian, qubitization-based construction, later literature places it within a wider program of programmable matrix-function implementation. At the constructional level, recursive QET/QSVT replaces difficult global high-degree synthesis by repeated application of an analytically solvable low-degree kernel. For the matrix sign function, the recursion

$\alpha \ge \|A\|_2$9

yields an analytically obtained parameter set composed of only $\Pi^\dagger U \Pi = A/\alpha.$0 different values for arbitrarily small error. QET-U, by contrast, adapts eigenvalue transformation to the Hamiltonian-evolution input model $\Pi^\dagger U \Pi = A/\alpha.$1, implementing $\Pi^\dagger U \Pi = A/\alpha.$2 with one ancilla qubit and no multi-qubit control operations in its short-depth version. Positive-side QET modifies the admissible domain for positive semidefinite matrices by using $\Pi^\dagger U \Pi = A/\alpha.$3 and $\Pi^\dagger U \Pi = A/\alpha.$4, and is used to obtain $\Pi^\dagger U \Pi = A/\alpha.$5 depth in a quantum conjugate-gradient method, at the cost of worse total query complexity (Mizuta et al., 2023, Dong et al., 2022, Toyoizumi et al., 2024).

For non-normal matrices, several papers develop broader eigenvalue-transformation frameworks that are no longer instances of the Hermitian GQET theorem. Lap-LCHS represents functions with a Laplace transform

$\Pi^\dagger U \Pi = A/\alpha.$6

and implements

$\Pi^\dagger U \Pi = A/\alpha.$7

through linear combinations of Hamiltonian simulations, enabling transformations such as $\Pi^\dagger U \Pi = A/\alpha.$8 and $\Pi^\dagger U \Pi = A/\alpha.$9 without explicitly inverting $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$0 (An et al., 2024). Contour-integral-based quantum eigenvalue transformation instead uses Cauchy’s formula

$\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$1

to reduce matrix functions to weighted sums of shifted resolvents, and gives a sampling-based LCU estimator for observables using only $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$2 additional qubits (Jiang et al., 17 Jan 2026). A still broader Weyl-calculus approach expresses analytic $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$3 for $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$4 through Fourier approximations on the numerical range, yielding discrete LCHS formulas

$\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$5

and claiming $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$6 query complexity scaling together with an ansatz-free convex optimization procedure for the coefficients (Ni et al., 29 Jun 2026).

A plausible implication is that “generalized quantum eigenvalue transformation” now functions at two levels in the literature: narrowly, as the Hermitian GQSP-based polynomial theorem of GQET, and more broadly, as a family of quantum functional-calculus methods that target eigenvalues rather than singular values and need not be reducible to standard QSVT.

6. Distinction from generalized eigenvalue problems

A recurrent source of confusion is the similarity between the phrase “generalized quantum eigenvalue transformation” and the large literature on quantum algorithms for generalized eigenvalue problems $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$7. These are different subjects. In the variational generalized-eigenvalue paper, the task is the Hermitian generalized eigenvalue problem with $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$8 positive definite, recast as minimization or maximization of the generalized Rayleigh quotient. The algorithm is a sequential variational optimizer in which the update of a single parameterized single-qubit gate reduces to a $\mathcal R_\Pi = -(\mathbb 1 - 2\Pi\Pi^\dagger),$9 generalized eigenvalue problem. The paper explicitly does not implement QSVT/QET-style block-encoding transformations; the “generalized eigenvalue” there is the problem being solved, not the transformation primitive (Sato et al., 2023).

The same distinction holds for phase-estimation and ODE-based generalized-eigenproblem algorithms. One paper reduces $\mathcal R_\Pi U$0 with Hermitian $\mathcal R_\Pi U$1 and positive-definite $\mathcal R_\Pi U$2 to the Hermitian problem $\mathcal R_\Pi U$3, so that ordinary QPE can be applied efficiently only when the transformed operator remains sparse (Parker et al., 2020). Another paper derives generalized eigenvalues from the ODE

$\mathcal R_\Pi U$4

whose solution encodes generalized eigenvalues as frequencies; a quantum linear-system solve and Fourier transform then recover the spectrum. That paper explicitly notes that it is not a direct QSVT/GQET construction (Shao et al., 2020).

More recent non-Hermitian generalized-eigenproblem work is likewise adjacent but not identical to GQET. VQGE uses generalized Schur decomposition and variationally searches for unitary matrices $\mathcal R_\Pi U$5 and $\mathcal R_\Pi U$6 that triangularize $\mathcal R_\Pi U$7 and $\mathcal R_\Pi U$8, after which generalized eigenvalues are read off as diagonal ratios $\mathcal R_\Pi U$9 (Li et al., 7 Jul 2025). A separate fault-tolerant algorithm scans a parameterized residue matrix family with QPE and amplitude amplification, locating generalized eigenvalues as minima of the singular-value landscape rather than by a direct polynomial spectral transform (Rajchel-Mieldzioć et al., 16 Jun 2025).

The essential boundary is therefore straightforward. GQET, in the strict sense, is a spectral transformation primitive acting on an encoded Hermitian matrix via GQSP and qubitization. Quantum algorithms for generalized eigenvalue problems may also manipulate spectra, but they typically do so through reduction to Hermitian eigenproblems, variational optimization, generalized Schur forms, ODE discretization, or spectral-landscape search rather than through the GQET polynomial transformation itself.