Quantum Power Lanczos Techniques

• Quantum Power Lanczos is a set of Krylov-subspace methods that use iterative polynomial filtering to extract extremal eigenvalues and eigenstates.
• It integrates diverse approaches like Chebyshev polynomial filtering, block recurrences, and multi-reference techniques, enhancing algorithmic stability and convergence.
• Its applications span quantum chemistry, nuclear physics, and condensed-matter simulations, providing practical strategies for improved eigenvalue approximations.

to data. Since user gave many paper details, we can synthesize comprehensive article with citations. Need maybe use arXiv search tool to confirm papers. Let's search "Quantum Power Lanczos" and key titles.

Searching arXiv for recent and foundational papers on Quantum Power Lanczos and closely related quantum Lanczos/Krylov methods. Quantum Power Lanczos denotes a family of Krylov-subspace methods in which repeated application of a Hamiltonian, transfer matrix, or propagator generates a power basis, and a Rayleigh–Ritz or Lanczos extraction is then used to approximate extremal or low-lying eigenvalues and eigenvectors. In the recent quantum-algorithm literature, the term is used explicitly for generalized quantum signal processing implementations of power iteration, power Lanczos, inverse iteration, and folded-spectrum methods, while several earlier works realize the same Krylov/polynomial-filtering idea without adopting the label. Across these variants, the defining structure is the replacement of direct large-scale matrix storage by subspace construction from iterated operators such as $H^k$, $e^{-i\Delta t H}$, $e^{-\Delta\tau H}$, transfer matrices, or Chebyshev polynomials, followed by a small projected eigenvalue problem or an equivalent filtered analysis (Khinevich et al., 15 Jul 2025, Kirby et al., 2022, Bowman, 2023, Abbott et al., 21 Mar 2025).

1. Conceptual and mathematical core

The classical basis of Quantum Power Lanczos is the Krylov subspace

$$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$

together with the Lanczos three-term recurrence

$$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$

or, in equivalent Rayleigh–Ritz language, the diagonalization of $H$ projected into that Krylov space. In the GQSP-based formulation, Quantum Power Lanczos uses a low-order polynomial filter

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$

followed by power iteration, while the transfer-matrix and filtered Rayleigh–Ritz literature emphasizes that Lanczos is equivalent to Rayleigh–Ritz applied to Krylov subspaces and that the projected spectrum consists of Ritz values and Ritz vectors (Khinevich et al., 15 Jul 2025, Wagman, 2024, Abbott et al., 21 Mar 2025).

A central conceptual point is that the “power” in Quantum Power Lanczos need not mean literal powers of $H$ implemented as nonunitary operators. Several constructions replace $H^k$ by bases that span the same polynomial space. The exact block-encoding approach uses $\{T_k(H)\}$, where $e^{-i\Delta t H}$0 are Chebyshev polynomials, and proves that

$e^{-i\Delta t H}$1

so the resulting Krylov space is identical to the classical power Krylov space. Real-time QLanczos instead uses the non-orthogonal basis $e^{-i\Delta t H}$2, and QITE-based variants use repeated imaginary-time steps to generate a filtered power basis (Kirby et al., 2022, Bowman, 2023, Yeter-Aydeniz et al., 2020).

The same Krylov principle also appears in non-gate-based many-body methods. Variational Monte Carlo “Lanczos steps” use

$e^{-i\Delta t H}$3

and neural-network quantum-state implementations use the same polynomial ansatz or an explicitly orthonormal Lanczos recursion on top of a variational seed. This suggests that Quantum Power Lanczos is best understood as a methodological class rather than a single circuit template: a polynomially filtered Krylov construction followed by an optimized spectral extraction (Becca et al., 2014, Chen et al., 2022).

2. Modes of Krylov-space generation

One major axis of variation is the choice of generator used to populate the Krylov basis. In real-time QLanczos, the basis states are

$e^{-i\Delta t H}$4

with overlap and Hamiltonian matrices

$e^{-i\Delta t H}$5

and the generalized eigenvalue problem

$e^{-i\Delta t H}$6

The multi-reference extension replaces a single seed by $e^{-i\Delta t H}$7 references,

$e^{-i\Delta t H}$8

so that the subspace dimension becomes $e^{-i\Delta t H}$9 (Bowman, 2023).

In operator-based quantum Lanczos recursion, the basis is built directly by the three-term recursion rather than by explicit propagator powers. The scalar recurrence is

$e^{-\Delta\tau H}$0

with

$e^{-\Delta\tau H}$1

The block version introduces a supervector

$e^{-\Delta\tau H}$2

and the block recurrence

$e^{-\Delta\tau H}$3

with

$e^{-\Delta\tau H}$4

Diagonalizing the resulting block-tridiagonal matrix yields simultaneous Rayleigh–Ritz approximations to multiple excited states (Baker, 2021).

A different construction is exact qubitized Lanczos. There, a block-encoding $e^{-\Delta\tau H}$5 with $e^{-\Delta\tau H}$6 and reflection $e^{-\Delta\tau H}$7 yields

$e^{-\Delta\tau H}$8

so powers of the qubitization iterate generate Chebyshev moments rather than time-evolved states. The required moments are

$e^{-\Delta\tau H}$9

from which the overlap and projected Hamiltonian matrices follow by closed-form identities (Kirby et al., 2022).

The broader literature further generalizes the generator from Hamiltonians to transfer matrices and Euclidean propagators. For a transfer matrix $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$0, the relevant Krylov space is $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$1, and the effective mass is identified as a power-iteration estimator, whereas Lanczos corresponds to a Krylov projection with faster convergence. In correlator-based formulations, Hankel matrices built from $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$2 or $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$3 supply the inner products needed for the same projected spectral analysis (Wagman, 2024, Abbott et al., 21 Mar 2025).

3. Quantum-computing realizations

On gate-based quantum hardware, Quantum Power Lanczos has been realized through several distinct primitive sets. QLR and QBLR use linear combinations of unitaries and oblivious amplitude amplification to apply operators, together with state-preserving quantum counting to estimate the Lanczos coefficients without destroying the reference state. In this setting, the $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$4th Krylov vector is written as

$$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$5

and coefficients are estimated from expectation values such as $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$6 and $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$7 on $$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$8. After assembling the tridiagonal or block-tridiagonal projection, diagonalization yields Ritz energies and the coefficients needed for state preparation operators

$$\mathcal{K}_d(H,|\psi^{(0)}\rangle) = \operatorname{span}\{ |\psi^{(0)}\rangle, H|\psi^{(0)}\rangle, H^2|\psi^{(0)}\rangle, \ldots, H^{d-1}|\psi^{(0)}\rangle \},$$9

Applying $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$0 to the initial state prepares the desired ground or excited Ritz state (Baker, 2021).

A closely related implementation obtains Lanczos coefficients from a single prepared ground state by state-preserving quantum counting. There, quantum phase estimation is run before and after the application of the relevant operator sequence, a pointer qubit records whether the energy registers agree, and only that pointer is measured. If the outcome indicates a mismatch, a recovery procedure restores the original state. The resulting coefficients feed a continued-fraction representation of the one-body Green’s function,

$$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$1

with $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$2, and the same machinery also supports a ground-state algorithm via classical diagonalization of the Lanczos tridiagonal and reconstruction operator $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$3 (Baker, 2020).

The GQSP framework unifies quantum power iteration, Quantum Power Lanczos, inverse iteration, and folded-spectrum methods within a single circuit model. A block-encoding

$$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$4

is combined with 0-controlled signal operators and single-qubit SU(2) rotations so that a degree-$$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$5 polynomial $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$6 is synthesized with $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$7 queries. In this language, the variational Lanczos state

$$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$8

is prepared by exact polynomial synthesis, and subsequent power iteration applies $$\beta_{k+1} |v_{k+1}\rangle = H |v_k\rangle - \alpha_k |v_k\rangle - \beta_k |v_{k-1}\rangle,$$9 to that improved initial vector. The reported molecular benchmarks state that Quantum Power Lanczos converges faster and more reliably than standard Quantum Power Iteration, while Quantum Inverse Iteration outperforms existing inverse iteration variants based on time-evolution operators (Khinevich et al., 15 Jul 2025).

The exact-and-efficient Lanczos construction removes both Trotterization and explicit time evolution. It measures Chebyshev moments on a quantum device, reconstructs the overlap and Hamiltonian matrices classically,

$H$0

$H$1

and then performs the orthogonalization and Ritz extraction entirely in the measured $H$2-dimensional subspace. The construction is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise (Kirby et al., 2022).

4. Block, multi-reference, and filtered extensions

A defining extension of Quantum Power Lanczos is the use of enlarged subspaces to target multiple states, resolve degeneracies, or stabilize noisy spectral extraction. In QBLR, block size $H$3 is the number of target excitations, each Lanczos step estimates the $H$4 matrices $H$5 and $H$6, and the effective Hamiltonian becomes block tridiagonal. The method is stated to resolve degeneracies more robustly than scalar Lanczos, because growing a block Krylov basis with $H$7 columns per step retains orthogonality across multiple targeted excitations and mitigates near-degenerate mixing (Baker, 2021).

Multi-reference QLanczos implements the same principle with multiple initial vectors rather than a block three-term recurrence. Using $H$8 references enlarges the subspace to $H$9, and the overlap and Hamiltonian matrices become block-structured in the reference and time-step indices. The reported numerical simulations state that using multiple reference states leads to faster convergences or higher accuracy for a fixed number of real-time iterations, and that in noisy scenarios $g(H) = 1 + C_1 H + \cdots + C_k H^k,$0 can restore iteration counts comparable to the noiseless single-reference case when $g(H) = 1 + C_1 H + \cdots + C_k H^k,$1 uniform noise is injected into the measured quantities (Bowman, 2023).

A separate but closely related generalization appears in filtered Rayleigh–Ritz and transfer-matrix analyses of correlation functions. There, the small projected problem is constructed from Hankel matrices,

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$2

or their block analogues, and the generalized eigenvalue problem

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$3

is interpreted as Rayleigh–Ritz on a Krylov subspace generated by time shifts. The filtered formulation identifies a Hermitian subspace, removes spurious states induced by statistical noise, and thereby retains the optimality guarantees of Rayleigh–Ritz on the filtered subspace (Abbott et al., 21 Mar 2025).

The bootstrap generalization of the Cullum–Willoughby method serves a related function in transfer-matrix Lanczos. For each bootstrap sample, one computes the real Ritz values of the tridiagonal projection, compares them to the spectrum of a deflated matrix obtained by deleting the first row and column, and classifies a Ritz value as spurious if it lies within an automatically determined threshold of the deflated spectrum, if it is complex, or if $g(H) = 1 + C_1 H + \cdots + C_k H^k,$4. The retained non-spurious largest $g(H) = 1 + C_1 H + \cdots + C_k H^k,$5 defines the ground-state estimate. This filtering is paired with robust bootstrap medians and two-sided residual bounds (Wagman, 2024).

These variants establish a recurrent pattern. Whether the enlarged subspace is created by blocks, multiple references, or correlator matrices, the goal is the same: enrich the polynomial span seen by Rayleigh–Ritz while controlling the spurious-state problem that arises from degeneracies, near-linear dependence, or statistical noise. A plausible implication is that “Quantum Power Lanczos” now names a spectrum of subspace-engineering strategies rather than only the minimal scalar power/Lanczos pipeline.

5. Error behavior, conditioning, and resource scaling

The accuracy of Quantum Power Lanczos depends on both subspace truncation and the fidelity with which projected matrix elements are estimated. In QLR and QBLR, first-order eigenvalue perturbation gives a linear dependence of the ground-state energy on coefficient uncertainties. In the scalar case,

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$6

and in the block case

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$7

The numerical study reports a linear mean absolute error trend,

$g(H) = 1 + C_1 H + \cdots + C_k H^k,$8

with $g(H) = 1 + C_1 H + \cdots + C_k H^k,$9 the Gaussian noise width injected into the coefficients (Baker, 2021).

Conditioning of the overlap matrix is a second recurring issue. Real-time QLanczos uses

$H$0

discards $H$1, constructs

$H$2

and forms the effective Hamiltonian

$H$3

This removes near-singular directions before the final Hermitian diagonalization. The exact block-encoding Lanczos method also regularizes the overlap matrix by thresholding its poorly conditioned eigenspaces before solving the generalized eigenproblem (Bowman, 2023, Kirby et al., 2022).

Resource scaling depends strongly on the primitive used to realize the polynomial filter. In QBLR, estimating the $H$4 blocks $H$5 and $H$6 requires one quantum-counting expectation value per matrix element, so the cost scales as $H$7 on top of the baseline QLR cost for each step. In GQSP-based implementations, queries per application of a polynomial $H$8 scale as $H$9; for QPL, preparing the filtered initial state requires degree $H^k$0, applying $H^k$1 requires degree $H^k$2, and the combined filter has degree $H^k$3 (Baker, 2021, Khinevich et al., 15 Jul 2025).

Operator-application cost is another bottleneck. In QLR/QBLR, naive oblivious amplitude amplification for an operator using $H^k$4 ancilla qubits has cost $H^k$5. Two mitigation strategies are explicitly given: partition the Hamiltonian terms into groups of size $H^k$6, which reduces the total cost to

$H^k$7

and work with $H^k$8 while increasing $H^k$9 in larger steps than real-time evolution would allow, so that fewer Lanczos steps are needed at each increment (Baker, 2021).

Transfer-matrix Lanczos provides the clearest rigorous convergence and residual statements. For Hermitian $\{T_k(H)\}$0, the Kaniel–Paige–Saad bound gives exponentially faster convergence than power iteration in the near-continuum regime, and residual-based two-sided eigenvalue bounds take the form

$\{T_k(H)\}$1

with $\{T_k(H)\}$2 computed from the last Lanczos coefficient and the Ritz-vector edge component. The article further states that these two-sided bounds guarantee that excited-state effects cannot shift Lanczos results far outside their statistical uncertainties (Wagman, 2024).

6. Applications across quantum physics

In quantum chemistry, GQSP-based Quantum Power Lanczos has been benchmarked on molecular Hamiltonians. For H$\{T_k(H)\}$3 in cc-pVDZ with a Hartree–Fock initial state, QPI achieves chemical accuracy within $\{T_k(H)\}$4 iterations across bond lengths, while the one-step Lanczos-filtered state

$\{T_k(H)\}$5

matches CASCI energy immediately after optimizing $\{T_k(H)\}$6. For CH$\{T_k(H)\}$7 in cc-pVDZ, QPL with $\{T_k(H)\}$8–$\{T_k(H)\}$9 markedly improves convergence for both singlet and triplet sectors; for N$e^{-i\Delta t H}$00, the paper reports that QPL is most effective at refining good initial states such as partially converged UCCD rather than recovering strong static correlation from scratch (Khinevich et al., 15 Jul 2025).

In nuclear structure, real-time QLanczos has been used in numerical simulations for the low-lying eigenstates of $e^{-i\Delta t H}$01Ne, $e^{-i\Delta t H}$02Na, and $e^{-i\Delta t H}$03Na, comparing imaginary- and real-time evolution. The reported finding is that imaginary-time evolution converges in fewer iterations than real-time, but real-time still converges within tens of iterations and is naturally implementable on quantum hardware because it is unitary. Quantum circuit prototype simulations for $e^{-i\Delta t H}$04Be were carried out in both the spherical basis and Hartree–Fock basis, with the M-scheme spherical basis reported to yield lower depth circuits than the Hartree–Fock basis (Bowman, 2023).

In condensed-matter dynamics on NISQ hardware, the quantum Lanczos algorithm has been used to obtain all energy levels and corresponding eigenstates of the one-dimensional transverse Ising model on $e^{-i\Delta t H}$05 and $e^{-i\Delta t H}$06 spatial sites with periodic boundary conditions, enabling one- and two-particle transition amplitudes, particle numbers for spatial sites, and transverse magnetization as functions of time. The circuits were executed on IBM 5-qubit superconducting hardware, and the reported experimental results with readout error mitigation are in very good agreement with the values obtained using exact diagonalization (Yeter-Aydeniz et al., 2020).

For Green’s functions and ground-state preparation, state-preserving Lanczos recursion provides a continued-fraction representation directly from a single prepared ground-state wavefunction, with the explicit claim that the wavefunction does not need to be re-prepared at each iteration. The same counting-based architecture is proposed for ground-state determination by building the Lanczos tridiagonal classically and reconstructing the target state from the Ritz coefficients (Baker, 2020).

Outside digital quantum computing, power-Lanczos ideas remain active in large-scale many-body variational methods. In frustrated Heisenberg models on square and Kagome lattices, a few Lanczos steps on top of Gutzwiller-projected fermionic states lower the variational energy and support zero-variance extrapolation, with thermodynamic $e^{-i\Delta t H}$07 gaps reported as $e^{-i\Delta t H}$08 on the square lattice and $e^{-i\Delta t H}$09 on the Kagome lattice, both compatible with gapless spectra within error bars. Neural-network quantum states for the square-lattice $e^{-i\Delta t H}$10–$e^{-i\Delta t H}$11 model implement the same Krylov enhancement through symmetry-projected RBMs and Lanczos recursion, achieving state-of-the-art accuracy for the ground state in the reported $e^{-i\Delta t H}$12 and $e^{-i\Delta t H}$13 studies (Becca et al., 2014, Chen et al., 2022).

The conceptual reach of Lanczos-based power constructions extends further. Transfer-matrix Lanczos has been applied to the simple harmonic oscillator and to the LQCD proton mass, where it provides faster ground-state convergence than the effective mass and more accurate energy estimates than multi-state fits at small imaginary times, while the filtered Rayleigh–Ritz framework identifies an equivalence class containing Lanczos, Prony’s method, GPOF, and GEVP-based analyses. In cosmology, the generalized Lanczos algorithm has been used to construct an open two-mode squeezed state for curvature perturbations, with the resulting power spectrum reported to match that of the Bunch–Davies vacuum numerically across the observable range (Wagman, 2024, Abbott et al., 21 Mar 2025, Zhai et al., 27 May 2025).