ROQAM: Robust Quantum Arnoldi Method

• The paper’s main contribution is the development of ROQAM, a robust quantum Arnoldi method that preserves the upper-Hessenberg structure via orthogonal-polynomial recurrences to efficiently estimate Green’s functions.
• It leverages time-evolution operators to compute autocorrelation moments and reconstructs a compact projected matrix, reducing the cost of evaluating Green’s functions over extensive frequency grids.
• The method extends to finite temperatures through a thermofield-double construction, enabling both zero and finite temperature Green’s function evaluations with enhanced robustness against finite-precision errors.

Robust Quantum Arnoldi Method (ROQAM) is a quantum Krylov-subspace algorithm for estimating Green’s functions by returning a compact projected representation rather than isolated frequency-by-frequency estimates. Its defining feature is a formulation in terms of orthogonal polynomials, which preserves the upper-Hessenberg structure of the projected matrices despite finite-precision estimation. In the formulation introduced for single-particle Green’s functions, the quantum subroutine estimates autocorrelation moments of a time-evolution operator, after which a small projected matrix is reconstructed classically and reused to evaluate the Green’s function over intervals of real frequencies, many discrete points, or sets of complex frequencies. The same framework is extended to finite temperature through a thermofield-double construction that uses only a single Krylov subspace (Nelson et al., 21 May 2026).

1. Green’s functions, resolvents, and the target quantity

ROQAM is motivated by applications in which one needs a functional representation of a Green’s function over many frequencies rather than a single value at one frequency. In this setting, pointwise methods such as matrix inversion or QSVT can estimate $G(\omega)$ at a given $\omega$, but repeated estimation over many $\omega$-values becomes expensive. Arnoldi-type projection instead produces a small reduced matrix from which the function can be evaluated classically at arbitrary frequencies (Nelson et al., 21 May 2026).

At zero temperature, the method is formulated for an $n$-qubit Hamiltonian $H$ with ground-state eigenpair $(\ket{\psi_0},E_0)$. The particle and hole Green’s functions are

$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$

$$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$

and

$$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$

The associated spectral function is

$$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$

This formulation makes the computational objective explicit: estimate expectation values of resolvents of $\omega$0, with $\omega$1 acting as a broadening parameter controlling spectral resolution. The same projected representation can then support dense real-frequency grids, many discrete points, or imaginary-axis evaluations.

For diagonal and off-diagonal matrix elements, the initial state is chosen according to the Green’s-function component of interest. For the zero-temperature particle part, ROQAM uses

$\omega$2

while the hole part uses

$\omega$3

Off-diagonal elements are obtained by additional iterations with linear combinations such as

$\omega$4

with analogous annihilation-operator initial states for the backward Green’s function.

2. Arnoldi projection and the orthogonal-polynomial construction

ROQAM is a quantum adaptation of classical Arnoldi. Given an operator $\omega$5 and a starting state $\omega$6, the relevant Krylov space is

$\omega$7

The projected-matrix approximation used throughout is

$\omega$8

so once $\omega$9 is known, the resolvent can be evaluated by small dense linear algebra rather than further quantum calls (Nelson et al., 21 May 2026).

A direct quantum Arnoldi implementation based on explicitly estimating projected matrix elements is not the route taken in ROQAM. The central construction is instead an orthogonal-polynomial recursion,

$\omega$0

$\omega$1

$\omega$2

with basis vectors represented as

$\omega$3

The projected matrix elements are defined by

$\omega$4

ROQAM does not apply this recursion directly to a block-encoded $\omega$5. The paper explains that working with a $\omega$6 block-encoding effectively gives $\omega$7, and if QSVT is used to form Arnoldi/Lanczos orthogonal polynomials in $\omega$8, then polynomial subnormalization grows like $\omega$9 at depth $n$0, making estimation expensive. To avoid this, ROQAM performs Arnoldi on the time-evolution operator

$n$1

builds the projected matrix $n$2, and then defines the projected Hamiltonian by

$n$3

With $n$4, the projected entries reduce to autocorrelation moments. In particular,

$n$5

so the iteration is determined by estimates of

$n$6

The quantum computer estimates these moments, while the classical post-processing reconstructs the orthogonal polynomials, the upper-Hessenberg projected matrix $n$7, the projected Hamiltonian $n$8, and finally the Green’s function approximation. For example, for the diagonal zero-temperature particle Green’s function,

$n$9

3. Robustness: structural preservation, finite precision, and failure modes

The term “robust” in ROQAM refers first to structure preservation. A naive quantum Arnoldi implementation would estimate basis overlaps and projected entries separately, and finite-precision errors could destroy the exact upper-Hessenberg structure of the projected matrix. In ROQAM, the orthogonal-polynomial formulas enforce

$H$0

by construction, independent of estimation errors. This does not remove all numerical error, but it prevents a central structural failure mode of direct noisy reconstruction (Nelson et al., 21 May 2026).

Finite-precision errors enter through the estimated moments $H$1, each known only up to some $H$2. These perturb the recursively generated orthogonal polynomials, the projected matrix, and the final quantity $H$3. The paper does not provide a new tight theorem specifically proving a finite-precision ROQAM bound, and it explicitly notes that deriving tight a priori bounds is difficult. It also states that a rigorous backward-stability result would increase confidence in heuristic parameter choices, but such a proof is not given.

The timestep $H$4 is a second robustness parameter. The no-aliasing condition is

$H$5

where $H$6 is the largest-magnitude eigenvalue having support on $H$7. If $H$8 is too large, aliasing invalidates the logarithm inversion. If $H$9 is too small, then $(\ket{\psi_0},E_0)$0, Arnoldi vectors become nearly linearly dependent, their norms become small, and normalization amplifies errors. The paper does not provide a closed-form optimal rule and states that heuristics are likely needed.

A further limitation appears at low iteration depth. Even noiselessly, when $(\ket{\psi_0},E_0)$1, the projected matrix $(\ket{\psi_0},E_0)$2 need not be unitary although $(\ket{\psi_0},E_0)$3 is, so

$(\ket{\psi_0},E_0)$4

need not be exactly Hermitian. The resulting approximate Green’s function can therefore exhibit poles slightly off the real axis, producing unphysical artifacts at low depth. The post-processing remedies proposed are projection of $(\ket{\psi_0},E_0)$5 to the nearest unitary via SVD,

$(\ket{\psi_0},E_0)$6

or projection of $(\ket{\psi_0},E_0)$7 to the nearest Hermitian,

$(\ket{\psi_0},E_0)$8

One of the paper’s distinctive observations is that later moments need not be estimated as accurately as early ones. Numerical tests indicate that the allowed estimation error can increase with depth, and an “aggressive” budget used in the supplement is

$(\ket{\psi_0},E_0)$9

The stated intuition is that higher-$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$0 moments appear in fewer places in the projected matrix, so coarse estimation of late moments has limited impact on the final result.

4. Outputs, many-frequency reuse, and the finite-temperature extension

The practical output of ROQAM is a small projected representation tailored to the chosen initial state: first the projected unitary $$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$1, and then the projected Hamiltonian

$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$2

Once $$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$3 is available, the Green’s function can be evaluated classically through expressions of the form

$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$4

with $$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$5 or the corresponding thermal argument. This is the central amortization mechanism: the expensive quantum stage is executed once, and the resulting reduced model supports evaluation over dense real-frequency grids, imaginary-axis grids, interpolation across intervals, and spectral-function computation (Nelson et al., 21 May 2026).

The finite-temperature extension is based on a thermofield-double state,

$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$6

which the supplement also interprets as

$$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$7

Using vectorization identities, the thermal Green’s functions are rewritten as pure-state expectations of resolvents involving the doubled-space generator $$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$8: $$G^+_{pq}(\omega) = \bra{\psi_0}a_{p}\left[ (\omega + i\gamma) - (H - E_0) \right]^{-1}a_{q}^{\dag}\ket{\psi_0},$$9

$$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$0

Operationally, the thermal algorithm replaces the ground state $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$1 by $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$2, replaces $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$3 by $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$4, and then applies the same ROQAM machinery. The paper emphasizes that this thermal version uses only a single Krylov subspace, unlike stochastic trace-estimation approaches that require many random starting vectors and many separate Krylov constructions. It also notes that the finite-temperature version requires no estimate of eigenvalues of $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$5, unlike the zero-temperature formulation that uses $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$6.

5. Resource model, empirical performance, and practical scope

ROQAM is analyzed through a practical resource model rather than a single theorem-style asymptotic complexity bound. The quantum computer estimates the moments

$$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$7

and the classical computer performs the reconstruction of $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$8, computes $$G^-_{pq}(\omega) = \bra{\psi_0}a_{p}^{\dag}\left[ (\omega + i\gamma) + (H - E_0) \right]^{-1}a_{q}\ket{\psi_0},$$9, and evaluates the Green’s function at as many frequencies as desired. Using iterative quantum amplitude estimation (IQAE), the supplement states that the average number of generalized Grover-iterate calls needed for additive error $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$0 and failure probability $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$1 obeys

$$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$2

and the effective total additive error for each moment is budgeted as

$$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$3

The principal comparison in the paper is against pointwise QSVT matrix inversion. Because QSVT estimates the Green’s function at each $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$4 separately and the effective condition number

$$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$5

can strongly affect polynomial degree, the total cost scales poorly when an entire interval is needed. ROQAM’s advantage is that one quantum run yields a compact projected representation that is then reused classically across the full interval (Nelson et al., 21 May 2026).

For a single-impurity Anderson model, the paper estimates aggregate $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$6-gate counts for achieving below $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$7 mean relative error in the spectral function. It reports that ROQAM is roughly two orders of magnitude cheaper than even the single hardest QSVT frequency point among the tested instances, while producing a converged estimate over the entire frequency interval. The abstract states the broader conclusion that ROQAM outperforms pointwise estimation via quantum singular value transformation by multiple orders of magnitude. These resource estimates do not include the cost of preparing $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$8 or obtaining $$G_{pq}(\omega) = G^+_{pq}(\omega) + G^-_{pq}(\omega).$$9, so they are subroutine comparisons rather than full end-to-end resource counts.

The method’s practical limitations are explicit. There are no sharp predictive bounds for the optimal timestep $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$0, the required moment precisions $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$1, or the necessary iteration depth $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$2. Convergence depends strongly on the broadening $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$3: if $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$4 is too small, ROQAM becomes close to an eigensolver trying to locate all poles, which requires deeper iteration and higher precision; larger $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$5 improves convergence but lowers spectral resolution. The method also assumes access to efficient Hamiltonian simulation or block-encoding and to preparation of $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$6 or thermofield-double states, with state preparation explicitly recognized as outside the paper’s scope.

6. Historical and methodological context

ROQAM belongs to a broader Arnoldi lineage in which robustness is achieved by spectral transformation, projected reduced models, or carefully structured recursions. The following works supply the most immediate context.

Work	Main contribution	Relation to ROQAM
"Acceleration of the Arnoldi method and real eigenvalues of the non-Hermitian Wilson-Dirac operator" (Bergner et al., 2011)	Polynomial transformations and a peeling construction accelerate restarted Arnoldi for low-lying real eigenvalues of a non-Hermitian operator	Supplies a classical spectral-filtering viewpoint; it does not discuss quantum computation
"Quantum Arnoldi and conjugate gradient iteration algorithm" (Shao, 2018)	Early quantum Arnoldi for general matrices, with a complexity of $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$7 for obtaining the Hessenberg matrix	Foundational precursor, but the paper explicitly states that it did not concern numerical stability
"Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems" (Minganti et al., 2021)	Arnoldi basis generation from time evolution under $$A_{pq}(\omega) = -\pi^{-1}\mathrm{Im}\!\left[G_{pq}(\omega)\right].$$8 or a Floquet map, with residual-based convergence checks	A ROQAM-like open-system instance using matrix-free evolution and a contractive spectral transformation
"Convergence analysis and parameter estimation for the iterated Arnoldi-Tikhonov method" (Bianchi et al., 2023)	Iterated reduced-space regularization with explicit total-error decomposition and a parameter-choice rule; numerical results show robustness with respect to the regularization parameter	Classical evidence that robustness can come from balancing subspace approximation, reduced-model processing, and regularization

Among these antecedents, the non-quantum Wilson–Dirac work is relevant because it formalizes polynomial spectral transformation as a way to make non-Hermitian eigenproblems more targetable and more robust for Krylov-type eigensolvers. The early quantum Arnoldi work is relevant because it already framed Arnoldi vectors as polynomial transforms of the initial state, but it explicitly left stability largely untreated. The Arnoldi-Lindblad construction shows that a matrix-free Arnoldi scheme based on time evolution can remain effective for non-Hermitian open-system generators. The iterated Arnoldi-Tikhonov analysis, although developed for inverse problems rather than Green’s functions, suggests a broader methodological lesson: robustness in projected Krylov methods depends not only on the subspace itself but also on how the reduced model is reconstructed, regularized, and post-processed.

Within that lineage, ROQAM’s distinctive contribution is to combine time-evolution-based Krylov data with an orthogonal-polynomial formulation that preserves the upper-Hessenberg structure under finite-precision estimation, while shifting the computational objective from isolated values to a reusable functional representation of the Green’s function.