Quantum Power Iteration Methods

• Quantum Power Iteration is a family of quantum and hybrid quantum–classical algorithms that adapt the classical power iteration for amplifying dominant eigenstate components via spectral filtering.
• It employs variants including inverse iteration, shifted unitary maps, and QSVT-based techniques to target ground and excited states while optimizing resource scaling and convergence rates.
• Implementations range from variational and tensor-network approaches to stochastic and Krylov methods, demonstrating applications in molecular dynamics, optimization problems, and many-body quantum systems.

Quantum power iteration denotes a family of quantum and hybrid quantum–classical procedures that transplant classical power iteration, inverse iteration, and related spectral filtering schemes into quantum state space. Across the literature, the target transform may be a direct power $H^n$, a shifted inverse $(H-\omega I)^{-1}$, a shifted unitary map such as $I-U$, or a generalized cooling operator $\alpha_n(-\tau H)$; repeated application amplifies the eigencomponent associated with a dominant, shifted, or filtered eigenvalue and supports ground-state preparation, excited-state targeting, and global optimization (He et al., 2020, Daskin, 2020, Kyaw et al., 2022, Patil et al., 26 Jun 2026).

1. Classical template and quantum reinterpretations

At its narrowest, power iteration is the recurrence

$x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$

which converges to the eigenvector with largest $|\lambda|$ under standard spectral assumptions. Inverse iteration replaces $A$ by $(A-\sigma I)^{-1}$,

$$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$

so that the eigenvalue closest to the shift $\sigma$ becomes dominant. Quantum inverse power iteration adopts exactly this shift-and-invert logic for Hamiltonians, with the idealized map $(H-\omega I)^{-1}$0 or $(H-\omega I)^{-1}$1, and the resulting energy error decays as $(H-\omega I)^{-1}$2 after $(H-\omega I)^{-1}$3 inverse steps (He et al., 2020).

The phrase also covers direct quantum analogues of shifted power iteration. For a unitary $(H-\omega I)^{-1}$4, one can iterate with $(H-\omega I)^{-1}$5 rather than $(H-\omega I)^{-1}$6, so that the effective eigenvalues are $(H-\omega I)^{-1}$7, and the convergence rate is governed by $(H-\omega I)^{-1}$8. A single ancilla, two Hadamards, and a controlled-$(H-\omega I)^{-1}$9 realize the map $I-U$0 conditioned on a measurement outcome, yielding a direct quantum adaptation of the shifted power method for unitary matrices (Daskin, 2018).

A third line of work generalizes power iteration by exploiting extra structure. The extended power method for commuting operators $I-U$1 and $I-U$2 iterates with

$I-U$3

so that a chosen eigenvalue $I-U$4 of $I-U$5 can be pre-selected while iteratively converging to the corresponding eigenvalue $I-U$6 of $I-U$7. This gives a power-method-like route to excited states, degeneracy resolution, and convergence acceleration through the spectral shift $I-U$8 (Berger, 2015).

The same template admits stochastic and inexact formulations. In the inexact-power-iteration view of FCIQMC and FRI, the exact update $I-U$9 is replaced by

$\alpha_n(-\tau H)$0

with unbiased or biased error models and variance bounds of the form

$\alpha_n(-\tau H)$1

This recasts quantum many-body Monte Carlo ground-state projection as a form of randomized power iteration on $\alpha_n(-\tau H)$2 (Lu et al., 2017).

Variant	Representative transform	Representative use
Direct dominant-eigenvalue amplification	$\alpha_n(-\tau H)$3, $\alpha_n(-\tau H)$4, $\alpha_n(-\tau H)$5	unitary eigenpairs, QUBO
Shift-and-invert / inverse iteration	$\alpha_n(-\tau H)$6	ground and excited states
Generalized cooling / filtering	$\alpha_n(-\tau H)$7, $\alpha_n(-\tau H)$8	global optimization, filtered search
Krylov generalizations	$\alpha_n(-\tau H)$9	Arnoldi, CG, multireference subspaces

2. Non-unitary spectral transforms and circuit realizations

The central technical obstacle is that useful power-iteration maps are typically non-unitary. One solution is to rewrite the inverse Hamiltonian as an integral of unitaries: $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$0 assuming strictly positive spectrum after an energy shift. In the continuous-variable implementation, a qumode ancilla with quadratures $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$1 is coupled through

$x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$2

and the overlap $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$3 realizes an approximation to $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$4. Finite squeezing produces a state error $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$5, an energy error $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$6, and success probability scaling $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$7; increasing $x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$8 improves accuracy but lowers postselection rate (He et al., 2020).

A different construction starts from real-time dynamics. Since

$x^{(k+1)}=\frac{A x^{(k)}}{\|A x^{(k)}\|},$9

central finite differences express $|\lambda|$0 as a linear combination of short-time evolutions. Replacing exact dynamics by a symmetric Suzuki–Trotter formula yields

$|\lambda|$1

with operator error

$|\lambda|$2

Because the construction is Hermitian and even in $|\lambda|$3, Richardson extrapolation can suppress the leading systematic terms without increasing per-circuit depth (Seki et al., 2020).

Block-encoding and QSVT provide a fault-tolerant route. For an $|\lambda|$4-sparse Hermitian matrix $|\lambda|$5 with eigenvalues in $|\lambda|$6, an $|\lambda|$7-approximate block-encoding of $|\lambda|$8 can be built in time

$|\lambda|$9

and QSVT then applies a polynomial $A$0 to the spectrum using $A$1 calls to the block-encoding. In the improved quantum power method, a rank-1 operator built from $A$2 is transformed by an exponential-type polynomial $A$3 that maps an exponentially small singular value to a near-constant one, eliminating the $A$4 postselection penalty and giving overall runtime

$A$5

The same QSVT viewpoint underlies recent excited-state QIPI, where a shift $A$6 is used to approximate $A$7 through either a Chebyshev inverse or an eigenstate-filtering polynomial; the EF version is reported to be substantially more robust because the symmetry of the filtering polynomial avoids divergence with respect to $A$8 and suppresses off-target eigenstates in closely spaced spectra (Nghiem et al., 2024, Patil et al., 26 Jun 2026).

3. Variational, hybrid, and optimization-oriented power algorithms

In NISQ-oriented work, the power method often appears in variational form. For QUBO problems encoded into a diagonal Ising Hamiltonian

$A$9

the variational quantum power method replaces direct iteration by a control-qubit circuit that implements $(A-\sigma I)^{-1}$0 with $(A-\sigma I)^{-1}$1. Conditioned on measuring the ancilla in $(A-\sigma I)^{-1}$2, the data register undergoes

$(A-\sigma I)^{-1}$3

and the optimization objective is

$(A-\sigma I)^{-1}$4

In simulations on random QUBO instances up to 21 parameters, the mean number of iterations needed for the solution state’s probability to reach at least $(A-\sigma I)^{-1}$5 grows approximately linearly with problem size; about half the tested instances reached the exact optimum within 100 iterations, while the rest converged to near-optimal strings (Daskin, 2020).

A more explicitly imaginary-time-like family is QIPA. Here the non-unitary “power operator” is a concatenated exponential oracle

$(A-\sigma I)^{-1}$6

with the paper focusing on the double-exponential case

$(A-\sigma I)^{-1}$7

McLachlan projection onto a parametrized ansatz yields a linear system

$(A-\sigma I)^{-1}$8

whose solution updates the circuit parameters. In the reported benchmarks—$(A-\sigma I)^{-1}$9 dissociation, flux-tunable transmon ground-state search, and biprime factorization—QIPA converges faster than QITE in iteration count under the same time-step discretization, although evaluating $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$0 requires higher powers of $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$1 and therefore larger gate counts per step (Kyaw et al., 2022).

A different hybridization strategy is the quantum random power method. Here the quantum device estimates only a fixed number of matrix elements of a filtered polynomial $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$2, and the classical iteration uses a randomized sparse update

$$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$3

with $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$4. The algorithm is proved to converge with probability one to an approximation of the ground-state subspace, the per-iteration classical complexity is constant in $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$5 for fixed sampling parameters, and the same framework admits systematic and sampling noise through a perturbed matrix $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$6 (Ko et al., 2024).

The same optimization viewpoint appears in purely classical tensor-network form. The iterative power algorithm with quantics tensor trains applies

$$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$7

and converges to Dirac delta distributions or Dirac combs on the global minimizers of $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$8. Because QTT is treated as a classical analogue of quantum superposition and MPS structure, this construction serves as a template for optimization-oriented quantum power iteration based on imaginary-time projection without kinetic terms (Soley et al., 2021).

4. Krylov, tensor-network, and generalized subspace extensions

Power iteration is also the primitive behind quantum Krylov methods. In the superposition-of-time-evolved-states approach, the real target is not only $$x^{(k+1)}=\frac{(A-\sigma I)^{-1}x^{(k)}}{\|(A-\sigma I)^{-1}x^{(k)}\|},$$9 but the whole block Krylov space

$\sigma$0

followed by a Rayleigh–Ritz solve of the generalized eigenproblem $\sigma$1. In the reported Heisenberg and Hubbard examples, the ground-state energy error decays approximately exponentially with the Krylov depth $\sigma$2, and combining the method with VQE reference states substantially improves both energy and fidelity (Seki et al., 2020).

Quantum Arnoldi and quantum conjugate gradient push this further. Arnoldi builds an orthonormal basis of a Krylov space and a small Hessenberg matrix $\sigma$3; the quantum algorithm uses an HHL-style polynomial-in-$\sigma$4 subroutine, swap tests, and a new linear-combination-of-states method to achieve final complexity

$\sigma$5

For SPD systems, the corresponding quantum conjugate gradient algorithm attains

$\sigma$6

viewing CG as a polynomial iteration in $\sigma$7 whose coefficients are updated classically while states proportional to $\sigma$8 are synthesized quantumly (Shao, 2018).

Tensor-network-based quantum optimization gives yet another realization. A discretized objective function is encoded as a diagonal MPO $\sigma$9, then approximated by a unitary MPO $(H-\omega I)^{-1}$00 obtained by minimizing

$(H-\omega I)^{-1}$01

over unitary tensor cores. With ancillas initialized and post-selected in $(H-\omega I)^{-1}$02, one application of $(H-\omega I)^{-1}$03 realizes a single power iteration on the system register, and $(H-\omega I)^{-1}$04 repetitions produce $(H-\omega I)^{-1}$05. The compiled CNOT count scales as $(H-\omega I)^{-1}$06 and the circuit depth for $(H-\omega I)^{-1}$07 quantum power iterations as $(H-\omega I)^{-1}$08, where $(H-\omega I)^{-1}$09 is the chosen unitary-MPO bond dimension (Akshay et al., 2024).

The commuting-operator extension occupies a related niche. By transforming the simultaneous-eigenvalue problem for $(H-\omega I)^{-1}$10 into the generalized iteration

$(H-\omega I)^{-1}$11

the method makes a target quantum number $(H-\omega I)^{-1}$12 dominant in the effective spectrum. The resulting power iteration can pre-select symmetry sectors, compute excited states, and disentangle degenerate manifolds when $(H-\omega I)^{-1}$13 resolves them (Berger, 2015).

5. Convergence, resource scaling, and separation claims

Resource scaling depends strongly on how the non-unitary map is realized. In CV-assisted inverse iteration, first-order Trotterization of $(H-\omega I)^{-1}$14 gives per-iteration gate count

$(H-\omega I)^{-1}$15

total gate count

$(H-\omega I)^{-1}$16

and, after including postselection and energy-estimation sampling, overall time complexity

$(H-\omega I)^{-1}$17

The same analysis makes the squeezing tradeoff explicit: achieving energy error $(H-\omega I)^{-1}$18 through finite squeezing requires $(H-\omega I)^{-1}$19, whereas the success probability per inverse iteration scales as $(H-\omega I)^{-1}$20 (He et al., 2020).

The QSVT-based improvement of the quantum power method changes the situation qualitatively. Earlier approaches that explicitly prepared $(H-\omega I)^{-1}$21 paid an exponential $(H-\omega I)^{-1}$22 cost through tiny postselection probabilities. By embedding a rank-1 operator built from $(H-\omega I)^{-1}$23 into a block-encoding and applying an exponential-type QSVT polynomial, the singular value is flattened to a near-constant, removing dependence on $(H-\omega I)^{-1}$24 from the runtime and leaving only polylogarithmic dependence on system dimension (Nghiem et al., 2024).

QRPM exposes a different scaling profile. Its per-iteration classical complexity is constant in the Hilbert-space dimension for fixed $(H-\omega I)^{-1}$25, its required quantum circuit depth is independent of the initial overlap, and it has no or a square-root dependence on the spectral gap, depending on the polynomial-filtering construction. When the filtering polynomial corresponds to a sparse matrix, the iteration complexity scales linearly in the Hilbert-space dimension, and the quantum depth remains tied to estimating only a fixed number of matrix elements of $(H-\omega I)^{-1}$26 (Ko et al., 2024).

Claims of exponential improvement over other hybrid methods have been examined critically. For QIPA$(H-\omega I)^{-1}$27, the proposed separation from varQITE was summarized through

$(H-\omega I)^{-1}$28

The separation criteria require simultaneously

$(H-\omega I)^{-1}$29

The subsequent analysis shows that these conditions force super-polynomial eigenvalue scales, and the algorithmic error term then blows up super-polynomially or exponentially, so the claimed exponential speedup is practically unachievable even though polynomial enhancement remains possible (Czégel et al., 8 Feb 2025).

6. Applications, practical constraints, and current outlook

The application range is broad. CV-assisted inverse iteration was demonstrated numerically on molecular hydrogen, a three-qubit transverse-field Ising model, and a three-qubit Kitaev ring; for $(H-\omega I)^{-1}$30, with $(H-\omega I)^{-1}$31, cutoff 20 Fock states, and $(H-\omega I)^{-1}$32 inverse iterations, the energy follows exact diagonalization across the dissociation curve, and chemical accuracy is achieved within $(H-\omega I)^{-1}$33 even in a less favorable spectral-ratio setting (He et al., 2020). Variational QIPA was applied to $(H-\omega I)^{-1}$34 dissociation, transmon ground-state search, and biprime factorization, where the double-exponential oracle consistently reached accurate solutions in fewer time steps than QITE under matched discretization (Kyaw et al., 2022). The variational quantum power method targeted QUBO instances up to 21 variables, while QRPM was validated on TFIM, XXZ, and Hubbard models under both systematic and sampling noise (Daskin, 2020, Ko et al., 2024).

Excited-state targeting has become a central theme. Shift-and-invert QIPI already permits arbitrary-eigenstate targeting in principle by choosing $(H-\omega I)^{-1}$35 so that $(H-\omega I)^{-1}$36 is small (He et al., 2020). The recent QSVT-based EF construction makes that goal explicit: it targets arbitrary excited states through a shift $(H-\omega I)^{-1}$37, compares Cheb-inv with EF filters, and reports improved convergence on $(H-\omega I)^{-1}$38, LiH, and BeH$(H-\omega I)^{-1}$39, together with logical $(H-\omega I)^{-1}$40-gate estimates for fault-tolerant settings (Patil et al., 26 Jun 2026).

Practical constraints remain method-specific but recurring. Inverse-iteration schemes depend critically on initial overlap, squeezing or ancilla quality, and postselection success; variational schemes depend on ansatz expressivity, stable estimation of the matrices entering $(H-\omega I)^{-1}$41, and the accuracy of Hadamard-test data; tensor-network block-encodings require low-rank MPO or MPS structure and tolerate postselection only when ancilla success is not prohibitively small (He et al., 2020, Kyaw et al., 2022, Akshay et al., 2024). The same pattern appears in randomized methods: the fidelity lower bound for QRPM is controlled by the norm of the aggregate perturbation $(H-\omega I)^{-1}$42, and the bound requires $(H-\omega I)^{-1}$43 to remain below half of the filtered spectral gap (Ko et al., 2024).

The present literature therefore does not define a single algorithm so much as a design space. The phrase encompasses direct shifted-power circuits on unitaries, inverse-Hamiltonian filters, variational imaginary-time-like flows, QSVT polynomial filters, stochastic matrix-element iterations, and tensor-network block-encoded realizations. What unifies these approaches is the same spectral logic: engineer a transform $(H-\omega I)^{-1}$44 or $(H-\omega I)^{-1}$45 so that the desired eigenspace becomes dominant under repeated application, then realize that transform with the resources available on a quantum device (Daskin, 2018, Nghiem et al., 2024, Patil et al., 26 Jun 2026).