Quantum Inverse Power Iteration (QIPI)

• QIPI is a family of quantum eigenstate filtering techniques that amplify the eigencomponent nearest a chosen shift using inverse or inverse-like functions, closely mirroring classical inverse power iteration.
• The approach encompasses diverse implementations—ranging from Fourier-based simulation and continuous-variable qumode methods to block-encoded QSVT schemes—each tailored for different hardware models and spectral targeting.
• Practical insights include tradeoffs between integration accuracy, postselection success, and resource scaling, with performance dependent on Hamiltonian conditioning and ansatz quality.

Quantum Inverse Power Iteration (QIPI) denotes a family of quantum eigensolver and state-preparation techniques that reproduce the spectral-filtering logic of classical inverse power iteration: a trial state is transformed by an inverse or inverse-like function of a Hamiltonian, most commonly $(H-\omega I)^{-1}$ or $H^{-k}$, so that the eigencomponent nearest a chosen shift, or the component associated with the smallest positive eigenvalue, is amplified relative to the rest. In the literature, the term does not refer to a single canonical algorithm. It encompasses Fourier-decomposed inverse iteration on programmable quantum simulators, continuous-variable implementations, block-encoding and generalized quantum signal processing (GQSP) realizations, quantum singular value transformation (QSVT) filtering schemes, and variational quantum iterative power algorithms (QIPA) that generalize imaginary-time-like propagation through positive oracle functions (Kyriienko, 2019, He et al., 2020, Khinevich et al., 15 Jul 2025, Patil et al., 26 Jun 2026, Kyaw et al., 2022).

1. Classical structure and terminological scope

In its linear-algebraic form, inverse power iteration targets the eigenstate whose eigenvalue is closest to a shift. For an eigenproblem $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$, the shifted iteration is

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$

Starting from $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$, one obtains

$$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$

so if $E_\ell$ is the eigenvalue closest to $\lambda$, the unwanted components decay like

$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$

This shift-invert mechanism is the common backbone of QIPI formulations that target either ground states or interior excited states (Zhao et al., 2024).

The same inverse-iteration logic also appears outside linear matrix problems. A nonlinear analogue for $p$-ground states defines iterates through

$H^{-k}$0

with monotone decrease of the Rayleigh quotient and convergence, after rescaling, to a $H^{-k}$1-ground state. That construction is purely classical PDE analysis, not a quantum algorithm, but it shows that inverse iteration is fundamentally a variational spectral-filtering paradigm rather than a matrix-specific routine (Hynd et al., 2015).

The literature uses several overlapping labels for quantum realizations of this paradigm.

Label in the literature	Core transform	Characteristic regime
Quantum inverse iteration / Q-Inv / QuIPI	Fourier or integral approximation of $H^{-k}$2 or $H^{-k}$3 via sums or integrals of $H^{-k}$4	Ground-state estimation on simulators; postprocessed overlaps
GQSP-based QII	Polynomial approximation to $H^{-k}$5 on a block encoding	Trotter-free inverse filtering near a chosen shift
QSVT-based EF-QIPI	Eigenstate filtering polynomial around $H^{-k}$6	Arbitrary excited-state targeting
Variational QIPA / QIPA$H^{-k}$7	Positive oracle $H^{-k}$8, including generalized imaginary-time-like propagation	NISQ-oriented hybrid optimization

This multiplicity of meanings is material rather than cosmetic. Some constructions are explicitly shift-invert methods, some approximate inverse powers through time-evolution integrals, and some replace the literal inverse by a monotone positive oracle. This suggests that QIPI is best regarded as a family of inverse spectral filters whose members share an amplification principle but differ substantially in hardware model, convergence analysis, and resource scaling (Kyriienko, 2019, Khinevich et al., 15 Jul 2025, Patil et al., 26 Jun 2026, Kyaw et al., 2022).

2. Fourier-based inverse iteration on programmable quantum devices

A central early formulation writes the nonunitary inverse power as a weighted combination of unitary real-time evolutions. In the programmable-simulator setting, the $H^{-k}$9-th inverse power is represented as

$\hat H|\psi_k\rangle = E_k|\psi_k\rangle$0

and, after discretization,

$\hat H|\psi_k\rangle = E_k|\psi_k\rangle$1

The ground-state energy estimate is then reconstructed from overlap measurements between the initial state and propagated states, rather than from a deterministic implementation of $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$2 itself. In this formulation the dominant practical resource is the maximal propagation phase $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$3, which functions as an evolution-time or gate-budget surrogate (Kyriienko, 2019).

The same Fourier logic underlies later numerical studies of the quantum inverse algorithm, denoted Q-Inv. There, $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$4 is approximated either with trapezoidal integration or with Gauss-Legendre quadrature, and its performance is compared to repeated application of an approximate $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$5 inverse, denoted I-Iter, as well as to an exact inverse obtained by LU decomposition. The numerical analysis shows that Q-Inv energy estimates usually improve with $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$6 only up to a system- and discretization-dependent optimum, after which the error rises because cutoff and quadrature errors accumulate. The study reports that Gauss-Legendre is markedly more efficient for the $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$7-integration, while trapezoidal integration is more effective for the oscillatory $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$8-integration, leading to a hybrid Legendre–Trapezoidal recommendation. It also uses chemical accuracy, $\hat H|\psi_k\rangle = E_k|\psi_k\rangle$9, as the reference tolerance, and identifies the difficult $$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$0 case as one where pure Q-Inv does not reach the expected error threshold and benefits from a Q-Inv+I-Iter continuation strategy (Cainelli et al., 2024).

Within this Fourier class, several benchmark domains recur. For molecular hydrogen and beryllium hydride, inverse iteration converges rapidly in the idealized setting, but the approximate implementation depends strongly on Hamiltonian conditioning and on $$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$1. For the Bose–Hubbard model, the method is used not only for energies but also for correlation functions. A recurring structural feature is that the algorithm converts ground-state preparation into a set of dynamical experiments plus classical recombination, which is particularly natural on analog or programmable simulators that already expose Hamiltonian dynamics as a native primitive (Kyriienko, 2019).

3. Continuous-variable realizations and qumode-assisted inverse Hamiltonians

A distinct realization introduces a continuous-variable ancilla, or qumode, to encode the integral weights of the inverse Hamiltonian. The key identity is

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$2

which motivates a system–qumode entangling unitary

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$3

With the ideal resource state

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$4

and postselection onto

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$5

one obtains

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$6

In this construction, the qumode wavefunction in momentum space is the coefficient distribution of the continuous linear combination of coherent Hamiltonian evolutions (He et al., 2020).

Finite squeezing replaces the ideal states by Gaussian-smeared half-line states. The output after entangling and postselecting becomes

$$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$7

and for large $$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$8 it approaches the inverse action with state error $$|\phi^{(i)}\rangle = (\hat H-\lambda I)^{-1}|\psi^{(i-1)}\rangle,\qquad |\psi^{(i)}\rangle = \frac{|\phi^{(i)}\rangle}{\langle \phi^{(i)}|\phi^{(i)}\rangle^{1/2}}.$$9. The same analysis gives an energy error $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$0, a success probability $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$1, and therefore the squeezing requirement

$|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$2

to obtain energy precision $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$3. The qumode-assisted method thus exposes an explicit tradeoff between inverse approximation quality and postselection overhead (He et al., 2020).

The benchmarks include $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$4, a three-qubit transverse-field Ising model, and a three-qubit Kitaev ring. The method reproduces the bond-dissociation curve of $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$5, captures the phase transition structure of the Kitaev ring near $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$6, and exhibits Trotter energy error scaling as $|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$7 in the Ising example. The analysis also treats bosonic loss on the qumode, depolarizing noise on qubits, and zero-noise extrapolation. In this sense, the qumode formulation occupies a middle ground between fully discrete Fourier summation and later block-encoding approaches: it preserves the integral representation of the inverse explicitly, but moves the coefficient synthesis into a physical ancilla mode rather than a classical quadrature table (He et al., 2020).

4. Block-encoding, GQSP, and QSVT formulations

A later line of work recasts inverse iteration as a polynomial transformation of a block-encoded Hamiltonian. In the GQSP framework, quantum inverse iteration is called Quantum Inverse Iteration (QII) and targets

$|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$8

The inverse power is approximated by truncating the Taylor series

$|\psi^{(0)}\rangle=\sum_k c_k^{(0)}|\psi_k\rangle$9

followed by conversion to a Chebyshev form compatible with GQSP. The convergence condition is stated as $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$0. Query complexity is especially simple: the number of queries to the encoded unitary equals the polynomial degree used in the method. The framework uses one signal-processing qubit, one capitalization qubit if needed, and $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$1 qubits for the LCU block encoding, avoids Suzuki–Trotter decomposition, and obtains the desired nonunitary transform through postselection, with low success probability identified as a principal drawback (Khinevich et al., 15 Jul 2025).

The numerical benchmarks in this GQSP setting compare QII to earlier inverse-iteration variants based on time-evolution operators. For $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$2 in cc-pVDZ with active space $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$3, truncation degree $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$4 is reported to achieve chemical accuracy in only one iteration for all bond lengths. For LiH, QII reaches chemical accuracy in 3 iterations while Q-Inv and I-Iter need more than 10; for BeH$$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$5, QII reaches chemical accuracy in 1 iteration while the other methods again need more than 10; and for square $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$6, QII with $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$7 reaches chemical accuracy in 2 iterations, although some competing methods can overtake it at longer iteration counts. The framework also notes that a complex shift $$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$8 can regularize the pole and enlarge the convergence radius, although that option is not used in the numerical examples (Khinevich et al., 15 Jul 2025).

QSVT-based QIPI pushes the same shift-invert principle further toward arbitrary excited states. There the shifted and scaled Hamiltonian is

$$|\psi^{(n)}\rangle = N_n\sum_k (E_k-\lambda)^{-n}c_k^{(0)}|\psi_k\rangle,$$9

and two approximation strategies are contrasted. The first, Cheb-inv, approximates $E_\ell$0 directly by a truncated Chebyshev series and is found to be numerically unstable and highly sensitive to $E_\ell$1. The second, eigenstate filtering (EF), constructs a symmetric even-degree polynomial

$E_\ell$2

with worst-case suppression

$E_\ell$3

Because the filter depends on $E_\ell$4, off-target components are suppressed symmetrically on both sides of the shift, and EF-based QIPI is reported to avoid divergence with respect to the choice of $E_\ell$5 while remaining effective in closely spaced spectra (Patil et al., 26 Jun 2026).

In this QSVT setting, one iteration costs $E_\ell$6 queries to the block encoding, and the total query complexity is given as

$E_\ell$7

with amplitude amplification. The fault-tolerant resource analysis isolates QSVT phase-synthesis cost in terms of T-count, giving

$E_\ell$8

and reporting that precision around $E_\ell$9 corresponds to about 30 T gates per rotation. Numerical simulations cover $\lambda$0 on 2 qubits, LiH on 10 qubits, and BeH$\lambda$1 on 12 qubits, with low-overlap trial states satisfying $\lambda$2. For $\lambda$3, degree $\lambda$4 reaches chemical accuracy in one iteration for almost all initial overlaps; for BeH$\lambda$5, query counts can rise to about 460 at small overlaps. The principal claim is not merely faster convergence relative to ordinary power iteration, but robust access to arbitrary excited states (Patil et al., 26 Jun 2026).

5. Variational QIPA, generalized imaginary time, and the separation controversy

A separate branch of the literature frames QIPI through variational quantum iterative power algorithms. In this formulation, the target state is

$\lambda$6

where $\lambda$7 is any strictly increasing positive function. Standard imaginary-time evolution is recovered with the ordinary exponential, while more general concatenated exponential oracle functions are defined recursively by

$\lambda$8

with the calculations using the double-exponential choice obtained for $\lambda$9 and $$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$0. McLachlan’s variational principle yields a linear system

$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$1

which is solved classically by conjugate gradients with tolerance $$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$2, while the matrix elements are estimated by Hadamard tests. In this NISQ-oriented setting, QIPA is presented as a generalized imaginary-time-like hybrid algorithm with shallow circuits and demonstrations on $$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$3 ground-state optimization, a flux-tunable transmon Hamiltonian, and biprime factorization. The factoring experiments report speedups of up to 50% in iteration count depending on the instance and the ansatz (Kyaw et al., 2022).

The same terminology later became the object of a detailed complexity-theoretic critique. There, Quantum Iterative Power Algorithms are interpreted as a quantum generalization of a classical iterative power algorithm with tensor-train or MPS compression replaced by a variational quantum state representation. Two variants are distinguished: a simple exponential oracle, stated to be effectively equivalent to varQITE, and a double exponential function, denoted QIPA$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$4. The lower-bound comparison is

$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$5

with $$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$6 encoding the unique solution and $$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$7 the next-largest eigenvalue. Exponential separation is formulated as varQITE requiring exponentially many iterations while QIPA$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$8 remains polynomial, and the resulting criteria are

$$\left(\frac{|E_\ell-\lambda|}{|E_i-\lambda|}\right)^n.$$9

$p$0

$p$1

A stronger appendix result states that the second-largest eigenvalue must be super-polynomial in $p$2 for the claimed exponential separation to hold (Czégel et al., 8 Feb 2025).

The same work then introduces a preprocessing step

$p$3

which preserves eigenvectors and the ratio $p$4, but scales the absolute gap according to

$p$5

If the original absolute gap is $p$6, enforcing the favorable QIPA$p$7 criterion requires $p$8 of order $p$9. The critique is that precisely this scaling causes the error to blow up, since

$H^{-k}$00

and the stated QIPA$H^{-k}$01 error bound is worse than varQITE’s by an additional $H^{-k}$02 term. The conclusion is that the apparent implication $H^{-k}$03 would arise only if approximation errors were ignored; once the error is included, no complexity collapse follows, and the promised worst-case exponential separation is practically unachievable (Czégel et al., 8 Feb 2025).

The same analysis nonetheless reports a practically relevant polynomial enhancement on a small 7-node weighted MaxCut instance. After mapping the cut cost to an Ising-type Hamiltonian and applying the scaling $H^{-k}$04 with $H^{-k}$05, an implementation based on Kyaw et al.’s code, integrated into Qiskit and run on IBM devices, is reported to make QIPA$H^{-k}$06 converge faster than varQITE and to reach close to $H^{-k}$07, the true ground-state value. The claim is explicitly limited to practical speed improvement on a small instance rather than to a worst-case exponential separation (Czégel et al., 8 Feb 2025).

6. Applications, limitations, and current interpretation

Across formulations, QIPI has been applied to molecular Hamiltonians such as $H^{-k}$08, LiH, BeH$H^{-k}$09, and square $H^{-k}$10; to lattice and many-body models such as the Bose–Hubbard chain, the transverse-field Ising model, and the Kitaev ring; to hardware-oriented Hamiltonians such as the flux-tunable transmon; and to nontraditional optimization encodings including biprime factorization and MaxCut. One strand emphasizes programmable simulators and analog dynamics, another fault-tolerant block-encoding architectures, and another hybrid NISQ workflows based on variational state families and Hadamard-test estimation (Kyriienko, 2019, He et al., 2020, Khinevich et al., 15 Jul 2025, Patil et al., 26 Jun 2026, Kyaw et al., 2022, Czégel et al., 8 Feb 2025).

The principal limitations are equally recurrent. Fourier and time-evolution methods are sensitive to condition number, integration cutoffs, and quadrature design; Q-Inv can improve initially with inverse power $H^{-k}$11 and then worsen as numerical integration error accumulates. Continuous-variable realizations incur a squeezing–success-probability tradeoff, with postselection probability $H^{-k}$12. GQSP and QSVT methods require a block encoding and a nonzero initial overlap, while postselection and amplitude amplification can dominate practical depth. GQSP-based QII additionally assumes a good estimate of $H^{-k}$13 and the convergence condition $H^{-k}$14. Variational QIPA depends strongly on ansatz quality, time-step choice, and stability of the linear solve built from noisy Hadamard-test data. In the double-exponential QIPA$H^{-k}$15 setting, Hamiltonian upscaling can improve iteration counts only by simultaneously degrading error behavior (He et al., 2020, Cainelli et al., 2024, Khinevich et al., 15 Jul 2025, Kyaw et al., 2022, Czégel et al., 8 Feb 2025).

Two misconceptions are therefore explicitly contradicted by the published record. The first is that QIPI denotes a single, standardized algorithm; the literature instead presents multiple inequivalent inverse-filtering schemes under overlapping labels such as quantum inverse iteration, Q-Inv, QuIPI, QII, and QIPA. The second is that inverse-power-based quantum methods have already established a generic worst-case exponential advantage over standard variational imaginary-time approaches. The available analysis supports strong application-dependent improvements, including one- or few-iteration convergence in favorable chemical examples and practical polynomial enhancement on small optimization instances, but it does not support an error-controlled worst-case exponential separation for the criticized QIPA$H^{-k}$16 regime (Khinevich et al., 15 Jul 2025, Patil et al., 26 Jun 2026, Czégel et al., 8 Feb 2025).

A plausible synthesis is that QIPI has matured from a ground-state-oriented inverse-Hamiltonian approximation into a broader framework for spectral filtering, with particularly strong recent emphasis on arbitrary excited-state targeting through block-encoding and QSVT. At the same time, the family remains sharply bifurcated by implementation model: simulator-based Fourier methods trade nonunitary inverses for overlap reconstruction, continuous-variable methods trade them for qumode postselection, and signal-processing methods trade them for polynomial approximation degree and success probability. The modern significance of QIPI lies less in a single asymptotic claim than in the range of inverse spectral transformations it makes available to quantum hardware under different resource assumptions.