Eigenstate Filtering in Quantum Systems

• Eigenstate filtering is a spectral-transform technique that reshapes quantum state amplitudes to amplify target eigenstates while suppressing off-target components.
• It utilizes methods like QSP/QSVT, minimax polynomial constructions, and QPE-based windowing to achieve sharp spectral selectivity and improve state-preparation success.
• Practical implementations balance spectral sharpness, overlap amplification, and success probability, with applications in ground-state, excited-state targeting, and steady-state estimation.

Searching arXiv for recent and foundational papers on eigenstate filtering to ground the article. I’ll look up the cited arXiv papers and closely related work on eigenstate filtering. Searching arXiv for “eigenstate filtering quantum singular value transformation quantum phase estimation”. Eigenstate filtering (EF) is the use of a spectral transform to reshape the amplitudes of a quantum state in the eigenbasis of an operator, typically a Hamiltonian, so that selected eigenstates are preserved or amplified while off-target components are suppressed. In the literature summarized here, EF appears as projector approximation $f(H)\approx P$, as polynomial filtering implemented by QSP or QSVT, as windowed filtering within QPE, as filtered-state preparation preceding high-precision phase estimation, and as measurement-conditioned filtering for zero-mode sectors of Hermitian embeddings. Its applications include ground-state preparation, arbitrary excited-state targeting, low-energy spectral simulation, and direct steady-state estimation in open quantum systems (Lin et al., 2019, Sakuma et al., 2 Jul 2025, Lee et al., 5 Oct 2025, Patil et al., 26 Jun 2026, Yeo et al., 28 Jun 2026).

1. Conceptual scope and operator formulation

At its most general, EF starts from an initial state expanded in an eigenbasis,

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$

and applies a bounded spectral filter $f(H)$ so that the normalized output

$$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$

has larger overlap with a target eigenspace than the input. In projector language, the problem is to construct an operator $f(H-\lambda_* I)$ such that $f(H-\lambda_* I)\approx P_{\lambda_*}$, where $P_{\lambda_*}$ is the projector onto the eigenspace of a known eigenvalue $\lambda_*$. In filtered-state preparation, the same idea is expressed through the success probability

$p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$

and the overlap amplification of the desired eigenstate after post-selection (Lin et al., 2019, Lee et al., 5 Oct 2025).

The target subspace can be a single eigenvalue, an interval, a low-energy band, or a known zero sector. Accordingly, EF is realized as low-pass, band-pass, or single-point filtering. A recurring design constraint is $|f(x)|\le 1$ on the relevant spectral domain, because practical implementations via block-encoding, QSP, QSVT, or GQSP require bounded transforms. The same abstraction also covers QPE-based filters, where the spectral response is encoded indirectly through ancilla-window amplitudes and thresholding rather than by an explicit polynomial in $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$0 (Sakuma et al., 2 Jul 2025).

Realization	Spectral action	Representative papers
QSP/QSVT polynomial filtering	$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$1 or band-pass polynomial response	(Lin et al., 2019, Patil et al., 26 Jun 2026)
QPE-based window filtering	Ancilla thresholding induces $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$2	(Sakuma et al., 2 Jul 2025)
Filtered-state preparation for QPE	$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$3 amplifies overlap before QPE	(Lee et al., 5 Oct 2025)
Measurement-conditioned filtering	Repeated successful rounds multiply eigencomponents by filter factors	(Yeo et al., 28 Jun 2026)

A common misconception is to identify EF only with polynomial QSVT constructions. The contemporary literature uses the term more broadly: QPE filters, Gaussian and Krylov filters, Kaiser-window filters, and Rodeo filters are all treated as EF because each implements an energy-dependent transfer function that selectively preserves a target spectral region (Sakuma et al., 2 Jul 2025, Lee et al., 5 Oct 2025, Yeo et al., 28 Jun 2026).

2. Polynomial EF, minimax filters, and excited-state targeting

The most explicit projector-style formulations of EF are polynomial. In the QSP-based construction for a Hermitian $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$4, one shifts and rescales the spectrum so that the target eigenvalue is mapped to $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$5 in a normalized operator $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$6, then implements a real even polynomial that equals $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$7 at the origin and is small outside a gap window. The 2019 minimax construction uses

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$8

with $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$9 the Chebyshev polynomial of the first kind. For $f(H)$0, this polynomial is even, satisfies $f(H)$1, obeys $f(H)$2 on $f(H)$3, and suppresses the stopband $f(H)$4 as

$f(H)$5

It also solves the minimax problem over degree-$f(H)$6 polynomials normalized at $f(H)$7, establishing degree-optimal stopband suppression within polynomial EF (Lin et al., 2019).

The 2026 QIPI work transfers this minimax construction to arbitrary excited-state targeting by replacing a direct approximation of $f(H)$8 with a symmetric eigenstate filtering polynomial implemented by QSVT. There the shifted and scaled Hamiltonian is

$f(H)$9

and the EF objective is to build $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$0, where $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$1 is the eigenvalue closest to $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$2. For even degree $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$3, the filter is again

$$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$4

with $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$5 and off-target suppression bounded by $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$6 on $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$7. The paper emphasizes that the even symmetry $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$8 is crucial: unlike direct approximations to $$|\phi_f\rangle=\frac{f(H)|\phi_0\rangle}{\|f(H)|\phi_0\rangle\|}$$9, the filter is non-divergent at the origin, remains centered at the desired shift, and becomes better conditioned as $f(H-\lambda_* I)$0 approaches the target eigenvalue (Patil et al., 26 Jun 2026).

This symmetry underlies the contrast with Chebyshev inverse approximations. The QIPI study reports that low-degree Cheb-inv can amplify the wrong eigenstates or exhibit pseudo-convergence, whereas EF consistently amplifies the eigenstate closest to the shift and avoids divergence with respect to $f(H-\lambda_* I)$1. Numerical simulations on molecular Hamiltonians of H$f(H-\lambda_* I)$2, LiH, and BeH$f(H-\lambda_* I)$3 show improved convergence and enhanced access to higher excited states relative to other quantum power methods, while logical resource estimates indicate that high target-state amplification can be achieved with modest polynomial degrees in fault-tolerant settings (Patil et al., 26 Jun 2026).

3. QPE-based EF and spectral window engineering

QPE-based EF realizes filtering through a phase register rather than through an explicit polynomial in $f(H-\lambda_* I)$4. Given

$f(H-\lambda_* I)$5

one prepares an ancilla-window state

$f(H-\lambda_* I)$6

runs QPE, and keeps outcomes $f(H-\lambda_* I)$7. The resulting filter factor is

$f(H-\lambda_* I)$8

so each eigenstate weight is effectively multiplied by $f(H-\lambda_* I)$9. In coherent EF with QAA or in classical postselection EF, this constitutes a non-destructive filter with respect to the relative amplitudes inside the retained energy window (Sakuma et al., 2 Jul 2025).

The behavior of $f(H-\lambda_* I)\approx P_{\lambda_*}$0 is controlled entirely by the ancilla coefficients $f(H-\lambda_* I)\approx P_{\lambda_*}$1. With the rectangular window $f(H-\lambda_* I)\approx P_{\lambda_*}$2, the tails satisfy $f(H-\lambda_* I)\approx P_{\lambda_*}$3, producing strong sidelobes and Gibbs oscillations. With the sine window,

$f(H-\lambda_* I)\approx P_{\lambda_*}$4

the tails improve to $f(H-\lambda_* I)\approx P_{\lambda_*}$5. With the Kaiser window,

$f(H-\lambda_* I)\approx P_{\lambda_*}$6

the mainlobe is sharp and the sidelobes are flat and exponentially suppressed, with leakage $f(H-\lambda_* I)\approx P_{\lambda_*}$7 and $f(H-\lambda_* I)\approx P_{\lambda_*}$8. The paper derives the transition-width relation

$f(H-\lambda_* I)\approx P_{\lambda_*}$9

and therefore the query scaling

$P_{\lambda_*}$0

for Kaiser-based QPE filtering. It also shows numerically that the number of queries required for Kaiser-window-based filtering is comparable to that for QETU with optimized phase angles (Sakuma et al., 2 Jul 2025).

The same analysis reframes Gibbs suppression through $P_{\lambda_*}$1-factors induced by the window choice. Rectangular windows produce large oscillations near the cutoff; sine windows reduce them; Kaiser windows yield a nearly monotonic, smoothed step. This matters operationally because EF errors are precisely leakage across the intended spectral boundary. In application to low-energy spectral simulation, a two-step algorithm—coarse EF followed by fine-grid QPE—was benchmarked on the density of states of antiferromagnetic type-II MnO in a one-particle approximation. For a cutoff near $P_{\lambda_*}$2 Hartree, rectangular-window EF left visible ripples, whereas Kaiser-window EF produced much cleaner separation, and the error in the $P_{\lambda_*}$3 region was reduced by about six orders of magnitude (Sakuma et al., 2 Jul 2025).

4. Filtered-state preparation, FQPE, and adaptive filter families

A central development in recent EF theory is the treatment of filtering as a preprocessing stage for QPE. In the filtered-state framework, $P_{\lambda_*}$4 is chosen so that

$P_{\lambda_*}$5

has much larger overlap with the target eigenstate than $P_{\lambda_*}$6. The basic trade-off is explicit: $P_{\lambda_*}$7 and therefore

$P_{\lambda_*}$8

Strong overlap amplification thus necessarily reduces success probability. The point of FQPE is not to remove this trade-off, but to exploit it in the regime where state-preparation depth is much smaller than the high-precision QPE depth (Lee et al., 5 Oct 2025).

Within this framework, several filter families are analyzed. Low-pass filters based on error-function approximations, Gaussian band-pass filters

$P_{\lambda_*}$9

Chebyshev-type minimax filters, and Krylov-subspace filters all fit the template

$\lambda_*$0

with polynomial realizations handled by QSVT and trigonometric realizations by GQSP. For Gaussian and low-pass filters, the required degree scales as $\lambda_*$1; for the Chebyshev-type minimax analysis in that paper, trigonometric filters scale as $\lambda_*$2 and polynomial filters as $\lambda_*$3. Krylov filters are obtained by solving a generalized eigenproblem in a basis $\lambda_*$4, yielding adaptive filters that depend on both the Hamiltonian and the reference state (Lee et al., 5 Oct 2025).

The numerical evidence reported for Fermi–Hubbard models focuses on high-precision regimes where standard QPE is strongly limited by poor initial overlap. In those experiments, FQPE reduces total runtime by more than two orders of magnitude, with overlap amplification exceeding a factor of one hundred. The 7-site 1D case shows a minimum relative cost of approximately $\lambda_*$5 at $\lambda_*$6, while a modified Krylov filter achieves a cost reduction factor of approximately $\lambda_*$7 at $\lambda_*$8. The same study also records an important caution: a naive Krylov filter with $\lambda_*$9 can drive the overlap close to $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$0 while collapsing the success probability to $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$1, making FQPE more expensive than standard QPE. EF performance is therefore governed not only by spectral selectivity but also by the balance between passband gain and post-selection cost (Lee et al., 5 Oct 2025).

5. Hardware-oriented EF and adiabatic preconditioning

A distinct branch of EF research addresses noisy digital devices by reducing both the required filter degree and the effective post-selection overhead. In the filter-enhanced adiabatic approach, a digitized adiabatic evolution first prepares $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$2, and a shallow QETU filter is then applied to refine it. The filter has the form

$p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$3

and is implemented with a single ancilla through the QETU sequence

$p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$4

Conditioned on measuring the ancilla in $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$5, the output is proportional to $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$6, with success probability

$p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$7

Because $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$8 is roughly set by the weight of the state inside the pass band, adiabatic preprocessing directly improves filtering viability (Karacan et al., 26 Mar 2025).

The paper’s central practical device is spectral profiling. One scans $p_f=\langle \phi_0|f(H)^\dagger f(H)|\phi_0\rangle$9 and $|f(x)|\le 1$0 while monitoring the ancilla success probability, thereby identifying a narrow occupied spectral window $|f(x)|\le 1$1 for the adiabatic output. This allows the final filter to operate on a compressed spectral region rather than on the full spectrum. In the reported implementations, the final ground-state filter uses the lowest non-trivial QETU degree $|f(x)|\le 1$2, while deeper filters are reserved for the offline profiling step, where only a single-qubit success probability is estimated and the noise burden is correspondingly milder (Karacan et al., 26 Mar 2025).

The benchmarks combine numerical simulations with experiments on the Quantinuum H1-1 quantum computer. For the 8-qubit periodic Heisenberg model, AQC+F was compared to purely adiabatic preparation at matched native two-qubit gate counts, and the hybrid protocol yielded significantly smaller relative energy errors. For 2D TFIM instances up to $|f(x)|\le 1$3, AQC+F achieved at least an order-of-magnitude improvement in ground-state infidelity over AQC alone for all but the very smallest gate counts. The practical interpretation given in the paper is that moderate adiabatic depth concentrates spectral weight into the lowest few eigenstates, after which a very low-degree EF step can remove the residual excited-state contamination more efficiently than continuing the adiabatic evolution on noisy hardware (Karacan et al., 26 Mar 2025).

6. Zero-mode and sector filtering in open quantum systems

EF also extends beyond closed-system Hamiltonian eigenproblems. For Markovian open quantum systems, the steady state $|f(x)|\le 1$4 satisfies $|f(x)|\le 1$5. After vectorization, the Liouvillian becomes a non-Hermitian matrix $|f(x)|\le 1$6, and a Hermitian embedding

$|f(x)|\le 1$7

is introduced on an auxiliary qubit and the Liouville-space register. Under the assumptions of a unique NESS and a two-dimensional zero sector, $|f(x)|\le 1$8 has zero-eigenvalue vectors

$|f(x)|\le 1$9

and the filtering task becomes projection onto $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$00. The relevant gap is

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$01

the separation between the zero sector and the rest of the embedding spectrum (Yeo et al., 28 Jun 2026).

The implementation studied in this setting is Rodeo filtering. For successful rounds with times $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$02, an eigenmode of $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$03 with eigenvalue $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$04 acquires the factor

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$05

When the target energy is the known value $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$06, the zero modes satisfy $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$07, while nonzero modes are suppressed stochastically. With Gaussian-distributed times of rms scale $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$08, the paper derives

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$09

which yields $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$10 successful rounds and total controlled-evolution depth

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$11

For a QPE-based filter onto the same zero bin, by contrast, the required depth scales as

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$12

The paper therefore identifies a change in the target-error dependence from power-law to logarithmic when QPE-based projection is replaced by Rodeo filtering (Yeo et al., 28 Jun 2026).

Two broader EF lessons follow. First, known-center filtering is structurally easier than generic eigenvalue targeting: the open-system steady-state problem avoids the spectral search that would otherwise be required to center the filter. Second, restartability can be an algorithmic resource. In Rodeo, failure is detected at each round and later controlled evolutions are skipped, whereas QPE pays the full circuit depth before failure is known. At the same time, the paper is explicit about limitations: the method assumes a unique steady state, a well-separated zero sector, and manageable post-selection overhead; small $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$13, imperfect mid-circuit measurement, and embedding overhead all degrade practicality (Yeo et al., 28 Jun 2026).

7. Trade-offs, misconceptions, and research directions

The modern EF literature is unified by three recurring trade-offs. The first is between spectral sharpness and implementation cost: sharper filters require larger degree, more time steps, or more filtering rounds. The second is between overlap amplification and success probability: the filtered-state framework makes this quantitative through $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$14. The third is between asymptotic optimality and device-level practicality: degree-optimal minimax polynomials may still be less useful on noisy hardware than low-degree filters combined with adiabatic preconditioning or other overlap-boosting strategies (Lin et al., 2019, Lee et al., 5 Oct 2025, Karacan et al., 26 Mar 2025).

Several controversies or apparent contradictions in EF are resolved by keeping the target task explicit. Rectangular-window QPE filtering is not intrinsically “wrong”; it is a valid EF realization whose difficulty is Gibbs-induced leakage near spectral cutoffs. Direct inverse constructions are not uniformly inferior to filtering polynomials, but the excited-state QIPI study shows that they become numerically unstable and hyperparameter-sensitive in precisely the interior-spectrum regime where symmetric EF remains well-conditioned. Krylov-based filters can outperform non-adaptive Gaussian filters, but only when optimization penalizes vanishing success probability; otherwise they may produce excellent overlap amplification with unusable post-selection rates (Sakuma et al., 2 Jul 2025, Patil et al., 26 Jun 2026, Lee et al., 5 Oct 2025).

The immediate research trajectory suggested by these works is not a single canonical EF algorithm but a design space. The papers explicitly point to adaptive choices of $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle,$15 and degree, alternative minimax or multi-window filters, integration with optimized qubitization or QROAM block encodings, multistage filtering for spectroscopy, extensions to excited-state and interval filtering, and more detailed analyses of noise robustness. A plausible implication is that EF is best understood not as one algorithmic primitive but as a family of spectral transfer mechanisms whose optimal realization depends on whether the objective is projector approximation, eigenvalue readout, subspace concentration, or zero-sector isolation (Patil et al., 26 Jun 2026, Sakuma et al., 2 Jul 2025, Lee et al., 5 Oct 2025, Yeo et al., 28 Jun 2026).