Hamiltonian-Filtered States: Methods & Insights

• Hamiltonian-filtered states are defined as states obtained by applying a function of the Hamiltonian to an initial state, focusing the spectral weight within a desired energy window.
• They incorporate various filtering mechanisms—including Gaussian, Lorentzian, polynomial, and QPE-based filters—that offer different trade-offs in spectral selectivity and implementation complexity.
• These constructions support diverse applications such as state preparation, effective Hamiltonian derivation, compressed subspace identification, and metrological analyses in quantum systems.

Searching arXiv for recent and foundational papers on Hamiltonian-filtered states and closely related formulations. arXiv search query: Hamiltonian filtered states energy-filtered random-phase state quantum Gaussian filter phase estimation based filtering thermal filtered state Parseval frame graphene effective Hamiltonian edge states Hamiltonian-filtered states are states selected by acting on an initial state with a function of a Hamiltonian, most explicitly in the form $f(\hat{\mathcal H})|\psi\rangle$, so that spectral weight is concentrated in a chosen energy or quasienergy window. In the arXiv literature, however, the expression is not used in a single uniform sense. It covers Gaussian, Lorentzian, cosine, polynomial, and QPE-induced filters; Gibbs states viewed as exponentially filtered states; parent-Hamiltonian constructions whose ground state is the filtered state; Parseval-frame compressions of a physical Hilbert space; and, in a distinct but closely related sense, effective Hamiltonians that isolate the subspace relevant for edge physics without reproducing the full Brillouin zone (Seki et al., 2022, He et al., 2021, Irmejs et al., 2023, García-Pintos et al., 2024, Deshpande et al., 2016). This suggests a family of constructions organized by how the Hamiltonian itself selects, suppresses, or compresses degrees of freedom.

1. Spectral-functional definitions

The most explicit definition appears in work on the energy-filtered random-phase state, where a Hamiltonian-filtered state is exactly $f(\hat{\mathcal H})|\phi_r\rangle$ with

$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$

so that each eigencomponent $|E_n\rangle$ is multiplied by the Gaussian weight $e^{-\frac12(E_n-E)^2\tau^2}$ (Seki et al., 2022). A closely related Gaussian construction uses

$e^{-(\hat H-\mu \hat I)^2/\sigma^2},$

with the filtered state

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$

again making the spectral reweighting completely explicit (He et al., 2021). A Lorentzian variant defines

$$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$

so that the energy-basis probabilities are suppressed by

$\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$

rather than by a Gaussian (Irmejs et al., 2023). In the QSP formulation, the filter is an even polynomial $R_\ell(x;\Delta)$ chosen so that $f(\hat{\mathcal H})|\phi_r\rangle$0 and $f(\hat{\mathcal H})|\phi_r\rangle$1 is exponentially small on $f(\hat{\mathcal H})|\phi_r\rangle$2, thereby approximating the spectral projector onto a target eigenspace (Lin et al., 2019).

A different but mathematically immediate filtered-state viewpoint appears for Gibbs states: $f(\hat{\mathcal H})|\phi_r\rangle$3 which is a state obtained by filtering the Hamiltonian spectrum with the exponential weight $f(\hat{\mathcal H})|\phi_r\rangle$4 (García-Pintos et al., 2024). In QPE-based filtering the state transformation is mediated by an ancilla register rather than by directly writing $f(\hat{\mathcal H})|\phi_r\rangle$5, but the effective action is again spectral: the weight of each eigencomponent is renormalized by

$f(\hat{\mathcal H})|\phi_r\rangle$6

after postselecting ancilla outcomes $f(\hat{\mathcal H})|\phi_r\rangle$7 (Sakuma et al., 2 Jul 2025).

Construction	Defining filter or mechanism	Representative paper
Gaussian energy filter	$f(\hat{\mathcal H})|\phi_r\rangle$8	(Seki et al., 2022)
Quantum Gaussian filter	$f(\hat{\mathcal H})|\phi_r\rangle$9	(He et al., 2021)
Lorentzian filtered product state	$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$0	(Irmejs et al., 2023)
Polynomial eigenstate filter	$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$1	(Lin et al., 2019)
QPE low-pass filter	$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$2	(Sakuma et al., 2 Jul 2025)
Gibbs thermal filter	$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$3	(García-Pintos et al., 2024)

These constructions share the same structural feature: the Hamiltonian eigenbasis is the basis in which the filter acts diagonally or effectively diagonally. What differs is the filter profile, the implementation primitive, and whether the object of interest is a pure state, a mixed state, or a postselected subsystem state.

2. Spectral filter constructions and algorithmic realizations

A major implementation route rewrites the filter as a Fourier transform of real-time evolution. For the Gaussian operator,

$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$4

the identity

$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$5

makes the filter compatible with quantum simulation, and the corresponding filtered state becomes “essentially a Fourier transform of a time-evolved state” (Seki et al., 2022). The quantum Gaussian filter uses the same idea in a different notation: $$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$6 which is then discretized into a finite linear combination of unitaries $$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$7 with coefficients

$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$8

(He et al., 2021).

QPE-based filtering uses a different control structure. The ancilla is initialized in

$$f(\hat{\mathcal H})=e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2},$$9

where the amplitudes $|E_n\rangle$0 define a window function, and inverse QFT plus ancilla postselection implement an effective low-pass filter. The choice of window function determines the leakage profile. For the rectangular window, $|E_n\rangle$1 in the tails and the resulting filter exhibits the Gibbs phenomenon. For the sine window, $|E_n\rangle$2. For the Kaiser window, the passband/stopband guarantees are controlled by a parameter $|E_n\rangle$3, with

$|E_n\rangle$4

and the paper reports 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).

QSP-based filtering realizes the filter as an optimal polynomial transformation of a block-encoded Hamiltonian. After shifting and rescaling to

$|E_n\rangle$5

the even polynomial $|E_n\rangle$6 solves the minimax problem

$|E_n\rangle$7

and yields

$|E_n\rangle$8

This gives a $|E_n\rangle$9-block-encoding of the target spectral projector with

$e^{-\frac12(E_n-E)^2\tau^2}$0

queries (Lin et al., 2019).

The implementation literature therefore splits between time-domain Fourier constructions, ancilla-threshold filters, and polynomial eigenvalue transformations. The common principle is spectral selectivity; the differences lie in how that selectivity is synthesized and in whether the output is normalized directly, postselected, or reconstructed from measured overlaps.

3. Prepared states, parent Hamiltonians, and compressed subspaces

One line of work turns the filtered state itself into the ground state of a new Hamiltonian. For a product state $e^{-\frac12(E_n-E)^2\tau^2}$1 and Lorentzian filter $e^{-\frac12(E_n-E)^2\tau^2}$2, the parent Hamiltonian

$e^{-\frac12(E_n-E)^2\tau^2}$3

has unique ground state

$e^{-\frac12(E_n-E)^2\tau^2}$4

and the paper proves a gap $e^{-\frac12(E_n-E)^2\tau^2}$5 before rescaling. Along the adiabatic path, the rescaled Hamiltonian satisfies

$e^{-\frac12(E_n-E)^2\tau^2}$6

leading to the rigorous runtime bound

$e^{-\frac12(E_n-E)^2\tau^2}$7

while numerics support a practical circuit depth

$e^{-\frac12(E_n-E)^2\tau^2}$8

(Irmejs et al., 2023).

A second construction prepares a nonorthogonal basis of raw Hamiltonian-filtered states by time propagation: $e^{-\frac12(E_n-E)^2\tau^2}$9 Approximate eigenstates are then learned by solving a generalized eigenvalue problem in that basis,

$e^{-(\hat H-\mu \hat I)^2/\sigma^2},$0

so the filter is synthesized a posteriori as the trigonometric polynomial $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$1 (Cohn et al., 2021). In that work, compressed double-factorized Hamiltonians are used to generate the basis states cheaply, while a more converged Hamiltonian is used for projected matrix elements.

A third formulation begins from a physical projection rather than from an explicit spectral window. If $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$2 and $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$3 are projections of an orthonormal family, then $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$4 form a Parseval frame and the physical Hamiltonian is

$e^{-(\hat H-\mu \hat I)^2/\sigma^2},$5

Here the filter is the projector $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$6, the filtered states are states in the physical subspace $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$7, and the operator becomes a compression of the diagonal multiplication operator to the admissible coefficient range $e^{-(\hat H-\mu \hat I)^2/\sigma^2},$8 (Bagarello et al., 2020).

These constructions are not equivalent, but they share a common structural move: the target state space is made accessible by Hamiltonian design. In one case the filtered state is a unique ground state, in another it is a learned linear combination of propagated states, and in the Parseval-frame setting it is the state space that is compressed before the Hamiltonian is read off.

4. Thermodynamic and metrological interpretations

The energy-filtered random-phase state was introduced as a microcanonical thermal pure quantum state. With

$e^{-(\hat H-\mu \hat I)^2/\sigma^2},$9

the corresponding Gaussian-broadened microcanonical ensemble is

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$0

with effective energy window

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$1

Thermodynamic quantities follow from traces of $$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$2 and $$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$3, including

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$4

(Seki et al., 2022).

A distinct but complementary interpretation arises for Gibbs states. The thermal state

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$5

is itself a Hamiltonian-filtered state with exponential filter $$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$6. In that setting, the paper on Hamiltonian-parameter estimation shows that the information retained after filtering depends on the thermal variance of the parameter-coupling operator and on its noncommutativity with the full Hamiltonian. For a parameter $$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$7 coupled through $$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$8, the quantum Fisher information obeys

$$\ket{\psi_f} = \frac{1}{\sqrt C} \sum_{j=0}^{2^N-1} a_j\, e^{-(\lambda_j-\mu)^2/\sigma^2}\ket{\lambda_j},$$9

and a sharper pair of bounds involves the Wigner–Yanase skew information,

$$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$0

$$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$1

This makes precise the statement that commuting operators are metrologically favored because the filter does not hide sensitivity in incompatible eigenspaces (García-Pintos et al., 2024).

The most direct answer to the question of thermality of filtered pure states is given in the Floquet-based analysis of narrow filters. There, Hamiltonian-filtered states are built from the cosine filter

$$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$2

or its truncated version, and are mapped to effective Floquet-filtered states. Under Floquet ETH, the deviation of local observables from thermal values is bounded by the square root of the filter width: $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$3 At the same time, the Rényi entropies remain sharply nonuniform in $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$4: for $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$5 they scale only logarithmically in $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$6, while $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$7 is linear in $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$8 up to logarithmic corrections. A plausible implication is that local thermality and global entropic thermality separate sharply in filtered-state constructions (Yang et al., 7 Jul 2026).

5. Effective Hamiltonians, Hamiltonian engineering, and extended meanings

Not all work closest to the topic uses Hamiltonian-filtered states in the strict spectral sense. In graphene, the $$F_L(E,\delta)=\left(1+i\,\delta^{-1}(H-E)\right), \qquad \ket{\Phi} \propto F_L^{-1}\ket{\Psi},$$9-point effective Hamiltonian

$\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$0

is described as the paper’s “closest analogue” to Hamiltonian-filtered states because the effective Hamiltonian isolates the subspace of states relevant for edge physics by expanding about the time-reversal-invariant $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$1 point and retaining the two-band structure that encodes the edge-band inversion. The paper is explicit that this is not “spectral filtering” in the numerical linear-algebra sense and not an explicit projection operator onto edge states; rather, it is a local effective Hamiltonian that filters the problem down to the edge-relevant subspace by construction (Deshpande et al., 2016).

The same paper also shows that boundary engineering can move the edge-state crossing between the 1D time-reversal-invariant momenta $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$2 and $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$3 without changing the bulk spectrum. For zigzag ribbons, a boundary mass term $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$4 with $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$5 produces a real-space band inversion and midgap edge states near $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$6, while $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$7 shifts the edge states to $\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$8. This use of the Hamiltonian to isolate and relocate an edge sector is conceptually close to filtering, but it belongs to effective-field-theory and boundary-condition engineering rather than to spectral windowing (Deshpande et al., 2016).

An even broader use of the idea appears in filtered Hamiltonian engineering for spin networks. There the object being filtered is not the state but the Hamiltonian terms themselves. Collective pulse blocks, free evolution under a static field gradient, cycle repetition, and a time-domain weighting function produce an average Hamiltonian

$\frac{|c_n|^2}{1+\delta^{-2}(e_n-E)^2}$9

where the Bragg-grating-like filter

$R_\ell(x;\Delta)$0

preserves or suppresses couplings according to the accumulated gradient phase (Ajoy et al., 2012). The paper explicitly states that this is not “filtering of states” in the ordinary energy-eigenstate sense; rather it is filtering of Hamiltonian terms so that only a chosen interaction graph remains. The resulting effective Hamiltonian then determines which states and transport modes are dynamically supported.

These examples broaden the encyclopedia sense of the term. In one case the Hamiltonian filters the relevant subspace of a continuum model; in the other it filters interaction terms in average-Hamiltonian theory. Both usages are adjacent to, but distinct from, direct spectral filtering $R_\ell(x;\Delta)$1.

6. Conceptual boundaries, tradeoffs, and recurrent distinctions

The literature repeatedly emphasizes that “Hamiltonian-filtered states” is not a synonym for a single algorithm. Gaussian, Lorentzian, cosine, polynomial, Gibbs, QPE, parent-Hamiltonian, Parseval-frame, and effective-Hamiltonian formulations all appear, but they answer different questions. Some are explicit spectral projectors or smooth spectral selectors; some are state-preparation mechanisms; some are low-energy or microcanonical approximations; some are compressed descriptions of a physical Hilbert space; and some filter Hamiltonian terms or subspaces rather than states directly (Seki et al., 2022, Irmejs et al., 2023, Bagarello et al., 2020, Ajoy et al., 2012).

Several tradeoffs recur across these constructions. Narrower filters improve spectral selectivity but increase time range, circuit depth, query count, or sensitivity to finite-size structure. In the Gaussian microcanonical setting, larger $R_\ell(x;\Delta)$2 narrows the energy window but can make the window narrower than the local level spacing; in QGF, smaller $R_\ell(x;\Delta)$3 or lower $R_\ell(x;\Delta)$4 strengthen filtering but shrink normalization and amplify statistical and hardware errors; in Lorentzian product-state filtering, the rigorous adiabatic cost scales as $R_\ell(x;\Delta)$5; in QPE filtering, better stopband suppression requires larger ancilla cost or more elaborate window preparation; and in the thermality analysis, polynomially narrow filters suffice for local thermal behavior but not for full thermal Rényi scaling (Seki et al., 2022, He et al., 2021, Irmejs et al., 2023, Sakuma et al., 2 Jul 2025, Yang et al., 7 Jul 2026).

A second recurrent distinction concerns overlap and accessibility. Projection and filter methods are efficient only when the initial state already has support in the target sector. The Lorentzian parent-Hamiltonian construction assumes a product state near the target energy; QFD requires nonzero overlap of the reference state with the target eigenspace; QSP eigenstate filtering assumes a known target eigenvalue and a lower bound on the spectral gap; and the QGF complexity remains overlap-limited through the parameter $R_\ell(x;\Delta)$6 (Irmejs et al., 2023, Cohn et al., 2021, Lin et al., 2019, He et al., 2021).

A final distinction concerns what is directly supported by the cited works and what is extrapolation. It is directly supported that many papers realize states or subspaces of the form $R_\ell(x;\Delta)$7, or effective analogues obtained by projection or Hamiltonian engineering. It would be an extrapolation to identify all of them with one universal formalism. The most precise encyclopedia-level conclusion is therefore that Hamiltonian-filtered states form a technical umbrella for constructions in which the Hamiltonian, or a controlled deformation or compression of it, selects the physically relevant sector by spectral weighting, postselection, projection, or effective reduction.