--- title: 'Hamiltonian-Filtered States: Methods & Insights' url: https://www.emergentmind.com/topics/hamiltonian-filtered-states type: topic --- # Hamiltonian-Filtered States: Methods & Insights 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 [2207.01782][2112.06026][2312.13892][2401.10343][1603.04329]. 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 \(e^{-\frac12(E_n-E)^2\tau^2}\) [2207.01782]. 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 [2112.06026]. 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 [2312.13892]. In the QSP formulation, the filter is an even polynomial \(R_\ell(x;\Delta)\) chosen so that \(R_\ell(0;\Delta)=1\) and \(|R_\ell(x;\Delta)|\) is exponentially small on \(\mathcal D_\Delta=[-1,-\Delta]\cup[\Delta,1]\), thereby approximating the spectral projector onto a target eigenspace [1910.14596]. A different but mathematically immediate filtered-state viewpoint appears for Gibbs states: \[ \rho_\beta(\vec\mu)=\frac{e^{-\beta H(\vec\mu)}}{Z_\beta(\vec\mu)}, \] which is a state obtained by filtering the Hamiltonian spectrum with the exponential weight \(f(E)=e^{-\beta E}\) [2401.10343]. In QPE-based filtering the state transformation is mediated by an ancilla register rather than by directly writing \(f(H)\), but the effective action is again spectral: the weight of each eigencomponent is renormalized by \[ |C_\mu|^2 \to |C_\mu|^2 \mathcal R_\mu, \qquad \mathcal R_\mu=\sum_{y=0}^{y_c}P_\mu(y), \] after postselecting ancilla outcomes \(y\le y_c\) [2507.01361]. | Construction | Defining filter or mechanism | Representative paper | |---|---|---| | Gaussian energy filter | \(e^{-\frac12(\hat{\mathcal H}-E)^2\tau^2}\) | [2207.01782] | | Quantum Gaussian filter | \(e^{-(\hat H-\mu \hat I)^2/\sigma^2}\) | [2112.06026] | | Lorentzian filtered product state | \(\left(1+i\delta^{-1}(H-E)\right)^{-1}\) | [2312.13892] | | Polynomial eigenstate filter | \(R_\ell(\bar H;\bar\Delta)\approx P_\lambda\) | [1910.14596] | | QPE low-pass filter | \(\mathcal R_\mu=\sum_{y=0}^{y_c}P_\mu(y)\) | [2507.01361] | | Gibbs thermal filter | \(e^{-\beta H}/Z_\beta\) | [2401.10343] | 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, \[ \hat{G}_\tau(E)=e^{-(\hat{\mathcal H}-E)^2\tau^2}, \] the identity \[ \hat{G}_\tau(E)= \frac{1}{2\sqrt{\pi}\tau} \int_{-\infty}^{\infty} dt\, e^{-t^2/(4\tau^2)} e^{iEt}\hat U(t), \qquad \hat U(t)=e^{-i\hat{\mathcal H}t}, \] makes the filter compatible with quantum simulation, and the corresponding filtered state becomes “essentially a Fourier transform of a time-evolved state” [2207.01782]. The quantum Gaussian filter uses the same idea in a different notation: \[ e^{-(\hat H-\mu \hat I)^2/\sigma^2} = \frac{\sigma}{2\sqrt\pi} \int_{-\infty}^{\infty} e^{-\sigma^2 y^2/4}\,e^{i\mu y}\,e^{-i\hat H y}\,dy, \] which is then discretized into a finite linear combination of unitaries \(e^{-i\hat H t_y}\) with coefficients \[ b_{\mu,\sigma,y} = \frac{\sigma}{2\sqrt\pi}\, e^{-(y\Delta_y \sigma)^2/4}\, e^{i\mu (y\Delta_y)} \] [2112.06026]. QPE-based filtering uses a different control structure. The ancilla is initialized in \[ \sum_{j=0}^{N-1} a_j |j\rangle, \] where the amplitudes \(a_j\) 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, \(P_\mu(y)=O(y^{-2})\) in the tails and the resulting filter exhibits the Gibbs phenomenon. For the sine window, \(P_\mu(y)=O(y^{-4})\). For the Kaiser window, the passband/stopband guarantees are controlled by a parameter \(\alpha\), with \[ N \approx \frac{2\pi}{\delta T}\,2\alpha = O\!\left(\delta^{-1}\log \epsilon^{-1}\right), \] 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 [2507.01361]. QSP-based filtering realizes the filter as an optimal polynomial transformation of a block-encoded Hamiltonian. After shifting and rescaling to \[ \bar H=\frac{H-\lambda I}{\alpha+|\lambda|}, \] the even polynomial \(R_\ell(x;\Delta)\) solves the minimax problem \[ \underset{p(x)\in\mathbb{P}_{2\ell}[x],\,p(0)=1}{\mathrm{minimize} \max_{x\in\mathcal D_\Delta}|p(x)|, \] and yields \[ \|R_\ell(\bar H;\bar\Delta)-P_\lambda\| \le 2e^{-\sqrt{2}\ell\bar\Delta}. \] This gives a \((1,m+2,\epsilon)\)-block-encoding of the target spectral projector with \[ \mathcal{O}\!\left(\frac{\alpha}{\Delta}\log\frac1\epsilon\right) \] queries [1910.14596]. 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 \(\ket{\Psi}\) and Lorentzian filter \(F_L(E,\delta)=1+i\delta^{-1}(H-E)\), the parent Hamiltonian \[ \mathcal H = \sum_i F_L^\dagger P_i F_L \] has unique ground state \[ \ket{\mathrm{GS}_{\mathcal H}}=\frac{1}{\mathcal N}F^{-1}\ket{\Psi}=\ket{\Phi}, \] and the paper proves a gap \(\Delta\ge 1\) before rescaling. Along the adiabatic path, the rescaled Hamiltonian satisfies \[ \Delta(s)\ge \frac{1}{1+s^2\delta^{-2}}, \] leading to the rigorous runtime bound \[ T=O(N^3\delta^{-4}), \] while numerics support a practical circuit depth \[ D=O(N^2\delta^{-4}) \] [2312.13892]. A second construction prepares a nonorthogonal basis of raw Hamiltonian-filtered states by time propagation: \[ |\Psi_m\rangle = e^{- i \Delta t\, m \hat H}\,|\Psi_0\rangle. \] Approximate eigenstates are then learned by solving a generalized eigenvalue problem in that basis, \[ |\Phi_I\rangle = \sum_{m=0}^{M-1} c_{mI} |\Psi_m\rangle = \left(\sum_{m=0}^{M-1} c_{mI} e^{-i m \Delta t \hat H}\right)|\Psi_0\rangle = f_I(\hat H)|\Psi_0\rangle, \] so the filter is synthesized a posteriori as the trigonometric polynomial \(f_I(\hat H)\) [2104.08957]. 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 \(\mathcal H_{\mathrm{ph}}=P\mathcal H\) and \(\varphi_n=Pe_n\) are projections of an orthonormal family, then \(\{\varphi_n\}\) form a Parseval frame and the physical Hamiltonian is \[ H_{\mathrm{ph}}=PHP, \qquad H_{\mathrm{ph}}f=\sum_n E_n\langle\varphi_n,f\rangle\varphi_n. \] Here the filter is the projector \(P\), the filtered states are states in the physical subspace \(P\mathcal H\), and the operator becomes a compression of the diagonal multiplication operator to the admissible coefficient range \(R(\theta_\varphi)\subset \ell^2(J)\) [2010.05043]. 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 \[ \hat G_\tau(E)=e^{-(\hat{\mathcal H}-E)^2\tau^2}, \qquad |\psi_{\tau,r}(E)\rangle=\hat G_\tau(E)^{1/2}|\phi_r\rangle, \] the corresponding Gaussian-broadened microcanonical ensemble is \[ \hat\rho_\tau(E)=\frac{\hat G_\tau(E)}{[\hat G_\tau(E)]} = \frac{\sum_n e^{-(E_n-E)^2\tau^2}|E_n\rangle\langle E_n|}{\sum_n e^{-(E_n-E)^2\tau^2}, } \] with effective energy window \[ \delta E=\frac{\sqrt\pi}{\tau}, \qquad \sigma_\tau(E)\sim \frac{1}{\sqrt2\,\tau}. \] Thermodynamic quantities follow from traces of \(\hat G_\tau(E)\) and \(\hat{\mathcal H}\hat G_\tau(E)\), including \[ S_\tau(E)=\ln([\hat G_\tau(E)]), \qquad \beta_\tau(E)=2\tau^2\big(\mathcal E_\tau(E)-E\big) \] [2207.01782]. A distinct but complementary interpretation arises for Gibbs states. The thermal state \[ \rho=\frac{1}{Z_\beta}e^{-\beta H} \] is itself a Hamiltonian-filtered state with exponential filter \(f(E)=e^{-\beta E}\). 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 \(\mu_l\) coupled through \(A_l\), the quantum Fisher information obeys \[ F_{ll}\le \beta^2(\Delta A_l)^2, \] and a sharper pair of bounds involves the Wigner–Yanase skew information, \[ F_{ll}\le 2.4\,c_2\,\beta^2\left((\Delta A_l)^2-\frac12\|[\sqrt\rho,A_l]\|_2^2\right), \] \[ F_{ll}\ge 0.8\,\beta^2\left((\Delta A_l)^2-\frac12\|[\sqrt\rho,A_l]\|_2^2\right). \] This makes precise the statement that commuting operators are metrologically favored because the filter does not hide sensitivity in incompatible eigenspaces [2401.10343]. 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 \[ P^{\cos}_{\delta_H}(H,E):=\cos^{2M^2}\!\left[\frac{T}{2}(H-E)\right], \] 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: \[ \left| \bra{\psi_{\delta_H}}\hat O\ket{\psi_{\delta_H}} - \mathrm{tr}(\rho_{\rm th}(\psi)\hat O) \right| = O(\sqrt{\delta_H}). \] At the same time, the Rényi entropies remain sharply nonuniform in \(\alpha\): for \(\alpha>1\) they scale only logarithmically in \(1/\delta\), while \(S_1\) is linear in \(1/\delta\) up to logarithmic corrections. A plausible implication is that local thermality and global entropic thermality separate sharply in filtered-state constructions [2607.06847]. ## 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 \(M\)-point effective Hamiltonian \[ H_M(\mathbf{k}) \] 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 \(M\) 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 [1603.04329]. The same paper also shows that boundary engineering can move the edge-state crossing between the 1D time-reversal-invariant momenta \(\Lambda=0\) and \(\Lambda=\pi\) without changing the bulk spectrum. For zigzag ribbons, a boundary mass term \(b\,\sigma_z\) with \(b_e<0\) produces a real-space band inversion and midgap edge states near \(k_\parallel=0\), while \(b_e>0\) shifts the edge states to \(|k_\parallel|\gtrsim 1\). 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 [1603.04329]. 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 \[ \bar H = \sum_{i