Filtered-State Preparation

• Filtered-state preparation is a quantum state-engineering method that uses selective maps to eliminate undesired components while amplifying target sectors.
• It leverages diverse strategies—spectral, local, optical, and measurement-based—to produce thermal, finite-energy, and ground-state approximations in complex systems.
• Different implementations, from coherent time evolution to QSP filters and adiabatic plus feedback techniques, balance overlap amplification with success probability.

Filtered-state preparation denotes a class of quantum state-engineering procedures in which an easily preparable input is transformed by a selective map that suppresses unwanted components while retaining, amplifying, or isolating a target sector. The selectivity may act in the energy basis, in a Fock basis, on measurement-conditioned branches, through local POVMs, or through structural redundancies in a compilation tree. In quantum simulation, this yields finite-energy, thermal, and ground-state approximations; in optics, it produces nonclassical states by deleting specific number components or by interference-based extraction; in communication and ensemble settings, it reshapes mixed-state structure while preserving operational constraints such as PPTness or high-temperature NMR conditions (Lee et al., 5 Oct 2025, Irmejs et al., 2023, Meher et al., 2019).

1. Formal notions and unifying abstractions

In the spectral formulation, filtered-state preparation starts from

$$\hat H=\sum_{i=0}^{d-1} E_i \ket{E_i}\bra{E_i},\qquad \ket{\phi_0}=\sum_i \gamma_i \ket{E_i},$$

and applies a bounded filter $f(\hat H)$ to obtain

$$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$

The associated success probability is

$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$

and the post-filter target overlap obeys

$$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$

This formulation makes filtered-state preparation a controlled trade-off between overlap amplification and success probability (Lee et al., 5 Oct 2025).

Other subfields use the same basic logic without a Hamiltonian spectral filter. In local filtration for entangled mixed states, the state update is

$$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$

with postselection on successful local outcomes (Mishra et al., 2020). In optical number-state filtering, a coherent state is modified by deleting selected Fock components; for a single removed number state $\ket{m}$,

$$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$

so the filter literally removes a basis component from the superposition (Meher et al., 2018).

This variety implies that the common structure is not a specific primitive such as QSVT, adiabatic evolution, or postselection. Rather, the recurring operation is selective elimination of degrees of freedom that are operationally irrelevant to the target.

2. Energy filtering, finite-energy states, and thermality

A prominent many-body instantiation filters a product state at a target energy using a Lorentzian operator. For a local Hamiltonian $H$ and a product state $\ket{\Psi}$ with mean energy $f(\hat H)$0, the filter is

$f(\hat H)$1

and the filtered state is

$f(\hat H)$2

The same work constructs a parent Hamiltonian

$f(\hat H)$3

whose unique ground state is the filtered product state, proves that this parent Hamiltonian is gapped with $f(\hat H)$4, and gives an adiabatic preparation bound $f(\hat H)$5. For local Hamiltonians and product states, the filtered energy variance scales as $f(\hat H)$6, and the practical circuit-depth estimate is $f(\hat H)$7. The favorable variance scaling is explicitly tied to filtering at the product-state mean energy $f(\hat H)$8; when $f(\hat H)$9, the method is much less effective (Irmejs et al., 2023).

A later analysis asks how narrow a filter must be for the resulting state to be thermal. In the Hamiltonian setting, the canonical cosine filter is

$$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$0

and its truncated form is a weighted sum of time evolutions. In the Floquet setting, the corresponding filter is

$$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$1

so filtered states become superpositions of time-evolved copies of the initial state. Under Floquet ETH, the deviation between filtered and thermal local expectations is bounded by $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$2, and the Hamiltonian analogue scales as $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$3. The same analysis reproduces distinct Rényi-$$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$4 entropy scalings: for $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$5, the entropy is only logarithmic in $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$6, whereas for $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$7 it can scale linearly in $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$8 for the truncated filter and as $$\ket{\phi_f}=\frac{f(\hat H)\ket{\phi_0}}{\|f(\hat H)\ket{\phi_0}\|}.$$9 for the exact cosine filter (Yang et al., 7 Jul 2026).

These results rule out a common oversimplification: narrow filtering does not by itself guarantee full thermal behavior. The thermodynamic interpretation depends on filter width, on whether the Floquet or Hamiltonian picture is used, and on which entropic diagnostic is considered.

3. Ground-state filtering and algorithmic realizations

A unified ground-state framework treats filtered-state preparation as application of a bounded function

$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$0

with either polynomial bases $$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$1 or trigonometric bases $$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$2. The framework makes explicit the central trade-off

$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$3

so overlap amplification generally lowers success probability. Within this setting, Gaussian filters and Krylov-subspace-based filters are developed, and a filtered variant of quantum phase estimation, FQPE, is constructed. Numerical experiments on Fermi–Hubbard models report overlap amplification exceeding a factor of one hundred and total runtime reduction by more than two orders of magnitude in the high-precision regime (Lee et al., 5 Oct 2025).

A distinct implementation route uses a Power-Cosine QSP filter built directly from coherent time evolution. One ancilla qubit, Hadamards, and controlled-$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$4 produce the non-unitary step

$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$5

and repeating it $$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$6 times yields

$$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$7

On an eigenstate $$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$8, the amplitude envelope is $$p_f=\bra{\phi_0}f(\hat H)^\dagger f(\hat H)\ket{\phi_0} =\sum_i |\gamma_i|^2 |f(E_i)|^2,$$9, so excited components are exponentially suppressed. The method uses a single ancilla together with mid-circuit measurement and reset, has circuit-depth scaling $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$0, and is positioned as implementationally simpler than block-encoding-based QSVT at the cost of quadratic gap dependence (Jo, 23 Feb 2026).

On noisy digital hardware, filter-enhanced adiabatic quantum computing combines digitized adiabatic preparation with a QETU/QSVT-type eigenstate filter. The adiabatic stage biases the input toward the low-energy sector, allowing use of the lowest non-trivial QETU degree, $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$1, and thereby reducing filter depth. In experiments on the Quantinuum H1-1 system, the filter block contributed a fixed cost of 88 native two-qubit gates, and AQC+F consistently outperformed plain AQC in the 8-qubit Heisenberg model. In statevector simulations of the $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$2D TFIM, AQC+F yielded infidelities at least an order of magnitude smaller than AQC at comparable two-qubit gate count, except at the very lowest gate counts (Karacan et al., 26 Mar 2025).

Across these approaches, the implementation models differ sharply—qubitization/QSVT, coherent time evolution with MCMR, and adiabatic preconditioning followed by postselected filtering—but the objective is the same: reduce residual excited-state weight more cheaply than would be possible by high-precision preparation from an unstructured input.

4. Dissipative, measurement-based, and feed-forward filtering

Filtered-state preparation need not be closed-system or purely unitary. In engineered dissipative adiabatic preparation, a filtered reservoir induces predominantly downward transitions in the instantaneous eigenbasis while leaving the instantaneous ground state dark: $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$3 Near a minimal avoided crossing, the effective model

$$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$4

suppresses the excited-state population to

$$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$5

In the strong-relaxation regime $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$6, the runtime scaling improves from the closed-system $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$7 law to $$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$8. At finite temperature, upward transitions introduce the thermal floor

$$|\gamma_{f0}|^2=\frac{1}{1+R_f}, \qquad R_f:=\sum_{i>0}\left|\frac{\gamma_i f(E_i)}{\gamma_0 f(E_0)}\right|^2.$$9

so speedup persists only when the allowed error exceeds that floor (Zhou et al., 4 Jun 2026).

Measurement and classical feedback provide another non-unitary route. In the self-learning protocol for AKLT preparation, a parameterized unitary is followed by projective ancilla measurement and conditional feedback $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$0. The method treats measurement as a filtering step that branches the dynamics and then learns feedback that maps distinct branches back to the same target state. The work identifies “measurement-induced local minima,” in which the measurement distribution collapses toward a delta function and the filtering mechanism becomes ineffective. A mitigation schedule that starts with update frequency 100 and linearly decreases it to 5 over the first $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$1 epochs, together with ancilla regularization, improves training; with a bidirectional RNN, the reported average infidelity is around $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$2 after $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$3 epochs (Puente et al., 2024).

The LAQCC model uses shallow quantum layers, intermediate measurements, and $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$4 classical feed-forward to realize filtering on marked subspaces. If a constant-depth unitary flags a known constant-fraction subset $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$5, the model can prepare

$$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$6

by exact amplitude amplification. This mechanism underlies preparation of uniform superpositions over arbitrary $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$7, Dicke states for $$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$8, and many-body scar states obtained by imposing a Fibonacci no-adjacent-$$\rho \rightarrow \rho'=\frac{L\rho L^\dagger}{\operatorname{Tr}(L\rho L^\dagger)}, \qquad L=L_A\otimes L_B,$$9 constraint (Buhrman et al., 2023).

These works show that “filtering” can mean branch selection and error erasure rather than spectral projection. The target state is reached by continuously removing leaked population, by correcting measurement-conditioned branches, or by amplifying a flagged combinatorial sector.

5. Optical and bosonic filtered states

In quantum optics, filtered-state preparation often means explicit removal of chosen Fock components. Number-state filtered coherent states are obtained by deleting $\ket{m}$0 from the coherent-state expansion, which sets the corresponding detection probability to zero and alters the photon statistics away from Poissonian. The strongest deviation from the coherent state occurs near

$\ket{m}$1

where Wigner negativity and entanglement potential are also largest. The vacuum-filtered case $\ket{m}$2 is especially notable: it is almost like photon addition in suitable limits, but is reported to be more resilient against dissipation than photon addition and to perform better than the photon-added coherent state in a two-way quantum key distribution protocol (Meher et al., 2018).

For interferometry, single- and multi-number-state filtered coherent states improve phase sensitivity through enhanced QFI. The analysis of Mach–Zehnder inputs shows that suitable filtering can beat the standard quantum limit, and for single-number-state filtering the best sensitivity occurs when the removed number satisfies $\ket{m}$3. Filtering more than one suitable Fock component decreases the figure of merit $\ket{m}$4, corresponding to better performance relative to the Heisenberg limit. In the limiting even and odd coherent-state cases, the asymptotic phase sensitivity equals that of squeezed vacuum. The same study emphasizes that improvement in phase sensitivity is not in direct proportion to the nonclassicality measured by Wigner negativity (Meher et al., 2019).

A different optical route uses a nonlinear Mach–Zehnder interferometer with a Kerr arm. In the weak-input regime, destructive interference between photon-number-dependent paths can cancel the leading two-photon amplitude, giving a highly antibunched output with $\ket{m}$5. The simple two-unit device acts as a nonlinear filter for weak coherent inputs, while a cascaded NMZI generalizes this to deterministic extraction of Fock states and arbitrary superpositions from strong coherent states. For a target $\ket{m}$6, a chain with $\ket{m}$7 units yields fidelities and purities exceeding $\ket{m}$8 once $\ket{m}$9, and in the large-$$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$0, large-$$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$1 limit the fidelity approaches

$$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$2

The same work explicitly distinguishes filtration, which blocks undesired components, from extraction, which actively pulls out the target state from a classical resource (Zou et al., 2015).

The optical literature therefore uses “filtered-state preparation” in two related but distinct senses: deletion of basis amplitudes from a prescribed superposition, and interferometric rejection of unwanted photon-number paths.

6. Local, ensemble, and structured filtration

In high-temperature NMR, filtered-state preparation appears as preparation of pseudo-pure states by averaging away unwanted populations while preserving the ground-state population. Controlled-transfer gates implement partial averaging of diagonal populations, and crush gradients or temporal averaging remove off-diagonal terms. For the two-qubit homonuclear case, a repeated $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$3 network converges asymptotically to the pseudo-pure state with error

$$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$4

after $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$5 repetitions. The method is presented as analytically tractable, scalable, and robust when the initial state is diagonal (Kawamura et al., 2010).

In quantum communication, local filtration is used to increase the one-way distillable key rate of PPT bound entangled states while preserving PPTness. Because the filter is local, it cannot change PPT to NPT, but it can reshape the matrix blocks relevant to privacy squeezing and the Devetak–Winter lower bound. For three state families, numerical optimization over diagonal local filters yields positive effective key rate $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$6 after filtration, even when the unfiltered one-way key rate is negative over much of the parameter range. The filtered states remain bound entangled but become key-distillable in the sense analyzed there (Mishra et al., 2020).

A compiler-oriented usage appears in structured quantum state preparation for real-valued states with separable or partially disentangled structure. There, separability manifests as repetition patterns in the uniformly controlled-gate multiplexer tree. Detecting a repetition at power-of-two distance identifies redundant controls; removing them halves the gate count in the affected multiplexer, and if all controls are removed the multiplexer depth collapses to one. The simplification algorithm has worst-case cost $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$7 for a multiplexer of size $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$8, and for states with an entangled component of size $$|\psi(\alpha,m)\rangle = \frac{1}{N_m}\left(I-|m\rangle\langle m|\right)|\alpha\rangle = \frac{1}{N_m}\left(|\alpha\rangle - C_m|m\rangle\right),$$9, the best-case circuit depth becomes $H$0 rather than scaling with the total qubit count. For states with more than 12 qubits, the reported compilation/transpilation time is at least an order of magnitude better than previous methods (Carvalho et al., 2024).

Taken together, these directions broaden the meaning of filtered-state preparation beyond spectral filtering. The same logic governs population equalization in ensemble processors, resource enhancement under LOCC constraints, and elimination of redundant branches in multiplexer-based state compilers. A plausible implication is that the most stable cross-domain definition of the field is selective removal of irrelevant structure, with the target state emerging as the normalized survivor of that removal.