Quantum Folded Spectrum Method

• Quantum Folded Spectrum Method (QFSM) is a quantum excited-state preparation technique that uses a polynomial spectral filter to target eigenvalues near a specified energy shift.
• It leverages generalized quantum signal processing (GQSP) and block-encoding to synthesize an exact Chebyshev polynomial filter without variational optimization.
• The approach achieves chemical accuracy in molecular benchmarks by amplifying eigencomponents near the target energy while suppressing others, though its success is sensitive to spectral crowding.

Quantum Folded Spectrum Method (QFSM) is a quantum excited-state preparation method that implements the classical folded-spectrum idea within generalized quantum signal processing (GQSP). For a target energy estimate $\epsilon_m$, it replaces direct minimization of an excited-state objective by a polynomial spectral filter built from the shifted-square operator, so that eigencomponents with energies nearest $\epsilon_m$ are preferentially retained while others are suppressed. In the GQSP formulation, QFSM is realized from a block-encoding of a normalized Hamiltonian, synthesized as an exact finite-degree polynomial in the qubitized signal operator, and used either as a repeated degree-2 filter or as a single-shot degree-$2n$ filter. Within the unified framework of quantum power methods, it is the component specifically aimed at obtaining excited states without variational optimization, Suzuki–Trotter decomposition, or long-time phase estimation (Khinevich et al., 15 Jul 2025).

1. Spectral definition and folded transformation

QFSM starts from a Hermitian Hamiltonian $H$ on $m$ qubits with spectral decomposition

$H|\psi_i\rangle = E_i|\psi_i\rangle,$

where the eigenvectors form an orthonormal basis. To make the spectrum compatible with Chebyshev-polynomial synthesis, the Hamiltonian is rescaled by the LCU norm

$$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$

and the normalized operator

$\mathfrak h := H/\alpha$

has spectrum in $[-1,1]$.

The folded-spectrum principle targets eigenvalues near a chosen shift. Classically one considers

$f(H;\sigma) := (H-\sigma I)^2,$

so eigenvalues close to $\epsilon_m$0 are mapped to small values while distant eigenvalues are mapped to larger values. QFSM adopts the equivalent shifted-square logic but uses a power filter

$\epsilon_m$1

with $\epsilon_m$2 near the desired excited energy and $\epsilon_m$3 chosen so that the folded spectrum is bounded. In normalized variables,

$\epsilon_m$4

The resulting spectral mapping is

$\epsilon_m$5

This acts as a band-pass filter centered at $\epsilon_m$6: if $\epsilon_m$7 is close to $\epsilon_m$8, then $\epsilon_m$9 is small and the gain remains near $2n$0; otherwise the component is suppressed. In this sense, QFSM is a folded-spectrum power iteration in which the target excited eigenvector becomes the dominant component under repeated or synthesized polynomial application (Khinevich et al., 15 Jul 2025).

2. Block-encoding, qubitization, and GQSP realization

The GQSP implementation assumes a standard LCU-based block-encoding. There exists a unitary $2n$1 on ancilla-plus-system qubits such that

$2n$2

In the qubitized rotation form,

$2n$3

so powers of $2n$4 embed the Chebyshev polynomials $2n$5 in the top-left block.

The controlled signal operator used by GQSP is

$2n$6

The ancilla structure is correspondingly modest: one signal qubit for GQSP, $2n$7 ancillas for LCU PREPARE/SELECT where $2n$8 is the number of Pauli strings in the LCU expansion, and optionally one additional ancilla for capitalization used for numerical stabilization of angle-finding. In the worst case $2n$9, so the LCU ancilla count scales as $H$0.

GQSP synthesizes the target polynomial through alternating applications of $H$1 and single-qubit rotations

$H$2

The sequence

$H$3

yields a $H$4 block with top-left block $H$5 and bottom-left block $H$6, provided $H$7 and $H$8 satisfy $H$9 on the unit circle. A key feature of GQSP is that it can synthesize complex polynomials directly rather than only the parity-restricted real polynomials of standard QSP; for QFSM the filter itself is real-valued, but the same machinery is used throughout the unified framework (Khinevich et al., 15 Jul 2025).

3. Polynomial construction and algorithmic workflow

For QFSM the implemented filter is

$m$0

which has degree

$m$1

in $m$2. By binomial expansion,

$m$3

so the folded filter is exactly a finite Chebyshev polynomial and can be synthesized by GQSP without truncation. The complementary polynomial $m$4 and the rotation angles $m$5 are obtained with Prony’s method, optionally stabilized by capitalization. For folded-spectrum filters, the remaining implementation error is numerical phase precision; in simulation the filter is implemented to machine precision.

The reported excited-state preparation workflow consists of the following steps:

1. Input: a Pauli-expanded Hamiltonian $m$6, an initial state $m$7, a target energy estimate $m$8, a filter constant $m$9, and an integer $H|\psi_i\rangle = E_i|\psi_i\rangle,$0.
2. Scaling and block-encoding: compute $H|\psi_i\rangle = E_i|\psi_i\rangle,$1, define $H|\psi_i\rangle = E_i|\psi_i\rangle,$2, realize $H|\psi_i\rangle = E_i|\psi_i\rangle,$3, and set $H|\psi_i\rangle = E_i|\psi_i\rangle,$4.
3. Filter definition: form $H|\psi_i\rangle = E_i|\psi_i\rangle,$5 and expand it in the Chebyshev basis.
4. GQSP synthesis: scale to ensure $H|\psi_i\rangle = E_i|\psi_i\rangle,$6 on $H|\psi_i\rangle = E_i|\psi_i\rangle,$7, construct $H|\psi_i\rangle = E_i|\psi_i\rangle,$8, carve the rotation angles, and build the alternating $H|\psi_i\rangle = E_i|\psi_i\rangle,$9-rotation circuit.
5. Filter application: prepare the signal qubit, LCU ancillas, and $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$0; run the GQSP circuit; postselect on the ancilla state; and normalize the output. Amplitude amplification may be added to increase success probability.
6. Iteration strategy: either repeatedly apply the degree-2 folded operator $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$1, or synthesize the degree-$$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$2 filter once.
7. Validation: estimate $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$3 of the prepared state and compare it to $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$4 or to a reference such as CASCI.

If

$$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$5

then after one application,

$$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$6

The postselection success probability per application is approximately

$$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$7

Convergence is therefore driven by spectral selectivity rather than by classical parameter optimization; low success probability remains a generic drawback of non-unitary state-preparation methods (Khinevich et al., 15 Jul 2025).

4. Query complexity, ancillas, and convergence behavior

In the reported framework, the number of queries to the block-encoded unitary equals the polynomial degree. A synthesized application of the QFSM filter with degree $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$8 uses $$\alpha := \|H\|_{\ell_1} = \sum_k |\alpha_k|,\qquad H=\sum_k \alpha_k P_k,$$9 controlled applications of $\mathfrak h := H/\alpha$0, interleaved with single-qubit rotations and $\mathfrak h := H/\alpha$1 gates. If capitalization is used to stabilize angle-finding, the circuit is effectively doubled and one extra ancilla is added. When QFSM is realized by repeated application of the degree-2 folded operator, the total query count scales linearly with the number of iterations; when realized as a single-shot degree-$\mathfrak h := H/\alpha$2 filter, the queries scale as $\mathfrak h := H/\alpha$3.

The ancilla and depth profile is correspondingly structured. The signal register requires one qubit. The LCU implementation requires $\mathfrak h := H/\alpha$4 ancillas. Capitalization adds one optional ancilla. Circuit depth is dominated by the $\mathfrak h := H/\alpha$5 controlled-$\mathfrak h := H/\alpha$6 calls together with the PREPARE/SELECT depth of the LCU. Because the method uses block-encoding and qubitization rather than product-form simulation, no Suzuki–Trotter decomposition is used; there is no Trotter error, and gate counts are predictable from the degree and LCU cost.

Selectivity and robustness are controlled primarily by $\mathfrak h := H/\alpha$7, the shift $\mathfrak h := H/\alpha$8, and the parameter $\mathfrak h := H/\alpha$9. Higher degree yields narrower selectivity and faster suppression of non-target components. The paper recommends choosing $[-1,1]$0 so that $[-1,1]$1 lies in $[-1,1]$2 on the relevant spectral region. Large $[-1,1]$3 suppresses out-of-band components more aggressively but can reduce success probability. The normalization $[-1,1]$4 changes the mapped squared distances and therefore the effective passband.

The method requires a reasonably accurate energy estimate of the target excited state. Poor $[-1,1]$5 and poor initial overlap hinder convergence, particularly in near-degenerate regions. Closely spaced energies around $[-1,1]$6 can attract the iteration toward multiple targets simultaneously, reducing selectivity and slowing convergence. The reported practical recommendation is to use symmetry-adapted initial states with large target overlap and to obtain $[-1,1]$7 from a reasonable classical estimate such as small active-space CASCI. Numerical errors associated with Prony’s method and capitalization are small in the reported simulations, but classical power-method pathologies still appear: in $[-1,1]$8 singlet runs, loss of orthogonality and divergence after about 25 iterations were observed due to floating-point effects (Khinevich et al., 15 Jul 2025).

5. Chemical benchmarks and observed performance

The reported numerical study evaluates QFSM on molecular Hamiltonians and treats it as an excited-state preparation routine rather than as a variational solver. The principal benchmarks are $[-1,1]$9 in a cc-pVDZ $f(H;\sigma) := (H-\sigma I)^2,$0 active space and ethylene $f(H;\sigma) := (H-\sigma I)^2,$1 in a cc-pVDZ $f(H;\sigma) := (H-\sigma I)^2,$2-only $f(H;\sigma) := (H-\sigma I)^2,$3 space.

System	Setup	Reported outcome
$f(H;\sigma) := (H-\sigma I)^2,$4	Symmetry-adapted guesses $f(H;\sigma) := (H-\sigma I)^2,$5, $f(H;\sigma) := (H-\sigma I)^2,$6; CASCI shifts	$f(H;\sigma) := (H-\sigma I)^2,$7 and $f(H;\sigma) := (H-\sigma I)^2,$8 reached chemical accuracy across bond lengths
$f(H;\sigma) := (H-\sigma I)^2,$9	Same model	$\epsilon_m$00 and $\epsilon_m$01 were slower and did not reach chemical accuracy at $\epsilon_m$02 and $\epsilon_m$03 within 50 iterations
Ethylene	Four low-lying states along C=C torsion	At $\epsilon_m$04, all except $\epsilon_m$05 reached chemical accuracy within 50 iterations
Ethylene	Same model	At $\epsilon_m$06, none of the states reached chemical accuracy within 50 iterations; up to $\epsilon_m$07, both $\epsilon_m$08 states did

These results identify two empirical features. First, QFSM can reach excited states to chemical accuracy within tens of iterations when the shift and initial state are well chosen. Second, the dominant failure mode is spectral crowding near the target energy: near-degenerate states around the shift weaken selectivity, as seen in the $\epsilon_m$09 $\epsilon_m$10 states at short bond lengths and in the $\epsilon_m$11 torsion point of ethylene. The reported interpretation is therefore not that QFSM universally resolves difficult excited-state manifolds, but that it can do so without variational optimization when physically informed initial states and reliable energy shifts are available (Khinevich et al., 15 Jul 2025).

6. Relation to variational folded-spectrum methods and broader folded-spectrum practice

QFSM should be distinguished from folded-spectrum VQE. In FS-VQE, the cost function is

$\epsilon_m$12

and the algorithm directly minimizes the expectation value of the squared shifted Hamiltonian. That formulation preserves the eigenvectors of $\epsilon_m$13 and targets the state whose eigenvalue is closest to $\epsilon_m$14, but it incurs explicit $\epsilon_m$15 measurement overhead. In the reported FS-VQE study, $\epsilon_m$16 on 4 qubits had 15 Pauli terms grouped into 5 QWC groups for $\epsilon_m$17, while $\epsilon_m$18 had 24 terms grouped into 9 QWC groups; for LiH on 12 qubits in STO-3G, $\epsilon_m$19 had 631 terms and 136 QWC groups, whereas $\epsilon_m$20 had 25,542 terms and 2,216 QWC groups (Tazi et al., 2023). By contrast, the GQSP QFSM formulation emphasizes that it avoids explicit construction and measurement of $\epsilon_m$21, avoids classical optimization, and avoids variational pathologies such as barren plateaus, while introducing instead the non-unitary overhead of postselection and the need for good initial overlap and accurate shift placement (Khinevich et al., 15 Jul 2025).

The same folded-spectrum idea also appears in other quantum settings. In the shortest vector problem, the target is a nontrivial first excited state of a problem Hamiltonian, and a folded-spectrum operator

$\epsilon_m$22

is used so that the first excited state of the original Hamiltonian becomes the ground state of the folded Hamiltonian. That work combines folded spectrum with one-hot encoding and quantum imaginary-time or variational updates, indicating that the shift-and-square construction is not specific to molecular excited states (Mizuno et al., 2024).

Outside quantum algorithms, folded-spectrum transformations remain standard for interior-eigenvalue extraction in large-scale computational physics. Partitioned folded-spectrum eigensolvers for Kohn–Sham problems and matrix-free inverse-Krylov plus folded-spectrum schemes for polarization propagators both use shift-and-square mapping to convert interior states into extremal ones while avoiding full diagonalization (Briggs et al., 2015, Martin et al., 10 Dec 2025). This suggests that QFSM is best understood as a quantum polynomial-filter realization of a long-standing interior-spectrum strategy: it retains the folded-spectrum objective, but implements it through block-encoding, Chebyshev synthesis, and GQSP rather than through classical diagonalization or variational expectation minimization.