Gaussian-Filtered Quantum Phase Estimation

• The paper demonstrates that applying a Gaussian filter amplifies the target eigenstate’s overlap in quantum phase estimation.
• The methodology employs spectral filtering via block-encoding and polynomial approximations to balance filter sharpness with preparation cost.
• Empirical tests on Fermi-Hubbard models show significant overlap amplification, reducing the query complexity in high-precision regimes.

Gaussian-filtered quantum phase estimation denotes a family of phase-estimation constructions in which a Gaussian profile is used to localize spectral or phase information before, during, or instead of textbook quantum phase estimation. In the formulation most directly identified with the term, a Gaussian spectral filter

$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right]$$

is applied to an input state $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$ so that the desired eigencomponent is amplified prior to a final high-precision QPE stage (Lee et al., 5 Oct 2025). Related uses of Gaussian filtering include truncated Gaussian time-sampling for multiple-eigenvalue search (Ding et al., 2024), Gaussian spin states whose measurement update acts as a local Gaussian phase filter (Pezzè et al., 2020), and linearized continuous-time quantum filtering for small interferometric phase shifts (Gough, 2016).

1. Spectral-filter formulation

In filtered QPE, the starting point is a normalized Hamiltonian $H$ with spectrum in $[-1,1]$ and a reference state

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

The Gaussian filter is centered at $\mu$, ideally close to the target energy $E_0$, and has bandwidth set by $\sigma$:

$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$

Acting on the reference state gives the unnormalized filtered state

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$

and the normalized post-selected state is

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$0

The corresponding block-encoding requirement is a unitary $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$1 on $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$2 ancilla qubits such that

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$3

that is, a $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$4 block-encoding of $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$5 (Lee et al., 5 Oct 2025).

The central quantity is the overlap amplification achieved after filtering. The success probability of post-selection is

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$6

while the amplified ground-state overlap is

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$7

so that

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

If the Gaussian is peaked at $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$9 with $H$0 and satisfies $H$1 outside the main lobe $H$2, then

$H$3

hence

$H$4

This makes explicit how excited-state suppression improves the effective input to the subsequent QPE stage (Lee et al., 5 Oct 2025).

The same formalism also makes explicit the trade-off between overlap amplification and preparation cost. The filtered success probability obeys

$H$5

Accordingly, sharper filtering can improve the final overlap but may reduce post-selection probability, so the advantage of Gaussian-filtered QPE is governed by the balance between overlap amplification and the query cost of implementing the filter.

2. Implementations and the two-stage protocol

The Gaussian filter can be realized in several ways. In the trigonometric realization, one uses

$H$6

with $H$7, where $H$8 is the desired spectral resolution, approximately $H$9. This is implemented via generalized quantum signal processing, with depth $[-1,1]$0 calls to $[-1,1]$1. In the polynomial realization, the Gaussian is expanded in Chebyshev polynomials,

$[-1,1]$2

with $[-1,1]$3, and is implemented by QSVT or qubitization using $[-1,1]$4 calls to the qubitization oracle. A third route uses Krylov-subspace filters: choosing a basis $[-1,1]$5, forming

$[-1,1]$6

solving

$[-1,1]$7

and taking the minimizer $[-1,1]$8 to define

$[-1,1]$9

This yields an adaptive filter concentrated on the low-lying spectrum (Lee et al., 5 Oct 2025).

The filtered-QPE protocol is organized in three stages. Stage 1 is optional coarse estimation: one runs standard low-precision QPE of depth $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$0 and shots $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$1 to obtain $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$2 and $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$3 within error $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$4. Stage 2 prepares the filtered state by fixing

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$5

with

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$6

implementing a block-encoding of order

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$7

and repeating the post-selection procedure until success, requiring approximately $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$8 attempts. Stage 3 then runs standard QPE on $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle.$9 with depth $\mu$0 and number of shots

$\mu$1

The overall success probability satisfies

$\mu$2

(Lee et al., 5 Oct 2025).

A notable aspect of this construction is that the Gaussian filter is not itself the final estimator. It is a state-preparation primitive inserted before the standard phase-readout stage, and its value lies in improving the overlap properties of the state on which high-precision QPE is executed.

3. Complexity and empirical performance

For filtered QPE, the total expected cost in queries to $\mu$3 or to qubitization oracles is

$\mu$4

using the identity

$\mu$5

By contrast, standard QPE has cost

$\mu$6

The point of the Gaussian filter is therefore to replace the direct $\mu$7 dependence by a two-term cost in which the overlap penalty is paid primarily at the filter-preparation stage (Lee et al., 5 Oct 2025).

A parameter choice is obtained by balancing filter preparation and final QPE depth through $\mu$8. This gives

$\mu$9

where $E_0$0 is the Lambert $E_0$1-function. Under the conditions

$E_0$2

Theorem 1 shows

$E_0$3

The corresponding corollary gives

$E_0$4

so that in the high-precision regime $E_0$5, the $E_0$6 term dominates and the dependence on $E_0$7 is reduced (Lee et al., 5 Oct 2025).

Numerical experiments were carried out on 1D $E_0$8-site and 2D $E_0$9-site Fermi-Hubbard Hamiltonians with Bravyi-Kitaev tapering and spectrum normalized to $\sigma$0, using reference Néel product states. The reported parameter range includes initial overlaps

$\sigma$1

and spectral gaps

$\sigma$2

For Gaussian FQPE, both trigonometric and polynomial filters yielded overlap amplification factors

$\sigma$3

best cost-reduction factors of approximately $\sigma$4 for $\sigma$5, and still $\sigma$6 for $\sigma$7. The worst case over prior errors up to $\sigma$8 remained below unity in the high-precision regime. Krylov filters with dimension $\sigma$9 gave similar overlap amplification but very small success probabilities unless modified; with a small penalty $$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$0, the modified KSD procedure recovered

$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$1

and a cost reduction of about $$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$2 (Lee et al., 5 Oct 2025).

4. Related Gaussian-filtered phase-estimation paradigms

The broader literature uses Gaussian filtering in several technically distinct ways (Lee et al., 5 Oct 2025, Ding et al., 2024, Pezzè et al., 2020, Gough, 2016).

Method	Gaussian object	Salient feature
FQPE	$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$3	Filtered-state preparation followed by standard QPE
QMEGS	Truncated Gaussian time sampling and $$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$4	Hadamard-test filter-and-search for multiple eigenvalues
GSS cascade	Local Gaussian phase filter of width $$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$5	Cascade QPE with Gaussian spin states
Interferometric filtering	Linearized Gaussian quantum filter	Continuous nondemolition phase tracking

In Quantum Multiple Eigenvalue Gaussian filtered Search, the Gaussian enters through the truncated time-sampling distribution

$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$6

whose Fourier transform is the filter kernel

$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$7

The algorithm assumes access to controlled-$$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$8 and one ancilla qubit, uses Hadamard-test subroutines to obtain unbiased estimates of $$f(H)=\exp\!\left[-\frac{(H-\mu)^2}{2\sigma^2}\right].$$9, and then defines

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$0

On average,

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$1

so peaks of $$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$2 locate eigenvalues. The paper states that QMEGS is the first algorithm to simultaneously satisfy two properties: Heisenberg-limited scaling without any spectral gap assumption, and, under a positive energy gap and additional assumptions on the initial state, estimation of all dominant eigenvalues to $$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$3 accuracy with significantly reduced circuit depth compared to standard QPE. In the most favorable scenario, the maximal runtime can be as low as $$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$4 (Ding et al., 2024).

In the Gaussian-spin-state cascade, the Gaussian is built into the ancilla states themselves. A Gaussian spin state is

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$5

with squeezing parameter $$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$6. At step $$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$7, the outcome distribution can be interpreted through the filter function

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$8

which in the phase variable corresponds to a local Gaussian filter of width

$$|\tilde\phi_f\rangle=f(H)|\phi_0\rangle=\sum_i \gamma_i f(E_i)|E_i\rangle,$$9

The cascade yields

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$00

so that for $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$01, $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$02, and in the full-cascade limit $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$03, with total time $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$04 (Pezzè et al., 2020).

The interferometric work of Gough uses “Gaussian” in a different sense: a continuous-time quantum filter becomes exactly linear and Gaussian after linearization around a small phase offset. For homodyne detection and $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$05,

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$06

and the conditional state estimate and variance,

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$07

satisfy

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$08

with

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

for constant coherent amplitude. This is exactly the Kalman-Bucy form, and the mean-square estimation error tends to zero under continuous homodyne probing (Gough, 2016).

5. Conceptual distinctions and common misconceptions

The coexistence of these constructions suggests that “Gaussian-filtered quantum phase estimation” is not a single algorithmic template but a family of methods in which the Gaussian appears at different points of the estimation pipeline (Lee et al., 5 Oct 2025, Ding et al., 2024, Pezzè et al., 2020, Gough, 2016). In FQPE, the Gaussian acts directly on the Hamiltonian spectrum through $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$10. In QMEGS, the Gaussian is a time-domain sampling distribution whose Fourier transform yields a spectral search kernel. In the GSS cascade, the Gaussian is encoded in the spin-state amplitudes and induces a local Gaussian phase filter after measurement. In the interferometric setting, the Gaussian structure arises from a linearized conditional dynamics rather than from a discrete spectral filter.

A recurrent misconception is to equate overlap amplification with a free improvement in QPE complexity. The filtered-state formalism makes the opposite point explicit: overlap amplification and success probability are linked by

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

The advantage of FQPE comes from a favorable redistribution of cost between state preparation and final phase readout, not from eliminating the preparation burden altogether (Lee et al., 5 Oct 2025).

Another misconception is that Gaussian filtering necessarily requires a large QPE register or a quantum Fourier transform. QMEGS explicitly avoids both: it uses one ancilla, Hadamard-test circuits, and a classical grid search, with no Fourier transform or root-finding beyond the search over $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$12 (Ding et al., 2024). Conversely, filtered QPE in the sense of FQPE does use standard QPE as its final high-precision stage; the Gaussian filter is a preconditioning step, not a replacement for phase readout.

A further distinction concerns target structure. FQPE is organized around improving the overlap with a single target eigenstate, especially the ground state. QMEGS is formulated as a multiple-eigenvalue estimator for the set of dominant eigenvalues. The GSS cascade estimates an unknown phase $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$13 encoded by a controlled-$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$14 gate through successive residual-phase updates. The interferometric Kalman-Bucy filter tracks a small continuous phase fluctuation $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$15 under continuous nondemolition measurement. These are related estimation problems, but they are not interchangeable.

6. Limitations and open research directions

Gaussian-filtered QPE remains parameter sensitive. In the FQPE formulation, the filter center must satisfy $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$16 prior $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$17, and the width

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$18

must be chosen so that the Gaussian covers $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$19 while rejecting $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$20. The implementation error, including Trotter or imperfect qubitization error, must satisfy

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$21

to preserve the filter profile accurately. The reported advantage is therefore tied to the regime $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$22 and prior accuracy $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$23 (Lee et al., 5 Oct 2025).

QMEGS has a different set of open problems. The classical search over $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$24 grid points has cost $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$25, and multiscale or binary-search strategies are proposed as a possible route to polylogarithmic cost in $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$26. Adaptive tuning of $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$27, $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$28, $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$29, and $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$30 without prior spectral-gap or overlap information remains open. The analysis assumes ideal controlled-$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$31 access, and although the incorporation of realistic Hamiltonian-simulation errors or noise models is described as straightforward in many cases, detailed bounds are left for further study. The paper also identifies Gaussian-derivative, Heaviside-derivative, and Kaiser windows as possible alternative kernels (Ding et al., 2024).

The GSS cascade is robust against several noise sources, but its asymptotics change under strong depolarization. Full depolarization in the symmetric subspace imposes a minimum squeezing

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$32

As long as $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$33, the ideal Heisenberg scaling can be recovered; beyond that point, one can no longer increase $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$34 and instead repeats the best-squeezed step, yielding

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$35

(Pezzè et al., 2020).

The continuous-time interferometric filter is restricted to the linear Gaussian regime. Its exact Kalman-Bucy form requires $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$36 and neglects higher-order phase curvature. A nonzero coherent intensity $|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$37 is required; with vacuum input, the innovations vanish. Detector inefficiency, optical loss, and finite detector bandwidth are omitted, although the paper notes that such effects can be incorporated by standard quantum filtering extensions. Possible extensions include squeezed-state probes and adaptive feedback of the form

$|\phi_0\rangle=\sum_i \gamma_i |E_i\rangle$38

(Gough, 2016).

Taken together, these limitations clarify the present status of Gaussian filtering in phase estimation. The Gaussian profile is a versatile device for suppressing unwanted spectral or phase contributions, but its practical benefit depends on how precisely the filter can be centered, how cheaply it can be implemented, and whether the underlying estimation problem is discrete, multiple-eigenvalue, collective-spin, or continuous-time interferometric.