Quantum-Classical Hybrid Fourier Readout

• Quantum-Classical Hybrid Fourier Readout is a method that encodes continuous functions into low-order Fourier modes, reducing the need for extensive measurements.
• It uses QFT and fractional transforms to compress spectral information, achieving near-optimal readout efficiency in sensing and PDE solvers.
• Applications include quantum sensing with NV centers, integrated photonic systems, and hybrid quantum machine learning frameworks for efficient data recovery.

Quantum-Classical Hybrid Fourier Space Readout (FSR) denotes a family of readout protocols in which information encoded in a quantum state, a sensing register, or a Fourier-domain latent representation is transferred into Fourier or fractional-Fourier coordinates on quantum hardware and then recovered through a hybrid quantum-classical pipeline. In the CAE and PDE setting, the canonical problem is the recovery of a real-valued function $f$ from an amplitude-encoded state $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$, where direct real-space sampling requires $\Omega(N)$ measurements; FSR instead estimates a truncated set of Fourier coefficients and reconstructs $f$ classically at arbitrary target points (Huang et al., 28 Jul 2025). In QFT-based sensing, the same principle appears as direct digitization of accumulated phase into a finite register via $\mathrm{QFT}^\dagger$, yielding a single-shot Fourier-space word rather than a long time-domain trace (Vorobyov et al., 2020). Related realizations extend the concept to discrete fractional Fourier transforms in integrated photonics and to partitioned Fourier layers in scientific quantum machine learning (Weimann et al., 2015, Marcandelli et al., 11 Jul 2025, Papierz et al., 6 Apr 2026).

1. Spectral formulation and compression principle

The central FSR formulation begins with amplitude encoding of a discretized function on a uniform grid. For $N=2^n$ points and domain length $L$,

$$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$

so that

$$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$

The readout bottleneck is immediate: conventional real-space readout needs $O(N)$ repetitions to estimate $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$0 for each $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$1, and full quantum tomography is even more costly (Huang et al., 28 Jul 2025).

FSR replaces pointwise sampling by spectral compression. After applying $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$2, the quantum Fourier coefficients are

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$3

and, for sufficiently smooth $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$4, one keeps only the first $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$5 modes, with

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$6

when the continuous Fourier series coefficients decay like $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$7. Classical reconstruction is then performed from the truncated series, for example

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$8

Because continuous functions have rapidly decaying Fourier spectra, the hybrid strategy compresses the information into the first $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$9 Fourier coefficients, with $\Omega(N)$0 for smooth $\Omega(N)$1, and the readout cost becomes independent of the full grid size $\Omega(N)$2 (Huang et al., 28 Jul 2025).

The multidimensional formulation used in CFD and quantum PDE solvers is analogous. One writes $\Omega(N)$3, encodes the grid values into amplitudes, and reconstructs with a truncated Fourier series

$\Omega(N)$4

where $\Omega(N)$5. The reconstruction error splits into truncation and sampling terms, so the efficacy of FSR is tied to spectral decay rather than to any generic property of amplitude encoding (Huang et al., 25 Nov 2025).

2. Hybrid algorithmic pipeline and circuit primitives

The hybrid workflow in the function-recovery setting has three stages. First, a quantum state-preparation routine $\Omega(N)$6 prepares $\Omega(N)$7, typically with depth $\Omega(N)$8. Second, a Fourier-component readout stage applies $\Omega(N)$9, isolates the low-$f$0 sector, and measures it repeatedly. Third, classical post-processing reconstructs $f$1 at $f$2 target points by evaluating the truncated Fourier series in $f$3 arithmetic operations, or by an FFT of size $f$4 when $f$5 (Huang et al., 25 Nov 2025).

A concrete circuit construction uses an even extension to enforce real Fourier amplitudes. Starting from $f$6, one applies a Hadamard on an ancilla qubit, a controlled-Not fanout from the ancilla to the lower $f$7 qubits, and a controlled incrementer $f$8 on the lower $f$9 lines. After this extension, $\mathrm{QFT}^\dagger$0 is applied to the $\mathrm{QFT}^\dagger$1 data qubits, and one post-selects on the $\mathrm{QFT}^\dagger$2 most-significant qubits being $\mathrm{QFT}^\dagger$3 to isolate the first $\mathrm{QFT}^\dagger$4 coefficients. Absolute values are obtained by $\mathrm{QFT}^\dagger$5-basis measurement and histogramming of the lower $\mathrm{QFT}^\dagger$6 qubits, while signs are extracted by inserting a shift of the Fourier amplitudes by $\mathrm{QFT}^\dagger$7 via an LCU circuit and comparing $\mathrm{QFT}^\dagger$8 against $\mathrm{QFT}^\dagger$9 (Huang et al., 28 Jul 2025).

The resource analysis is correspondingly hybrid. Oracles $N=2^n$0, QFT, incrementer, and LCU all cost $N=2^n$1 gates, with ancilla overhead of 2 qubits. Sampling error on each of the $N=2^n$2 coefficients scales like $N=2^n$3, and the overall error is bounded by $N=2^n$4 when

$N=2^n$5

so no factor of $N=2^n$6 appears in the sample complexity (Huang et al., 28 Jul 2025). In the more general comparison framework, circuit depth per shot is $N=2^n$7, the number of shots is $N=2^n$8, and the classical cost is $N=2^n$9, with $L$0 (Huang et al., 25 Nov 2025).

3. QFT digitization in NV-center sensing

A distinct but closely related realization of FSR is the QFT-based sensing protocol implemented in a hybrid register built from a single nitrogen-vacancy center in diamond. The sensor qubit is the electron-spin $L$1 of a single NV center, coupled by hyperfine interaction to three nearby nuclear spins: one $L$2 with $L$3 and two weakly coupled $L$4 spins with $L$5. Together these form a 12-dimensional hybrid register, used both as long-lived quantum memory and as the processor for the Quantum Fourier Transform (Vorobyov et al., 2020).

The three nuclear spins are initialized into the equal-superposition state $L$6, using Hadamard gates on the qubits and a Chrestenson gate on the qutrit. A time-varying signal, either classical RF or nuclear-spin free precession, is converted into an accumulated phase $L$7 on the NV electron by dynamical decoupling or Ramsey-type sequences, repeated $L$8 times for bit $L$9. Each phase is then mapped from electron to nucleus by a controlled-NOT sequence so that, after three acquisitions, the register state is $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$0. The inverse transform $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$1 then converts the phase-encoded superposition into a computational-basis population $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$2, where $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$3 is the 3-bit digital representation of $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$4, and a single-shot readout of all three nuclear spins yields the Fourier-space digit (Vorobyov et al., 2020).

The readout statistics are sharply structured. The probability of obtaining digit $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$5 is

$$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$6

The Fisher information obeys

$$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$7

and in the long total-time limit $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$8 this yields $$x_j=\frac{jL}{N},\qquad \psi_j=\frac{f(x_j)}{A_N},\qquad A_N=\Bigl(\sum_{j=0}^{N-1}|f(x_j)|^2\Bigr)^{1/2},$$9, i.e. near-Heisenberg scaling in time. Experimentally, Fisher-information analysis and direct variance measurements confirm near-Heisenberg scaling in $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$0 up to $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$1, limited by nuclear $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$2, while imperfect gates reduce the Fourier-space peak amplitudes by a factor $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$3 per nuclear spin bit (Vorobyov et al., 2020).

The same platform also demonstrates demultiplexing of two weakly coupled target nuclear spins with hyperfine couplings $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$4 and $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$5. By choosing $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$6 such that $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$7, the stronger-coupled spin is mapped exclusively onto the least-significant bit and the weaker spin onto the most-significant bit, effectuating spatial separation in the hybrid Fourier domain. Subsequent correlation spectroscopy then yields two independent oscillations on each nuclear-spin readout, and classical FFT reveals resonances at $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$8 kHz and $$|\psi\rangle=\frac{1}{A_N}\sum_{j=0}^{N-1} f(x_j)\,|j\rangle.$$9 kHz. In this implementation, single-electron-spin Ramsey is limited to $O(N)$0, whereas the 3-bit FSR extends the range to $O(N)$1 with 8 distinct outputs, giving an $O(N)$2 improvement in dynamic range, while achieving the same spectral resolution as classical FFT of Ramsey fringes but in a single shot without the long-time correlation step of conventional NMR-CSI (Vorobyov et al., 2020).

4. Complexity, benchmarks, and CFD deployment

Within the broader readout literature for quantum PDE solvers, FSR is positioned against real-space sampling and approximate or QAE-based alternatives. The comparison study states that the Fourier space readout (FSR) and the proposed approximate real space readout (ARSR) methods are currently the most efficient and practical ones for the purpose of reconstructing continuous real-valued functions, whereas the quantum amplitude estimation based methods, especially in the Fourier space, are favorable for mid-term/far-term quantum devices (Huang et al., 25 Nov 2025).

The asymptotic differences are explicit. RSR costs $O(N)$3 shots. ARSR costs $O(N)$4 shots but requires $O(N)$5. FSR costs $O(N)$6 shots, with depth $O(N)$7, and preserves a readout complexity independent of the full grid size $O(N)$8. FSQAE costs $O(N)$9 oracle calls and is optimal in the large-$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$00, small-$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$01 regime if $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$02, though the same study characterizes it as impractical pre-2025 (Huang et al., 25 Nov 2025).

The numerical behavior is likewise domain-dependent. For 1D and 2D smooth functions, FSR benefits from spectral decay, whereas RSR exhibits the usual $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$03 penalty in sampling error. In the 2025 function-recovery formulation, 1D tests on $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$04 and a sum of Gaussians on $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$05 points show RMS error $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$06 with $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$07 modes and $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$08 shots, while RSR error scales as $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$09 (Huang et al., 28 Jul 2025). In the comparison study, 2D Gaussians and sinusoidal data exhibit the predicted FSR error tradeoff $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$10, with the smooth periodic regime recovering $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$11 behavior (Huang et al., 25 Nov 2025).

A full hybrid deployment appears in the 2D Burgers’ equation experiment based on time-stepwise readout. The solver uses $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$12 on $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$13 with $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$14, 25 steps, $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$15, and $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$16. At each step one prepares $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$17, applies a PITE step $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$18, post-selects a success flag, reads out Fourier coefficients with $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$19, reconstructs $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$20, and differentiates it classically to build the next operator. The reported relative $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$21 error at $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$22 is $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$23 with FSR versus $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$24 with RSR, the success probability per PITE step is $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$25, and the total shots are $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$26, all independent of grid size beyond the $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$27 depth of QFT (Huang et al., 25 Nov 2025).

5. Fractional-Fourier photonics and Fourier-neural-operator variants

The fractional-Fourier variant realizes FSR in integrated photonics rather than on a gate-based qubit register. Here the discrete fractional Fourier transform of order $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$28 is defined on $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$29 sites by

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$30

with kernel

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$31

where the $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$32 are eigenvectors of the $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$33 $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$34-matrix. At $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$35 one recovers the discrete Fourier transform, and at $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$36 the identity. The implementation uses a straight-line array of single-mode waveguides in fused silica with engineered nearest-neighbor couplings, and the propagation coordinate $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$37 plays the role of fractional order through $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$38. Because each $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$39-slice corresponds to a different $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$40, the device can provide simultaneous readout of all transformation orders. Reported metrics include classical transformation fidelity $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$41, quantum state fidelity $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$42, concurrence $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$43 before FSR and $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$44 after FSR, and suppression-law visibility $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$45 (Weimann et al., 2015).

In scientific quantum machine learning, the PHQFNO framework extends the Fourier-space split into a tunable partition between QPU and CPU resources. A hybridization ratio $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$46 selects quantum and classical channels, unary encoding loads each quantum row into a one-hot basis $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$47, a QFT acts on the assigned low modes, variational learning gates emulate Fourier-space multiplication, and readout proceeds through computational-basis probabilities

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$48

The reconstructed quantum Fourier coefficients are then concatenated with the classically processed spectrum and returned to physical space by a classical IFFT. On Burgers’ equation and incompressible Navier-Stokes benchmarks, PHQFNO recovers classical FNO accuracy, achieves higher accuracy than classical counterparts on incompressible Navier-Stokes, and shows improved stability under input noise (Marcandelli et al., 11 Jul 2025).

A further development appears in HQ-LP-FNO, where a hybrid Fourier layer uses a quantum branch as an FSR module for mode-shared spectral mixing. The pipeline performs a real/imaginary split of selected complex Fourier channels, robust scaling into angles, AngleEmbedding and QFT, a variational mixer $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$49, inverse QFT, and measurement of Pauli-$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$50 expectations

$|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$51

These measured expectations are decoded back to complex spectral channels and concatenated with a classical branch. The paper reports that HQ-LP-FNO reduces trainable parameters by $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$52 relative to a classical baseline while lowering phase-fraction mean absolute error by $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$53 and relative temperature MAE from $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$54 to $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$55. A sweep over the quantum-channel budget shows that a moderate VQC allocation yields the best temperature metrics across all tested configurations, and a noisy-simulator study under backend-calibrated noise from ibm-torino finds convergence near $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$56 shots (Papierz et al., 6 Apr 2026).

6. Advantages, limitations, and scope conditions

Across these implementations, the primary advantage of FSR is that it shifts the measurement burden from real-space amplitudes to a compressed spectral representation. In CAE and PDE readout, this means replacing $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$57 pointwise sampling by $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$58 measurements of low-order Fourier coefficients when $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$59 for smooth $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$60 (Huang et al., 28 Jul 2025). In NV-center sensing, the corresponding gain is single-shot digitization of phase in Fourier space, near-Heisenberg limited precision in sensing classical and quantum signals, and in situ demultiplexing of multi-frequency signals with exponentially enhanced dynamic range using only a single sensor spin and a small register of nuclear qubits (Vorobyov et al., 2020). In photonic and operator-learning settings, the same spectral principle enables simultaneous multi-order transforms, partitioned execution across quantum and classical hardware, and parameter-efficient mode-shared mixing (Weimann et al., 2015, Marcandelli et al., 11 Jul 2025, Papierz et al., 6 Apr 2026).

The limitations are equally systematic. The CAE formulation notes that sign determination for tiny coefficients can be noisy and that functions with very slow spectral decay require larger $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$61 (Huang et al., 28 Jul 2025). The sensing implementation requires high-fidelity control of multi-spin QFT, currently $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$62 per bit, and its spectral resolution is ultimately limited by nuclear-spin coherence times, with $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$63; moreover, protocol complexity scales as $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$64 two-qubit gates for an $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$65-bit register (Vorobyov et al., 2020). The photonic DFrFT platform is constrained by propagation loss, detector inefficiency, and coupling-coefficient precision, with fabrication challenges requiring precision better than $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$66 for $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$67 in higher-dimensional scaling (Weimann et al., 2015).

A recurrent misconception is that Fourier-space readout removes the readout bottleneck for arbitrary quantum states. The published analyses support a narrower statement: FSR is efficient when the target object is a continuous real-valued function or signal whose Fourier spectrum is concentrated in a small number of low-order modes, or when a sensing protocol is deliberately engineered so that $|\psi\rangle=\sum_{j=0}^{N-1}\psi_j |j\rangle$68 maps phase into a compact digital register (Huang et al., 28 Jul 2025, Huang et al., 25 Nov 2025). It is therefore not equivalent to full tomography, and it does not guarantee speedup for non-smooth functions, states with slow spectral decay, or architectures in which the transform itself is too noisy. A plausible implication is that FSR is best viewed not as a universal readout primitive, but as a spectral-compression strategy whose success depends on regularity, sparsity, and hardware-specific transform fidelity.