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Theory and Experimental Demonstration of Quantum Invariant Filtering

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Pith's one-line read Quantum Invariant Filtering turns a desired spectral filter into the continuous driving field that realizes it.

desk verdict A genuinely new way to go from frequency-domain filter specs to continuous control fields, with a solid experimental demo and a real but fixable linearization gap that the paper never quantifies. read the letter →

arxiv 2506.15805 v1 pith:EHERVA3X submitted 2025-06-18 quant-ph

Loris Maria Cangemi , Yoav Woldiger , Amikam Levy , Assaf Hamo This is my paper · ORCID
classification quant-ph
keywords quantuminvariantfilteringdynamicalinvariantscontroldecouplingfrequency-domainfilterdesignnitrogen-vacancycentersensingdual-bandpass
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes and demonstrates a method, Quantum Invariant Filtering, that reverses the usual order of quantum control design: instead of choosing a time-domain pulse sequence and then measuring its frequency response, the user starts with a desired spectral filter and derives the continuous Hamiltonian modulation that implements it. The claim is that any finite-impulse-response filter profile, whether single-band, multi-band, or phase-sensitive, can be mapped into an experimentally feasible drive on a qubit through the dynamical-invariant framework. On a single nitrogen-vacancy center in diamond, the method reproduces prescribed passbands, suppresses low-frequency noise, keeps coherence for milliseconds, and tolerates drive-amplitude errors far better than Carr-Purcell-Meiboom-Gill sequences. If correct, this gives quantum control and sensing a direct design path from the frequency domain to the waveform.

What carries the argument

The central object is the dynamical invariant $I(t)$ of a driven qubit, parametrized by two auxiliary angles $\alpha(t)$ and $\beta(t)$. Under the constraint $\alpha(t)=\pi$ and $\dot\alpha(t)=0$, the relative Lewis-Riesenfeld phase accumulated between invariant eigenstates is exactly $\beta(t)$, so the second-order response to a $z$-axis perturbation becomes a convolution with kernel $\cos\beta(t)=H(t)$. To make the target filter the response, one sets $\beta(t) = -\pi/2 + \arcsin H(t)$, so that the Hamiltonian control $\varepsilon(t)$ is obtained from the time derivative of the reversed impulse response. This identity, filter specification in frequency to inverse Fourier transform to auxiliary angle to drive field, is what carries the argument.

What would settle it

Scale a designed impulse response $H(t)$ to a maximum amplitude around $0.5$, measure $\langle\sigma_z(t_f)\rangle$ versus probe frequency, and compare with two predictions: the linearized kernel gives $|H(\omega)|^2$, while the exact $\arcsin$ construction shows a third-order replica at three times the center frequency whose height grows as $\max|H|^3/6$. A direct time-domain deconvolution of the realized kernel would also settle whether the linearization is acceptable.

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Extended reading notes

Core claim

The central discovery is that the final-time readout $\langle\sigma_z(t_f)\rangle$ of a qubit driven by a reverse-engineered invariant control is, to second order in the signal amplitude, the squared convolution of the input signal with a kernel $H(t)$: $\langle\sigma_z(t_f)\rangle \simeq \langle\sigma_z(t_f)\rangle_0 - \frac{\delta^2}{2}\left(\int f_{\mathrm{in}}(s)H(t_f-s)\,ds\right)^2$. Since $H(t)$ is generated from the inverse Fourier transform of the target filter $H(\omega)$, the measured contrast for a monochromatic probe directly reproduces $|H(\omega)|^2$. The construction fixes the auxiliary invariant angle to $\beta(t) = -\pi/2 + \arcsin H(t)$, and in the implemented linearized version $\beta(t) = -\pi/2 + H(t)$, the control field is $\varepsilon(t) = \partial_t H(t_f - t)$. Because the protocol starts from the spectral specification rather than from a pulse sequence, it can realize passband shapes and dual-band filters that pulsed dynamical decoupling cannot, and it does so with a smooth, continuous drive.

Load-bearing premise

The experimental filter uses the linearized auxiliary angle $\beta(t) = -\pi/2 + H(t)$ instead of the exact $\beta(t) = -\pi/2 + \arcsin H(t)$, so the realized spectral kernel is $\sin H(t)$ rather than $H(t)$, and the paper does not quantify how large $H(t)$ may be before the cubic error distorts the passband.

Editorial extensions

If this is right

  • A user can design quantum control for a qubit by specifying the spectral response directly, including multi-band passbands and phase offsets.
  • Noise concentrated outside the passband is suppressed, extending coherence much longer than Carr-Purcell-Meiboom-Gill sequences; the paper reports about two orders of magnitude.
  • The measurement of the final population yields both amplitude and phase of a signal near the filter frequency, enabling lock-in-style quantum sensing.
  • The same prescription applies to other qubit platforms such as superconducting qubits, trapped ions, and nuclear magnetic resonance.
  • Because the control is continuous, drive-amplitude miscalibration of $\pm 50\%$ still leaves $\langle Z\rangle$ above roughly $0.8$, unlike $\pi$-pulse sequences.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the exact $\arcsin$ mapping would produce kernel $\sin(\arcsin H(t))=H(t)$, but the linearized implementation uses $\sin H(t)\approx H(t)-H(t)^3/6$; for stronger filters the cubic term will create spectral replicas that the paper does not quantify.
  • Editorial inference: because the response is quadratic in the signal amplitude, very weak signals produce a quadratically suppressed readout, so practical sensitivity will require either larger $\delta$ or additional amplification; a dedicated calibration curve would show where this becomes limiting.
  • Editorial inference: by symmetry, the same invariant construction should work for perturbations along other axes, turning QIF into a vector-field filter, though the paper only demonstrates $z$-axis noise and signal coupling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript introduces Quantum Invariant Filtering (QIF), a method that reverse-engineers continuous qubit control fields from a target frequency-domain filter profile. Within the dynamical-invariant framework and a second-order Dyson expansion, the authors derive that the final-time expectation value <σz(tf)> responds as a convolution of the input signal with an impulse response H(t) (Eq. S39). An exact construction sets the auxiliary field to β(t) = −π/2 + arcsin H(t); the experiment instead uses the linearized form β(t) = −π/2 + H(t), yielding the control field ε(t) = ∂t H(tf − t). The method is demonstrated on a single NV center in diamond, with experiments showing tunable bandpass filtering, phase-sensitive detection, amplitude detection, dual-band filtering, and robustness/coherence comparisons against CPMG sequences.

Significance. The theoretical idea is elegant and, if properly supported, would constitute a broadly applicable framework for frequency-domain-designed quantum control and sensing. Strengths include a clean analytic mapping from a specified filter profile to a control Hamiltonian, the numerical cross-check in Fig. S1, an explicit construction recipe, and a diverse set of experimental demonstrations. The novelty is clear and the level of ambition is appropriate. However, the central 'arbitrary FIR' claim currently rests on an unquantified linearization of the auxiliary field, and the experimental evidence lacks error bars, uses normalization and a fitted phase offset, and the coherence comparison may be confounded by the use of a different NV center. These issues are fixable, but they are load-bearing for the paper's headline claims.

major comments (4)
  1. [Main text after Eq. (5); Supplementary S2 B, Eq. (S48)] The substitution β(t) = −π/2 + H(t) replaces the exact relation β(t) = −π/2 + arcsin H(t) from Eq. (S35). The realized second-order kernel is then cos β = sin H(t) ≈ H(t) − H(t)^3/6, so the implemented filter is |F[sin H]|^2 rather than |H(ω)|^2. For the dual-band kernel of Eq. (S56) with f1 = 1.5 MHz and f2 = 2.5 MHz, the leading third-order correction generates intermodulation products at |2f1 − f2| = 0.5 MHz and |2f2 − f1| = 3.5 MHz, both inside the measured band. The manuscript neither bounds max|H(t)| nor quantifies these replicas. Because the abstract's claim of arbitrary FIR mapping and the dual-band demonstration depend on the realized kernel, the authors should either implement the exact arcsin construction or provide a rigorous small-H error bound and demonstrate that the replicas are below the experimental sensitivity.
  2. [Fig. 4b; Supplementary S2 C, 'T1 and T2 Figure'] The supplement states that the T1 and T2 measurements were performed on a different NV center that was not polarized. If the QIF and CPMG decay curves in Fig. 4b were acquired on different NV centers, the reported factor-of-approximately-40 improvement in decay time cannot be attributed solely to the protocol. The authors should either measure all protocols on the same center or report per-center noise characterization (e.g., T2* or an independently measured noise spectral density) to make the coherence comparison valid.
  3. [Figs. 2b, 2e, 3b, 3f, 4a] Experimental points are presented without error bars, and the theoretical curves in Figs. 2b and 3f are explicitly normalized to the maximum experimental contrast. The normalization constant is a free parameter, so the agreement currently demonstrates qualitative line-shape correspondence rather than quantitative transfer-function fidelity. The authors should report statistical uncertainties (one million averages per point should allow tight error bars) and either show absolute, unnormalized data or state explicitly that only relative response is claimed.
  4. [Eqs. (S37)–(S39) and Fig. S1] The central response formula is second-order perturbative in the signal amplitude δ, and the numerical check in Fig. S1 shows that the perturbation deviates from exact dynamics at the response peaks even for δ = 0.5. The manuscript does not report the δ values used in the experimental scans of Figs. 2 and 3, nor does it give the domain of validity of Eq. (S39) as a function of δ·tf. Since Fig. 3d explicitly addresses signal amplitude extraction, the authors should state the experimental δ and verify that the quoted 'high fidelity' and amplitude calibration lie within the validated perturbative regime.
minor comments (4)
  1. [Fig. 1 caption] There is a typo: 'of the the QIF protocol' should read 'of the QIF protocol'.
  2. [Throughout main text] The symbol H is used both for the Hamiltonian (Eq. (1)) and for the impulse response (Eq. (5) and later). This is confusing; consider denoting the impulse response with a lowercase symbol such as h(t).
  3. [Abstract and Discussion] The phrase 'arbitrary finite-impulse responses' is broader than the class actually implemented: the construction requires a smooth continuous H(t) with H(0)=H(tf)=0, |H(t)|<1, and time symmetry about tf/2. The authors should qualify this claim.
  4. [Fig. 3c] The phase offset Δφ_signal ≈ π/8 is introduced as a fitted adjustment and then corroborated by electronic measurement; the fit uncertainty and the uncertainty of the electronic corroboration should be reported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the target filter H is the input, control fields are derived from it via the invariant formalism, and the measured NV response is an independent external benchmark.

full rationale

The paper's derivation chain is a design-and-verify construction rather than a circular one. The target frequency-domain filter H(ω) is chosen first; its impulse response H(t) is then inserted into the invariant auxiliary field via β(t)=−π/2+arcsin H(t) (Eq. S35), and Eq. (S39) shows that the second-order response is A(tf)=∫fin(s)H(s). Since H(t) was already defined as cos β(t), this relation is by construction—the paper is explicitly engineering H into the control, not fitting H from data. The central claim is that the physical qubit response, measured on a nitrogen-vacancy center, reproduces this designed spectral profile. The experimental validation uses only a normalization of the theoretical curve and a phase offset that is independently corroborated by direct electronic measurement; no spectral shape is fitted to the measured response. The noted experimental simplification β(t)=−π/2+H(t) (Eq. S48) replaces the exact arcsin mapping and yields a realized kernel sin H(t) rather than H(t), introducing an unquantified approximation for larger filter amplitudes. This is a correctness or approximation concern, not a circularity, because it does not make the output equivalent to the input by definition or by fitted parameters. Self-citations appear only for standard dynamical-invariant results (e.g., Eq. S5) and are not load-bearing for the filter construction, which rests on the external Lewis-Riesenfeld framework and standard FIR design. The comparison against CPMG is an external benchmark. Therefore no circular step meets the required evidentiary standard, and the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The derivation relies on standard invariant theory and on several idealizations (RWA, z-axis coupling, second-order perturbation). The experimental implementation adds a linearization whose error is unquantified. Three fitted or normalization constants (filter amplitude, response normalization, phase offset) control the absolute comparison between theory and data, though the spectral shape is not fitted.

free parameters (3)
  • Impulse response scaling amplitude A_H
    The designed FIR kernel H(t) is rescaled to keep |H(t)|<1 to satisfy invariant boundary conditions. The numerical value is not reported; it sets the filter strength and modifies the absolute response scale.
  • Circuit impedance phase offset Δϕ_signal = ≈ π/8
    The measured phase shift in Fig. 3c is fitted (brown fit) to align the sin² prediction with the data. The authors report an independent electronic measurement corroborating the value, which weakens the circularity but a free fit is present.
  • Normalization constant for theoretical filter response = normalized to max experimental contrast
    Designed filter responses are normalized to the experimental maximum in Figs. 2b, 2d, 3f, so absolute response magnitudes are not predicted from first principles.
assumptions (4)
  • standard math Lewis-Riesenfeld dynamical invariant theory (Eq. S2 provides a complete basis of conserved populations).
    Used throughout to express the propagator and compute the response; unproved background result.
  • domain assumption Qubit control in the rotating-wave approximation with Hamiltonian H(t)=Δ/2 σz - ε/2 σx and signal/noise coupling V=δ f_in σz/2.
    Assumed in Eqs. (S19)-(S21) and in the response calculation. Valid for on-resonance, weak driving, and z-axis noise.
  • domain assumption Second-order perturbation theory is sufficient to model the filter response (Eq. S39).
    The design uses the second-order Dyson series; truncation errors grow with δ·tf, as shown by deviations in Fig. S1 for δ=0.5.
  • ad hoc to paper The auxiliary field linearization β=-π/2+H(t) is a faithful experimental representation of the exact β=-π/2+arcsin H(t).
    Introduced for experimental convenience (S2 B). Valid only for |H(t)|<1 and small enough that sin H ≈ H; no error bound is provided.

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Cite this review

Pith. "Pith review of Theory and Experimental Demonstration of Quantum Invariant Filtering." pith.science (2026). https://pith.science/paper/EHERVA3X

@misc{pith2026250615805,
  author       = {Pith},
  title        = {Pith review of: Theory and Experimental Demonstration of Quantum Invariant Filtering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHERVA3X}},
  note         = {Machine review of arXiv:2506.15805}
}
read the original abstract

Quantum control protocols are typically devised in the time domain, leaving their spectral behavior to emerge only a posteriori. Here, we invert this paradigm. Starting from a target frequency-domain filter, we employ the dynamical-invariant framework to derive the continuous driving fields that enact the chosen spectral response on a qubit. This approach, Quantum Invariant Filtering (QIF), maps arbitrary finite-impulse responses, including multi-band and phase-sensitive profiles, into experimentally feasible Hamiltonian modulations. Implemented on a single nitrogen-vacancy center in diamond, the method realizes the prescribed passbands with high fidelity, suppresses noise, and preserves coherence for milliseconds, two orders of magnitude longer than Carr-Purcell-Meiboom-Gill sequences, while remaining robust to 50% drive-amplitude errors. Our results establish QIF as a broadly applicable framework for enhanced quantum control and sensing across diverse physical platforms, including superconducting qubits, trapped ions, and nuclear magnetic resonance systems.

Figures

Figures reproduced from arXiv: 2506.15805 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. b shows the first experimental realization of the Quantum Invariant Filter (QIF), implementing a bandpass frequency response centered at a target fre￾quency by applying the designed control field along the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: evaluates the robustness of the QIF method in the absence of a signal input, focusing on its tolerance to amplitude errors and performance over long evolution times. Fig. 4a presents the expectation value of the Pauli￾Z operator, ⟨Z⟩, measured at the end of the protoco…

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    A. D’Arrigo, G. Piccitto, G. Falci, and E. Paladino, Open-loop quantum control of small-size networks for high-order cumulants and cross-correlations sensing, Sci- entific Reports 14, 16681 (2024). 9 SUPPLEMENTARY INFORMATION S1. QIF THEORY Below, we present the theoretical fr...

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    Then, following Eq

    Parametrization of the invariant and boundary conditions To set the stage of our reverse-engineered protocols, we set h2(t) = 0 at each point in time. Then, following Eq. (S15), we can set k∗ = 3 and, using the conservation of the norm of I, we can write h1(t) = ˙I3(t) I2(t), ...

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    (S18) implies ∆( ta) = 0, i.e., (Ho(t),I (t)) are proportional to σx at the start and at the end of the protocol

    (S18) Notice also that Eq. (S18) implies ∆( ta) = 0, i.e., (Ho(t),I (t)) are proportional to σx at the start and at the end of the protocol. Furthermore, the boundary conditions set for β(t) ensure that Eqs. (S17) are well-posed for t = t0,t f. Except for the endpoint times, f...

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    (S7) into Eq

    First-order response By plugging Eq. (S7) into Eq. (S25), after some algebra we get the first-order correction to the state that reads |ψ(t)⟩≃| ψo(t)⟩ +|χ(1)(t)⟩ , |χ(1)(t)⟩ =−i h ⟨ϕ+(t0)|ψ(t0)⟩L11(t) +⟨ϕ−(t0)|ψ(t0)⟩L12(t) · ·ei(φ+(t)−φ+(t0))|ϕ+(t)⟩ + ⟨ϕ+(t0)|ψ(t0)⟩L21(t) +⟨ϕ−...

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    (S29) are considered as well

    Second-order response The design of the first-order response may not be sufficient to achieve significant filtering effects, so second-order terms in Eq. (S29) are considered as well. We thus take into account second-order terms, as written in Eq. (S25). Again, inserting the a...

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    Second-order response under FIR-modulated invariants In order to design the QIF, we restrict to the case of auxiliary fields described in Eqs. (S35). We stress that, although in this limit the invariant in Eq. (S16) reduces to a simple constant of the motion, this does not red...

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    (S39) reveals a clear convolution struc- ture

    FIR and convolution structure of the QIF response The second-order correction to the expectation value⟨σz(tf)⟩ derived in Eq. (S39) reveals a clear convolution struc- ture. Specifically, the deviation from the unperturbed outcome depends quadratically on a filtered signal ampl...

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    However, due to the truncation introduced in Eq

    Accuracy of the perturbation approach The time-dependent perturbation approach is a straightforward tool to model the basic mechanism of the QIF. However, due to the truncation introduced in Eq. (S25), it is not expected to recover the actual response to the external fields. T...

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    Indeed, the solution of Eq

    Magnus expansion The results presented in S1 E 1 can be equivalently derived by means of Magnus expansion [13, 49, 50]. Indeed, the solution of Eq. (S23) can be written as follows UI(t,t 0) =Te−i R t t0 VI(t′)dt′ =eΩ(t), (S43) 16 0 1 2 3 4 5 ω[MHz] −0.5 −0.4 −0.3 −0.2 −0.1 0.0...

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    This is accomplished by embedding a classical Finite Impulse Response (FIR) filter into the time-dependent control Hamiltonian via the invariant formalism

    Construction of the Band-Pass Filter for Experimental Implementation To realize the Quantum Invariant Filter (QIF) experimentally, we construct custom-designed band-pass filters that define the desired system response in the frequency domain. This is accomplished by embedding ...

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    The measured system response, quantified by the second-order amplitude A2(tf) (see Eq

    Experimental Characterization of Frequency and Phase Response Following the construction of the band-pass kernel H(t) described in the previous section, we now demonstrate experimentally that the QIF protocol faithfully reproduces its designed spectral and phase properties. Th...

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