Variational optical phase learning on a continuous-variable quantum compiler

Alberto M. Marino; Andrew Sornborger; Claire E. Marvinney; Matthew A. Feldman; Raphael C. Pooser; Seongjin Hong; Tyler Volkoff; Zoe Holmes

REVIEW 3 major objections 6 minor 88 references

Variational optical phase learning on a continuous-variable quantum compiler

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports an experimental continuous-variable quantum compiler that learns the phase of an optical gate using a two-mode squeezed state, with a 5.4-fold precision gain and a 3.6-fold faster convergence.

desk verdict Real first demonstration of a CV quantum compiler, but the 'genuine quantum advantage' claim lacks an equal-energy coherent-state control and the headline precision factor rests on N=5. read the letter →

arxiv 2502.10242 v1 pith:4JYFTPBS submitted 2025-02-14 quant-ph

Matthew A. Feldman , Tyler Volkoff , Seongjin Hong , Claire E. Marvinney , Zoe Holmes , Raphael C. Pooser , Andrew Sornborger , Alberto M. Marino This is my paper · ORCID

classification quant-ph

keywords continuous-variablequantumcomputingcompilingtwo-modesqueezedstatevariationalalgorithmopticalphaseestimationhomodynedetectionfour-wavemixingbarrenplateaus

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

This paper reports the first experimental continuous-variable quantum compiler: a circuit that learns an unknown optical phase shift by iteratively adjusting a control phase until its effect matches the target. The resource is a two-mode squeezed state of light, whose quantum correlations let the experiment tune the steepness of the cost landscape by changing the squeezing level. At the highest squeezing studied, the compiled phase estimate was 5.4 times more precise and reached convergence 3.6 times faster than at the lowest squeezing. The paper argues this improvement comes from the quantum correlations themselves, not from sending more photons, and that the same mechanism pushes phase precision toward Heisenberg scaling.

What carries the argument

The load-bearing object is the cost function $C(r,\varepsilon,\varepsilon',N'_B,\Delta\phi) = -\rho(0,r,\varepsilon,\varepsilon',N'_B,\Delta\phi) = -\sqrt{2/(\pi(\mathrm{tr}A+2A_{1,2}))}$, obtained by projecting the Wigner function of the noisy two-mode squeezed state onto the amplitude-difference quadrature $(q_1-q_2)/\sqrt{2}$ and taking its peak value at $X_-=0$; $A$ is the $2\times 2$ matrix in Eq. (4) depending on squeezing $r$, transmission efficiencies $\varepsilon,\varepsilon'$, seed excess noise $N'_B$, and phase difference $\Delta\phi$. This cost function is evaluated from homodyne measurement statistics and minimized by gradient descent on $\phi_c$. The squeezing parameter $r$ acts as a landscape dial: it controls the curvature $f(N)$ near the minimum, which enters the quadratic expansion $C \approx -1 + f(N)\Delta\phi^2/2$ and therefore sets both the achievable phase precision and the gradient magnitude available for training.

What would settle it

Lock the apparatus at each detuning and measure the amplitude-difference squeezing directly at $\Delta\phi=0$ with an independent characterization, then check whether the observed phase standard deviation and time-to-solution follow the $f(N)$-based scalings with that independently measured $r$; separately, run the compiler with a coherent-state probe of equal total photon number, because if its precision and speed match the squeezed-resource results, the claimed quantum advantage is falsified.

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Extended reading notes

Core claim

Using a two-mode squeezed state produced by four-wave mixing in a rubidium vapor cell, the authors implement a variational quantum compilation algorithm that learns the phase $\phi_0$ of a target optical phase gate. A control phase $\phi_c$ is adjusted by gradient descent to minimize a cost function defined as the negative peak value of the homodyne-measured marginal distribution of the amplitude-difference quadrature; minimizing this cost drives $\phi_c$ to $\phi_0$. The central discovery is that the squeezing parameter $r$ tunes the shape of this cost function: increasing $r$ narrows the minimum and steepens its slopes, which simultaneously raises the precision with which $\phi_0$ can be estimated and shortens the time-to-solution. Quantitatively, the measured phase standard deviation falls from 513.8 mrad at $r=0.18$ to 96 mrad at $r=0.74$ (a 5.4-fold precision gain), while convergence time falls from 470 to 130 iterations (a 3.6-fold speedup). In the ideal limit the cost function's curvature grows with $f(N)=\sqrt{N(N+1)}(2\sqrt{N(N+1)}+(2N+1))$, so the phase error scales as $\delta/N^2$ with the squeezed photon number $N$, which the paper presents as evidence for Heisenberg-like scaling and for a genuine quantum advantage.

Load-bearing premise

The whole quantitative story depends on modeling the four-wave-mixing resource as a two-mode squeezed state with one excess-noise parameter and noiseless attenuation; the fit of this model to the homodyne data supplies the $r$ values, the convergence thresholds, and the cost curves, and the paper's own Appendix D notes the model ignores distributed gain and loss in the medium, which would increase the effective input noise.

Editorial extensions

If this is right

At higher squeezing, the same compiler yields more precise phase estimates while converging in fewer iterations, so squeezing is a tunable resource rather than a fixed noise cost.
Because the cost curvature grows with $f(N)$, the phase error scales as $\delta/N^2$ in the ideal lossless case, approaching the Heisenberg limit set by squeezed-light interferometry.
The adaptive strategy of starting at low squeezing and increasing it during training can avoid the barren-plateau regime at high squeezing, making the compiler trainable at precision that would otherwise be inaccessible.
The same cost-function construction generalizes to learning other Gaussian unitary parameters from homodyne data, since Gaussian states are fully characterized by first and second quadrature moments.
The demonstrated quantum advantage, if it holds, means continuous-variable variational compilers can outperform equal-energy classical or coherent-state counterparts on phase learning.

Reading between the lines

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

Beyond the paper, a direct head-to-head test against an equal-energy coherent-state probe, rather than only the theoretical comparison in Eq. (8), would put the 'quantum correlations, not photon number' claim on firmer experimental footing.
Beyond the paper, the proposed low-to-high squeezing schedule could be tested by comparing random initialization with the adaptive schedule at the same final squeezing, checking whether barren-plateau avoidance actually delivers the projected precision gains.
Beyond the paper, the cost-function readout relies on one Gaussian marginal; extending it to multi-parameter Gaussian unitaries such as beam splitters would require tracking more than one marginal, and the paper's Appendix C notes that the simple moment-based cost is not universal.
Beyond the paper, if the single-excess-noise model underestimates distributed gain and loss in the medium, the inferred $r$ values may be systematically biased, and a more detailed model would likely change the quantitative precision scaling even if the qualitative trend survives.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is a real experimental implementation of a CV quantum compiler using two-mode squeezed light, the first I know of, and the cost-landscape narrowing with squeezing is clearly shown. But the paper's strongest claim—'genuine quantum advantage'—is not supported by the data, because there is no equal-energy coherent-state control. The reported 5.4x precision factor is also fragile: N=5 runs, no error bars, and the N=15 validation gives 4.15 under a different learning rate.

The experimental work itself is solid. The authors implement the TMSS-based cost function from Volkoff et al., show that the measured cost curves track the theory across three squeezing levels, and demonstrate that higher squeezing gives faster convergence and lower phase variance. The seeded-TMSS precision analysis around Eqs. (8)-(10) is a nice addition, showing interpolation between sub-SQL and Heisenberg-like scaling within that manifold. Credit is due for transparency: Appendix D admits that the model ignores distributed gain and loss, and the paper explicitly notes that the Eq. (3) cost function is meaningless for pure coherent states.

The soft spots are real but not fatal to the core demonstration. First, the 'genuine quantum advantage' claim in the Discussion outruns the evidence. Since increasing r also increases total photon number, the observed improvements could in principle come from energy rather than correlations. The paper needs a coherent-state baseline with the same photon budget, or a resource-accounted comparison, before claiming the correlations are doing the work. The fact that Eq. (3) can't be used for coherent states means the authors would need a different readout for the control, but that's an experimental challenge, not a get-out-of-jail card. Second, the headline precision factor is a point estimate from five runs with no uncertainty; the N=15 study gives 4.15 with a different learning rate, so the exact number should not be taken as precise. Third, the squeezing parameters come from a fitted noisy-TMSS model whose limitations are acknowledged; the fits look good and are constrained by independent squeezing measurements, so this is a moderate concern, not a fatal one.

Overall: this is a solid proof-of-principle experiment that deserves peer review. It should be published after the authors either pull the quantum advantage claim back to what is actually shown (tunable cost landscape, faster convergence, improved precision with squeezing) or add a proper classical baseline. I'd send it to a serious referee.

Referee Report

3 major / 6 minor

Summary. This paper reports an experimental continuous-variable (CV) quantum compiler that learns a single optical phase gate via gradient descent. The resource is a two-mode squeezed state (TMSS) generated by four-wave mixing in a truncated SU(1,1) interferometer. The authors derive a cost function from the amplitude-difference marginal of the Wigner function, fit their measured cost curves to a noisy-TMSS model with two free parameters, and show that increasing the effective squeezing parameter r narrows the cost landscape, speeds convergence (3.6-fold reduction in iterations to convergence), and improves the reported phase precision (5.4-fold between r=0.18 and r=0.74). They also present a theoretical analysis of the cost landscape and claim that the observed enhancements constitute a 'genuine quantum advantage' driven by quantum correlations rather than total photon number. The paper includes detailed appendices on data processing, parameter estimation, and robustness checks.

Significance. If supported, this would be a valuable first experimental demonstration of a squeezing-based CV variational quantum compiler, with a practical cost function, direct comparison to theoretical predictions, and a tunable landscape that connects to barren-plateau avoidance in the CV setting. The theoretical cost-function derivation is explicit, and Eq. (5) provides a parameter-free prediction of the lossless precision scaling given the squeezing parameter, which is a notable strength. The experimental methods are detailed, including noise modeling, AIC-based model comparison, and robustness tests over target/control phases. However, the central quantitative claims—the 5.4-fold and 3.6-fold factors and, especially, the asserted 'genuine quantum advantage'—are not fully supported by the data as presented, so the significance of the paper currently rests on the more modest demonstration of a controllable CV compiler.

major comments (3)

[Section IV] The claim of a 'genuine quantum advantage' over an equal-energy classical (coherent-state) resource is not supported by the data. No equal-energy coherent-state control experiment is reported, and the cost function defined in Eq. (3) is phase-insensitive for coherent states because the quadrature-noise marginal of a coherent state has constant variance independent of Δφ. The interpolating analysis in Eqs. (8)–(10) lives on the seeded-TMSS manifold and does not compare against an optimized coherent-state phase-estimation strategy with the same total photon number and measurement time. The observed 5.4-fold precision increase and 3.6-fold speedup could in principle arise from increased photon number, higher local-oscillator power, or improved signal-to-noise rather than from two-mode squeezing correlations. To substantiate the 'quantum correlations, not merely total photon number' statement, the authors should either add a resource-accounted comparison with a classical baseline using a phase-sensitive readout, or explicitly temper the claim to a demonstration of enhanced precision and speed with squeezed resources without asserting genuine quantum advantage.
[Section III, Table I, and Appendix B] The 5.4-fold precision factor rests on N=5 runs per squeezing value with no uncertainty reported on σΔφ, and the N=15 validation run in Appendix B yields a smaller factor of 4.15 with a modified learning rate. The paper does not provide error bars, confidence intervals, or a statistical test that the precision ratio differs from what could be obtained from noisy data. Please report the uncertainty on the σΔφ estimates (e.g., bootstrap or chi-square intervals) or present the factor as a qualitative trend rather than a precise quantitative claim. In addition, the claim that the measurements are 'unbiased' is not persuasive given the relatively large phase-difference means in Table I (e.g., 90 mrad for r=0.18) and especially in Table II (485 mrad for r=0.18), where the authors attribute the offset to the learning rate; this should be discussed quantitatively.
[Appendix D] The inferred squeezing parameters r=0.18, 0.35, and 0.74, and hence the theoretical cost curves and the quoted precision scaling, depend on a model that the authors themselves state neglects distributed gain and loss in the four-wave mixing medium and underestimates the effective input noise Nin. The fitting procedure constrains r through a relation r(Nin, ε') that is derived from this simplified model, so the reported ratios and the 'validated' scaling could shift under a different noise model. Please provide a sensitivity analysis showing how the extracted r values and the resulting precision/speed-up factors change under the alternative noisy-homodyne model (or a model that incorporates distributed loss), rather than only reporting the AIC difference.

minor comments (6)

[Section II A and II C] The text refers to the main experimental schematic as Fig. 4(a) and Fig. 4(b), but the schematic with panels (a) and (b) appears to be Fig. 1; the later Fig. 4 is the data-processing validation diagram. Please correct the cross-references.
[Section II A] There is a typo: 'homoydne detection' should be 'homodyne detection'.
[Appendix D] The sentence 'expressed as, expressed as' contains a duplicated phrase.
[Appendix E] The text 'over the thee runs' should read 'over the three runs'.
[Appendix B] The quantity σΔσ in the paragraph describing Table II should be σΔφ.
[Fig. 4] The figure title says 'complimentary data processing procedure'; 'complimentary' should be 'complementary'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the precision scaling is an analytic expansion of a self-contained cost model, and the reported precision/speedup factors are measured rather than fitted.

full rationale

Walking the derivation chain, the cost function C (Eq. 3) is obtained from the Wigner marginal of a noisy TMSS in Appendix C, and the ideal precision scaling Eqs. (5)-(6) follows by Taylor expansion of that cost function. This scaling is an analytic prediction for a given squeezing parameter; it is not fitted to the measured phase statistics. The experimental squeezing parameters r are estimated from independent amplitude-difference squeezing and loss measurements subject to a two-parameter fit of the cost model, but the reported 5.4x precision factor and 3.6x time-to-solution factor are measured quantities from the QCA runs (Table I), not values extracted from the fitted cost function. The paper explicitly notes that Eq. (3) is not applicable to coherent states, so the 'genuine quantum advantage' claim in Section IV lacks an equal-energy classical control and is therefore not established; however, that is a benchmark/comparison gap, not a case of a prediction being equivalent to its input by construction. Self-citations to Ref. [26] motivate the landscape-narrowing and barren-plateau discussion, but the cost-function derivation is self-contained in Appendix C and the barren-plateau instance is also shown experimentally, so the citations are not load-bearing in a circular way. No equation in the paper reduces to another by construction, and no fitted parameter is renamed as an independent prediction. Therefore no significant circularity is identified.

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

The central quantitative claims rest on three fitted or inferred parameters (r, N'_B, epsilon') and on a Gaussian resource-state model whose limitations are explicitly acknowledged in Appendix D. No new physical entities are introduced.

free parameters (3)

Effective squeezing parameter r = 0.18, 0.35, 0.74
Controlled experimentally via one-photon detuning, but the reported values are extracted by fitting the noisy-TMSS model to measured amplitude-difference squeezing; they anchor the precision and time-to-solution comparisons.
Probe excess noise N'_B (input noise Nin) = N'_B = 1.08, 1.08, 2.37 (Nin = 1.58, 1.58, 2.87)
Free parameter in the maximum-likelihood fit of the cost function; the measured lower bound is 0.71 but fitted values exceed it.
Probe transmissivity epsilon' = 0.69, 0.70, 0.59
Free parameter in the fit, bounded above by measured absorption; differs per squeezing setting and affects the inferred r via the model relation.

assumptions (4)

domain assumption The four-wave-mixing output is a Gaussian TMSS described by Eq C4: thermal excess noise in the probe, vacuum conjugate, noiseless attenuation channels, and no distributed gain or loss.
Used to derive the cost function Eq 3 and to extract r; Appendix D notes the model underestimates the effective Nin by ignoring distributed loss and gain.
domain assumption The peak of the X- marginal of the Wigner function is a faithful cost function with a unique global minimum at Delta phi = 0.
Invoked in Appendix C; requires the homodyne distribution to be Gaussian so that first and second moments identify the state.
domain assumption The weak coherent seed used for phase locking does not affect the cost function because the cost is computed from fluctuations about the mean.
Stated in Section II C; the displacement is filtered or centered out, but any residual effect on the measured variances is assumed negligible.
standard math Shot-noise calibration assumes a minimum-uncertainty quadrature variance of 1/2 for the vacuum.
Appendix A uses <Delta x_j^2> = 1/2 to convert measured voltages into quadrature values; this is a standard quantum optics convention.

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Cite this review

Pith. "Pith review of Variational optical phase learning on a continuous-variable quantum compiler." pith.science (2026). https://pith.science/paper/4JYFTPBS

@misc{pith2026250210242,
  author       = {Pith},
  title        = {Pith review of: Variational optical phase learning on a continuous-variable quantum compiler},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JYFTPBS}},
  note         = {Machine review of arXiv:2502.10242}
}

read the original abstract

Quantum process learning is a fundamental primitive that draws inspiration from machine learning with the goal of better studying the dynamics of quantum systems. One approach to quantum process learning is quantum compilation, whereby an analog quantum operation is digitized by compiling it into a series of basic gates. While there has been significant focus on quantum compiling for discrete-variable systems, the continuous-variable (CV) framework has received comparatively less attention. We present an experimental implementation of a CV quantum compiler that uses two mode-squeezed light to learn a Gaussian unitary operation. We demonstrate the compiler by learning a parameterized linear phase unitary through the use of target and control phase unitaries to demonstrate a factor of 5.4 increase in the precision of the phase estimation and a 3.6-fold acceleration in the time-to-solution metric when leveraging quantum resources. Our results are enabled by the tunable control of our cost landscape via variable squeezing, thus providing a critical framework to simultaneously increase precision and reduce time-to-solution.

Figures

Figures reproduced from arXiv: 2502.10242 by the authors.

**Figure 2.** FIG. 2. Cost function landscape for different degrees of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Theoretical (solid lines) and experimentally me [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Schematic representation of a complimentary data pr [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Examples of QCA runs used for precision analysis [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Comparison of experimental data (dots) and theo [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7. Convergence of the QCA algorithm for different initia [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: FIG. 8. Convergence behavior of the QCA algorithm with the [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Reference graph

Works this paper leans on

88 extracted references · 69 canonical work pages

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can then be used to compute the Wigner function via the Fourier transform of the characteristic function [46]. Given the correlations that are present between the amplitude diﬀerence and phase sum quadratures in a TMSS, to obtain a cost function that can be eﬃciently evaluated through homodyne detection, we choose to project the Wigner function onto the a...
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The distribution of such joint quadrature with eigenvalue X− = ( x1 − x2)/ √ 2 is obtained via ρ(X−, r, ε, ε′, N ′ B, ∆ φ) := 1 4 ∫ R3 dX+dy1dy2W (x1, y1, x2, y2), (2) where y1 and y2 are the eigenvalues of the phase quadra- tures of the respective modes and X+ = (x1 + x2)/ √ 2 is the eigenvalue of the quadrature sum, ( q1 + q2)/ √ 2. Finally, to obtain a...
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# CPU !",!#,$% HD & $' HD !

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

can then be used to compute the Wigner function via the Fourier transform of the characteristic function [46]. Given the correlations that are present between the amplitude diﬀerence and phase sum quadratures in a TMSS, to obtain a cost function that can be eﬃciently evaluated through homodyne detection, we choose to project the Wigner function onto the a...

[2] [2]

The distribution of such joint quadrature with eigenvalue X− = ( x1 − x2)/ √ 2 is obtained via ρ(X−, r, ε, ε′, N ′ B, ∆ φ) := 1 4 ∫ R3 dX+dy1dy2W (x1, y1, x2, y2), (2) where y1 and y2 are the eigenvalues of the phase quadra- tures of the respective modes and X+ = (x1 + x2)/ √ 2 is the eigenvalue of the quadrature sum, ( q1 + q2)/ √ 2. Finally, to obtain a...

[3] [3]

# CPU !",!#,$% HD & $' HD !

to obtain the cost function 3 a) ! "# CPU !",!#,$% HD & $' HD !" !# $% | ()**+,-. / | 01+ 23;456 7 *#287 b) | !" (0;1/2) FIG. 1. (a) Schematic of a continuous-variable (CV) quantum compiler. A parameteric ampliﬁer is weakly seeded with a noi sy probe beam modeled as a thermal state, ρPr, with no displacement, i.e. ρPr(0; Nin) = D(0)ρPr(Nin)D†(0) where D(α...

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arXiv 2018

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Although in both Eqs

is deﬁned in terms of the quadrature noise properties of the resource state and is not applicable to pure coherent states because it contains no information about ∆ φ in this case. Although in both Eqs. (

[6] [6]

and ( 8) the physical source of the precision advantage (namely quadrature squeez- ing) is the same as in homodyne readout of a Mach- Zehnder interferometer, the gradient-descent based al- gorithm learns φ0 by tuning φc and calculating the cost function, not by directly estimating the φ0 phase shift. Our demonstration of this approach in Section II C show...

[7] [7]

We reduce the number of ﬁt parameters by directly mea- suring the degree of squeezing and losses after the FWM process

and Eq.( 4), were obtained using a maximum likelihood ﬁt of C to the data. We reduce the number of ﬁt parameters by directly mea- suring the degree of squeezing and losses after the FWM process. This leaves only two ﬁtting parameters: the ex- cess noise for the input probe seed, N ′ B, and the probe transmissivity, ε′. Determining these parameters also al...

[8] [8]

reduces to the same quadratic scaling found in Eq. ( 5), −1 + 2N 2 ∆ φ2 + o(∆ φ4), (9) yielding ∆ φ2 ∝ δ/N 2 when the cost function is within δ of the minimum, thereby recovering Heisenberg scaling of the precision, where here N = sinh 2 r. In the opposite limit of negligible squeezing r → 0, Eq. ( 8) simpliﬁes to −1 + N 2 ∆ φ4 + o(∆ φ4), (10) where N = α...