Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor

Brian J. Lester; Christopher S. Wang; Isaac L. Chuang; Jacob C. Curtis; Jessica Freeze; Liang Jiang; Luigi Frunzio; Patrick H. Vaccaro; Robert J. Schoelkopf; S. M. Girvin

REVIEW 2 major objections 5 minor 1 cited by

Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A superconducting two-mode bosonic processor efficiently samples molecular vibronic spectra using single-shot photon-number-resolved detection.

desk verdict The single-bit FCF measurements are solid; the fifteen-photon sampling claim is ahead of its evidence. read the letter →

arxiv 1908.03598 v2 pith:7DAU7KTO submitted 2019-08-09 quant-ph

Christopher S. Wang , Jacob C. Curtis , Brian J. Lester , Yaxing Zhang , Yvonne Y. Gao , Jessica Freeze , Victor S. Batista , Patrick H. Vaccaro

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Isaac L. Chuang Luigi Frunzio Liang Jiang S. M. Girvin Robert J. Schoelkopf

This is my paper · ORCID

classification quant-ph

keywords bosonsamplingvibronicspectraFranck-Condonfactorssuperconductingcircuitsphoton-number-resolvingdetectionDoktorovtransformationcircuitquantumelectrodynamicsGaussianoperations

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 tries to establish that a naturally bosonic quantum processor—two microwave cavities in a superconducting circuit—can efficiently simulate the vibrational spectra of molecules, a task that maps onto multiphoton boson sampling. The authors implement the Doktorov unitary that connects vibrational states across an electronic transition and, crucially, demonstrate a single-shot photon-number-resolving detector that resolves up to 15 photons per mode, which removes the exponential overhead of one-bit-at-a-time readout. They then reprogram the processor to reproduce Franck-Condon spectra for four molecular processes, obtaining distances to ideal distributions as low as 0.019. A sympathetic reader would take the paper to show that superconducting bosonic hardware is a practical and scalable platform for this class of bosonic quantum simulations, including vibrationally hot initial states prepared as Fock states.

What carries the argument

The central object is the Doktorov transformation $\hat U_{\rm Dok} = \hat D(\alpha) \hat S^\dagger(\zeta') \hat R(U) \hat S(\zeta)$, which maps the pre-transition vibrational ladder operators to the post-transition ones. It is implemented by displacement, single-mode squeezing, and a two-mode beamsplitter produced by four-wave mixing with a coupler transmon. The other load-bearing mechanism is the QND photon-number-resolving detector: parity operators $\hat P_k$ that read the $k$th bit of the binary decomposition of the photon number are mapped onto an auxiliary transmon by optimal-control pulses, so four sequential measurements resolve $n = 0,\dots,15$ in each mode and return a single sample from the joint Fock-state distribution per run.

What would settle it

Prepare a cavity state with a known photon-number distribution, such as a displaced squeezed state, measure it with the single-bit scheme for every photon number up to 15, apply the paper's correction, and compare with the exact distribution; if the corrected error grows with photon number, the assumption of photon-number-independent auxiliary-qubit rates is false. The same check can be done by measuring the auxiliary qubit's Rabi and T1 rates conditioned on $n = 0,1,\dots,15$.

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Extended reading notes

Core claim

The central claim is that molecular vibronic spectra—the Franck-Condon factors $\left|\langle \vec n' | \hat U_{\rm Dok} | \vec n \rangle\right|^2$—can be extracted from a two-mode superconducting bosonic processor by directly realizing the Doktorov transformation $\hat U_{\rm Dok} = \hat D(\alpha) \hat S^\dagger(\zeta') \hat R(U) \hat S(\zeta)$ as a sequence of displacements, single-mode squeezes, and one beamsplitter, each generated by four-wave mixing on a Josephson coupler. The paper reports the first single-shot QND photon-number-resolving detector, in any platform, that resolves up to 15 photons per mode by reading the binary decomposition of the photon number onto an auxiliary transmon; this turns each run into a direct sample of the joint Fock-state distribution rather than a single bit of one joint occupation. With these tools, the measured spectra for photoelectron processes in H$_2$O, O$_3$, NO$_2$, and SO$_2$ sit close to the ideal Franck-Condon distributions, with total-variation distances $D$ from 0.019 to 0.105 for the corrected single-bit-extraction scheme and 0.075 to 0.209 for the scalable sampling scheme.

Load-bearing premise

The correction applied to the most accurate data assumes that the auxiliary qubit's rotation speed and its decay and heating rates are the same whether the cavity holds 0, 1, or 15 photons; if those rates actually change with photon number, every corrected Franck-Condon probability carries an unquantified bias.

Editorial extensions

If this is right

Single-shot photon-number-resolving measurement of up to 15 photons per mode makes the sampling scheme scalable: $N\log_2(n_{\max})$ binary QND measurements return one full sample from the joint distribution, instead of querying each of the exponentially many joint Fock states.
The native bosonic implementation of the Doktorov unitary costs $O(N^2)$ Gaussian operations and $O(N)$ circuit depth, compared with $O(N^2 n_{\max}^2 \log^3(1/\varepsilon))$ gates for a qubit-based algorithm at fixed error $\varepsilon$.
Because the same device reproduces the spectra of H$_2$O, O$_3$, NO$_2$, and SO$_2$ by changing only the Doktorov parameters, the processor is reprogrammable across molecules and initial vibrational states.
The ability to prepare non-Gaussian Fock states as inputs lets the simulator start from vibrationally excited ensembles, not just the ground state.
Adding a third cavity mode would extend the simulation to nonlinear triatomic molecules of $C_s$ symmetry, and a linear array of $N$ modes covers molecules with up to $3M-6$ vibrational degrees of freedom.

Reading between the lines

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

The reported sampling distances are noisier than the corrected single-bit distances; the paper leaves open whether deconvolution of the bit-wise readout errors, for example by unfolding methods, can close that gap, which is a direct testable next step.
The weakest premise is the photon-number independence of the auxiliary qubit's Rabi and decoherence rates used in the correction; a dedicated calibration at each $n = 0,\dots,15$ would either validate all reported $D$ values or expose a systematic bias in them.
The same hardware elements—Gaussian operations plus programmable self-Kerr nonlinearity—are enough to simulate time-domain vibrational dynamics and anharmonic bosonic models, though the paper only gestures toward those targets.
If the architecture scales as proposed, a many-cavity linear array would be a general-purpose simulator of harmonic vibrational structure for polyatomic molecules, with the measurement cost growing only logarithmically in the per-mode Hilbert-space dimension.

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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

First thing to know: the single-bit extraction results are the real work, and they are good; the scalable-sampling/PNR fifteen-photon claim is not yet demonstrated at the claimed fidelity. The paper says so itself, in Section IV.B, where it defers full characterization of the binary-decomposition measurement chain to future work. The abstract still calls it "high-fidelity single-shot PNR up to 15 photons," which overstates what is shown.

What is genuinely new: a two-mode superconducting bosonic processor that implements the Doktorov unitary and samples FCFs for four molecules, including non-Gaussian Fock inputs up to n=1,m=2. The QND parity measurement scheme itself—four sequential pulses resolving the binary decomposition—is a new hardware capability in this platform, even if the error budget is incomplete. The single-bit extraction data, corrected via Eq. S29 and compared against master-equation simulations with self-Kerr and loss, match the ab initio FCFs with distances 0.019 to 0.105. The Doktorov parameters come from independent CCSD(T) calculations, so there is no obvious circularity. No fitted parameter is doing the work.

Soft spots, in order of importance. First, the sampling chain: the D=0.075–0.209 values for the sampling scheme are raw, uncorrected, and include detector error indistinguishably. The paper directly says characterization is future work. So the headline scalability claim rests on an assertion, not a measurement. That should be fixed or the claim softened. Second, the correction model for single-bit extraction assumes the ancilla Rabi and decoherence rates are photon-number independent. That is plausible but untested over n=0..15; if wrong, the corrected probabilities carry a small unquantified bias. Minor relative to the first. Third, two modes is a small instance; the paper is honest about it.

Verdict: worth a serious referee. The experimental core is solid, the supplementary data is unusually complete, and the single-bit demonstration alone supports the paper's main physical point. A referee should require the authors to either add the measurement characterization or revise the abstract's fidelity claim. The paper would be stronger with both.

Referee Report

2 major / 5 minor

Summary. The paper reports an experimental implementation of the boson-sampling algorithm for molecular vibronic spectra on a two-mode superconducting bosonic processor. The authors prepare Fock states in two microwave cavities, apply displacement, squeezing, and beamsplitter operations corresponding to the Doktorov transformation for four triatomic molecules (H2O, O3, NO2, SO2), and detect the output photon-number distributions using two complementary readout schemes: a 'single-bit extraction' method based on selective ancilla pulses and a 'sampling' scheme based on sequential QND parity measurements that aim to resolve up to 15 photons per mode. The measured Franck-Condon factors are compared to ideal classical calculations, yielding distances D in the range 0.019-0.105 for the single-bit scheme and 0.075-0.209 for the sampling scheme. The central claims are (i) the demonstration of a scalable photon-number-resolving detection scheme and (ii) accurate simulation of molecular vibronic spectra in a superconducting platform.

Significance. If the claims are fully supported, this is an important experimental milestone in bosonic quantum simulation. The paper has clear strengths: the Doktorov parameters are computed with independent quantum-chemistry software (Gaussian/ezSpectrum) at the CCSD(T) level; the Gaussian operations are calibrated against physics-based models; the ideal Franck-Condon target distributions are computed classically without using the measured data; and the single-bit-extraction data agree with the ideal distributions with distances as low as 0.019, with full time-domain master-equation simulations (including self-Kerr and photon loss) provided as baselines. The demonstration of non-Gaussian input-state synthesis for vibrationally excited initial states is also a meaningful advance over previous trapped-ion emulations. However, the scalable sampling/PNR component, which is highlighted in the abstract, is not yet backed by a characterization of the measurement chain; this is the main gap that must be addressed in revision.

major comments (2)

[IV.B and abstract] The claim of 'for the first time in any platform, a high-fidelity single-shot photon number resolving detection scheme capable of resolving up to 15 photons per mode' is not supported by data in the manuscript. Section IV.B states that the task of characterizing the errors of the four sequential QND parity pulses is left to future work, and no confusion matrix or assignment-error analysis for the parity measurements is provided. Consequently, the sampling distances D reported in Table V and Fig. 3 are computed from raw counts without any correction protocol (Supplementary VI), so they do not separate detector error from simulator error. The estimate of measurement-induced error in Table I (κτ_meas ~ 10^-2-10^-1) indicates that these errors are not negligible. To substantiate the headline claim, the authors should provide a characterization of the sampling chain, e.g., by calibrating the bit-assignment probabilities for each Fock state up to 15 and reporting the resulting corrected distribution, or by explicitly qualifying the claim as 'demonstrated at the level of uncharacterized raw sampling'.
[Supplementary V, Eq. (S29)] The correction of the single-bit-extraction data relies on the assumption that the ancilla Rabi and decoherence rates have no photon-number dependence. The manuscript does not provide a test of this assumption across the relevant range n=0-15. If the Rabi frequency or the decoherence rates vary with cavity photon number, then the corrected Franck-Condon probabilities obtained from Eq. (S29) carry an unquantified systematic bias, and the quoted distances D for the single-bit scheme (e.g., D=0.049 for H2O and 0.019 for SO2) would be distorted. This is load-bearing because the single-bit results are the primary evidence for the accuracy of the two-mode simulation. The authors should either present a calibration of f and t as a function of photon number, or provide a theoretical and experimental argument that the variation is negligible at the reported level of precision.

minor comments (5)

[Table I] The notation for the sampling measurement outcome, |n′⟩ = |b3,b2,b1,b0⟩, uses commas inside the ket, which is nonstandard and potentially confusing; the authors should define b_i as the bits of the binary decomposition and write e.g. |n′⟩ = |b3 b2 b1 b0⟩.
[Fig. 3 caption] The caption states that statistical error bars for the sampling data are not visible; it would be helpful to report approximate error bar sizes in the caption or show them on a zoomed inset.
[Supplementary II.A, Eq. (S17)] In the expression for χ_iC, the argument of the absolute value is written as |g_i/δ_i|^2 with a missing subscript on δ; the authors should write δ_i to be consistent with the rest of the notation.
[Abstract] The phrase 'for the first time in any platform' is a strong priority claim. The authors should verify that no prior work has demonstrated single-shot photon-number resolution up to 15 photons in any platform, and add a citation or explicitly scope the claim to the circuit-QED platform.
[Supplementary VI] The table of numerical data would be easier to use if the columns for 'Single-bit extraction' and 'Sampling' included the number of runs N_runs used for each measurement, since the reported standard errors depend on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the simulated FCF targets are computed from independent quantum-chemistry data, and the hardware is calibrated against independently measured physics.

full rationale

The paper's central claim is that a two-mode superconducting processor can implement the Doktorov unitary and sample molecular vibronic spectra. The Doktorov parameters (Supplement Table II) are obtained from CCSD(T) calculations via Gaussian and ezSpectrum, independent of the experimental data; the ideal Franck-Condon distributions used as targets are classically computed from those same parameters. Using the same molecular parameters to program the unitary and to compute the benchmark is not circular: the experiment's output is not fed back into either the parameter derivation or the ideal-distribution calculation. Hardware operations are calibrated through independently measured cavity linewidths, dispersive shifts, gate rates, self-Kerr terms, and T1 values (Supplement Section III), with no fitted parameter extracted from the target FCFs. The single-bit extraction correction (Eq. S29) uses independently measured t and f offsets; the sampling-scheme D values are computed from raw counts with no correction in the sampling data processing. The paper's own statement that sampling-chain errors are left for future work, and the assumption of photon-number-independent ancilla rates, are verification and robustness gaps rather than circularity: they do not make any predicted quantity equal to an input by construction. Self-citations to prior Yale work on beamsplitter/squeezing and dispersive readout supply supporting experimental and theoretical tools, but the present paper calibrates its own operations and does not rely on those citations to define its predicted spectra. No equation in the paper reduces a prediction to a fitted input, to a renamed known result, or to a self-cited uniqueness claim, so no circular step is found.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central simulation rests on standard quantum-chemistry approximations and the established Doktorov/boson-sampling mapping, plus the QND measurement assumption of circuit QED. No new physical entities are introduced; the only fitted numbers are control and error-correction calibrations that do not enter the target FCF distribution except through the error-correction model.

free parameters (3)

eta (squeezing scaling parameter) = 47.6381, 28.9364, 34.7639, 26.4676 for H2O, O3-, NO2-, SO2
Chosen in the optimization of squeezing parameters (supplementary text I.C) to minimize the total squeezing while leaving the Doktorov unitary L invariant. It is a control optimization parameter rather than a physics parameter, and does not affect the target distribution by construction.
f_A, f_B, t_A, t_B (measurement error correction) = f_A approx 0.004-0.005, f_B approx 0.001-0.003, t_A approx 0.931-0.938, t_B approx 0.943-0.951 (Table V)
Probabilities of false-excited assignment (f) and true-excited assignment (t) per ancilla, calibrated from Rabi experiments, used in Eq. S29/S30 to correct single-bit extracted probabilities. They are error-model calibrations, not fit to the spectral target.
Gaussian operation rates (g_sq, g_BS, displacement amplitude) = g_sq approx 60 kHz; g_BS approx 2pi x 44 kHz; tau_alpha=1 = 72 ns
Calibrated rates of the Gaussian operations (Table III) extracted from population fits. They are measured, not assumed, so they do not add an ungrounded degree of freedom, but the FCF claim depends on the accuracy of these calibrations.

assumptions (6)

domain assumption Born-Oppenheimer separation and harmonic approximation for the vibrational potential energy surfaces
Section II states the vibrational eigenstates are obtained by expanding the PES to quadratic terms; the entire mapping to Doktorov transformations requires this harmonic model.
domain assumption Sudden approximation and Condon approximation: vibronic band intensities are proportional to Franck-Condon factors
Section II invokes these approximations to connect the simulated FCFs to photoelectron spectra intensities.
domain assumption The Doktorov transformation U_Dok = D(alpha) S-dagger(zeta') R(U) S(zeta) is the correct mapping between molecular vibrational modes
Taken from prior literature (Doktorov 1977; Huh 2015), not re-derived in this paper.
domain assumption C2v symmetry restricts the four molecules to the two-dimensional subspace of symmetric-stretching and bending modes
Section V states this restriction for all molecules; it is required for the two-mode mapping.
domain assumption The dispersive Hamiltonian H = -chi c-dagger c t-dagger t plus drive enables QND binary-valued measurements and sequential parity measurements project to a Fock state without disturbing the cavity beyond the intended projection
Section IV.B constructs the binary decomposition of photon number on this QND assumption.
ad hoc to paper Measurement-error correction assumes no photon-number dependence of ancilla Rabi and decoherence rates
Supplementary text V explicitly states 'we assume that there is no photon number dependence to either the Rabi or decoherence rates of the ancilla.' If false, the reported single-bit FCFs carry unquantified bias.

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Pith. "Pith review of Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor." pith.science (2026). https://pith.science/paper/7DAU7KTO

@misc{pith2026190803598,
  author       = {Pith},
  title        = {Pith review of: Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DAU7KTO}},
  note         = {Machine review of arXiv:1908.03598}
}

abstract

The efficient simulation of quantum systems is a primary motivating factor for developing controllable quantum machines. For addressing systems with underlying bosonic structure, it is advantageous to utilize a naturally bosonic platform. Optical photons passing through linear networks may be configured to perform quantum simulation tasks, but the efficient preparation and detection of multiphoton quantum states of light in linear optical systems are challenging. Here, we experimentally implement a boson sampling protocol for simulating molecular vibronic spectra [Nature Photonics $\textbf{9}$, 615 (2015)] in a two-mode superconducting device. In addition to enacting the requisite set of Gaussian operations across both modes, we fulfill the scalability requirement by demonstrating, for the first time in any platform, a high-fidelity single-shot photon number resolving detection scheme capable of resolving up to 15 photons per mode. Furthermore, we exercise the capability of synthesizing non-Gaussian input states to simulate spectra of molecular ensembles in vibrational excited states. We show the re-programmability of our implementation by extracting the spectra of photoelectron processes in H$_2$O, O$_3$, NO$_2$, and SO$_2$. The capabilities highlighted in this work establish the superconducting architecture as a promising platform for bosonic simulations, and by combining them with tools such as Kerr interactions and engineered dissipation, enable the simulation of a wider class of bosonic systems.

Figures

Figures reproduced from arXiv: 1908.03598 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 1.** Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png]

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png]

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png]

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

**Figure 7.** Figure 7: FIG. 7. Photoionization of neutral water to the ( [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

**Figure 8.** Figure 8: FIG. 8. Photoionization of the ozone anion to neutral ozone starting in the vibrationless state [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

**Figure 9.** Figure 9: FIG. 9. Photoionization of the ozone anion to neutral ozone starting with one quanta in the symmetric-stretching mode and [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

**Figure 10.** Figure 10: FIG. 10. Photoionization of the ozone anion to neutral ozone starting with one quanta in the symmetric-stretching mode and [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]

**Figure 11.** Figure 11: FIG. 11. Photoionization of nitrite to nitrogen dioxide starting in the vibrationless state [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 12.** Figure 12: FIG. 12. Photoionization of nitrite to nitrogen dioxide starting with one quanta in the symmetric-stretching mode and zero in [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]

**Figure 13.** Figure 13: FIG. 13. Photoionization of sulfur dioxide to the cation starting in the vibrationless state [PITH_FULL_IMAGE:figures/full_fig_p038_13.png]

**Figure 14.** Figure 14: FIG. 14. Photoionization of sulfur dioxide to the cation starting with zero quanta in the symmetric-stretching mode and one [PITH_FULL_IMAGE:figures/full_fig_p039_14.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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