Programmable nonlinear quantum photonic circuits

Domain: physics
Published: 2025-12-11
Editorial outcome: minor_revision

Abstract
The lack of interactions between single photons prohibits direct nonlinear operations in quantum optical circuits, representing a central obstacle in photonic quantum technologies. Here, we demonstrate multi-mode nonlinear photonic circuits where both linear and direct nonlinear operations can be programmed with high precision at the single-photon level. Nonlinear interaction is realised with a tunable quantum dot embedded in a nanophotonic waveguide mediating interactions between individual photons within a temporal linear optical interferometer. We demonstrate the capability to reprogramme the nonlinear photonic circuits and implement protocols where strong nonlinearities are required, in particular for quantum simulation of anharmonic molecular dynamics, thereby showcasing the new key functionalities enabled by our technology.

Introduction
Photons are carriers of quantum information with unique features for quantum technology: the insensitivity to stochastic noise, the high-speed operation and ability to realise long-distance transmission, and the manufacturability of photonic circuits used for quantum-information processing1. Progress in photonic quantum technologies has brought large-scale linear optical quantum photonic circuits consisting of thousands of optical components that process tens of photons2, to realise photonic devices operating in computational regimes that challenge the capabilities of conventional computers for specific quantum computing tasks3,4. However, the lack of direct photon-photon interactions prohibits the nonlinear operations that enable deterministic multi-photon entangling gates5,6 and near-term photonic quantum simulators7,8. Currently, it represents a significant challenge for developing quantum photonic devices at the scale required for practical applications.

One approach for implementing photon nonlinearities with linear optical circuits relies on measurement-induced operations to mediate photon-photon interactions9. This approach exploits the nonlinear nature of quantum measurements to reproduce the desired entangling operations10. However, because measurements are intrinsically probabilistic in quantum mechanics, such operations necessarily have a finite success probability, dependent on and heralded by the detection of compatible measurement outcomes. Multiplexing techniques11,12 and quantum error correction13 methods can be used to turn such probabilistic operations into near-deterministic ones or to tolerate their failures, respectively, but require enormous overheads in terms of hardware components, stringent performance and noise requirements, and classical co-processing14,15. As a consequence, while programmable linear optical circuits have been demonstrated in large-scale quantum experiments, near-deterministic nonlinear photonic circuits have so far been elusive for all-optical approaches.

Light-matter interactions in quantum emitters represent an alternative approach to implementing optical nonlinearities at the single-photon level16,17. Because photon-photon operations in such systems are mediated through direct interaction with a quantum emitter, this approach enables nonlinearities to be implemented deterministically6. Recent experiments have demonstrated strong photon-photon nonlinearities with quantum emitters in atomic18,19 and solid-state systems, including quantum dots (QDs)20–22 and colour centres23, validating the strong potential of this approach for enabling deterministic nonlinear photonic devices. Strong nonlinear interactions have also been demonstrated in other platforms, including ions, neutral atoms, and superconducting circuits, and have been used to implement quantum simulation24–26 and computing circuits27,28. Examples of bosonic devices used for quantum chemistry are summarised in ref. 29.

Here, we demonstrate a programmable nonlinear photonic quantum processor based on optical nonlinearities embedded in a programmable linear optical circuit. The tunable nonlinearities are implemented on a GaAs chip through the light-matter interaction between single photons and an InAs QD embedded in a photonic-crystal waveguide nanostructure. These nonlinear operations are combined with programmable linear optical circuits implemented with a temporal interferometer, enabling a single QD to perform multi-mode nonlinear operations at different times. We demonstrate the separate and simultaneous programmability of both linear and nonlinear operations, thereby showing how quantum interference and the joint spectral properties of the processed photons can be tuned by different linear and nonlinear control parameters of the circuit. By enabling reliable photon-photon nonlinearities and entangling gates in programmable photonic circuitry, the developed programmable nonlinear quantum photonic platform has potentially transformative applications, e.g., for synthesising advanced photonic quantum states18,20,30 or realising deterministic Bell-state analysers31 and quantum optical neural networks32. We illustrate this potential by realising a proof-of-concept quantum simulation of anharmonic molecular vibrational dynamics, exploiting the natural mapping of this chemistry problem to the nonlinear quantum photonics platform. Our work extends pioneering previous work based solely on linear optics and probabilistic measurement-induced nonlinearities33,34, thereby paving a way to scaling photonic quantum devices.

Results
Programmable linear and nonlinear photonic circuits
Figure 1a displays the schematics of the photonic circuit realising programmable nonlinear photonic operations between two tunable linear interferometers. The scheme employs temporal degrees of freedom and universal two-mode linear operations as performed with two Mach-Zehnder interferometers implemented via a self-stabilising time-bin interferometer with a 3 ns delay between the two time-bins (see Fig. 1b, and Supplementary Note 1 for more details). The temporal mode encoding allows scattering from the same QD at different times, allowing for scaling up to multiple temporal modes.

To achieve this configuration, the photons emerging from the first time-bin interferometer are coupled via a grating coupler35 from free space and into the GaAs chip. In the GaAs chip, the photon-photon interaction is mediated using an InAs QD, which is embedded in a two-sided GaAs photonic crystal waveguide (PCW, shown in Fig. 1b) to enhance light-matter interactions between the incoming photons and the quantum emitter (coupling strength parameter of \beta \simeq 88%)36. This strong light-matter coupling gives rise to nonlinear effects, which, in the two-photon case relevant to the present experiment, can be described as a Kerr-like interaction or a programmable nonlinear phase gate9. The details of the calculation can be found in the Supplementary Note 2, while refs. 16,37 describe multi-photon contributions. After interacting with the QD, the photons are outcoupled from the chip and directed back to the time-bin interferometer for the second linear operation. Finally, after the interferometers, the photons are collected via optical fibres, demultiplexed via fibre beam splitters, and detected with four superconducting nanowire single-photon detectors (SNSPDs) enabling pseudo-number-resolving photon counting.

Characterisation of the linear and nonlinear operations
First, the QD detuning, pulsewidth and linear phase are scanned to demonstrate the complete and simultaneous linear and nonlinear programmability of the circuit. The interplay between linear and nonlinear processes can be observed in the dynamics of the joint temporal intensity (JTI) of photons scattered in the nonlinear transformation and interfered in the linear circuits. We probe the dynamics in time-resolved photon-correlation measurements of photons emerging from the final Mach-Zehnder interferometer after they have scattered and interfered. Figure 2a displays the normalised two-photon JTI measured in the |02> configuration (both output photons in the second spatial mode) for different values of both the linear phase \phi and the nonlinearity control parameter \Delta. It is observed that, while no correlations are present in the absence of the nonlinearities (\Delta \gg \sigma), they arise when the QD is brought into resonance, leading to a JTI with pronounced correlations along the diagonal in addition to persistent sidelobes20. When the linear phase is scanned, the sidelobes interfere due to their emergence in the contributions to the JTI from the terms where only a single photon is present in each time-bin mode during the scattering process, which we label as the |11> term. The diagonal part is primarily due to terms with both photons in the same temporal mode upon scattering.

The quantum interference patterns are also affected by the interplay between the linear and nonlinear parameters. Figure 2b shows the interference fringe in the output configuration with both photons bunched in the first spatial mode, that is, |20>, obtained when scanning the linear phase \phi and setting the Mach-Zehnder interferometers to balanced beam splitters (\theta = \pi/2). In the absence of nonlinearities (i.e., \Delta \gg \sigma), an interference fringe visibility of 97.1% is observed (dark blue curve in Fig. 2b, see Supplementary Note 3 for details on the data analysis), which is reduced by the nonlinear phase and amplitude contributions realized when bringing the QD into resonance. We describe the nonlinear transformation of the scattering process through a nonlinear phase parameter \phi_{NL} and the scattering probability \ell according to
|1> \rightarrow \sqrt{\eta(1-\ell)} e^{i\phi} |1> (1)
|2> \rightarrow \sqrt{\eta^2} e^{i(2\phi + \phi_{NL})} |2>, (2)
where \phi and \eta are the linear phase and transmission parameters, respectively, and the response is truncated to two-photon components. We note that this description of the scattering process is a simplified parametrisation in which both a Kerr-like nonlinear phase and the finite overlap between the input and output spectra are combined into a single effective nonlinear phase parameter, \phi_{NL}. Also note that photon wave packets do not remain single-mode upon scattering. (see Supplementary Note 2 for details). To mitigate multi-mode effects, one possible solution is to employ time-lens dispersion to rectify the modal distortion38. Alternatively, the orthogonal properties after consecutive interactions with quantum emitters can be used to prepare one- and two-photon states in orthogonal modes and split them in a photon sorter-like experiment, as is investigated theoretically in ref. 39. Nevertheless, this simplified model captures well the effects of the nonlinearities observable in our device. The nonlinear phase parameter \phi_{NL} manifests itself by reducing the visibility in the interference fringes in Fig. 2b due to the nonlinear contributions between the two-photon terms and the |11> term.

Using this model for the nonlinear transformation, we extract the nonlinear phase parameter \phi_{NL} and nonlinear scattering probability \ell as a function of the nonlinearity control parameters by modelling the experimental photon statistics data (see Supplementary Note 2 and Supplementary Note 4 for more details on the model and analysis). Figure 2c shows the results as a function of detuning \Delta. Equivalent results for controlling the nonlinearities by tuning the bandwidth of the photons are shown in Supplementary Note 2. From the photon statistics, we can extract the applied nonlinear phase parameters up to \phi_{NL} \simeq \pi/4, indicating an extensive range of tunability of the nonlinearities embedded within the universal programmable two-mode linear interferometer. The finite nonlinear scattering probability \ell arises due to the scattering process being implemented in a two-sided waveguide, where scattered photons can also be reflected. Because the probability of having at least one photon reflected is higher for the |1> than for the |2> term36, this process results in a finite success probability, cf. Fig. 2d, which decreases when increasing the strength of the nonlinearity. Notably the success probability for the nonlinear approach is found to significantly outperform that of the measurement-based scheme based on linear optics (dashed black line in Fig. 2d) by Sparrow et al.33. Furthermore, full deterministic operation of the nonlinearity can be readily achieved by implementing either one-sided waveguides40 or chiral photon-emitter interaction41 (dotted purple line in Fig. 2d), for which a unity intrinsic scattering success probability in the full [0, \pi] range is achievable.

Photonic quantum simulation of anharmonic molecular dynamics
Reconfigurable nonlinear photonic circuits enable a wide range of novel applications in quantum information science. This includes the realisation of deterministic quantum gates enabling photon sorting and enhanced Bell state analysers31, the generation of advanced photonic quantum states37, and nonlinear boson sampling circuitry42. In the present work, we benchmark our platform in the context of photonic quantum simulation by simulating the anharmonic time evolution of vibrations in a water molecule. First proposed in ref. 33, this class of quantum simulation methods exploits the equivalence between phonons evolving in the vibrational modes of a molecule and photons evolving in the optical modes of a photonic circuit, as depicted in Fig. 3a. In fact, because phonons and photons are both bosonic particles, the quantum dynamics of vibrations in a molecule evolving under a transformation U will be described by the same input-output statistics as a photonic system evolving through a circuit implementing an equivalent optical transformation U. In such a mapping, the N = 3M - 6 (The vibrational modes are 3M - 5 for linear molecules.) vibrational modes of an M-atom molecule each correspond to an optical mode in the photon circuit.

In the harmonic approximation, the system is generally represented by N coupled harmonic oscillators, with couplings between the vibrational modes corresponding to linear operations on the associated optical modes. The evolution of molecular quantum dynamics in the harmonic approximation can thus be simply implemented via a linear optical system: input photons are prepared in the same state as the initial phononic state of the molecule, and the unitary transformation U is implemented by programming the linear circuit. The output state is then analysed via photon detection. However, the potential energy landscapes for most molecular systems of practical interest are far from harmonic, and higher-order corrections are required to better capture the system’s relevant properties.

The Hamiltonian of a harmonic oscillator in the normal (eigen) basis is given by H = \sum_i \hbar \omega_i a_i^y a_i where a unitary basis transformation U to a basis localised around the different vibrational modes a_i^y = \sum_k U_{ki} a_k^y leads to the Hamiltonian H = \sum_{i,j,k} \hbar \omega_i U_{ki} U_{ji} a_j^y a_k. The atomic potentials are, however, not harmonic, but the anharmonicity can be included in the model. By extending the Taylor expansion of the potential energy from a second-order harmonic description to include third and semi-diagonal fourth-order terms, the true molecule can be approximated better. Adding these extra terms under vibrational perturbation theory yields the anharmonic Hamiltonian H_a = H + \hbar \sum_{i \leq j} \chi_{ij} \omega_i \omega_j (a_i^y a_i + a_j^y a_j + 2a_i^y a_j^y a_i a_j), where H is the harmonic Hamiltonian. The Harmonic part can be simulated with linear optics, whereas the nonlinear “coupling term" is \chi_{ik}. This Hamiltonian is diagonalised to find the eigenenergies, which are listed in Supplementary Note 5, as well as the unitary transformation between the normal and local bases33,43.

In the mapping between vibrational and optical modes, such higher-order corrections correspond to nonlinear coupling terms between the modes, which in the first-order Trotter approximation correspond to single-mode nonlinearities embedded in between two linear optical transformations33. Such a circuit is exactly what we implement in our device, as shown in Fig. 1, where the ability to tune both linear and nonlinear operations enables the direct implementation of transformations corresponding to the anharmonic dynamics of molecules. In particular, we simulate the anharmonic vibrational dynamics in the H_2O (water) molecule, which has three vibrational eigenmodes: two high-energy symmetric and asymmetric stretch modes (shown in Fig. 3b, with more details in Supplementary Note 5), and a low-energy bending mode, which in this model is uncoupled from the others33,43. We describe the molecule in the basis of local modes and normal modes. The normal modes are localised around singular atoms or chemical bonds, and here the localised modes are represented by stretch modes of the individual left and right H atoms, as shown in Fig. 3b.

In this way, the time evolution of the molecular vibration dynamics of H_2O, including its anharmonic potential energy, can be simulated by tuning the linear and nonlinear parameters of the quantum photonic circuit.

The first linear operation implements a basis transformation of the localised modes into the vibrational eigenmodes of the molecule. The evolution over a given timestep is then simulated as follows. From the anharmonic eigenenergies of the vibrational modes, whose values are reported in Supplementary Note 5, we calculate the phases accumulated during the simulated evolution time by the configurations with one and two excitations on each eigenmode, up to simulated evolution times of 0.5 ps (see Fig. 3b). The relative phases between configurations with a different number of excitations in the same mode determine the nonlinear phase \phi_{NL} to be applied. In contrast, the relative phases between configurations with the same number of excitations correspond to the relative linear phase \phi between the modes. We then programme the detuning \Delta to match the required nonlinear phase according to the relation characterised in Fig. 2c. At the same time, the linear phase \phi is programmed by reconfiguring the linear time-bin interferometers. The second interferometer transforms the eigenmodes back into localised modes, and photon counting at the output provides the statistics of the excitations in the localised modes after the evolution. The results are shown in Fig. 3c, where we report the probabilities of the phonons in the final molecular state occupying different local modes (top panel) or being bunched in the same mode as the input local mode (bottom panel). The results closely follow the theoretical evolution of the anharmonic model (solid line), which differs significantly from the predictions within the harmonic approximation (dashed line). The applied nonlinear phase shift at each time step is shown by the colour of the data points. We note that the most significant differences between experiment and theory are found at the time instances where the most substantial nonlinear phase shift is required, which originate from imperfections in the scattering process due to a finite quantum cooperativity. This discrepancy occurs around 0.1–0.15 ps, which is where the required nonlinear phase is highest, approximately \pi/2. Due to the presence of minor imperfections of the devices, notably weak residual spectral diffusion, these extreme nonlinear phase shift values are out of range for our device (see Fig. 2c), which is the main reason for the discrepancy factor. Future experiments can overcome residual spectral imperfections by implementing feedback to the electrical bias of the quantum dot device at the millisecond timescale typical of spectral wandering in quantum dots44. Finally, at the points of highest nonlinear phase shift, the nonlinear loss is also highest, which contributes to the discrepancy. Future improvements to the device include reducing minor residual broadening processes to improve the quantum cooperativity of the QD photon-emitter interface45, enabling chiral operation in glide-plane photonic crystal waveguides46, and improving overall system efficiency for true deterministic operation.

Methods
Programmability of linear and nonlinear operations
We consider input states with two photons in the first spatial input mode, implemented by sending weak-coherent laser pulses in the first mode and conditioning on two-photon detection events at the SNSPDs. The input pulses are created by modulating a narrowband continuous-wave (CW) laser through a lithium niobate intensity-electro-optic-modulator (EOM) driven by an arbitrary waveform generator that generates Gaussian pulses of duration 300 to 700 ps at a repetition rate of 38.5 MHz, i.e., slower than twice the temporal distance between the two time bins in order to separate neighbouring experiments in time. The coherent state is attenuated to n < 0.1 photons per pulse on average at the QD to suppress noise due to multi-photon components. In- and out-coupling from the GaAs chip is achieved with an efficiency of approximately 60%, which can readily be improved further by optimised gratings35 or by utilising evanescent coupling to a tapered optical fibre62. Further details on the experimental apparatus can be found in Supplementary Note 1.

The nonlinear interaction arises from the scattering process between the light pulse and the quantum emitter. This interaction will depend on the number of photons in the incoming pulse. The strength and exact dynamics of the process can be tuned by controlling the detuning between light and emitter, and by changing the temporal width of the incoming pulse. When truncating to 2 photons, this interaction is Kerr-like. See Supplementary Note 2 for more details on scattering. To programme the photon-photon nonlinearities, we use two control parameters: (i) the spectral detuning \Delta between the QD resonance and the photon frequency, and (ii) the temporal bandwidth of the photons relative to the intrinsic linewidth, \sigma, of the QD resonance (corresponding to a lifetime 2\pi/\sigma = 311 ps for the QD used in this work). The spectral detuning (i) is tuned by controlling the DC voltage applied to the QD, which shifts its resonance due to the DC Stark effect63. The bandwidth (ii) is adjusted by changing the duration of the modulation applied to the EOM via the arbitrary waveform generator. In both cases, the interaction is strongest when the pulses are resonant with the QD and when the bandwidth of the pulses matches or is narrower than the QD linewidth. The linear operations are programmed by tuning the reflectivity and relative phase of the self-stabilising temporal interferometers by rotating the polarisation in each arm of the interferometer64.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Nielsen et al. present a method to introduce tunable nonlinear operations in photonic circuits. They use a quantum dot embedded in the quantum circuit to scatter photons, with the scattering depending on detuning of the photon and quantum dot. They demonstrate that their setup works across a variety of linear and nonlinear parameters, including for the vibrational dynamics of H2O.\n\nThis work is an important development that expands the toolbox of photonic circuits. Its potential uses span multiple fields of physics and chemistry, and it is suitable for the broad readership of Nature Communications. However, I have a few questions and comments to be addressed prior to publication.\n\nThe details of the experimental setup are beyond my expertise, but they look sound to me. My questions focus largely on the details of the anharmonic vibrational dynamics simulation.\n\nIn figure 3, the authors state that the \"data are fitted to the theoretical model\". If there is fitting, what are the degrees of freedom for the fits? On a related note, the timescales are presumably those of the molecule. That, however, is not immediately obvious since the simulator frequencies can differ.\n\nThe authors implement an \"anharmonic coupling parameter\" chi_ij. Is the parameter found from ab initio data or fitted to experiential data? In either case, the source of model parameters is not clear and should be cited appropriately.\n\nThe authors state assumptions that the bending mode is uncoupled from bond stretches, and that \"molecules are often excited in modes localized around a single atom or bond\". In fact, by symmetry the symmetric stretch and bending mode are weakly coupled, whereas (in the absence of vibronic coupling) the asymmetric stretch is uncoupled from the others. Furthermore, excitation of a small symmetric molecule such as H2O would typically not excite a local mode. While the model system is a valid one, the justifications for it need some modification.\n\nIt would be good if the authors expanded on the utility of simulations of vibrational dynamics. The primary application is likely for the prediction of vibrational spectra, which would require a method to find the autocorrelation function. Using this idea to study reaction dynamics (i.e. the cleavage and formation of bonds) is also appealing.\n\nA final minor comment: All appendices appear to have been moved to supplemental information, but the section labels are very mysterious.", "reviewer_2": "The manuscript of Kasper H. Nielsen et al reports on the realization of a two-mode programmable nonlinear quantum photonic circuit that integrates deterministic photon-photon interactions with off-chip reconfigurable linear optical operations. The nonlinearity is achieved via scattering of single photons off a quantum dot embedded in a photonic crystal waveguide, while the linear transformations are implemented using an external time-bin interferometry (path-unbalanced Mach-Zehnder interferometer) with active control of phase and reflectivity.\n\nThe circuit enables separate and joint programmability of linear and nonlinear operations by tuning the quantum dot detuning (voltage control) and the temporal profile of the input weak, coherent pulses. The authors characterize the response of the system by measuring joint temporal intensity correlations (using 2 detectors per spatial output mode of the time-bin interferometer, and so, implementing the reconstruction of the photon number states up to 2) and output photon statistics under varying configurations of the control parameters.\n\nAs a demonstration of functionality, the platform is used to implement a simulation of anharmonic vibrational dynamics in a water molecule, reproducing part of the results of Sparrow et al, Nature 557, 660 (2018) (which used heralded single photon sources and integrated photonic circuits). This is achieved by encoding vibrational eigenmodes into photonic modes and reproducing the required Hamiltonian evolution using a sequence of linear transformations and nonlinear phase shifts. The results are compared with theoretical predictions based on both harmonic and anharmonic models. The Authors also claim a much higher efficiency on the molecular dynamics simulation than Nature 557, 660 (2018).\n\nFollowing the previous results of the group, in particular, Nat. Phys. 18, 1191 (2022), Nat Commun 15, 7583 (2024), and Nat. Phys. 20, 1429 (2024), I think that the paper provides a significant step in harnessing the non-linear interactions produced by quantum dots, and generally, is interesting for the community in waveguide QED setting a clear direction to go towards the challenging task of scaling the system to implement relevant simulations.\n\nBefore accepting the paper for publication, the Authors are kindly requested to address the following questions, applying all the corresponding modifications in the main text (or supplementary).\n- There is a problem in the submission files main and supplementary material have problems with the sections and references indexing.\n- Comment about the applicability, scalability and relevance of the work. The Authors write several sentences:\n\"Our work extends pioneering previous work based solely on linear optics and probabilistic measurement-induced nonlinearities [32, 33], thereby paving a way to scaling photonic quantum devices\"\n\"Extending the platform to more than two modes is required for more complicated molecular simulations or Bell-state analyzer experiments.\"\n\"In this respect, the ability to perform direct nonlinear operations could significantly increase the computational complexity of the simulated problem, thereby reducing the number of photons and hardware performance requirements needed to reach the quantum advantage regime [42].\"\n\"However, the scalability of the nonlinearity itself is still an open question.\"\n\"The demonstrated capability to combine linear and nonlinear operations in a photonic device unlocks new opportunities for photonic quantum technologies.\"\nThe main text requires a quantitative discussion about the molecular complexity that their QD-waveguide device offers to simulate complex systems in the near and long term (supported by a table, involving parameters of the system, such as beta, phi_NL, number of identical sources, integrated MZIs on chip, losses, etc).\nConsidering this discussion, is this platform a candidate for reaching computational advantage over classical or quantum systems (photonic, such as Nature 557, 660 (2018) or other physical platforms)? I guess that the answer from the Authors will be yes, then, this should be supported by a critical, comparative analysis of current and potential performance of the system. The Authors did something similar in their previous publication Sci. Adv. 6, eabc8268 (2020). Now, Fig. 2d seems a bit weak/disconnected about the scalable and simulation capabilities prospects of their device. This comment is also resonant with the sentence: “the success probability for the nonlinear approach is found to significantly outperform that of the measurement-based scheme based on linear optics (dashed black line in Fig. 2d) by Sparrow et al”.\n- The Authors write: \"Future experiments can overcome residual spectral imperfections by implementing feedback to the electrical bias of the quantum dot device.\" Can the authors indicate what is the required time scale (<microseconds?) on voltage tuning to compensate for these imperfections and whether the spatial dimensions of their chip allow for such dynamical feedback? I could not find any reference about this feedback correction on the emitter spectrum.\nOther comments:\n- The authors are invited to explicitly indicate in the main text the delay used in the Mach-Zehnder interferometer. One has to check in the supplementary that it is: \"corresponding to a 3 ns delay between the two arm\".\n- The Authors write: \"The coherent state is attenuated to n < 0.1 photons per pulse on average at the QD to suppress noise due to multi-photon components\", for the sake of clarity, maybe the authors can point to the reference where this estimation is explained in a previous implementation, and/or give a brief description in the S.I.\n- Could the authors explain why in Fig 2d, below phi_NL=pi/16 the two data points of P1 and P2 probabilities drop simultaneously?\n- Figure 1a shows 1 detector per output spatial mode of the MZI, it would be helpful to make this drawing (2 detectors per spatial mode) compatible with the sentence in the caption of Fig. 1a: “The transmitted photons are finally recorded with pseudo-photon-number-resolution using SNSPDs”.\n- In Fig. 1c, the output of the second fiber loop presents a single output. I apologise for the following comment: it is not clear to me what the grey box represents, is it a “deterministically switchable” beamsplitter? If so, I have concerns about its fidelity to the experimental setup. The setup has a probabilistic interference of E-L modes. The output of this fiber-loop should lead to the pseudo-photon-number-resolution detection: could the authors help the reader to link the correlations done in Fig. 1a to those that should be done in the output of Fig. 1c?\n- The Authors write \"(both output photons in the second mode)\", when mentioning \"modes\" along the paper (main and supplementary), it would be helpful to the reader to explicit temporal (E-L) or spatial (a-b).", "reviewer_3": "Dear Editor,\nThe manuscript Programmable Nonlinear Quantum Photonic Circuits demonstrates the operation of a programmable photonic quantum circuit in the optical regime which implements tunable linear and non-linear operations at the single photon level. The linear operations are performed with two Mach-Zehnder interferometers, while the non-linearities are realized by means of a quantum dot (QD) embedded in a photonic crystal waveguide. Importantly, the non-linear gate introduces a Kerr-like non-linear phase at the single photon level that is tunable, i.e., its value can be dynamically changed by adjusting the spectral detuning Delta between the QD resonance and the photon frequency, as well as the temporal bandwidth of the photons relative to the linewidth of the QD resonance. To showcase the power of their platform, the authors realized the quantum simulation of the vibrational dynamics of phonons in the water molecule, whose Hamiltonian is anharmonic and thus requires nonlinear terms to be simulated. In previous works, like Sparrow et al. [32], the non-linearities necessary for this simulation were implemented through measurements and ancillary photons, producing a large overhead in terms of additional optical modes. However, in the present manuscript the authors carry out this simulation in a deterministic way thanks to the strong photon-photon non-linearities provided by the QD, resulting in a more scalable platform. Overall, the manuscript is very well written, clear for the most part, and the methodology and results seem sound and convincing. The application of the strong tunable photonic non-linearities at the single-photon level in the optical regime to these quantum simulation purposes is indeed groundbreaking and represents an important milestone for the field. As you could read from that statement, I am overall positive about the manuscript, however I have some questions that I would like the Authors to address, and maybe modify the text, before giving my final positive recommendation:\n\n● Regarding the deterministic nature of the protocol: it is somehow emphasized in the text, e.g., in the abstract (“ Deterministic nonlinear interaction is realized”) that the process is deterministic, since a lot of emphasis is made comparing it with the large overhead of probabilistic processes (see second paragraph of intro). I agree with that argumentation, however, after reading the text, I saw some subtleties that the authors may want to comment or emphasize in a new version of the text, to maybe be more honest about the novelty and/or limitations of the protocol:\n- As they rightly say, they use weakly attenuated coherent states to make the “simulation”. That means that they have to condition on two-photon detection events. This means that at least the initialization of the process is probabilistic. The authors acknowledge that, but they do not comment on whether this poses some limitations on the scalability when more photons have to be included in the simulation. Is this an issue for the simulations?\n- The key part of the protocol, once these states are generated, is how to imprint linear and non-linear phase shifts with their non-linear interferometer. In particular, how to extract the different control terms in the single and two-photon mappings of Eqs. 1-2, which is discussed in Figure 2. The authors emphasize the deterministic nature of the process, but at the same time include and discuss in Fig. 2d the existence of a finite success probability. It is true that they say that this can be overcome with chiral waveguides. Plus, the Authors recognize that in- and out-coupling from the GaAs chip is achieved with a fidelity of about 60%. Again, they explain how this number could be improved. However, I think it will be fairer to the reader to differentiate between what has been done experimentally in this work, and the potential outlook on how to circumvent these limitations. For me, the fact that they outperform the measurement-based schemes of Sparrow et al. is already interesting enough.\n- Another doubt that I have is about the comparison of the use of non-linear phase shifts for photonic quantum computing and simulation. In photonic quantum computing, one wants the spectral shape of the photons to remain unaltered when it scatters the emitters. However, here it seems from the JSI that, especially in the resonant cases, it gets distorted. Could the Authors elaborate why this does not matter here, and they can still talk about |1> and |2> as if they were single-mode wavepackets? In the Appendix, they mention it, and just define new operators “a’” when they do input/output theory, is it as simple as that? Would it be an issue when you want to feed these distorted photons again into the interferometer?\nIn these three points, I might be misunderstanding something, but if not, I'd like the Authors to consider clarifying their writing in the new version.\n\n● Regarding the scalability of the process. Speaking in simple terms, they use a well-known mapping of the vibrational modes of the H2O molecules to two harmonic oscillators with two phonons. The two harmonic oscillators are here the two arms of the interferometers, while the number of photons encode the number of phonons into the problem. Although a nice demonstration, it would be interesting to see a more serious discussion of the challenges (or not) of scaling the protocol to simulate larger molecules.\n\n● In the discussion section, they mention that to scale it would require “temporal encoding allowing for a highly resource-efficient realization since multiple nonlinear operations are encoded by reusing the same quantum emitter at different times”. However, this raises for me a question regarding the programmability of the parameters. In this work, they can reconfigure the circuit, e.g., the energy of the emitters, via DC Stark shift. This is also demonstrated in Appendix. I understand this can be done for the particular scenario considered here: input photons and analyze the output after only one pass through the non-linear interferometer. However, if one wanted to send back the photons with a feedback loop, would it be possible to change the parameters of the non-linear circuits before photons enter again in the interferometer? The same with the other parameters. What are the technological challenges that need to be solved for that? How many rounds can the photons be sent through the non-linear interferometer before losing coherence?\n\nApart from that, there are other minor corrections that the authors may want to address:\n\n● How does the nonlinear transformation of the scattering process in Eqs. (1-2) map to the standard Kerr non-linearity with Hamiltonian H = K * n^2 (with n = a\\dagger * a the number operator)? Maybe it can be made more explicit. Is this only valid for 2 photons? What happens for a larger photon number? Do you get a non-linearity of other types?\n\n● There is a mistake (the interrogation marks “??”) in the first paragraph of section “Programmable linear and nonlinear photonic circuits” in page 2: “...see Fig. 1b, and Appendix ?? for more details…”. In general, in all references to the Appendix there is the same issue.\n\n● In the top of the right column of page 6, it reads “these extreme nonlinear phaseshift values are our of range for our device (see Fig. Fig 2.c)”. Shouldn’t it read “out of range”?\n\n● There’s an erratum at the bottom of the left column in page 2 where it reads “…corresponding to the first mode and condition on two photon detection…”. Shouldn’t it read “conditioned”?\n\n● The Authors may want to expand the outlook including other potential use of their programmable non-linearities, such as quantum metrology, where there are recent works along this direction, e.g., Phys. Rev. Research 6, 013299 (2024), Nanophotonics 2025; 14(11): 2075–2085, arXiv:2410.11158.\n\n● The explanation of Fig. 2a in the main text is obscure for me: How does the relative sign of Delta and sigma affect the type of correlations that are observed? Where are the side lobes? Where can I see the |11> term? Maybe the authors can point out these features in the joint temporal intensity (JTI) patterns shown in Fig. 2a, using arrows or encircling them.\n\n● AWG is only used twice, does it need an acronym?"}, "author_response": "Response to Reviewers\n• The reviewer’s comments are presented in blue and quoted.\n• Our responses are in black\n• The updates in the resubmitted manuscript are colored red both here and in the revised version of the\nmanuscript.\nReviewer #1\nNielsen et al. present a method to introduce tunable nonlinear operations in photonic circuits. They use a quantum\ndot embedded in the quantum circuit to scatter photons, with the scattering depending on detuning of the photon\nand quantum dot. They demonstrate that their setup works across a variety of linear and nonlinear parameters,\nincluding for the vibrational dynamics of H2O.\nThisworkisanimportantdevelopmentthatexpandsthetoolboxofphotoniccircuits. Itspotentialusesspanmultiple\nfields of physics and chemistry, and it is suitable for the broad readership of Nature Communications. However, I\nhave a few questions and comments to be addressed prior to publication.\nThe details of the experimental setup are beyond my expertise, but they look sound to me. My questions focus\nlargely on the details of the anharmonic vibrational dynamics simulation.\nWethankReviewer#1fortheinsightfulreviewofourpaperandgreatlyappreciatethereviewer’srecognitionofthe\nimportance of our work. In the following, we address each point individually.\n1. In figure 3, the authors state that the ”data are fitted to the theoretical model”. If there is fitting, what are\nthe degrees of freedom for the fits? On a related note, the timescales are presumably those of the molecule.\nThat, however, is not immediately obvious since the simulator frequencies can differ.\nWe thank the reviewer for pointing out that phrase, which contains a potentially confusing statement. In the\nprevious version of the phrase, Fitting was used improperly. What we meant is that we compare the data\nobtained from the experiment to the ideal theoretical models by plotting them together on the same figure\n(Fig.3c). We have now modified the phrase to clarify this:\nThe data are compared to the theoretical model (solid lines), and compared also to the predictions using the\nharmonic approximation (dashed lines).\nThe theoretical models are obtained by the analytical time evolution of the state of the molecule, possible for\nthe small-sized system considered here. The timescale is set by the eigenvalues of the vibrational Hamiltonian,\nas the reviewer correctly points out. The timescale is mapped to the quantum simulator by the formula that\nis now written in Appendix E2, where we have added the following section to clarify these points:\nThe time evolution of the Hamiltonian in the eigenbasis is given by the unitary evolution\nU(t)=eiνt/h¯, (1)\nwhere the eigenvalues from equation (1) has to be converted from cm−1 to frequencies. The phase acquired by\neachFockstatecomponentthusdependsontheevolutiontimeanditsassociatedeigenvalue. Theanharmonicity\nis manifested by having a nonlinear offset ϕ in the acquired phase of the one- and two-photon components,\nNL\nwhich we write as:\nϕ →2ϕ+ϕ ,\n|20⟩ NL\nϕ →ϕ , (2)\n|02⟩ NL\nϕ →ϕ,\n|11⟩\nwhere, for a given evolution time t, we have ν t=ϕ . The linear and nonlinear phases are programmable as\n|i⟩ |i⟩\nexplained in the main text. Along with the degree of freedom of the global phase, these two parameters allow\nus to match the equations in equation 13 for an arbitrary time evolution.\n2. The authors implement an “anharmonic coupling parameter” χ . Is the parameter found from ab initio data\nij\nor fitted to experiential data? In either case, the source of model parameters is not clear and should be cited\nappropriately.\nWe thank the reviewer for this comment. The anharmonic Hamiltonian model we use is from Ref. [8], where\nit was obtained from ab initio data. We now specify this in the dedicated Appendix E:\nThe frequencies of the single excitations of the eigenmodes, obtained from ab initio calculations in Ref. [8], are\ngiven by...\n1\n3. The authors state assumptions that the bending mode is uncoupled from bond stretches, and that “molecules\nareoftenexcitedinmodeslocalizedaroundasingleatomorbond”. Infact,bysymmetrythesymmetricstretch\nand bending mode are weakly coupled, whereas (in the absence of vibronic coupling) the asymmetric stretch is\nuncoupledfromtheothers. Furthermore,excitationofasmallsymmetricmoleculesuchasH2Owouldtypically\nnotexcitealocalmode. Whilethemodelsystemisavalidone,thejustificationsforitneedsomemodification.\nWethankthereviewerforelaboratingonthedetailsofquantumchemistry. Themodelweused,whichdescribes\ncouplingbetweensymmetricandasymmetricvibrationalmodes,aswellastheanharmonicity,istakenfromthe\nliterature [11, 8]. The degree of accuracy with which it describes the vibrational processes in water molecules\nin practical experiments is not our core expertise, and we appreciate the insight from an expert in the field.\nThe main argument of our paper is to demonstrate the controllability of the linear and nonlinear operations\nin our setup, and the quantum simulation is mostly meant as a proof-of-concept demonstration of what new\ncapabilities are enabled by our technology. Ours is thus just an example, and we do not wish to claim that it\ngenerally models the most representative scenario.\nWe have now modified the manuscript to remove the parts highlighted by the reviewer, and now present the\nmodel in general terms without claiming that it represents typical scenarios in practice:\n... and a low-energy bending mode, which in this model is uncoupled from the others [11, 8]. We describe the\nmolecule in the basis of local modes and normal modes. The local modes are localized around singular atoms\nor chemical bonds, and here the localized modes are represented by stretch modes of the individual left and\nright H atoms, as shown in Fig. 3b.\n4. It would be good if the authors expanded on the utility of simulations of vibrational dynamics. The primary\napplication is likely for the prediction of vibrational spectra, which would require a method to find the auto-\ncorrelation function. Using this idea to study reaction dynamics (i.e., the cleavage and formation of bonds) is\nalso appealing.\nWe thank the reviewer for the remark about the outlook of bosonic devices used for quantum chemistry. We\nagree that potential applications, while still speculative at this point, could be of interest to the readers. In\nthe updated discussion, we now comment on these aspects, including citing a recent review paper that nicely\ndescribes the potential applications of bosonic quantum devices (Ref. [2]):\nExtensions to simulators at increasing scale with this technology could be useful to investigate, for instance,\nvibrational and vibronic spectra in larger molecular systems and compounds [2].\n5. A final minor comment: All appendices appear to have been moved to supplemental information, but the\nsection labels are very mysterious.\nWe thank the reviewer for the comment. This was a compilation error which has now been fixed.\nReviewer #2\nThemanuscriptofKasperH.Nielsenetalreportsontherealizationofatwo-modeprogrammablenonlinearquantum\nphotonic circuit that integrates deterministic photon-photon interactions with off-chip reconfigurable linear optical\noperations. The nonlinearity is achieved via scattering of single photons off a quantum dot embedded in a pho-\ntonic crystal waveguide, while the linear transformations are implemented using an external time-bin interferometry\n(path-unbalanced Mach-Zehnder interferometer) with active control of phase and reflectivity.\nThe circuit enables separate and joint programmability of linear and nonlinear operations by tuning the quantum\ndot detuning (voltage control) and the temporal profile of the input weak, coherent pulses. The authors characterize\nthe response of the system by measuring joint temporal intensity correlations (using 2 detectors per spatial output\nmode of the time-bin interferometer, and so, implementing the reconstruction of the photon number states up to 2)\nand output photon statistics under varying configurations of the control parameters.\nAs a demonstration of functionality, the platform is used to implement a simulation of anharmonic vibrational dy-\nnamics in a water molecule, reproducing part of the results of Sparrow et al, Nature 557, 660 (2018) (which used\nheraldedsinglephotonsourcesandintegratedphotoniccircuits). Thisisachievedbyencodingvibrationaleigenmodes\ninto photonic modes and reproducing the required Hamiltonian evolution using a sequence of linear transformations\nand nonlinear phase shifts. The results are compared with theoretical predictions based on both harmonic and\nanharmonic models. The Authors also claim a much higher efficiency on the molecular dynamics simulation than\nNature 557, 660 (2018).\nFollowingthepreviousresultsofthegroup,inparticular,Nat. Phys. 18,1191(2022),NatCommun15,7583(2024),\nand Nat. Phys. 20, 1429 (2024), I think that the paper provides a significant step in harnessing the non-linear\ninteractions produced by quantum dots, and generally, is interesting for the community in waveguide QED setting a\nclear direction to go towards the challenging task of scaling the system to implement relevant simulations.\napplying all the corresponding modifications in the main text (or supplementary).\nWethankReviewer#2forthereviewofourpaperandfortheirconstructivecomments. Weappreciatetherecognition\nof the quality and prospects of our work.\n1. There is a problem in the submission files main and supplementary material have problems with the sections\nand references indexing.\nWe thank the reviewer for the comment. This was a compilation error that has now been fixed.\n2. Comment about the applicability, scalability and relevance of the work. The Authors write several sentences:\n“Our work extends pioneering previous work based solely on linear optics and probabilistic measurement-\ninduced nonlinearities [32, 33], thereby paving a way to scaling photonic quantum devices” “Extending the\nplatformtomorethantwomodesisrequiredformorecomplicatedmolecularsimulationsorBell-stateanalyzer\nexperiments.” “In this respect, the ability to perform direct nonlinear operations could significantly increase\nthecomputationalcomplexityofthesimulatedproblem,therebyreducingthenumberofphotonsandhardware\nperformance requirements needed to reach the quantum advantage regime [42].” “However, the scalability of\nthenonlinearityitselfisstillanopenquestion.”“Thedemonstratedcapabilitytocombinelinearandnonlinear\noperations in a photonic device unlocks new opportunities for photonic quantum technologies.” The main text\nrequires a quantitative discussion about the molecular complexity that their QD-waveguide device offers to\nsimulatecomplexsystemsinthenearandlongterm(supportedbyatable,involvingparametersofthesystem,\nsuch as β, ϕ , number of identical sources, integrated MZIs on chip, losses, etc).\nNL\nConsidering this discussion, is this platform a candidate for reaching computational advantage over classical\nor quantum systems (photonic, such as Nature 557, 660 (2018) or other physical platforms)? I guess that\nthe answer from the Authors will be yes, then, this should be supported by a critical, comparative analysis\nof current and potential performance of the system. The Authors did something similar in their previous\npublicationSci. Adv. 6, eabc8268(2020). Now, Fig. 2dseemsabitweak/disconnectedaboutthescalableand\nsimulationcapabilitiesprospectsoftheirdevice. Thiscommentisalsoresonantwiththesentence: “thesuccess\nprobability for the nonlinear approach is found to significantly outperform that of the measurement-based\nscheme based on linear optics (dashed black line in Fig. 2d) by Sparrow et al”.\nWethankthereviewerforthecommentandforstimulatingtheelaborationonthescalabilityofthetechnology.\nFollowing the reviewer’s suggestion, we have now performed a detailed analysis of what system scales could\nbe accessible for our approach with current and near-term technologies. The analysis is reported in the newly\nadded Appendix F of the updated manuscript, where in Table F2 we report the estimated detection rates for\nconstructing programmable nonlinear circuits with quantum emitters and time-bin encoding, using our same\nscheme but for a larger number of modes and photons. The analysis is done assuming both components with\nthe same performance as in our experiment, as well as components with state-of-the-art performance. In this\nanalysis, we show that experiments with approximately 20 modes and processing > 10 photons should be\navailable with the current state-of-the-art technology. Nonlinear quantum photonic systems at such a scale\nare already challenging to simulate on classical computers using current methods for nonlinear boson sampling\nsimulation [10]. However, a rigorous assessment of the computational complexity for classically simulating\nnonlinear quantum photonic circuits is an open problem. More sophisticated methods could be designed for\nfaster classical simulations compared to current techniques, e.g., via recently developed techniques based on\ntensor networks [16]. We believe our work could be of significant interest to theorists to motivate further\ninvestigation of this problem.\nThese results and discussion are now reported in detail in Appendix F:\nAppendix F: Estimation of system scalability with current and near-term technologies\nundergoscatteringfromaQDtoimplementthenonlinearity, followedbyaClements-styleinterferometerandfinally\nphoton number detection. The depth of a the interferometer is on the order of the number of modes.\nTechnology β η η |α|2 # modes # photons Rate (HZ)\nMZI coupling\n2 3e+3\n• Two-sided photonic crystal waveguide\n0.9 0.5 0.6 0.1 2 3 4e+1\n• Efficiencies from this work\n5 <1e-1\n2 3e+4\n• One-sided photonic crystal waveguide [15]\n0.9 0.5 0.6 0.1 2 3 3e+2\n• Efficiencies from this work\n5 <1e-1\n2 8e+5\n2 3 4e+4\n5 3e+1\n0.1\n2 3e+6\n10 3 3e+5\n•One-sidedphotoniccrystalwaveguide 5 1e+3\n0.98 0.9 0.85\n• State of the art efficiencies [7, 19, 1] 10 7e+6\n2 10 20 3e+2\n30 <1e+1\n10 7e+7\n4 20 20 1e+6\n30 <1e+1\nTable1: Estimationofdetectionratesforthecurrent-andfutureexperimentsusingstate-of-the-artequipment[7,19,\n1]. β isthecouplingefficiencyoftheQDtothewaveguidemode,η istheefficiencyoftheinterferometer,η\nMZI coupling\nis the coupling from chip to interferometer, |α|2 is the mean photon number. The detection rates are obtained by\nassuming an input rate of 100MHz, and sampling from Poissonian distributions determined by input mean photon\nnumbers and system efficiency. For 2-modes, the number of samples is 1e9 and for all other configurations, it is 1e7.\nScaling the current platform to larger numbers of modes and photons is an interesting research direction. It\nwould be challenging with the current bulky implementation of the linear interferometer but integration of\nthe non-linear scattering into a time-bin interferometer like the one presented in Ref. [7], would be a viable\noption. The framework is shown in Fig.F14. Estimated photon count rates for different numbers of modes are\nsummarized in Table 2a. In order to make complexity estimates, we assume that the nonlinearity works as\na single-mode Kerr nonlinearity for simplicity. In practice, higher-order photon scattering will result in even\nmore multimode distortion of the wavepackets leading to partial distinguishability of the interference.These\neffects are neglected in the present discussion, as they represent a technical challenge that can be mitigated\nwith appropriate engineering strategies. The β-factor is the coupling of the QD to the waveguide, which\nwill lead to losses if less than one. The η is the transmission of the a singular MZI in the setup. See\nMZI\nthe number of modes. The η is the coupling between QD waveguide and interferometer. |α|2 is the\ncoupling\nmean photon number. For a two-sided device, 50% of the photons are lost due to propagation directionality.\nDue to this limitation, considering the same component performance as in our experiment, we estimate that\nnonlinear quantum photonic circuits with 5 photons in 2 modes are available, but scaling beyond that would\nbechallenging. Afirstmitigationforthelimitedefficiencycanbeobtainedbyconsideringsingle-sidedorchiral\ndevices for interfacing the light-matter interaction, where, as discussed in the main text, we can assume no\nlossfromthescatteringprocess. AsshowninTableF1, thisimprovesexpectedratessignificantly, butpresents\nstill limited scalability due to the dominant losses from the increased optical depth of the interferometer when\ntrying to increase the number of modes. To surpass these limitations and scale the number of modes, reducing\ncomponentlossesisakeyrequirement. Whenconsideringstate-of-the-artefficienciesfortheopticalcomponents,\nwe indeed see that experiments with > 10 modes and > 10 photons become available. When also increasing\ntheamplitudeoftheinputweak-coherentstatecomparedtoourimplementation,weestimatethatexperiments\nwith approximately 20 modes and >10 photons are available with current technology.\nWe note that nonlinear quantum photonic systems at such a scale are already challenging to simulate on\nclassical computers using current methods for nonlinear boson sampling simulation [10]. However, a rigorous\nassessment of the computational complexity for classically simulating nonlinear quantum photonic circuits is\nan open problem. More sophisticated methods could be designed for faster classical simulations compared to\ncurrent techniques, e.g., via recently developed techniques based on tensor networks [16].\n3. The Authors write: “Future experiments can overcome residual spectral imperfections by implementing feed-\nbacktotheelectricalbiasofthequantumdotdevice.”Cantheauthorsindicatewhatistherequiredtimescale\n(<microseconds?) onvoltagetuningtocompensatefortheseimperfectionsandwhetherthespatialdimensions\nof their chip allow for such dynamical feedback? I could not find any reference about this feedback correction\non the emitter spectrum.\nWe thank the reviewer for raising this relevant point. The spectral wandering induced by charge fluctuations\ntypically manifests as telegraph noise on the millisecond timescale, which is thus the timescale required for the\nelectrical feedback. This noise mitigation strategy has been experimentally demonstrated in quantum dots in\nthe Atatu¨re group (Ref. [4]), which is now cited in the updated sentence:\nFuture experiments can overcome residual spectral imperfections by implementing feedback to the electrical\nbias of the quantum dot device at the millisecond timescale typical of spectral wandering in quantum dots [4].\n4. The authors are invited to explicitly indicate in the main text the delay used in the Mach-Zehnder interfer-\nometer. One has to check in the supplementary that it is: “corresponding to a 3 ns delay between the two\narm”.\nWethankthereviewerforthisclarifyingcomments. Wenowexplicitlystatethetimedelayoftheinterferometer\nin the main text:\n... with a 3 ns delay between the two time-bins.\n5. The Authors write: “The coherent state is attenuated to n < 0.1 photons per pulse on average at the QD to\nsuppress noise due to multi-photon components”, for the sake of clarity, maybe the authors can point to the\nreference where this estimation is explained in a previous implementation, and/or give a brief description in\nthe S.I.\nWethankthereviewerforthevaluablesuggestion. Wehavecalculatedtheaveragephotonnumberperemitter\nlifetime based on the corresponding excitation power and included the results in Table A1. We added the\nfollowing section in the associated Appendix A to elaborate on it:\nThe optical powers reported in Table A1 have been calibrated to account for the efficiencies of all associated\nopticalcomponents. Fromthesecalibratedvalues, wedeterminethephotonflux-definedastheaveragephoton\nnumber per emitter lifetime [14]. As discussed previously, this value remains below 0.1, ensuring that the\ninteraction is maintained in the weak regime during the measurements. This is further supported by the\ntransmissionspectra,whichshowneitheranincreaseintheminimumtransmissionnoranylinewidthbroadening\nwith increasing input power.\n6. Could the authors explain why in Fig 2d, below ϕ = π/16 the two data points of P1 and P2 probabilities\nNL\ndrop simultaneously?\nWe thank the reviewer for stimulating clarification on the effects of noise in the data. Indeed, the mismatch\nbetween the data and theory for the data points at ϕ ≃ π/16 in Fig.2d is due to noise in our measure-\nNL\nments. The probabilities of one-photon (P1) transmission after scattering are estimated based on a resonant\ntransmission (See Fig.A5 in the supplementary). The output can sometimes be noisy due primarily to spectral\n5\ndiffusion, which causes the transmitted power to appear fluctuating rather than the expected smooth Voigt\nfunction. The estimation of P2 is determined from the nonlinear transmission and the value of P1. Thus,\nthey both appear to drop simultaneously. If we were to integrate the resonant transmission measurements for\nlonger timescales, these fluctuations would be reduced, and we would get cleaner data. This discussion is now\nincluded in the relevant Appendix C2 to clarify the mismatch for these few data points:\n7. Figure 1a shows 1 detector per output spatial mode of the MZI, it would be helpful to make this drawing (2\ndetectors per spatial mode) compatible with the sentence in the caption of Fig. 1a: “The transmitted photons\nare finally recorded with pseudo-photon-number-resolution using SNSPDs”.\nWethankthereviewerforthesuggestion. GiventhatFig.1aisquitedense,toavoidclutteringitexcessivelywith\nthe proposed circuitry, we added a label in the figure to state that the detectors are pseudo-number-resolving\nand directly refer to it in the figure caption:\nThe transmitted photons are finally recorded with pseudo-photon-number-resolution (pPNR) using SNSPDs.\n8. -InFig. 1c,theoutputofthesecondfiberlooppresentsasingleoutput. Iapologiseforthefollowingcomment:\nit is not clear to me what the grey box represents, is it a “deterministically switchable” beamsplitter? If so,\nI have concerns about its fidelity to the experimental setup. The setup has a probabilistic interference of E-L\nmodes. The output of this fiber-loop should lead to the pseudo-photon-number-resolution detection: could the\nauthors help the reader to link the correlations done in Fig. 1a to those that should be done in the output of\nFig. 1c?\nWe thank the reviewer for the comment. The grey boxes do indeed represent a programmable beam splitter.\nIts functioning and characterisation are detailed in Appendix A, where we measure a fidelity >97%. We have\nexpanded the appendix to clarify this point:\nA high contrast of 97.12%, which corresponds to the interference visibility, is observed at the zero phase ϕ ∼\n0\n380◦, corresponding to a >97% fidelity for the performed switching operation.\nFurthermore, we have expanded the caption of Fig. 1 to elaborate on the connection between the schematics\nin Fig. 1a and Fig. 1c:\nThe pairofopticalmodesrepresentedina)arehereencodedinthetemporaldegreeoffreedomofthephotons,\nwhere the first optical mode corresponds to an early (E) time-bin and the second to a late (L) time-bin.\n9. The Authors write “(both output photons in the second mode)”, when mentioning “modes” along the paper\n(main and supplementary), it would be helpful to the reader to explicit temporal (E-L) or spatial (a-b).\nWe thank the reviewer for this comment on clarity of the manuscript. Both main text and supplementary\nmateriel has been revised to specify which type of mode is considered to avoid ambiguity.\nReviewer #3\nThemanuscriptProgrammableNonlinearQuantumPhotonicCircuitsdemonstratestheoperationofaprogrammable\nphotonic quantum circuit in the optical regime which implements tunable linear and non-linear operations at the\nsingle photon level. The linear operations are performed with two Mach-Zehnder interferometers, while the non-\nlinearities are realized by means of a quantum dot (QD) embedded in a photonic crystal waveguide. Importantly,\nthe non-linear gate introduces a Kerr-like non-linear phase at the single photon level that is tunable, i.e., its value\ncan be dynamically changed by adjusting the spectral detuning Delta between the QD resonance and the photon\nfrequency, as well as the temporal bandwidth of the photons relative to the linewidth of the QD resonance. To\nshowcase the power of their platform, the authors realized the quantum simulation of the vibrational dynamics of\nphononsinthewatermolecule,whoseHamiltonianisanharmonicandthusrequiresnonlineartermstobesimulated.\nInpreviousworks,likeSparrowetal. [32],thenon-linearitiesnecessaryforthissimulationwereimplementedthrough\nmeasurements and ancillary photons, producing a large overhead in terms of additional optical modes. However, in\nthepresentmanuscripttheauthorscarryoutthissimulationinadeterministicwaythankstothestrongphotonpho-\nton non-linearities provided by the QD, resulting in a more scalable platform. Overall, the manuscript is very well\nwritten,clearforthemostpart,andthemethodologyandresultsseemsoundandconvincing. Theapplicationofthe\nstrong tunable photonic non-linearities at the single-photon level in the optical regime to these quantum simulation\npurposes is indeed groundbreaking and represents an important milestone for the field. As you could read from that\nstatement, I am overall positive about the manuscript, however I have some questions that I would like the Authors\nto address, and maybe modify the text, before giving my final positive recommendation:\nWethankthereviewerfortheirtimeandexpertiseinreviewingourwork. Wearegladtheyappreciateitsquality\nand potential impact. We address their constructive comments below and in the revised manuscript.\n6\n1. Regarding the deterministic nature of the protocol: it is somehow emphasized in the text, e.g., in the abstract\n(“ Deterministic nonlinear interaction is realized”) that the process is deterministic, since a lot of emphasis is\nmade comparing it with the large overhead of probabilistic processes (see second paragraph of intro). I agree\nwith that argumentation, however, after reading the text, I saw some subtleties that the authors may want\nto comment or emphasize in a new version of the text, to maybe be more honest about the novelty and/or\nlimitations of the protocol:\n(a) As they rightly say, they use weakly attenuated coherent states to make the “simulation”. That means\nthat they have to condition on two-photon detection events. This means that at least the initialization\nof the process is probabilistic. The authors acknowledge that, but they do not comment on whether this\nposes some limitations on the scalability when more photons have to be included in the simulation. Is\nthis an issue for the simulations?\nWethankthereviewerforthecomment. Wenotethat,inthecontextofthesimulation,ourconfiguration\nwithaweak-coherentstateattheinputdirectlycorrespondstothescenariowherethemolecularvibrational\nmodes are also initialized to a coherent state (and measured in the Fock basis). If such a scenario is\nrelevant, our initialisation thus represents an appropriate mapping and does not introduce limitations.\nHowever, we acknowledge that, in most practical contexts, having a molecule starting in a coherent\nvibrational mode might now be representative of realistic chemical scenarios. In such cases, the relevant\ninput states should be used, e.g., thermal states or Fock states. That would require a different state\ngeneration procedure, which could be more complex compared to our particular implementation, such as\nusing a single photon source rather than a simpler weak-coherent input. On the other hand, apart from\ntheimplementationbeingmorecomplex,weexpectthatthereisnofundamentallimitationthatwouldbe\nintroducedwhencombiningsuchdifferentinputsourceswithournonlinearquantumprocessortechnology.\nFor example, single-mode nonlinearities obtained by scattering single photons from a quantum emitter\nthrough a quantum dot have recently been shown [3].\nThis discussion has now been included in the discussion part of the manuscript:\nIn the context of the simulation, our configuration with a weak-coherent state at the input directly\ncorresponds to the scenario where the molecular vibrational modes are also initialized to a coherent\nstate (and measured in the Fock basis). If such a scenario is relevant, our initialisation thus represents\nan appropriate mapping and does not introduce limitations. However, we acknowledge that, in most\npracticalcontexts,havingamoleculestartinginacoherentvibrationalmodemightnowberepresentative\nof realistic chemical scenarios. In such cases, the relevant input states should be used, e.g., thermal\nstates or Fock states. That would require a different state generation procedure, which could be more\ncomplex compared to our particular implementation, such as using a single photon source rather than a\nsimpler weak-coherent input. On theother hand, apartfrom theimplementation beingmore complex, we\nexpect that there is no fundamental limitation that would be introduced when combining such different\ninput sources with our nonlinear quantum processor technology. For example, single-mode nonlinearities\nobtained by scattering single photons from a quantum emitter through a quantum dot have recently been\nshown [3].\n(b) Thekeypartoftheprotocol,oncethesestatesaregenerated,ishowtoimprintlinearandnon-linearphase\nshifts with their non-linear interferometer. In particular, how to extract the different control terms in the\nsingle and two-photon mappings of Eqs. 1-2, which is discussed in Figure 2. The authors emphasize the\ndeterministic nature of the process, but at the same time include and discuss in Fig. 2d the existence of a\nfinite success probability. It is true that they say that this can be overcome with chiral waveguides. Plus,\nthe Authors recognize that in- and out-coupling from the GaAs chip is achieved with a fidelity of about\n60%. Again, they explain how this number could be improved. However, I think it will be fairer to the\nreadertodifferentiatebetweenwhathasbeendoneexperimentallyinthiswork,andthepotentialoutlook\non how to circumvent these limitations. For me, the fact that they outperform the measurement-based\nschemes of Sparrow et al. is already interesting enough.\nWethankthereviewerforthecomment. Weagreewiththeirsuggestionandhavenowremovedreferences\nto the nonlinearities being deterministic to avoid confusion, and kept the discussion on how they can be\nmade near-deterministic in the outlook of the manuscript.\n(c) Another doubt that I have is about the comparison of the use of nonlinear phase shifts for photonic\nquantum computing and simulation. In photonic quantum computing, one wants the spectral shape of\nthe photons to remain unaltered when it scatters the emitters. However, here it seems from the JSI that,\nespecially in the resonant cases, it gets distorted. Could the Authors elaborate why this does not matter\nhere, and they can still talk about |1⟩ and |2⟩ as if they were single-mode wavepackets? In the Appendix,\nthey mention it, and just define new operators “a’” when they do input/output theory, is it as simple as\nthat? Would it be an issue when you want to feed these distorted photons again into the interferometer?\nchange during the scattering process in the supplementary, and shown how this maps to the simplified\nsingle-mode Kerr nonlinearity picture:\nConsequently, the nonlinear operations are implemented via scattering with the QD in each individual\nmode. As noted above, the input is a weak coherent state containing up to two photons, deliberately\nprepared with a Gaussian temporal profile. And the input state |20⟩ can be written in a separable form\nf (ω ,ω )=f (ω )f (ω ). Within the framework of second quantization, the state can be represented as\n2 1 2 1 1 1 2\nan integral over ω ,ω of the photons. Thus the full state before scattering is given by:\n1 2\n(cid:90) 1 (cid:16) (cid:17)\ndω dω f (ω )f (ω )ω √ aˆ† aˆ† ei2ϕ+aˆ† aˆ† +2aˆ† aˆ† eiϕ |0⟩. (3)\n1 2 1 1 1 2 2 2 2 0,ω1 0,ω2 1,ω1 1,ω2 0,ω1 1,ω2\nThe scattering dynamics depend on the photon number. For a single photon case (corresponding to\nthe cross terms where one photon occupies mode 0 and the other mode 1), the transformation is given\nby f (ω )f (ω ) → t (ω )t (ω )f (ω )f (ω ). In contrast, when both photons occupy the same mode,\n1 1 1 2 k 1 k 2 1 1 1 2\nthe interaction gives the transformation as f (ω )f (ω ) → t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ), was\n1 1 1 2 k 1 k 2 1 1 1 2 B 1 2\ndescribed in the section on input output theory above. Omitting the integrals for brevity, we can now\npropagate the scattered photons through the rest of the circuit.\n(cid:34)\n−N−→L √ 1 (cid:0) t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ) (cid:1)(cid:0) aˆ† aˆ† ei2ϕ+aˆ† aˆ† (cid:1)\n2 2 k 1 k 2 1 1 1 2 B 1 2 0,ω1 0,ω2 1,ω1 1,ω2\n(cid:35)\n+2t (ω )t (ω )f (ω )f (ω )aˆ†′ aˆ†′ eiϕ |0⟩\nk 1 k 2 1 1 1 2 0,ω1 1,ω2\n(cid:34)\n−B−→S √ 1 (cid:0) t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ) (cid:1)(cid:0) aˆ† aˆ† +aˆ† aˆ† +2aˆ† aˆ† (cid:1) ei2ϕ (4)\n4 2 k 1 k 2 1 1 1 2 B 1 2 0,ω1 0,ω2 1,ω1 1,ω2 0,ω1 1,ω2\n+ (cid:0) t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ) (cid:1)(cid:0) aˆ† aˆ† +aˆ† aˆ† −2aˆ† aˆ† (cid:1)\nk 1 k 2 1 1 1 2 B 1 2 0,ω1 0,ω2 1,ω1 1,ω2 0,ω1 1,ω2\n(cid:35)\n+2t (ω )t (ω )f (ω )f (ω ) (cid:0) aˆ† aˆ† −aˆ† aˆ† (cid:1) eiϕ |0⟩\nk 1 k 2 1 1 1 2 0,ω1 0,ω2 1,ω1 1,ω2\nOutput statistics of the circuit can be readily obtained from this equation. For example, the amplitude\nfor two photons in mode 0 is given by\n√ 1 (cid:2) (t (ω )t (ω )f (ω )f (ω )+f (ω ,ω )) (cid:0) ei2ϕ+1 (cid:1) +2t (ω )t (ω )f (ω )f (ω )eiϕ(cid:3) aˆ† aˆ† |0⟩,\nk 1 k 2 1 1 1 2 B 1 2 k 1 k 2 1 1 1 2 0,ω1 0,ω2\n(5)\nwhich is derived from this work [17] for the case of ϕ = 0. This expression allows us to calculate the\nprobability of finding two photons in any of the output configurations (20), (02) and (11). For example,\nthe probability of the (20) output configuration is given by:\n(cid:34)\nP = 1 (cid:0) 2+ei2ϕ+e−i2ϕ(cid:1) |t (ω )t (ω )f (ω )f (ω )+f (ω ,ω )|2\n20 16 k 1 k 2 1 1 1 2 B 1 2\n+4|t (ω )t (ω )f (ω )f (ω )|2\nk 1 k 2 1 1 1 2\n(6)\n+2t∗(ω )t∗(ω )f∗(ω )f∗(ω ) (cid:0) t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ) (cid:1)(cid:0) eiϕ+e−iϕ(cid:1)\nk 1 k 2 1 1 1 2 k 1 k 2 1 1 1 2 B 1 2\n(cid:35)\n+2t (ω )t (ω )f (ω )f (ω ) (cid:0) t∗(ω )t∗(ω )f∗(ω )f∗(ω )+f∗(ω ,ω ) (cid:1)(cid:0) e−iϕ+eiϕ(cid:1) .\nk 1 k 2 1 1 1 2 k 1 k 2 1 1 1 2 B 1 2\nwhere we still omit the integrals for brevity. There are 3 types of overlap integrals between different\nwave-packets.\nP =|t (ω )t (ω )f (ω )f (ω )+f (ω ,ω )|2, (7)\n2 k 1 k 2 1 1 1 2 B 1 2\nP =|t (ω )t (ω )f (ω )f (ω )|2, (8)\n1 k 1 k 2 1 1 1 2\nP =t∗(ω )t∗(ω )f∗(ω )f∗(ω )(t (ω )t (ω )f (ω )f (ω )+f (ω ,ω ))=r eiθint (9)\nint k 1 k 2 1 1 1 2 k 1 k 2 1 1 1 2 B 1 2 int\ni)P denotes the transmission probability of the two-photon state after interacting with the QD. This\n2\nprocessinvolvestwointeractionmechanisms: elasticscattering,whichpreservestheenergyofeachphoton,\nand inelastic scattering, which conserves only the total energy. The latter part is labeled as f (ω ,ω )\nB 1 2\ncorresponds to the two-photon bound state and accounts for the Kerr-like nonlinear phase. ii) P is\n1\nthe transmission probability of the one-photon state after scattering and is given by the square of the\n8\nsingle-photon transmission probability. iii) Interference between two-photon and one-photon states. As\ndiscussed before in the case of inelastic two-photon interactions with the QD, the scattered two-photon\nwave packet is no longer identical to the single-photon wave packet. In addition, because the two-photon\nnonlinear interaction is not uniform across the photon frequencies, the scattered wave packet becomes\ndistorted, generating multiple entangled temporal modes instead of a single mode. As a consequence, the\ninterference is only partial. We have introduced the parametrization of the overlap between the input\nand output states from the scattering process, i.e., the overlap integral between the one- and two-photon\nstates (Eq. 9), using a polar decomposition as:\nr ≤1, θ ∈[0,2π] (10)\nint int\nwhichisageneraldescriptionofanyoverlapintegralbetweennormalizedquantumstates. Ifthestatesare\nidentical, the overlap will simply be unity. Using the definitions above, we can rewrite 6, which denotes\nthe two-photon probability for bunching in the upper mode:\n20 16 tt t int int\nIn regards the the reviewers last comment, it is true that scattering multiple times can be an issue due to the\nadded distortion arising from each scattering. There are ways to counter these effects. Techniques such as\ntime-lens-based fiber dispersion have been theoretically and experimentally explored to reshape photon wave\npackets [5]. Alternatively one can use the distortion of multiple scatterings as a feature rather than a problem,\nwhich is demonstrated (theoretically) in Ref. [18].\n2. In these three points, I might be misunderstanding something, but if not, I’d like the Authors to consider\nclarifying their writing in the new version.\n(a) Regarding the scalability of the process. Speaking in simple terms, they use well-known mapping of the\nvibrationalmodesoftheH2Omoleculestotwoharmonicoscillatorswithtwophonons. Thetwoharmonic\noscillators are here the two arms of the interferometers, while the number of photons encode the number\nofphononsintotheproblem. Althoughanicedemonstration,itwouldbeinterestingtoseeamoreserious\ndiscussion of the challenges (or not) of scaling the protocol to simulate larger molecules.\nWe thank the reviewer for the comment and for stimulating the elaboration on the scalability of the\ntechnology. Following the reviewer’s suggestion, we have now performed a detailed analysis of what\nsystem scales could be accessible for our approach with current and near-term technologies. The analysis\nis reported in the newly added Appendix F of the updated manuscript, where in Table F2 we report the\nestimated detection rates for constructing programmable nonlinear circuits with quantum emitters and\ntime-bin encoding, using our same scheme but for a larger number of modes and photons. The analysis is\ndone assuming both components with the same performance as in our experiment, as well as components\nwithstate-of-the-artperformance. Inthisanalysis,weshowthatexperimentswithapproximately20modes\nand processing > 10 photons should be available with the current state-of-the-art technology. Nonlinear\nquantum photonic systems at such a scale are already challenging to simulate on classical computers\nusing current methods for nonlinear boson sampling simulation [10]. However, a rigorous assessment of\nthe computational complexity for classically simulating nonlinear quantum photonic circuits is an open\nproblem. More sophisticated methods could be designed for faster classical simulations compared to\ncurrent techniques, e.g., via recently developed techniques based on tensor networks [16]. We believe our\nwork could be of significant interest to theorists to motivate further investigation of this problem.\nThese results and discussion are now reported in detail in Appendix F:\nAppendix F: Estimation of system scalability with current and near-term technologies\nundergoscatteringfromaQDtoimplementthenonlinearity, followedbyaClements-styleinterferometerandfinally\nphoton number detection. The depth of a the interferometer is on the order of the number of modes.\nTechnology β η η |α|2 # modes # photons Rate (Hz)\nMZI coupling\n2 3e+3\n• Two-sided photonic crystal waveguide\n0.9 0.5 0.6 0.1 2 3 4e+1\n• Efficiencies from this work\n5 <1e-1\n2 3e+4\n• One-sided photonic crystal waveguide [15]\n0.9 0.5 0.6 0.1 2 3 3e+2\n• Efficiencies from this work\n5 <1e-1\n2 8e+5\n2 3 4e+4\n5 3e+1\n0.1\n2 3e+6\n10 3 3e+5\n•One-sidedphotoniccrystalwaveguide 5 1e+3\n0.98 0.9 0.85\n• State of the art efficiencies [7, 19, 1] 10 7e+6\n2 10 20 3e+2\n30 <1e+1\n10 7e+7\n4 20 20 1e+6\n30 <1e+1\nTable2: Estimationofdetectionratesforthecurrent-andfutureexperimentsusingstate-of-the-artequipment[7,19,\n1]. β isthecouplingefficiencyoftheQDtothewaveguidemode,η istheefficiencyoftheinterferometer,η\nMZI coupling\nis the coupling from chip to interferometer, |α|2 is the mean photon number. The detection rates are obtained by\nassuming an input rate of 100MHz, and sampling from Poissonian distributions determined by input mean photon\nnumbers and system efficiency. For 2-modes, the number of samples is 1e9 and for all other configurations, it is 1e7.\nScalingthecurrentplatformtolargernumbersofmodesandphotonsisaninterestingresearchdirection. It\nwouldbechallengingwiththecurrentbulkyimplementationofthelinearinterferometerbutintegrationof\nthenon-linearscatteringintoatime-bininterferometerliketheonepresentedinRef. [7],wouldbeaviable\noption. TheframeworkisshowninFig.F14. Estimatedphotoncountratesfordifferentnumbersofmodes\nare summarized in Table 2a. In order to make complexity estimates, we assume that the nonlinearity\nworks as a single-mode Kerr nonlinearity for simplicity. In practice, higher-order photon scattering will\nresult in even more multimode distortion of the wavepackets leading to partial distinguishability of the\ninterference.These effects are neglected in the present discussion, as they represent a technical challenge\nthat can be mitigated with appropriate engineering strategies. The β-factor is the coupling of the QD to\nthe waveguide, which will lead to losses if less than one. The η is the transmission of the a singular\nMZI\noptical depth is equal to the number of modes. The η is the coupling between QD waveguide and\ncoupling\ninterferometer. |α|2 is the mean photon number. For a two-sided device, 50% of the photons are lost\ndue to propagation directionality. Due to this limitation, considering the same component performance\nas in our experiment, we estimate that nonlinear quantum photonic circuits with 5 photons in 2 modes\nare available, but scaling beyond that would be challenging. A first mitigation for the limited efficiency\ncan be obtained by considering single-sided or chiral devices for interfacing the light-matter interaction,\nwhere, as discussed in the main text, we can assume no loss from the scattering process. As shown\nin Table F1, this improves expected rates significantly, but presents still limited scalability due to the\ndominantlossesfromtheincreasedopticaldepthoftheinterferometerwhentryingtoincreasethenumber\nofmodes. Tosurpasstheselimitationsandscalethenumberofmodes,reducingcomponentlossesisakey\nrequirement. Whenconsideringstate-of-the-artefficienciesfortheopticalcomponents, weindeedseethat\nexperiments with > 10 modes and > 10 photons become available. When also increasing the amplitude\nof the input weak-coherent state compared to our implementation, we estimate that experiments with\napproximately 20 modes and >10 photons are available with current technology. We note that nonlinear\nquantum photonic systems at such a scale are already challenging to simulate on classical computers\nusing current methods for nonlinear boson sampling simulation [10]. However, a rigorous assessment of\nthe computational complexity for classically simulating nonlinear quantum photonic circuits is an open\nproblem. More sophisticated methods could be designed for faster classical simulations compared to\ncurrent techniques, e.g., via recently developed techniques based on tensor networks [16].\n(b) In the discussion section, they mention that to scale it would require “temporal encoding allowing for a\nhighly resource-efficient realization since multiple nonlinear operations are encoded by reusing the same\nquantumemitteratdifferenttimes”. However,thisraisesformeaquestionregardingtheprogrammability\nof the parameters. In this work, they can reconfigure the circuit, e.g., the energy of the emitters, via DC\nStark shift. This is also demonstrated in the Appendix. I understand this can be done for the particular\nscenarioconsideredhere: inputphotonsandanalyzetheoutputafteronlyonepassthroughthenon-linear\ninterferometer. However,ifonewantedtosendbackthephotonswithafeedbackloop,woulditbepossible\nto change the parameters of the non-linear circuits before photons enter again in the interferometer? The\nsame with the other parameters. What are the technological challenges that need to be solved for that?\nHowmanyroundscanthephotonsbesentthroughthenon-linearinterferometerbeforelosingcoherence?\nWe thank the reviewer for the comment regarding fast-reconfiguration of the nonlinear circuit. The\nnonlinaerity can be tuned in the millisecond time-scale which was demonstrated in Ref. [4], to overcome\nspectral wandering. This will most likely be to slow for the rates at which one would operate such an\nexperiment. We note, however, that tunability of the nonlinaerity might not be required if the linear part\nis tunable. In classical Neural Networks, nonlinear activation function between layers are typically static.\nThe nonlinearities in the proposal for quantum optical neural networks were also static [12]. Photonic\nintegrated circuits [13] can operate in the MHz regime, which match the typical lifetimes of quantum\ndots. In case of using a Trotterisation approach for quantum simulation, where one would concatenate\nnon-commuting elements, these are fixed meaning reconfigurability of the nonlinearity is also not needed.\nWeacknowlegde, thattheremaybeapplicationwherefasttunabilityintheMHz-regimeisrequired. This\nis in principle possible with the Stark tuning, but our current devices do not support such speeds as they\nare optimized for stability and low noise.\n3. Apart from that, there are other minor corrections that the authors may want to address:\n(a) How does the nonlinear transformation of the scattering process in Eqs. (1-2) map to the standard Kerr\nnon-linearity with Hamiltonian H = K ∗n2 (with n = a† ∗a the number operator)? Maybe it can be\nmade more explicit. Is this only valid for 2 photons? What happens for a larger photon number? Do you\nget a non-linearity of other types?\nWe thank the reviewer for the comment. We have expanded the appendix to more clearly show how we\ngo from the scattering nonlinearity to a simplified Kerr nonlinearity, and emphasized that this derivation\nis valid in the 2-photon regime. Larger photon numbers are still unexplored, with some theoretical efforts\nbeing made by post-selecting the output modes to be single-mode [6].\nFurthermore, we have added the following paragraph after the details on the QD scattering to compare\nto the single-mode Kerr nonlinearity.\nThe time evolution of the Hamiltonian in the eigenbasis is given by the unitary evolution\nU(t)=eiνt/h¯, (12)\nwheretheeigenvaluesfromequation(1)hastobeconvertedfromcm−1 tofrequencies. Thephaseacquired\nby each Fock state component thus depends on the evolution time and its associated eigenvalue. The\n11\nanharmonicity is manifested by having a nonlinear offset ϕ in the acquired phase of the one- and\nNL\ntwo-photon components, which we write as:\nϕ →2ϕ+ϕ ,\n|20⟩ NL\nϕ →ϕ , (13)\n|02⟩ NL\nϕ →ϕ,\n|11⟩\nwhere,foragivenevolutiontimet,wehaveν t=ϕ . Thelinearandnonlinearphasesareprogrammable\n|i⟩ |i⟩\nasexplainedinthemaintext. Alongwiththedegreeoffreedomoftheglobalphase,thesetwoparameters\nallow us to match the equations in equation 13 for an arbitrary time evolution. In terms of operations of\nsingle-mode phaseshifters and Kerr nonlinearity, the operators are given by:\nϕ=anˆ,ϕ extNL=knˆ(nˆ−1) (14)\nt\n(b) There is a mistake (the interrogation marks “??”) in the first paragraph of section “Programmable linear\nand nonlinear photonic circuits” in page 2: “...see Fig. 1b, and Appendix ?? for more details...”. In\ngeneral, in all references to the Appendix there is the same issue.\nWe thank the reviewer for the comment. This was a compilation error which has now been fixed.\n(c) In the top of the right column of page 6, it reads “these extreme nonlinear phaseshift values are our of\nrange for our device (see Fig. Fig 2.c)”. Shouldn’t it read “out of range”?\nWe thank the reviewer for the comment. The typo has been corrected in the manuscript.\n(d) There’s an erratum at the bottom of the left column in page 2 where it reads “...corresponding to the\nfirst mode and condition on two photon detection...”. Shouldn’t it read “conditioned”?\nWe thank the reviewer for the feedback. This has been corrected in the manuscript.\n(e) The Authors may want to expand the outlook including other potential use of their programmable non-\nlinearities, suchasquantummetrology, wheretherearerecentworksalongthisdirection, e.g., Phys. Rev.\nResearch 6, 013299 (2024), Nanophotonics 2025; 14(11): 2075–2085, arXiv:2410.11158.\nWe thank the reviewer for bringing up this insightful perspective on the potential applications of our\nprogrammable nonlinear circuit. We have expanded the discussions of this relevant point in the outlook\nsection.\nAndincorporatingsuchprogrammablenonlinearcircuitsintoquantummetrologyarchitecturesmayenable\nhigh-fidelity quantum gates, opening new avenues for precision-enhanced quantum technologies [9].\n(f) The explanation of Fig. 2a in the main text is obscure for me: How does the relative sign of Delta and\nsigma affect the type of correlations that are observed? Where are the side lobes? Where can I see the\n|11⟩ term? Maybe the authors can point out these features in the joint temporal intensity (JTI) patterns\nshown in Fig. 2a, using arrows or encircling them.\nWe thank the reviewer for the clarifying comments. ∆/σ denotes the laser detuning relative to the QD\nemission linewidth. The sign +(-)indicates red (blue) detuning. The response is symmetric. We have\nhighlighted the side lobes on the JTI figure in the supplementary and added the figure for the |11⟩ term.\nWe have added the following discussion to the the supplementary to clarify the observations made from\nthe JTIs.\nDC Stark tuning is used to spectrally align the QD, the positive (negative) ∆ corresponding to blue\n(red) detuning relative to the QD resonance. As expected, when |∆| approaches zero, the bunching\neffect becomes more pronounced, whereas for large detuning the bunching vanishes, as evident in the\nupper panel of the JTI. Fig.B7 displays the JTI patterns recorded from a single spatial mode, i.e., lower\nmode of the interferometer labeled |02⟩ in the main text. The observed pattern consists of two distinct\nfeatures: bunching along the diagonal, which originates from the contribution of two-photon states, and\nlobes that arise from single-photon states of different time-bin. The intensity of bunching is modulated\nby the applied linear phase, while the lobe feature is also phase-dependent, since the linear phase governs\nthe interference between different time-bin. This results in prominent lobes at ϕ=0 due to constructive\ninterference. Fig.B8 presents the correlation patterns for two different spatial modes labeled as |11⟩. In\nthis configuration, lobes are not observed, as no single-photon states of different time-bin are involved.\nThe experimental data and theoretical model shows less agreement compared to the bunching case. We\nbelieve this is due distinguishability between the first and second scattering, which impacts the (11)\nphoto-statistics more than the (20) and (02) relative to the amplitude.\n(right)JTIforphotonsregisteredinthelowermodeoftheinterferometerforvariousvaluesofthelinearphaseϕ,the\nrelative detuning ∆/σ, and the input pulse duration; of 300 ps for a) and b), 500 ps for c) and d), and 700 ps for e)\nand f). The bunching observed along the diagonal originates from the contribution of two-photons accommodating\nthe same temporal mode upon scattering, resulting in the bound state contribution. Additionally, the distinct lobes\nindicatedbythedashedwhitecirclesarisefromcontributionsofsinglephotonsindifferenttime-binofearlyandlate\nupon scattering.\n(right) JTI for a configuration in which photons are recorded from both the upper and lower modes of the interfer-\nometer for various values of the linear phase ϕ, the relative detuning ∆/σ, and the input pulse duration; of 300 ps\nfor a) and b), 500 ps for c) and d), and 700 ps for e) and f). The experimental data and theoretical model shows\nless agreement compared to the bunching case. 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In: npj Quantum Information 9.1 (2023), p. 3. url:\nhttps://www.nature.com/articles/s41534-023-00676-x.\n[11] Chris Sparrow et al. “Simulating the vibrational quantum dynamics of molecules using photonics”. In: Nature\n557.7707 (2018), pp. 660–667. doi: 10.1038/s41586-018-0152-9. url: https://doi.org/10.1038/s41586-\n018-0152-9.\n[12] Gregory R. Steinbrecher et al. “Quantum optical neural networks”. In: npj Quantum Information 5.1 (2019),\np. 60. doi: 10.1038/s41534-019-0174-7. url: https://doi.org/10.1038/s41534-019-0174-7.\n[13] PatrikI.Sundetal.“High-speedthin-filmlithiumniobatequantumprocessordrivenbyasolid-statequantum\nemitter”. In: Science Advances 9.19 (2023), eadg7268. doi: 10.1126/sciadv.adg7268. eprint: https://www.\nscience.org/doi/pdf/10.1126/sciadv.adg7268. url: https://www.science.org/doi/abs/10.1126/\nsciadv.adg7268.\n[14] HenriThyrrestrupetal.“Quantumopticswithnear-lifetime-limitedquantum-dottransitionsinananophotonic\nwaveguide”. In: Nano letters 18.3 (2018), pp. 1801–1806.\n[15] Ravitej Uppu et al. “Scalable integrated single-photon source”. In: Science Advances 6.50 (2020), eabc8268.\ndoi: 10.1126/sciadv.abc8268. eprint: https://www.science.org/doi/pdf/10.1126/sciadv.abc8268.\nurl: https://www.science.org/doi/abs/10.1126/sciadv.abc8268.\n[16] Jonas Vinther and Michael J. Kastoryano. “Variational tensor network simulation of Gaussian boson sampling\nand beyond”. In: Phys. Rev. A 112 (2 Aug. 2025), p. 022605. doi: 10.1103/z463-7gqy. url: https://link.\naps.org/doi/10.1103/z463-7gqy.\n[17] D. Witthaut, M. D. Lukin, and A. S. Sørensen. “Photon sorters and QND detectors using single photon\nemitters”. In: Europhysics Letters 97.5 (2012), p. 50007. doi: 10.1209/0295-5075/97/50007. url: https:\n//dx.doi.org/10.1209/0295-5075/97/50007.\n[18] FanYangetal.“DeterministicPhotonSortinginWaveguideQEDSystems”.In:Phys. Rev. Lett.128(21May\n2022), p. 213603. doi: 10.1103/PhysRevLett.128.213603. url: https://link.aps.org/doi/10.1103/\nPhysRevLett.128.213603.\n[19] Beibei Zeng et al. “Cryogenic packaging of nanophotonic devices with a low coupling loss < 1 dB”. In:\nApplied Physics Letters123.16(Oct.2023),p.161106.issn:0003-6951.doi:10.1063/5.0170324.url:https:\n//doi.org/10.1063/5.0170324.\n• The reviewer’s comments are presented in blue and quoted.\n• Our responses are in black\n• The updates in the resubmitted manuscript are colored red both here and in the revised version of the\nmanuscript.\nReviewer #1\nThank you to the authors for providing their responses and corrections. My main concerns have been addressed by\nthe authors, and I believe that the manuscript is now suitable for publication in Nature Communications.\nIhaveoneminoroutstandingconcernaboutthedescriptionoftheH2Omodel. Theauthorsnowcorrectlystatethat\nthe bending mode is uncoupled from the others ”in this model”. However, two sentences later the authors have left\nthe sentence ”Because these localized modes are superpositions of the stretch eigenmodes, they are uncoupled from\nthebendingmode,andthesystemcanthusbemodeledbyrestrictingtothesetwohigh-energyvibrationalmodes...”\nAs stated previously, this justification is incorrect and can likely be removed entirely.\nWe thank the reviewer for bringing this to our attention, and for the recommendation for publication. The\nsentence has now been removed from the manuscript.\nReviewer #2\nThe Authors have addressed all the comments, I recommend publication.\nWe thank the reviewer for the recommendation for publication.\nReviewer #3\nWe have reviewed the Authors’ response to our comments. Overall, they have satisfactorily responded to all the\npoints raised in the first round. Several of our comments have been clarified through additional analyses in the\nAppendices, and we appreciate the effort the authors have put into improving the manuscript.\nThat said, we believe the following points should be made more explicit in the Main Text before the manuscript is\naccepted, so that some key limitations of the current approach are not underplayed:\nWe thank the reviewer for their time to again review our updated manuscript.\n• Point 1a: The authors acknowledge that using coherent states as input corresponds to the specific situation\nin which the vibrational modes of the simulated H2O molecule are initialized in coherent states. However,\nin section ”Photonic quantum simulation of anharmonic molecular dynamics” of their manuscript they state\nthat ”We consider the H2O molecule initialized with two excitations (phonons) on the left stretch mode,\ncorrespondingtoaninitialphotonicstatewithtwophotonsonthefirstopticalmode”. Althoughweappreciate\nthat the Authors have added a new clarifying paragraph in the revised discussion section of their manuscript\n(”Inthecontextofthesimulation,ourconfigurationwithaweak-coherentstateattheinputdirectlycorresponds\nto the scenario where the molecular vibrational modes are also initialized to a coherent state (and measured in\nthe Fock basis)...”), the mentioned phrase in section ”Photonic quantum simulation of anharmonic molecular\ndynamics” can potentially be misleading, as it does not mention the need for post-selecting two-photon events\nwhen coherent states are used.\nWe thank the reviewer for the comment. We have now added the following clarification to this section, so it is\nclear that we base our experiment on weak coherent input.\nWe consider input states with two photon in the first spatial mode, implemented bu sending a weak-coherent\nlaser pulse in the first mode and conditioning on two-photon detection events at the SNSPDs.\n• Point1c: WethanktheAuthorsfortherevisedAppendixB2inwhichtheyincludeadetailedderivationonhow\nthephotonicwavepacketschangeuponscattering. However,wethinkthatitwouldberelevanttoincludeinthe\nmanuscript a brief comment on how the distortion of the wavepackets can affect photonic quantum computing\nand simulation, especially when the output photons are re-injected in the interferometer in a feedback loop.\nActually, the Authors already commented on this issue at the end of their reply to point 1c. It would be good\nto include that reply in the manuscript.\nWe thank the reviewer for the comment. We have added a sentence to the main text elaborating ways of\ncountering multi-mode effects by reshaping the wavepackets.\nTo mitigate multi-mode effects, one possible solution is to employ time-lens dispersion to rectify the modal\ndistortion [2]. Alternatively, the orthogonal properties after consecutive interactions with quantum emitters\ncan be used to prepare one- and two-photon states in orthogonal modes and split them in a photon sorter-like\nexperiment, as is investigated theoretically in Ref. [6].\n1\n• Point2a: WethanktheAuthorsforincludingthenewAppendixFinwhichtheyanalyzethescalabilityoftheir\nplatform. However, this new Appendix is not referenced in the Main Text. In our view, this is a sufficiently\nimportant outlook of the work to deserve a reference in the Main Text.\nWe thank the reviewer for point out the lack of reference in the main text. This has now been added to the\ndiscussion.\nFor more details on the scalability and computational complexity of nonlinear boson sampling circuits, see\nSupplementary Note 6.\n• Point 2b: Regarding the speed at which the tunable non-linearity can be reconfigured and the possibility\nof implementing a feedback loop, we thank the Authors for their clear response. However, we think that\nthis response is relevant enough to be included either in the discussion section of the manuscript or in the\nAppendices.\nWe thank the reviewer for this comment. The discussion from the response has now been added to the main\ntext, with further elaboration in the supplementary information.\nMain Text: The experiment presented in this article features independent programmability of its linear and\nnonlinear operations, although this is currently limited to slow speeds. Fast reprogrammability (in the MHz\nregime) may not be essential for all applications of nonlinear optical circuits, such as quantum simulation.\nHowever,thepotentialforhigh-speedoperationexists,asQD-basednonlinearoperationenablesfasttunability\n[3]. FastlinearreconfigurationsareachievablewithbulkEOMsorintegratedphotonics[5]. SeeSupplementary\nNote 7 for more details.\nSupplementaryNote7: (NotethisisnothighlightedasredinthesupplementaryPDF):Changingthenonlinear\noperation fundamentally corresponds to tuning the detuning (∆) between the light and the QD. This can be\nrealisedeitherbyscanningthelaserfrequencyrelativetotheQDresonancefrequency,wherecurrenttechnology\nenables tuning speed up to several hundred kilohertz, or by shifting the QD resonance frequency via DC Stark\ntuning. The achievable tuning speed in the latter case is limited by the RC time constant of the metallic\ngate, which is determined by its geometry. Tuning speeds in the MHz range have already been demonstrated,\nand operation at GHz regime is attainable with locally gated devices [3]. The nonlinearity can be tuned\non the millisecond time scale, as demonstrated in Ref. [1], to overcome spectral wandering. This will most\nlikely be to slow for the rates at which one would operate such an experiment. We note, however, that\ntunability of the nonlinearity might not be required if the linear part is tunable. In classical Neural Networks,\nnonlinearactivationfunctionsbetweenlayersaretypicallystatic. Thenonlinearitiesintheproposalforquantum\noptical neural networks were also static [4]. Photonic integrated circuits [5] can operate at MHz rates, which\nmatches the typical lifetimes of quantum dots. In the case of using a Trotterisation approach for quantum\nsimulation, where one would concatenate non-commuting elements, these are fixed, meaning reconfigurability\nof the nonlinearity is also not needed.\nAsidefromthis,theauthorshaveprovidedsatisfactoryresponsestoourcomments,includingthescalabilityanalysis,\nthe discussion of nonlinear tunability, the mapping to Kerr nonlinearities, and the clarifications regarding the joint\ntemporal intensity plots. In summary, we find the work interesting and suitable for publication in Nature Commu-\nnications, provided the authors revise the manuscript to more explicitly acknowledge the current limitations of their\napproach, particularly regarding the points 1a, 1c, 2a, and 2b.\nWe thank the reviewer for the positive comments regarding our work.\n[1] JackHansometal.“Frequencystabilizationofthezero-phononlineofaquantumdotviaphonon-assistedactive\nfeedback”. In: Applied Physics Letters 105.17 (2014).\n[2] F. Kappe et al. “Chirped Pulses Meet Quantum Dots: Innovations, Challenges, and Future Perspectives”. In:\nAdvanced Quantum Technologies 8.2 (2025), p. 2300352. doi: 10.1002/qxute.202300352.\n[3] Freja T. Pedersen et al. “Near Transform-Limited Quantum Dot Linewidths in a Broadband Photonic Crystal\nWaveguide”. In: ACS Photonics 7.9 (Sept. 2020), pp. 2343–2349. doi: 10.1021/acsphotonics.0c00758. url:\nhttps://doi.org/10.1021/acsphotonics.0c00758.\n[4] Gregory R. Steinbrecher et al. “Quantum optical neural networks”. In: npj Quantum Information 5.1 (2019),\np. 60. doi: 10.1038/s41534-019-0174-7. url: https://doi.org/10.1038/s41534-019-0174-7.\n[5] Patrik I. Sund et al. “High-speed thin-film lithium niobate quantum processor driven by a solid-state quantum\nemitter”. In: Science Advances 9.19 (2023), eadg7268. doi: 10.1126/sciadv.adg7268. eprint: https://www.\nscience.org/doi/pdf/10.1126/sciadv.adg7268. url: https://www.science.org/doi/abs/10.1126/\nsciadv.adg7268.\n[6] Fan Yang et al. “Deterministic Photon Sorting in Waveguide QED Systems”. In: Phys. Rev. Lett. 128 (21 May\n2022), p. 213603. doi: 10.1103/PhysRevLett.128.213603. url: https://link.aps.org/doi/10.1103/\nPhysRevLett.128.213603.\n3"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank you to the authors for providing their responses and corrections. My main concerns have been addressed by the authors, and I believe that the manuscript is now suitable for publication in Nature Communications.\n\nI have one minor outstanding concern about the description of the H2O model. The authors now correctly state that the bending mode is uncoupled from the others \"in this model\". However, two sentences later the authors have left the sentence \"Because these localized modes are superpositions of the stretch eigenmodes, they are uncoupled from the bending mode, and the system can thus be modeled by restricting to these two high-energy vibrational modes...\" As stated previously, this justification is incorrect and can likely be removed entirely.", "reviewer_2": "The Authors have addressed all the comments, I recommend publication.", "reviewer_3": "Dear Editor,\n\nWe have reviewed the Authors’ response to our comments. Overall, they have satisfactorily responded to all the points raised in the first round. Several of our comments have been clarified through additional analyses in the Appendices, and we appreciate the effort the authors have put into improving the manuscript.\n\nThat said, we believe the following points should be made more explicit in the Main Text before the manuscript is accepted, so that some key limitations of the current approach are not underplayed:\n\n-Point 1a: The authors acknowledge that using coherent states as input corresponds to the specific situation in which the vibrational modes of the simulated H2O molecule are initialized in coherent states. However, in section \"Photonic quantum simulation of anharmonic molecular dynamics\" of their manuscript they state that \"We consider the H2O molecule initialized with two excitations (phonons) on the left stretch mode, corresponding to an initial photonic state with two photons on the first optical mode\". Although we appreciate that the Authors have added a new clarifying paragraph in the revised discussion section of their manuscript (\"In the context of the simulation, our configuration with a weak-coherent state at the input directly corresponds to the scenario where the molecular vibrational modes are also initialized to a coherent state (and measured in the Fock basis)...\"), the mentioned phrase in section \"Photonic quantum simulation of anharmonic molecular dynamics\" can potentially be misleading, as it does not mention the need for post-selecting two-photon events when coherent states are used.\n\n-Point 1c: We thank the Authors for the revised Appendix B2 in which they include a detailed derivation on how the photonic wavepackets change upon scattering. However, we think that it would be relevant to include in the manuscript a brief comment on how the distortion of the wavepackets can affect photonic quantum computing and simulation, especially when the output photons are re-injected in the interferometer in a feedback loop. Actually, the Authors already commented on this issue at the end of their reply to point 1c. It would be good to include that reply in the manuscript.\n\n-Point 2a: We thank the Authors for including the new Appendix F in which they analyze the scalability of their platform. However, this new Appendix is not referenced in the Main Text. In our view, this is a sufficiently important outlook of the work to deserve a reference in the Main Text.\n\n-Point 2b: Regarding the speed at which the tunable non-linearity can be reconfigured and the possibility of implementing a feedback loop, we thank the Authors for their clear response. However, we think that this response is relevant enough to be included either in the discussion section of the manuscript or in the Appendices.\n\nAside from this, the authors have provided satisfactory responses to our comments, including the scalability analysis, the discussion of nonlinear tunability, the mapping to Kerr nonlinearities, and the clarifications regarding the joint temporal intensity plots.\n\nIn summary, we find the work interesting and suitable for publication in Nature Communications, provided the authors revise the manuscript to more explicitly acknowledge the current limitations of their approach, particularly regarding the points 1a, 1c, 2a, and 2b."}, "author_response": "We thank the reviewer for bringing this to our attention, and for the recommendation for publication. The sentence has now been removed from the manuscript.\n\nWe thank the reviewer for the recommendation for publication.\n\nWe thank the reviewer for their time to again review our updated manuscript.\n\n• Point 1a: The authors acknowledge that using coherent states as input corresponds to the specific situation in which the vibrational modes of the simulated H2O molecule are initialized in coherent states. However, in section ”Photonic quantum simulation of anharmonic molecular dynamics” of their manuscript they state that ”We consider the H2O molecule initialized with two excitations (phonons) on the left stretch mode, correspondingtoaninitialphotonicstatewithtwophotonsonthefirstopticalmode”. Althoughweappreciate that the Authors have added a new clarifying paragraph in the revised discussion section of their manuscript (”Inthecontextofthesimulation,ourconfigurationwithaweak-coherentstateattheinputdirectlycorresponds to the scenario where the molecular vibrational modes are also initialized to a coherent state (and measured in the Fock basis)...”), the mentioned phrase in section ”Photonic quantum simulation of anharmonic molecular dynamics” can potentially be misleading, as it does not mention the need for post-selecting two-photon events when coherent states are used.\n\nWe thank the reviewer for the comment. We have now added the following clarification to this section, so it is clear that we base our experiment on weak coherent input.\n\nWe consider input states with two photon in the first spatial mode, implemented bu sending a weak-coherent laser pulse in the first mode and conditioning on two-photon detection events at the SNSPDs.\n\n• Point1c: WethanktheAuthorsfortherevisedAppendixB2inwhichtheyincludeadetailedderivationonhow thephotonicwavepacketschangeuponscattering. However,wethinkthatitwouldberelevanttoincludeinthe manuscript a brief comment on how the distortion of the wavepackets can affect photonic quantum computing and simulation, especially when the output photons are re-injected in the interferometer in a feedback loop. Actually, the Authors already commented on this issue at the end of their reply to point 1c. It would be good to include that reply in the manuscript.\n\nWe thank the reviewer for the comment. We have added a sentence to the main text elaborating ways of countering multi-mode effects by reshaping the wavepackets.\n\nTo mitigate multi-mode effects, one possible solution is to employ time-lens dispersion to rectify the modal distortion [2]. Alternatively, the orthogonal properties after consecutive interactions with quantum emitters can be used to prepare one- and two-photon states in orthogonal modes and split them in a photon sorter-like experiment, as is investigated theoretically in Ref. [6].\n\n• Point2a: WethanktheAuthorsforincludingthenewAppendixFinwhichtheyanalyzethescalabilityoftheir platform. However, this new Appendix is not referenced in the Main Text. In our view, this is a sufficiently important outlook of the work to deserve a reference in the Main Text.\n\nWe thank the reviewer for point out the lack of reference in the main text. This has now been added to the discussion.\n\nFor more details on the scalability and computational complexity of nonlinear boson sampling circuits, see Supplementary Note 6.\n\n• Point 2b: Regarding the speed at which the tunable non-linearity can be reconfigured and the possibility of implementing a feedback loop, we thank the Authors for their clear response. However, we think that this response is relevant enough to be included either in the discussion section of the manuscript or in the Appendices.\n\nWe thank the reviewer for this comment. The discussion from the response has now been added to the main text, with further elaboration in the supplementary information.\n\nMain Text: The experiment presented in this article features independent programmability of its linear and nonlinear operations, although this is currently limited to slow speeds. Fast reprogrammability (in the MHz regime) may not be essential for all applications of nonlinear optical circuits, such as quantum simulation. However,thepotentialforhigh-speedoperationexists,asQD-basednonlinearoperationenablesfasttunability [3]. FastlinearreconfigurationsareachievablewithbulkEOMsorintegratedphotonics[5]. SeeSupplementary Note 7 for more details.\n\nSupplementaryNote7: (NotethisisnothighlightedasredinthesupplementaryPDF):Changingthenonlinear operation fundamentally corresponds to tuning the detuning (∆) between the light and the QD. This can be realisedeitherbyscanningthelaserfrequencyrelativetotheQDresonancefrequency,wherecurrenttechnology enables tuning speed up to several hundred kilohertz, or by shifting the QD resonance frequency via DC Stark tuning. The achievable tuning speed in the latter case is limited by the RC time constant of the metallic gate, which is determined by its geometry. Tuning speeds in the MHz range have already been demonstrated, and operation at GHz regime is attainable with locally gated devices [3]. The nonlinearity can be tuned on the millisecond time scale, as demonstrated in Ref. [1], to overcome spectral wandering. This will most likely be to slow for the rates at which one would operate such an experiment. We note, however, that tunability of the nonlinearity might not be required if the linear part is tunable. In classical Neural Networks, nonlinearactivationfunctionsbetweenlayersaretypicallystatic. Thenonlinearitiesintheproposalforquantum optical neural networks were also static [4]. Photonic integrated circuits [5] can operate at MHz rates, which matches the typical lifetimes of quantum dots. In the case of using a Trotterisation approach for quantum simulation, where one would concatenate non-commuting elements, these are fixed, meaning reconfigurability of the nonlinearity is also not needed.\n\nAsidefromthis,theauthorshaveprovidedsatisfactoryresponsestoourcomments,includingthescalabilityanalysis, the discussion of nonlinear tunability, the mapping to Kerr nonlinearities, and the clarifications regarding the joint temporal intensity plots. In summary, we find the work interesting and suitable for publication in Nature Communications, provided the authors revise the manuscript to more explicitly acknowledge the current limitations of their approach, particularly regarding the points 1a, 1c, 2a, and 2b.\n\nWe thank the reviewer for the positive comments regarding our work."}]

Decision letter