Programmable nonlinear quantum photonic circuits

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 tasks^3,4. However, the lack of direct photon-photon interactions prohibits the nonlinear operations that enable deterministic multi-photon entangling gates^5,6 and near-term photonic quantum simulators^7,8. 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 techniques^11,12 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-processing^14,15. 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 level^16,17. 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 atomic^18,19 and solid-state systems, including quantum dots (QDs)^20,21,22 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 simulation^24,25,26 and computing circuits^27,28. 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 states^18,20,30 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 nonlinearities^33,34, 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 β ≃ 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.

Fig. 1: Schematics of the programmable nonlinear quantum photonic circuit.

a The programmable nonlinear photonic circuit comprises tunable QD nonlinear transformations interleaved between two controllable linear optical circuits implemented via time-bin Mach-Zehnder interferometers that couple a pair of temporal optical modes. The photons are injected into the circuit in the first spatial mode and subsequently interfere and interact via linear and nonlinear operations, respectively. The transmitted photons are finally recorded with pseudo-photon-number-resolution (pPNR) using SNSPDs. b SEM image of the GaAs nanophotonic device, which is used to implement the nonlinear interaction by scattering the photons off an InAs QD embedded in a photonic crystal waveguide. Grating couplers are used for in- and out-coupling of photons from the device to interconnect it with the linear parts of the circuit. The electric field across the QD is controlled with electrodes (highlighted in yellow) that allow tuning of the nonlinear interaction by controlling the detuning Δ between the QD resonance and the incoming photons via the DC Stark effect. c Schematic of the linear optical setup. The pair of optical modes represented in a) are here encoded in the temporal degree of freedom of the photons, where the first optical mode corresponds to an early (E) time-bin and the second to a late (L) time-bin. The linear operations are implemented in a self-stabilising time-bin interferometer that couples the E and L modes and applies a relative linear phase ϕ between them. Photons are injected and retrieved from the GaAs device hosted in a 1.6K cryostat, where nonlinearities are implemented by scattering off the QD at the two different times E and L. The two temporal modes are recombined in the time-bin interferometer, where the output interference is affected by both linear and nonlinear phases. The nonlinear interaction significantly changes the properties of the photons, as illustrated by the theoretical joint temporal intensity (JTI) shown with (right inset) and without (left inset) nonlinear interactions for different settings of ϕ.

Characterisation of the linear and nonlinear operations

First, the QD detuning, pulse width 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 $\left\vert 02\right\rangle$ configuration (both output photons in the second spatial mode) for different values of both the linear phase ϕ and the nonlinearity control parameter Δ. It is observed that, while no correlations are present in the absence of the nonlinearities (Δ ≫ σ), they arise when the QD is brought into resonance, leading to a JTI with pronounced correlations along the diagonal in addition to persistent side lobes²⁰. When the linear phase is scanned, the side lobes 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 $\left\vert 11\right\rangle$ term. The diagonal part is primarily due to terms with both photons in the same temporal mode upon scattering.

Fig. 2: Programmability of the linear and nonlinear photonic operations.

a Experimental normalised joint temporal intensity data obtained via time-resolved correlation measurements of the output photons. The columns correspond to the linear-phase settings of the interferometer, while the rows correspond to the different QD detunings to control the nonlinearity strength. All measurements are taken with a 700 ps long input pulse. b Measurements of interference in the circuit for different nonlinearities. The measured probability for the output configuration $\left\vert 20\right\rangle$ is shown as a function of the linear phase ϕ for different values of the detuning Δ that controls the strength of the nonlinearity. Error bars represent one standard deviation and are calculated assuming Poissonian photon statistics. c Characterisation of the nonlinear phase and scattering probability parameters as a function of detuning Δ. Black (purple) data represent the nonlinear phase (scattering probability) as extracted from fitting the measured photon statistics to the nonlinear transformation model described in the main text. Inset: representation of the detuning Δ between the incoming photons (labelled as γ) and the QD used to control the nonlinearity. d Comparison of the transmission probability after the scattering process for one-photon (yellow) and two-photon (blue) components as a function of the implemented nonlinear phase φ_NL. Markers are values calculated from the experimental data, while the solid lines represent theoretical values that model the experimentally characterised system parameters. Analogue curves are also shown for the cases where scattering in a chiral or single-sided waveguide is employed (dotted purple line, assuming a unit β), and for the optimised measurement-based linear optical scheme proposed by ref. ³³ (dashed black line).

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, $\left\vert 20\right\rangle$, obtained when scanning the linear phase ϕ and setting the Mach-Zehnder interferometers to balanced beam splitters (θ = π/2). In the absence of nonlinearities (i.e., Δ ≫ σ), 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 φ_NL and the scattering probability ℓ_NL according to

$$\left\vert 1\right\rangle \mapsto \sqrt{\eta (1-{\ell }_{{{{\rm{NL}}}}})}{{{{\rm{e}}}}}^{i\phi }\left\vert 1\right\rangle$$

(1)

$$\left\vert 2\right\rangle \mapsto \sqrt{{\eta }^{2}}{{{{\rm{e}}}}}^{i(2\phi+{\varphi }_{{{{\rm{NL}}}}})}\left\vert 2\right\rangle,$$

(2)

where ϕ and η 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, φ_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 φ_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 $\left\vert 11\right\rangle$ term.

Using this model for the nonlinear transformation, we extract the nonlinear phase parameter φ_NL and nonlinear scattering probability ℓ_NL 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 Δ. 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 φ_NL ≃ π/4, indicating an extensive range of tunability of the nonlinearities embedded within the universal programmable two-mode linear interferometer. The finite nonlinear scattering probability ℓ_NL 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 $\left\vert 1\right\rangle$ than for the $\left\vert 2\right\rangle$ 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, π] 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.

Fig. 3: Proof-of-concept quantum simulation of vibrational dynamics in H₂O with a programmable nonlinear photonic quantum circuit.

a Illustration of molecular potential of vibrational degrees of freedom that, in general, is anharmonic (depicted as a solid line), i.e., a harmonic approximation (dashed line) is not sufficient. b The quantum vibrational dynamics of the molecule are mapped onto a photonic system by associating optical modes with vibrational modes, and single optical excitations (photons) with single vibrational excitations (phonons). The associated vibrational dynamics for an anharmonic Hamiltonian ${\hat{H}}_{a}$ (upper inset) can be described in the localised basis (depicted on left and right) by converting to the normal basis (shown in the middle) via a matrix transformation U, evolving in the normal basis for a time t, and then converting back to the localised basis via U^†. In the photonic implementation, the linear parts of the circuit perform the basis changes and the harmonic evolution. Subsequently, the photonic nonlinear interactions (depicted as yellow boxes) implement the anharmonic part of the evolution. c Experimental results for the quantum simulation of the anharmonic dynamics of the H₂O molecule initialised with two photons in the same localised stretch mode. The top and bottom panels report the output occupancy for configurations in which the excitations emerge in separate or the same localised stretch mode, respectively. The data are compared with the theoretical model (solid lines) and with predictions from the harmonic approximation (dashed lines). The data markers are colour-coded according to the strength of the nonlinear phase (scale bar as inset) associated with each evolution time step.

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 $H={\sum }_{i}\hslash {\omega }_{i}{a}_{i}^{{{\dagger}} }{a}_{i}$ where a unitary basis transformation U to a basis localised around the different vibrational modes ${a}_{i}^{{{\dagger}} }\to {\sum }_{k}{U}_{ki}{a}_{k}^{{{\dagger}} }$ leads to the Hamiltonian $H={\sum }_{i,j,k}\hslash {\omega }_{i}{U}_{ki}\overline{{U}_{ji}}{a}_{j}^{{{\dagger}} }{a}_{k}.$ 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 ${H}_{a}=H+\hslash {\sum }_{i\le j}\frac{{\chi }_{ij}}{2}\sqrt{{\omega }_{i}{\omega }_{j}}({a}_{i}^{{{\dagger}} }{a}_{i}+{a}_{j}^{{{\dagger}} }{a}_{j}+2{a}_{i}^{{{\dagger}} }{a}_{j}^{{{\dagger}} }{a}_{i}{a}_{j}),$ where H is the harmonic Hamiltonian. The Harmonic part can be simulated with linear optics, whereas the nonlinear “coupling term" is χ_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 bases^33,43.

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 H₂O (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 others^33,43. 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 H₂O, 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 time step 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 φ_NL to be applied. In contrast, the relative phases between configurations with the same number of excitations correspond to the relative linear phase ϕ between the modes. We then programme the detuning Δ to match the required nonlinear phase according to the relation characterised in Fig. 2c. At the same time, the linear phase ϕ 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 − 0.15ps, which is where the required nonlinear phase is highest, approximately π/2. Due to the presence of minor imperfections of the devices, notably weak residual spectral diffusion, these extreme nonlinear phaseshift 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.

Discussion

We have demonstrated a programmable nonlinear photonic circuit with tunable nonlinearities generated through photon-sensitive scattering with a quantum emitter. The versatility of this technology was tested by implementing an exemplary application - the analogue quantum simulation of vibrational quantum dynamics in a molecule. The implementation relies on temporal encoding, allowing for a highly resource-efficient realisation since multiple nonlinear operations are encoded by reusing the same quantum emitter at different times. Extensions of simulators at increasing scale with this technology could be useful for investigating, for instance, vibrational and vibronic spectra in larger molecular systems and compounds²⁹.

The demonstrated capability to combine linear and nonlinear operations in a photonic device unlocks new opportunities for photonic quantum technologies. These include implementing deterministic Bell-state and quantum non-demolition measurements^31,47 and deterministic photonic entangling gates⁴⁸, tasks that are fundamentally constrained in conventional linear optical approaches, posing significant limitations when developing resource-efficient photonic quantum computing architectures^49,50 or quantum repeater architectures⁵¹. Extending the platform to more than 2 modes is required for more complex molecular simulations or Bell-state analyser experiments. This can be realised either in the spatial or the temporal domain. The former requires tuning different quantum dots into mutual resonance, as demonstrated in ref. ⁵², enabling nonlinear operation in different spatial modes. The latter applies encoding across multiple time-bins^3,53, which is highly resource-efficient since it relies on numerous scattering from a single quantum emitter. Using nonlinear quantum photonic circuits to further scale up analogue quantum simulations of vibrational dynamics to complex molecular or even biological systems represents an important research direction, potentially opening a path to earlier practical applications of photonic quantum simulators. In the context of the simulation, our configuration with a weak-coherent state at the input corresponds to the scenario in which the molecular vibrational modes are also initialised to a coherent state (and measured in the Fock basis). If such a scenario is relevant, our initialisation thus represents an appropriate mapping and does not introduce limitations. However, we acknowledge that, in most practical contexts, having a molecule starting in a coherent vibrational mode may no longer be representative of realistic chemical scenarios. In such cases, the relevant input states should be used, e.g., thermal states or Fock states. That would require a different state-generation procedure, which could be more complex than our particular implementation, such as using a single-photon source rather than a simpler weak-coherent input. On the other hand, beyond the implementation being more complex, we expect no fundamental limitations when combining such different input sources with our nonlinear quantum processor technology. For example, single-mode nonlinearities obtained by scattering single photons from a quantum emitter through a quantum dot have recently been shown⁵⁴. Our resource-efficient approach is particularly appealing for scaling nonlinear circuits to target larger molecular systems, and low-loss time-bin linear interferometers with hundreds of modes have already been realized^3,55. Operating such nonlinear circuits at sufficiently large scale can enable quantum simulations of complex molecules with a computational advantage over classical simulation techniques. 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 regime⁴². For more details on the scalability and computational complexity of nonlinear boson sampling circuits, see Supplementary Note 6. Another promising direction would be to configure the programmable nonlinear quantum photonic circuit as a quantum neural network³² and train the system to synthesize advanced quantum states of light, e.g., multi-photon entangled graph states for photonic quantum computing^49,50 or one-way quantum repeaters⁵⁶, or non-Gaussian quantum states such as Schrodinger cat states or GKP states for continuous-variable quantum information processing^57,58. Incorporating such programmable nonlinear circuits into quantum metrology architectures may enable high-fidelity quantum gates, opening new avenues for precision-enhanced quantum technologies⁵⁹. 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, the potential for high-speed operation exists, as QD-based nonlinear operation enables fast tunability⁶⁰. Fast linear reconfigurations are achievable with bulk EOMs or integrated photonics⁶¹. See Supplementary Note 7 for more details.

We have presented frameworks to scale up our platform from two to multiple modes with nonlinearities in all modes. However, the scalability of the nonlinearity remains an open question. This would allow for simulation of more complicated Hamiltonians with terms of higher than quartic order, and is an interesting direction of research for both linear and nonlinear circuits.

Our experiment is the first step towards unlocking such opportunities for photonic quantum hardware using programmable nonlinear photonic circuits.

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 Δ between the QD resonance and the photon frequency, and (ii) the temporal bandwidth of the photons relative to the intrinsic linewidth, σ, of the QD resonance (corresponding to a lifetime 2π/σ = 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.