To provide an experimental demonstration of the nonlinearity of the resource states produced via the ABS scheme, we employed a state-of-the-art set-up comprising several interconnected stages. The QOLOSSUS-PRO photonic machine has a demultiplexed quantum-dot-based source (Fig. 2a), interfaced with either an 8-mode or a 12-mode programmable integrated interferometer17,40,41,42,43 (Fig. 2b). For more details of the experimental apparatus, refer to ‘Experimental apparatus’ in Methods.
Fig. 2: Experimental scheme used to verify the nonlinear behaviour of the ABS scheme via QOLOSSUS-PRO.
a, Experimental scheme to generate multi-photon input states. A quantum-dot single-photon source is interfaced with a time-to-space demultiplexing stage to generate synchronized multi-photon states with up to four photons distributed over distinct spatial modes. b, Experimental layout of the reconfigurable photonic processing stage. The multi-photon states are injected into programmable photonic integrated circuits (PICs) with either 8 or 12 spatial modes, which are configured to implement the ABS interferometric configurations employed in this work. c, Experimental scheme for the realization of adaptive measurements and real-time feed-forward operations. A three-photon input state is first processed by an initial unitary U0 implemented on the eight-mode PIC. Six modes are used in the adaptive measurement of one photon, and the remaining two-mode, two-photon state is routed to the adaptive processing stage. The latter undergoes polarization conversion and active phase stabilization, assisted by an auxiliary stabilization photon counter-propagating through the chip, which provides feedback to a proportional–integral–derivative control loop acting on a piezo-electric element. To enable real-time feed-forward, the polarization-encoded state is delayed with a 240-m-long optical fibre compensated with a polarization controller. The state is converted back to spatial encoding via a displaced Sagnac interferometer. An electro-optic phase modulator applies at this stage a measurement-conditioned phase shift depending on the outcome p measured by an avalanche photodiode and processed by a logical unit, effectively implementing the adaptive transformation V(p). Finally, the output state is analysed using a pseudo-number-resolving detection scheme exploiting fibre beam splitters. PC, polarization controller; PID, proportional–integral–derivative; PM, phase modulator.
As mentioned in ‘Witnessing nonlinear dynamics’, within an ABS scheme, the minimal requirement for observing nonlinear dynamics going beyond linear optics is to have \({n}^{{\prime} } > 1\) photons distributed on \({m}^{{\prime} } > 1\) output modes.
To benchmark the experimental platform and validate its ability to implement ABS protocols, we performed two kinds of experiments in this non-trivial regime.
The first employs real-time adaptivity and effectively implements a single-stage ABS experiment that probes nonlinearity in the case with \({n}^{{\prime} }=2\) output photons and \({m}^{{\prime} }=2\) output modes. In this experiment, genuine feed-forward control is implemented following the scheme shown in Fig. 2c and described in detail in Supplementary Section 2. We briefly summarize it below. After the intermediate detection of a single photon (r = 1) in k = 6 modes, the two undetected outputs are converted into polarization-encoded modes through a path-to-polarization converter with active phase stabilization (Fig. 2c). This procedure is crucial for phase-stable propagation outside the photonic chip. Then, the two-photon polarization state is temporally delayed using a 240-m-long optical fibre, which provides sufficient time to process the measurement outcome and apply the corresponding adaptive operation. The photons are then converted back into spatial modes using a polarization-to-path interferometric stage (a displaced Sagnac interferometer), during which an adaptive phase shift is applied in real time to one interferometer arm via a fast electro-optic modulator. The applied phase depends explicitly on the detected measurement outcome, which determines the voltage applied to the electro-optic modulator within a predefined evenly spaced range, thereby realizing an outcome-conditioned unitary transformation V(p). The corresponding ABS configuration is illustrated in Fig. 3a, in particular in the upper left diagram for \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\). The green arrow indicates the real-time implementation of adaptivity.
Fig. 3: Implementation of ABS experiments with an increasing number of output photons and modes.
a, Diagrams of the implemented ABS experiments with m = 8 modes and increasing numbers of output modes \({m}^{{\prime} }\) and output photons \({n}^{{\prime} }\). In the (2, 2) case, the experiment implements real-time adaptivity, which is represented by the solid green arrow. Data from the real-time adaptive experiment are also highlighted in green in c. In all other configurations, adaptivity is emulated via post-selection (dashed arrows). In this case, the experiments have n = 4 input photons and detect r photons in the adaptive measurement modes, thus resulting in an output state of \({n}^{{\prime} }\) photons in \({m}^{{\prime} }\) modes. b, Summary table reporting, for each configuration, the number of sampled input transformations U0, the number M of measurement outcomes and corresponding adaptive unitaries (pi, Vi), and the average value of 1 − TVDsim, where TVDsim is the total variation distance with respect to the numerical simulations (with a model accounting for experimental imperfections), averaged over all sampled instances of U0 and Vi. Configurations in which bunching events are also considered to be valid adaptive measurement outcomes pi are marked with an asterisk: \({({n}^{{\prime} },{m}^{{\prime} })}^{* }\). c, Violin plots showing the distribution of the values for 1 − TVDsim, taken as a metric of the quality of the overall implementation, for the different output configurations \(({n}^{{\prime} },{m}^{{\prime} })\). Each data point corresponds to a distinct experimentally sampled adaptive configuration, defined by a sampled input interferometer U0 and a measurement-conditioned adaptive evolution (pi, Vi). For each \(({n}^{{\prime} },{m}^{{\prime} })\) configuration, the total sample size is given by the total number of sampled adaptive instances reported in b, namely \({N}_{{\rm{tot}}}={\sum }_{x}\#{U}_{0}^{(x)}{M}^{(x)}\), where the sum includes both standard and starred configurations when present. The violin plots report the full distribution of the sampled values. Central markers and error bars indicate mean values ± standard deviations.
The second kind of experiment implements single-stage ABS architectures featuring output states with \({n}^{{\prime} }=\{2,3\}\) photons across \({m}^{{\prime} }=\{3,4,5\}\) modes. In these cases, the experiments are performed by emulating adaptivity through post-selection, which allows the exploration of more complex output configurations within the same single-stage ABS paradigm. In these experiments, the programmable device with m = 8 modes is programmed into two blocks, the first implementing a fixed and randomly extracted unitary transformation U0 and the other an adaptive unitary Vi acting on \({m}^{{\prime} }\) modes. Diagrams of the implemented configurations are given in Fig. 3a. The dashed arrows indicate that adaptivity is emulated via post-selection. Note that, here and thereafter in the post-selection experiments, the interferometer configuration for the adaptive unitary Vi is chosen according to a fixed, deterministic rule, which is not optimized for any specific task but rather serves as a generic prescription to probe the ABS framework. Specifically, the phases defining Vi are chosen according to a fixed mapping between the ordered measurement outcomes pi, with i = 1, …, M, and a set of phase values, for example, θi = ϕi = iπ/(M + 1). Post-selection ensures a consistent correspondence between each adaptive measurement outcome and the applied unitary Vi.
The full output photon-count distributions are reconstructed in the experiment via a pseudo-number-resolving technique that also detects events where more than one photon ends up in the same mode, following the data analysis procedure described in ref. 40. We note that we record only events in which the number of photons detected at the output of the interferometer is equal to the number of photons injected at its input.
To assess the accuracy of the experimental implementation, we use the total variation distance \(\mathrm{TV}{{\rm{D}}}_{\mathrm{sim}}=\frac{1}{2}\parallel {\bf{P}}-{{\bf{P}}}_{\mathrm{sim}}{\parallel }_{1}\) computed between P, the measured distribution, and Psim, obtained with a numerical simulation of the experiment that incorporates realistic noise sources affecting the apparatus. Specifically, Psim is numerically computed using a model that captures the dominant non-idealities of the experimental platform, namely partial photon distinguishability and multi-photon emission from the source but assuming that both the unitary transformation U0 and the adaptive evolution Vi are correctly implemented with no errors. Further details of the numerical model employed are provided in Supplementary Section 3. The table in Fig. 3b lists for each \(\{{m}^{{\prime} },{n}^{{\prime} }\}\) configuration the number of sampled U0 instances, the number M of different possible adaptive measurement outcomes pi and adaptive unitaries Vi and also the average values \({\varDelta }_{\mathrm{TVD}}=1-\mathrm{TV}{{\rm{D}}}_{\mathrm{sim}}\). The full distributions of ΔTVD across all the implemented ABS instances are shown in Fig. 3c. On average, values of ΔTVD in excess of 0.85 are obtained, thus confirming the high accuracy of the implementation and the overall control of the experimental apparatus.
As a subsequent step, the methods previously described to probe the emergence of nonlinear-optical dynamics within the ABS paradigm are applied. For this purpose, we performed new experimental acquisitions.
First, we consider two experimental implementations yielding two-photon, two-mode output states \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\), as shown in Fig. 4a,b. The first one is an extension of the real-time adaptive ABS experiment introduced before, with n = 3 input photons and r = 1 measured photon. The second implementation has n = 4 input photons and r = 2 measured photons, and in this case, adaptivity is emulated through post-selection. In both implementations, after a fixed evolution U0, measurement events with r = {1, 2} photons are recorded. Each configuration is associated with a different adaptive unitary Vi. Then, after the adaptive block, the projective measurements needed to perform a complete tomography of the output state are implemented, as detailed in Supplementary Section 1. For the real-time adaptive experiment, this is performed by exploiting the polarization-encoded representation of the modes, which enables the implementation of the required tomographic projections T through a set of wave plates located after the adaptive stage, as described in Supplementary Section 1.
Fig. 4: Experimental results for two-photon, two-mode ABS with full output state tomography.
a,b, Diagrams of the experimentally implemented schemes yielding a two-photon output state in two modes, \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\). a, Real-time adaptive ABS protocol with n = 3 input photons and r = 1 measured photon. An initial eight-mode unitary U0 is followed by an intermediate measurement yielding an outcome pi, which conditions the application of an adaptive two-mode unitary Vi via active feed-forward (solid green arrow). b, Post-selected implementation with n = 4 input photons and r = 2 measured photons. The adaptivity of the operation Vi, conditioned on the outcome pi, is emulated in post-selection (dashed arrow). In both cases, the final two-photon output state is characterized via full quantum state tomography, implemented through a set of tomographic projections T acting on the output modes. c,d, Fidelity kernel matrices Kij computed from the reconstructed output states obtained for a fixed choice of U0 and different adaptive unitaries Vi. c, Kernel matrix for the real-time adaptive experiment, constructed from the six reconstructed output states. d, Kernel matrix for the post-selected experiment, constructed from the 15 reconstructed output states. e,f, Minimum distances \(1-F(\widehat{\rho },{\widehat{\rho }}^{| 11\rangle })\) and \(1-F(\widehat{\rho },{\widehat{\rho }}^{| 20\rangle })\) between the reconstructed output states and the sets of states obtainable via linear-optical evolution from the input states \(| 1,1\rangle\) and \(| 2,0\rangle\), respectively. e, Distances for the six states in the real-time adaptive experiment. f, Distances for the 15 post-selected output states. In all cases, the distance is approximated by sampling a representative set of \({\mathcal{O}}(1{0}^{4})\) linear-optical states for each input. Data are presented as mean values ± standard deviation, obtained from Monte Carlo resampling of the experimentally measured tomographic counts assuming Poissonian counting statistics. For each reconstructed output state, the distances were evaluated over N = 100 resampled datasets generated from the measured photon-count distributions.
To quantify the accuracy of the reconstruction, the state fidelity between the experimentally reconstructed states \({\widehat{\rho }}_{i}\) and the corresponding states obtained from a numerical simulation that accounts for the dominant noise sources of the experimental apparatus is evaluated. For the real-time adaptive experiment with n = 3, we obtained an average fidelity \({\overline{F}}_{n=3}=0.83(6)\). For the post-selected implementation with n = 4 input photons, the obtained average fidelity was \({\overline{F}}_{n=4}=0.91(1)\).
The output state distribution in the output Fock space is shown via the pairwise fidelity kernel matrices (Fig. 4c,d), computed as \({K}_{ij}={({\rm{Tr}}\sqrt{\sqrt{{\widehat{\rho }}_{i}}{\widehat{\rho }}_{j}\sqrt{{\widehat{\rho }}_{i}}})}^{2}\). We note that the distribution of ABS output states in Hilbert space can be substantially reshaped by different associations between measurement outcomes and adaptive operations, as discussed in Supplementary Section 4.
Given the knowledge of a given ABS output state, one can numerically evaluate its minimal distance with respect to the set of states obtainable via an equivalent passive linear-optical dynamics, as described in ‘Passive separability and emergence of optical nonlinearities’. To do so, ~104 random states were numerically sampled from the sets of passive separable states \({\widehat{{\boldsymbol{\sigma }}}}_{| 1,1\rangle }\) and \({\widehat{{\boldsymbol{\sigma }}}}_{| 2,0\rangle }\). As shown in Fig. 4e,f, the values obtained for the minimized distances are, in most cases, greater than zero, even when considering the associated experimental errors. Overall, this demonstrates that, even in a small-scale ABS implementation, the observed dynamics can hardly be traced back to equivalent linear BS evolutions. For \({m}^{{\prime} } > 2\), we discuss a similar approach based on the trace distance in Supplementary Section 5, which we apply to the output distributions obtained from post-selection experiments in these regimes.
As a final step, we experimentally implemented ABS schemes with \({m}^{{\prime} }=\{2,3,4\}\) and \({n}^{{\prime} }=\{2,3\}\), focusing on the measurement of the Lie algebraic quantity of equation (5).
First, we consider again the real-time adaptive set-up introduced above and also extended configurations with up to \({n}^{{\prime} }=3\) output photons and \({m}^{{\prime} }=4\) output modes. The corresponding interferometer diagrams are illustrated in Fig. 5a,b, respectively. For the post-selection regime, we show as a representative case the set-up with \({n}^{{\prime} }=2\) output photons and \({m}^{{\prime} }=4\) output modes. In all cases considered, the final layers of the interferometer were employed to implement the transformations required to measure the observables \({\widehat{O}}_{i}\) used to compute the quantity \(I({\widehat{\rho }}_{{\rm{ABS}}})\) for a given output state \({\widehat{\rho }}_{{\rm{ABS}}}\). Specifically, we used a suitable configuration of both balanced and cross beam splitters to couple non-adjacent modes40. A schematic representation of the corresponding interferometric architecture is shown in the lower right parts of Fig. 5a,b. In this way, the average number of photons output \(\langle {\widehat{n}}_{j}\rangle\) was experimentally measured and used to estimate \({\rm{Tr}}({\widehat{O}}_{i}{\widehat{\rho }}_{{\rm{ABS}}})\). Note that, in general, measuring Lie-invariant observables requires fewer measurements than full state tomography. When \({n}^{{\prime} }={m}^{{\prime} }=2\), the observables required to evaluate \(I(\widehat{\rho })\) are a subset of the tomographic measurements, and thus, no further measurements are required in this case.
Fig. 5: Lie invariants as witnesses of nonlinearity in ABS.
a–c, Conceptual diagrams of the ABS protocols employed to probe Lie invariants. a, Real-time adaptive ABS configuration with n = 3 input photons and r = 1 measured photon, yielding an output state with \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\). An intermediate measurement outcome p conditions the application of an adaptive unitary V via active feed-forward. The final output state is characterized by directly measuring the expectation values of selected Lie observables. b, Single-stage adaptive configuration with n = 4 input photons. A measurement outcome p conditions a single adaptive unitary V, shown here for a representative case with \({m}^{{\prime} }=4\) output modes. c, Two-stage adaptive configuration with n = 4 input photons. Two sequential measurement outcomes (p(1), p(2)) condition two adaptive unitaries V(1) and V(2). In these latter two cases, adaptivity is emulated through post-selection. In all cases, the lower right parts illustrate the linear-optical transformations that need to be implemented to measure the expectation values of specific \({O}_{j}^{z}\), \({O}_{jk}^{x}\) and \({O}_{jk}^{y}\) observables. In the real-time adaptive experiment, these unitaries were implemented using polarization-encoded wave plates, as described in Supplementary Section 1. In the post-selection experiments, they were implemented directly on the integrated photonic chip. d–f, Experimental results for the measured Lie-invariant quantities \(I(\widehat{\rho })\) obtained from ABS experiments. d, Measured Lie invariants obtained in the real-time adaptive experiment with n = 3 input photons and output configuration \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\). e,f, Histograms of measured Lie-invariant quantities \(I(\widehat{\rho })\) obtained from ABS experiments. In the single-stage configuration (e), invariants are reported for different output photon and mode numbers \(({n}^{{\prime} },{m}^{{\prime} })=\{(2,3),(2,4),(3,3),(3,4)\}\). In the two-stage adaptive configuration (f), invariant values are shown for the probed case \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\). In all cases, the histograms include results from different output states generated by permuting the association of the adaptive unitary to the measurement outcomes. Vertical dashed lines indicate the theoretical values for \(I(\widehat{\rho })\) from Fock states with the corresponding number of photons and modes, as well as the theoretical minimum and maximum values allowed in each configuration. We also show results obtained from simulating random ABS output states within the corresponding configuration.
Figure 5d shows the measured values of \(I({\widehat{\rho }}_{{\rm{ABS}}})\) for the real-time adaptive experiments with \({n}^{{\prime} }=2\) and \({m}^{{\prime} }=2\). Figure 5e depicts the distribution of the measured values of \(I({\widehat{\rho }}_{{\rm{ABS}}})\) for several emulated ABS configurations defined via the parameters \(({n}^{{\prime} },{m}^{{\prime} })\). Note that the post-selection regime in these experiments allows one to consider permutations of the association between measurement outcomes and adaptive unitaries, as explained in Supplementary Section 6, thereby increasing the available statistics. For comparison, the values of \(I(\widehat{\rho })\) associated with the relevant passively separable states in each configuration are reported, together with the minimum and maximum value that can be obtained in each configuration. The latter quantities were computed following the procedure reported in ref. 40 and are, respectively, \(I{(\widehat{\rho })}_{\min }={n}^{{\prime} 2}/{m}^{{\prime} }\) and \(I{(\widehat{\rho })}_{\max }={n}^{{\prime} 2}\).
Note that the measured Lie invariant \(I({\widehat{\rho }}_{{\rm{ABS}}})\) can substantially deviate from any Lie-invariant value obtainable in an equivalent passive BS dynamics with \({n}^{{\prime} }\) photons in \({m}^{{\prime} }\) modes, which shows that the resources \({\widehat{\rho }}_{{\rm{ABS}}}\) obtained via ABS can be obtained only in a regime going beyond linear optics. We emphasize that the nonlinearity affecting the invariant \(I(\widehat{\rho })\) originates from the measurement operation. In fact, different values of \(I(\widehat{\rho })\) are observed for different outcomes pi. However, for a fixed measurement outcome pi, the value of \(I(\widehat{\rho })\) remains unchanged when permuting the association between pi and different adaptive unitaries Vi. This reflects that, in the single-stage configuration, the conditional evolution induces a linear transformation of the output state. This is discussed in detail in Supplementary Section 6.
All the analysis presented so far has been for a single adaptive stage after the application of the first unitary U0. Indeed, interesting dynamics arise when several adaptive stages are cascaded. We studied this regime in a set-up where two sequential measurement outcomes (p(1), p(2)) condition two subsequent adaptive unitaries V(1) and V(2). This configuration, schematically illustrated in Fig. 5c and described in detail in Supplementary Section 6, results in an output state with \(({n}^{{\prime} },{m}^{{\prime} })=(2,2)\), which is again analysed by measuring Lie-invariant observables. Experimentally, this configuration is implemented on a 12-mode integrated photonic processor. We exploited the increased interferometric depth to include several sequential adaptive stages within a single device. Figure 5f shows the experimentally obtained distribution of \(I({\widehat{\rho }}_{{\rm{ABS}}})\) in this configuration. In contrast to the single-stage case, the measured invariant values now depend on the choice of the adaptive unitaries, as detailed in Supplementary Section 6. Although V(2) acts only as a final linear evolution layer, V(1) reshapes the state before the subsequent measurement, thus effectively changing the nonlinear dynamics.
Furthermore, from data obtained in the \({m}^{{\prime} }=\{2,3\}\) scenarios, the spectrum of the state density matrices projected onto the subalgebra of linear-optical Hamiltonians can also be computed. This offers an alternative criterion for showing that the states obtained in the ABS regime are nonlinear, as discussed in Supplementary Section 7.
Finally, note that even if ABS could be employed to access a set of states whose preparation is forbidden with only linear optics, there still exists a regime in which such dynamics is not efficient, for example, when k scales linearly with m. Notably, when k and r are allowed to increase, the strategy proposed in ref. 17 is to consider mixed states formed by averaging over all (or subsets of) adaptive measurement outcomes. Although the main analysis is focused here on pure states, the mixed-state scenario is investigated in Supplementary Section 8, which confirms that signatures of dynamics beyond linear optics persist in this regime.