Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M -mode noisy circuit with detected modes D and undetected modes U , the complexity of our algorithm is O(M2∏i∈UCi2∏i∈DCi) , rather than O(M2∏i∈D∪UCi2) , where Ci is the Fock cutoff of mode i . As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.
Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M -mode noisy circuit with detected modes D and undetected modes U , the complexity of our algorithm is O(M2∏i∈UCi2∏i∈DCi) , rather than O(M2∏i∈D∪UCi2) , where Ci is the Fock cutoff of mode i . As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.
Gottesman-Kitaev-Preskill (GKP) states are essential non-Gaussian resources for fault-tolerant photonic quantum computing, enabling logical qubit encoding with intrinsic robustness against errors. Several approaches to GKP state preparation have been explored, including measurement-based protocols in circuit QED and trapped-ion systems, cat-state breeding, and photon-subtraction schemes. However, these methods are either restricted to specific platforms or require deep non-Gaussian resource chains with exponentially low success probabilities. Gaussian Boson Sampling (GBS) offers a compelling all-photonic alternative by generating non-Gaussian states through measurement-induced nonlinearity, without the need for matter-based ancilla or active feedforward. Nevertheless, its practical implementation is limited by the exponential computational cost of evaluating matrix hafnians-#P-complete functions that govern photon-number probabilities. To address this challenge, we introduce a two-stage Histogram Gradient Boosting surrogate pipeline that predicts, without any hafnian computation, the optimal heralding pattern, circuit fidelity, and post-selection probability for candidate GBS circuits, while reserving exact quantum simulation exclusively for surrogate-selected candidates. Trained on circuit configurations across 3-5 optical modes, the surrogate achieves 90.0% GKP-detection accuracy on a held-out set, representing a 23.7 percentage-point improvement over the baseline, with a fidelity mean absolute error of 0.032 and a log-scale post-selection probability R2=0.837, reducing the total simulation burden by approximately 90%.
We propose a framework to design and optimize generic photonic quantum circuits composed of Gaussian objects (pure and mixed Gaussian states, Gaussian unitaries, Gaussian channels, Gaussian measurements) as well as non-Gaussian effects such as photon-number-resolving measurements. In this framework, we parametrize a phase space representation of Gaussian objects using elements of the symplectic group (or the unitary or orthogonal group in special cases), and then we transform it into the Fock representation using a single linear recurrence relation that computes the Fock amplitudes of any Gaussian object recursively. We also compute the gradient of the Fock amplitudes with respect to phase space parameters by differentiating through the recurrence relation. We can then use Riemannian optimization on the symplectic group to optimize M M -mode Gaussian objects, avoiding the need to commit to particular realizations in terms of fundamental gates. This allows us to “mod out” all the different gate-level implementations of the same circuit, which now can be chosen after the optimization has completed. This can be especially useful when looking to answer general questions, such as bounding the value of a property over a class of states or transformations, or when one would like to worry about hardware constraints separately from the circuit optimization step. Finally, we make our framework extendable to non-Gaussian objects that can be written as linear combinations of Gaussian ones, by explicitly computing the change in global phase when states undergo Gaussian transformations. We implemented all of these methods in the freely available open-source library MrMustard, which we use in three examples to optimize the 216-mode interferometer in Borealis, and 2- and 3-modes circuits (with Fock measurements) to produce cat states and cubic phase states.
Photonic qubits play an important role in quantum information processing. However, to date, efficient high-speed single photon sources and photon number resolving (PNR) detectors are still difficult to make. Here, we use phase-randomized coherent states and threshold detectors, combined with decoy-state post-processing, to simulate single photon sources and PNR detectors and perform various quantum information processing tasks. We propose a simple linear regression method to predict Fock state statistics based on observable statistics from coherent-state probes, combined with random/structured sampling of probe settings. Compared with traditional decoy-state post-processing based on linear programming and Cartesian product grid sampling, our method enables drastically improved accuracy and much reduced probe resources. It works particularly well for unknown unitary circuits of interest with fixed-size Hilbert space dimensions. For instance, it can accurately simulate 4-photon interference in linear optics with just 32 input/output probe settings and 6-photon interference with 256 probe settings. Moreover, it also works for non-unitary (such as lossy, nonlinear, or arbitrary) quantum channels. This enables a wide variety of applications that can be implemented with easily attainable coherent light sources and threshold detectors, such as characterization of small-scale linear optical quantum computing circuits or quantum sensing/metrology devices.
Generation of high-fidelity photonic non-Gaussian states is a crucial ingredient for universal quantum computation using continuous-variable platforms, yet it remains a challenge to do this efficiently. We present a general framework for a probabilistic production of multimode non-Gaussian states by measuring a few modes of multimode Gaussian states via photon-number-resolving detectors. We use Gaussian elements consisting of squeezed displaced vacuum states and interferometers, the only non-Gaussian elements consisting of photon-number-resolving detectors. We derive analytic expressions for the output Wigner function, and the probability of generating the states in terms of the mean and the covariance matrix of the Gaussian state and the photon detection pattern. We find that the output states can be written as a Fock-basis superposition state followed by a Gaussian gate, and we derive explicit expressions for these parameters. These analytic expressions show exactly what non-Gaussian states can be generated by this probabilistic scheme. Further, it provides a method to search for the Gaussian circuit and measurement pattern that produce a target non-Gaussian state with optimal fidelity and success probability. We present specific examples such as the generation of cat states, ON states, Gottesman-Kitaev-Preskill states, NOON states, and bosonic-code states. The proposed framework has potentially far-reaching implications for the generation of bosonic error-correction codes that require non-Gaussian states and resource states for the implementation of non-Gaussian gates needed for universal quantum computation, among other applications requiring non-Gaussianity. The tools developed here could also prove useful for the quantum resource theory of non-Gaussianity.
Photonic quantum computing has gained significant interest in recent years due to its potential for scaling to large numbers of qubits. A critical requirement for fault-tolerant quantum computation is the reliable generation of non-Gaussian quantum states, typically achieved using Gaussian operations and photon-number-resolving detectors. However, the probabilistic nature of quantum measurement typically results in low success rates for state preparation. Conventionally, these circuits are optimized to herald a single specific target outcome, thereby disregarding the potential utility of alternative measurement patterns generated by the same physical setup. In this work, we propose and demonstrate a multi-outcome optimization strategy that increases the overall acceptance probability by allowing a single circuit to produce useful quantum states across several measurement patterns. To evaluate this approach, we apply the framework to the generation of Gottesman-Kitaev-Preskill core states, Schrodinger cat states, binomial codes, and cubic phase states using both two-mode and three-mode Gaussian circuits. We demonstrate that the success probability can be enhanced through two distinct mechanisms: first, by simultaneously targeting a diverse set of useful resource states, and second, by aggregating degenerate outcomes to maximize the production rate of a single target state.
To understand quantum optics experiments, we must perform calculations that consider the principal sources of noise, such as losses, spectral impurity and partial distinguishability. In both discrete and continuous variable systems, these can be modeled as mixed Gaussian states over multiple modes. The modes are not all resolved by photon-number measurements and so require calculations on coarse-grained photon-number distribution. Existing methods can lead to a combinatorial explosion in the time complexity, making this task unfeasible for even moderate sized experiments of interest. In this work, we prove that the computation of this type of distributions can be done in exponential time, providing a combinatorial speedup. We develop numerical techniques that allow us to determine coarse-grained photon number distributions of Gaussian states, as well as density matrix elements of heralded non-Gaussian states prepared in the presence of spectral impurity and partial distinguishability. These results offer significant speed-up and accuracy improvements to validation tests of both Fock and Gaussian boson samplers that rely on binned probability distributions.