Gaussian Boson Sampling Research Articles

A universal programmable Gaussian boson sampler for drug discovery

Gaussian boson sampling (GBS) has the potential to solve complex graph problems, such as clique finding, which is relevant to drug discovery tasks. However, realizing the full benefits of quantum enhancements requires large-scale quantum hardware with universal programmability. Here we have developed a time-bin-encoded GBS photonic quantum processor that is universal, programmable and software-scalable. Our processor features freely adjustable squeezing parameters and can implement arbitrary unitary operations with a programmable interferometer. Leveraging our processor, we successfully executed clique finding on a 32-node graph, achieving approximately twice the success probability compared to classical sampling. As proof of concept, we implemented a versatile quantum drug discovery platform using this GBS processor, enabling molecular docking and RNA-folding prediction tasks. Our work achieves GBS circuitry with its universal and programmable architecture, advancing GBS toward use in real-world applications.

Post-selection in noisy Gaussian boson sampling: part is better than whole

Gaussian boson sampling (GBS) is originally proposed to show quantum advantage with quantum linear optical elements. Recently, several experimental breakthroughs based on GBS pointing to quantum computing supremacy have been presented. However, due to technical limitations, the outcomes of GBS devices are influenced severely by photon loss. Here, we present a practical method to reduce the negative effect caused by photon loss. We first show with explicit formulas that a GBS process can be mapped to another GBS processes. Based on this result, we propose a post-selection method which discards low-quality data according to our criterion to improve the performance of the final computational results, say part is better than whole. As an example, we show that the post-selection method can turn a GBS experiment that would otherwise fail in a ‘non-classicality test’ into one that can pass that test. Besides improving the robustness of computation results of current GBS devices, this post-selection method may also benefit the further development of GBS-based quantum algorithms.

A Quadratic Speedup in the Optimization of Noisy Quantum Optical Circuits

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.

Classical models may be a better explanation of the Jiuzhang 1.0 Gaussian Boson Sampler than its targeted squeezed light model

Recently, Zhong et al. [1, 2] performed landmark Gaussian boson sampling experiments with up to 144 modes using threshold detectors. The authors claim to have achieved quantum computational advantage with the implementation of these experiments, named Jiuzhang 1.0 and Jiuzhang 2.0. Their experimental results are validated against several classical hypotheses and adversaries using tests such as the comparison of statistical correlations between modes, Bayesian hypothesis testing and the Heavy Output Generation (HOG) test. In this work we propose an alternative classical hypothesis for the validation of these experiments. We use the probability distribution of mixtures of coherent states sent into a lossy interferometer; these input mixed states, which we term squashed states, have vacuum fluctuations in one quadrature and excess fluctuations in the other. We find that for configurations in the high photon number density regime, the comparison of statistical correlations does not tell apart the ground truth of the experiment (two-mode squeezed states sent into an interferometer) from our alternative hypothesis. On the other hand, the Bayesian test indicates that, for all configurations excepting Jiuzhang 1.0, the ground truth is a more likely explanation of the experimental data than our alternative hypothesis. A similar result is obtained for the HOG test: for all configurations of Jiuzhang 2.0, the test indicates that the experimental samples have higher ground truth probability than the samples obtained from our alternative distribution; for Jiuzhang 1.0 the test is inconclusive. Our results provide a new hypothesis that should be considered in the validation of future GBS experiments, and shed light into the need to identify proper metrics to verify quantum advantage in the context of threshold GBS. Additionally, they indicate that a classical explanation of the Jiuzhang 1.0 experiment, lacking any quantum features, has not been ruled out.

Page curves and typical entanglement in linear optics

Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.

Solving Graph Problems Using Gaussian Boson Sampling.

Gaussian boson sampling (GBS) is not only a feasible protocol for demonstrating quantum computational advantage, but also mathematically associated with certain graph-related and quantum chemistry problems. In particular, it is proposed that the generated samples from the GBS could be harnessed to enhance the classical stochastic algorithms in searching some graph features. Here, we use Jiǔzhāng, a noisy intermediate-scale quantum computer, to solve graph problems. The samples are generated from a 144-mode fully connected photonic processor, with photon click up to 80 in the quantum computational advantage regime. We investigate the open question of whether the GBS enhancement over the classical stochastic algorithms persists-and how it scales-with an increasing system size on noisy quantum devices in the computationally interesting regime. We experimentally observe the presence of GBS enhancement with a large photon-click number and a robustness of the enhancement under certain noise. Our work is a step toward testing real-world problems using the existing noisy intermediate-scale quantum computers and hopes to stimulate the development of more efficient classical and quantum-inspired algorithms.

Measuring the Schmidt number of parametric down conversion by exploiting photon distribution

The Schmidt number quantifies the number of modes and is mainly used as a measure for the quality of entanglement. We theoretically compute the photon distribution of type-I spontaneous parametric down conversion (SPDC) with an arbitrary Schmidt number. The photon distribution is used for a novel method to measure the Schmidt number. This method requires only two on–off single-photon detectors with no photon number or temporal resolution. The method works in the strong pumping regime where high photon numbers are non-negligible. We experimentally demonstrate the method for type-II SPDC. The easy and fast measurement of the Schmidt number has a broad range of applications from the calibration of strong pump SPDC and entanglement sources to multi-photon quantum interference and Gaussian boson sampling.

Multi-qubit BEC trap for atomic boson sampling

We suggest a new platform for examination of quantum statistical phenomena in many-body interacting systems – a multi-qubit Bose-Einstein-condensate (BEC) trap. Ideally, it is suitable for demonstrating manifestations of quantum supremacy of such systems through atomic boson sampling of excited-state occupations. This is wholly different than the linear-interferometer platform which uses Gaussian boson sampling of non-interacting non-equilibrium photons rather than interacting atoms in equilibrium. We present a basic model of the multi-qubit BEC trap and illustrate its essential properties by means of numerical simulations.

Comparison of Performances for Quantum and Conventional Algorithms: Shor’s algorithm and Boson Sampling

Quantum computing has been extensive popularity discussed recently. The foundation of quantum study is the investigation of how quantum computing shows its superiority compared with conventional algorithms. On this basis, this paper focuses on the comparison of the performances of quantum and conventional algorithms about the same relevant area. Two classic problems, factorization of large integer and boson sampling, are chosen to show the specific difference of related quantum and conventional algorithms towards the same problem. This paper specifically compares the performance of Shor's algorithm and other advanced traditional integer factorization algorithms for prime factorization problems and compares the processing capabilities of traditional algorithms and Gaussian Boson sampling for sampling problems. According to the analysis, for the same research problem, quantum algorithms show strong advantages over traditional algorithms in many performances, e.g., time efficiency, algorithm adaptability. It leads to a deeper understanding of quantum superiority and a further explanation of the significance of the development of quantum computing.

Non-linear Boson Sampling

Boson Sampling is a task that is conjectured to be computationally hard for a classical computer, but which can be efficiently solved by linear-optical interferometers with Fock state inputs. Significant advances have been reported in the last few years, with demonstrations of small- and medium-scale devices, as well as implementations of variants such as Gaussian Boson Sampling. Besides the relevance of this class of computational models in the quest for unambiguous experimental demonstrations of quantum advantage, recent results have also proposed the first applications for hybrid quantum computing. Here, we introduce the adoption of non-linear photon–photon interactions in the Boson Sampling framework, and analyze the enhancement in complexity via an explicit linear-optical simulation scheme. By extending the computational expressivity of Boson Sampling, the introduction of non-linearities promises to disclose novel functionalities for this class of quantum devices. Hence, our results are expected to lead to new applications of near-term, restricted photonic quantum computers.

Investigation of Gaussian boson sampling under phase noise of the light source

Gaussian boson sampling is one of the main promising approaches to realizing the quantum computational advantage, which also offers potential applications such as in dense subgraphs problem and quantum chemistry. However, the inevitable noise in experiment may weaken the quantum advantage of Gaussian boson sampling. Photon loss and photon partial indistinguishability are two major sources of noise. Their influence on the complexity of Gaussian boson sampling has been extensively studied in previous work. However, the phase noise of the input light source, a noise which is suitable for tailored for Gaussian boson sampling, has not been studied so far. Here, we investigate the phase noise of the input light source in Gaussian boson sampling through numerical simulation. We use the Monte Carlo method to calculate the output probability distribution under phase noise approximately. It is found that the phase noise of the light source can cause the input state to change from a Gaussian state into a non-Gaussian mixed state. For a given phase noise level, the fidelity of the non-Gaussian mixed state and the noise-free ideal state decreases monotonically as the mean photon number of input increases. Meanwhile, owing to the phase noise the deviation of the output probability distribution gradually increases with the number of detected photons increasing. Furthermore, the phase noise results in the capability of heavy sample generation (HOG), significantly decreasing. Finally, it is found that Gaussian boson sampling with photon loss is more tolerant to phase noise than the lossless case given that the mean photon number of input is the same. Our study is helpful in suppressing the phase noise in large-scale Gaussian boson sampling experiments.

Atomic boson sampling in a Bose-Einstein-condensed gas

We consider quantum statistical physics of many-body equilibrium fluctuations in an interacting Bose-Einstein-condensed (BEC) gas. We find a universal analytic formula for a characteristic function (Fourier transform) of a joint probability distribution for the particle occupation numbers in a BEC gas and discuss $\ensuremath{\sharp}\mathrm{P}$-hardness of computing this distribution. The latter is done by means of the Hafnian master theorem generalizing the classical permanent master theorem of MacMahon. We suggest an atomic boson sampling in the many-body interacting systems as an alternative to a widely studied Gaussian boson sampling of photons. We outline a multiqubit BEC trap, formed by a set of the single-qubit potential wells, as a convenient model for studying atomic boson sampling.

Multi-Qubit Bose-Einstein Condensate Trap for Atomic Boson Sampling.

We propose a multi-qubit Bose-Einstein-condensate (BEC) trap as a platform for studies of quantum statistical phenomena in many-body interacting systems. In particular, it could facilitate testing atomic boson sampling of the excited-state occupations and its quantum advantage over classical computing in a full, controllable and clear way. Contrary to a linear interferometer enabling Gaussian boson sampling of non-interacting non-equilibrium photons, the BEC trap platform pertains to an interacting equilibrium many-body system of atoms. We discuss a basic model and the main features of such a multi-qubit BEC trap.

The Complexity of Bipartite Gaussian Boson Sampling

Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof. Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.

Certification of Gaussian Boson Sampling via graphs feature vectors and kernels

Gaussian Boson Sampling (GBS) is a non-universal model for quantum computing inspired by the original formulation of the Boson Sampling (BS) problem. Nowadays, it represents a paradigmatic quantum platform to reach the quantum advantage regime in a specific computational model. Indeed, thanks to the implementation in photonics-based processors, the latest GBS experiments have reached a level of complexity where the quantum apparatus has solved the task faster than currently up-to-date classical strategies. In addition, recent studies have identified possible applications beyond the inherent sampling task. In particular, a direct connection between photon counting of a genuine GBS device and the number of perfect matchings in a graph has been established. In this work, we propose to exploit such a connection to benchmark GBS experiments. We interpret the properties of the feature vectors of the graph encoded in the device as a signature of correct sampling from the true input state. Within this framework, two approaches are presented. The first method exploits the distributions of graph feature vectors and classification via neural networks. The second approach investigates the distributions of graph kernels. Our results provide a novel approach to the actual need for tailored algorithms to benchmark large-scale Gaussian Boson Samplers.

Experimentally Finding Dense Subgraphs Using a Time-Bin Encoded Gaussian Boson Sampling Device

Gaussian boson sampling is a quantum computing concept based on drawing samples from a multimode nonclassical Gaussian state using photon-number resolving detectors. It was initially posed as a near-term approach to achieve quantum advantage, and several applications have been proposed since, including the calculation of graph features. For the first time, we use a time-bin encoded interferometer to implement Gaussian boson sampling experimentally and extract samples to enhance the search for dense subgraphs in a graph. Our results indicate an improvement over classical methods for subgraphs of sizes three and four in a graph containing ten nodes. In addition, we numerically explore the role of imperfections in the optical circuit and on the performance of the algorithm.2 MoreReceived 21 June 2022Revised 26 August 2022Accepted 29 August 2022DOI:https://doi.org/10.1103/PhysRevX.12.031045Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBoson samplingOptical quantum information processingQuantum information processing with continuous variablesQuantum simulationQuantum Information

Distinguishability in quantum interference with multimode squeezed states

Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of $n$ pairs of photons, whose normalization $q_{2n}$ is the probability that $2n$ photons interfere as indistinguishable. For two pairs of photons $q_{4}=(1+ 2\mathbb{P} )/3$, where $\mathbb{P} $ is the purity of the squeezed states ($K=1/\mathbb{P} $ is the Schmidt number). For a fixed purity $\mathbb{P}$, the probability $q_{2n}$ decreases exponentially fast in $n$. For example, in the experimental Gaussian boson sampling of H.-S.~Zhong \textit{et al} [Science \textbf{370}, 1460 (2020)], the achieved purity $\mathbb{P}\approx 0.938$ for the average number of photons $2n\ge 43$ gives $q_{2n}\lesssim 0.5$, i.e., close to the middle line between $n$ indistinguishable and $n$ distinguishable pairs of photons. In derivation of all the results the first-order quantization representation based on the particle decomposition of the Hilbert space of identical bosons serves as an indispensable tool. The approach can be applied also to the generalized (non-Gaussian) squeezed states, such as those recently generated in the three-photon parametric down-conversion.

Benchmarking 50-Photon Gaussian Boson Sampling on the Sunway TaihuLight

Boson sampling is expected to be an important milestone that will demonstrate quantum computational advantage (or quantum supremacy). This work establishes the benchmarking of Gaussian boson sampling (GBS) with threshold detection based on the Sunway TaihuLight supercomputer. To achieve the best performance and provide a competitive scenario for future quantum computing studies, the selected simulation algorithm is fully optimized based on a set of innovative approaches, including a parallel framework with almost perfect load balance and an instruction-level optimizing scheme based on a shortest-path-based instruction scheduling. In addition, data precision is carefully processed by an integer-instruction-based and multiple-precision fixed-point implementation, including 128- and 256-bit precison mode, which can be appropriately selected based on an adaptive precision optimizing scheme. Based on these methods, a highly efficient parallel quantum sampling algorithm is designed. The largest run enables us to obtain one Torontonian function of a <inline-formula><tex-math notation="LaTeX">$100\times 100$</tex-math></inline-formula> submatrix from 50-photon GBS within 20 hours in 128-bit precision and 2 days in 256-bit precision. To our knowledge, this was the largest quantum computing simulation based on Boson Sampling by using modern supercomputers.

Applications of the Lambert–Tsallis Wq function in quantum photonic Gaussian boson sampling

Applications of the Lambert–Tsallis Wq function in quantum photonic Gaussian boson sampling

Experimental Demonstration of Gaussian Boson Sampling with Displacement

Gaussian boson sampling (GBS) is a quantum sampling task in which one has to draw samples from the photon-number distribution of a large-dimensional nonclassical squeezed state of light. In an effort to make this task intractable for a classical computer, experiments building GBS machines have mainly focused on increasing the dimensionality and squeezing strength of the nonclassical light. However, no experiment has yet demonstrated the ability to displace the squeezed state in phase space, which is generally required for practical applications of GBS. In this work, we build a GBS machine that achieves the displacement by injecting a laser beam alongside a two-mode squeezed vacuum state into a 15-mode interferometer. We focus on two new capabilities. Firstly, we use the displacement to reconstruct the multimode Gaussian state at the output of the interferometer. Our reconstruction technique is in situ and requires only three measurement settings regardless of the state dimension. Secondly, we study how the addition of classical laser light in our GBS machine affects the complexity of sampling its output photon statistics. We introduce and validate approximate semiclassical models that reduce the computational cost when a significant fraction of the detected light is classical.4 MoreReceived 1 February 2022Accepted 19 April 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020336Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBoson samplingPhoton statisticsQuantum simulationQuantum states of lightQuantum tomographyQuantum InformationAtomic, Molecular & Optical