An atomic boson sampler

Boson sampling, a restricted model of quantum computing, involves sampling from the distribution resulting from the interference of identical bosons under programmable, non-interacting dynamics. Efficient classical simulation of this process is believed to be intractable, motivating experimental efforts using photons. However, photon-based approaches face challenges in generating and evolving specific photon numbers with low loss. This paper explores an alternative using ultracold atoms, which are more amenable to preparation in Fock states and offer advantages in terms of state preparation, evolution control, and detection. The use of ultracold atoms in a two-dimensional optical lattice allows for deterministic preparation of large ensembles of atoms, their evolution under precisely controlled dynamics, and high-fidelity detection of their final positions. This study aims to overcome limitations in previous atom-based approaches by using advanced techniques such as optical tweezers for rapid state preparation, minimizing loss during evolution, and enabling high-fidelity detection with atom-resolved imaging. The ability to perform these processes with high speed and high fidelity is a key advancement that enables the large-scale demonstration presented in this paper.

Previous boson sampling experiments have primarily utilized photons, with increasing numbers of photons being used in increasingly larger interferometers. These experiments have encountered challenges related to photon loss during propagation through the interferometer, making it necessary to employ probabilistic techniques such as postselection to obtain a valid sample. The postselection procedure typically sacrifices many runs of the experiment in order to ensure that the resulting sampling problem maintains its computational complexity. Other experiments have circumvented this issue by utilizing readily accessible non-classical states of light, but often at the cost of requiring additional assumptions for the hardness of classical simulation. Alternative approaches using atoms have been proposed and demonstrated with small numbers of particles; however, these experiments have also been limited by lower state preparation and detection fidelities and longer preparation times, hindering the demonstration of large-scale boson sampling. Previous demonstrations have shown two-atom interference using atoms trapped in optical tweezers; this paper aims to scale such demonstrations to higher atom numbers by using tools that have been developed to rapidly image, optically cool, and deterministically rearrange individual atoms trapped in optical tweezers. Using the tweezers to implant atoms in a tunnel-coupled optical lattice allows for both fast state preparation, and the required dynamics for boson sampling.

The experiment uses a two-dimensional tunnel-coupled optical lattice to trap and manipulate ⁸⁸Sr atoms. State preparation involves loading atoms into an optical tweezer array, imaging their positions, rearranging them into desired patterns using optical tweezers, and cooling them to the motional ground state via resolved sideband cooling. The atoms then evolve under a programmable single-particle Hamiltonian, realizing a quantum walk. After a variable evolution time, the atom positions are measured via site-resolved fluorescence imaging. The many-body evolution is described by the permanent of a matrix related to the single-particle unitary, which is computationally hard to calculate classically for larger numbers of atoms. Direct verification of the boson sampling distribution is not feasible for the system sizes studied; therefore, the authors employ several targeted tests to validate the experiment's performance. These tests include stringent tests of atom indistinguishability using up to eight atoms, characterization of the single-particle unitary through one- and two-particle quantum walks, and observation of bunching features for a range of atom numbers and effective particle statistics. The indistinguishability is assessed using Hong-Ou-Mandel (HOM) interference experiments. A maximum likelihood estimation method is used to characterize the single-particle unitary by performing a fit on the experimental data obtained from the one and two particle quantum walk experiments. The indistinguishability of the atoms is characterized by comparing the coincidence probability of atoms to that for distinguishable particles. The authors also introduce and utilize generalized bunching, a measure of the probability that all atoms appear in a specific subset of lattice sites, to probe the indistinguishability of larger atom numbers. In addition, the authors use time labeling techniques to study partially distinguishable atoms, allowing for comparison with classical simulations. The experimental setup and procedure are carefully calibrated, and multiple steps are taken to account for various sources of error, including loss of atoms during evolution, detection infidelity, and fluctuations in the applied unitary evolution. This is achieved by a multi-step algorithm for atom rearrangement, using advanced techniques for imaging and feedback, calibrations of the unitary using spectroscopic measurements and maximum likelihood estimation, and incorporation of error models into the simulations.

The experiments demonstrate boson sampling with up to 180 atoms distributed among ~1000 sites. A lower bound of 97.1±1.5% indistinguishability is inferred from HOM experiments, with an estimated value of 99.5±0.5% after additional modeling of the lattice potential. Measurements of full bunching and clouding for up to eight atoms show good agreement with theoretical predictions for indistinguishable bosons. The single-particle unitary is directly characterized using a maximum likelihood fitting procedure, achieving greater precision than previous methods. Generalized bunching measurements for up to 180 atoms exhibit a clear separation between distinguishable and bosonic behaviors, consistent with low thermal occupation of the out-of-plane motional degree of freedom. Experiments with partially distinguishable atoms, created by introducing time labels, allow for comparison with classical simulations, further supporting the bosonic nature of the system. The authors observed that the distribution of the observed fraction of surviving atoms after the quantum walk is sensitive to the effects of interference due to parity projection and is consistent with simulations of partially distinguishable atoms. While direct verification of computational complexity is impossible at this scale, the results provide strong evidence that the experiment samples from an extremely large state space and are not feasible to simulate classically. The low loss, which is not observed to scale with evolution time, high state preparation and detection fidelity, and many lattice sites enable studies of boson sampling that require fewer assumptions for the hardness of classical simulation. This contrasts with previous demonstrations of boson sampling where the survival probability decays exponentially with evolution time, which can be exploited in classical simulation methods. Although the atoms are fundamentally bosonic composite particles, they may not behave bosonically on the lattice due to other degrees of freedom (DOFs). In the experiment, the indistinguishability is 99.5±1.5%.

The findings address the research question by demonstrating a scalable platform for boson sampling using ultracold atoms. The high fidelity of state preparation, low loss during evolution, and high-fidelity detection, combined with the large number of atoms used in the experiments, provide strong evidence that the system is indeed sampling from an extremely large state space and is not feasible to simulate classically. The results significantly advance the field of boson sampling, showcasing an alternative to photon-based approaches and overcoming limitations of previous atom-based methods. The ability to directly characterize the unitary evolution and to perform experiments with partially distinguishable atoms provides further confidence in the results and allows for validation through comparison with classical simulations. This approach opens new avenues for exploring complex quantum phenomena and advancing quantum computation. The diverse set of tests implemented, leveraging the programmability of the input states and the ability to realize different unitaries, provides a more comprehensive characterization of the system than previous efforts. The flexible programmability using optical tweezers suggests the possibility of stronger certifications of high-order interference at the core of boson sampling and studies of dynamical phase transitions in sample complexity.

This paper presents a significant advancement in large-scale boson sampling using ultracold atoms. The unique combination of tools for rapid assembly, evolution, and detection of atoms in a tunnel-coupled optical lattice, along with rigorous benchmarking techniques, enables the study of boson sampling with a scale and fidelity previously unattainable. The results strongly suggest sampling from an extremely large state space, although direct verification remains a challenge. Future work will focus on improving the efficiency of protocols for calibrating the applied unitary for large systems and on exploring stronger certifications of computational hardness. The platform also holds promise for simulating interacting Hubbard models and exploring new approaches to quantum computation.

While the results strongly suggest that the experiment samples from a computationally hard distribution, direct verification of this claim is currently not feasible. The family of unitaries that can be applied in the current system is restricted, and this could in principle be taken advantage of by efficient classical simulations. Further, the dominant source of distinguishability in the experiment is attributed to imperfect cooling in the direction normal to the lattice, which could be improved in future experiments. The current experimental limitations in data rates make it difficult to precisely verify the behavior of more than 5 atoms in bunching measurements, and the effects of parity projection are significant at higher particle numbers.