Two recent landmark experiments have performed Gaussian boson sampling (GBS) with a non-programmable linear interferometer and threshold detectors on up to $144$ output modes (see Refs.~\onlinecite{zhong_quantum_2020,zhong2021phase}). Here we give classical sampling algorithms with better total variation distance than these experiments and a computational cost quadratic in the number of modes. Our method samples from a distribution that approximates the single-mode and two-mode ideal marginals of the given Gaussian boson sampler, which are calculated efficiently. One implementation sets the parameters of a Boltzmann machine from the calculated marginals using a mean field solution. This is a $2$nd order approximation, with the uniform and thermal approximations corresponding to the $0$th and $1$st order, respectively. The $k$th order approximation reproduces marginals and correlations only up to order $k$ with a cost exponential in $k$ and high precision, while the experiment exhibits higher order correlations with lower precision. This methodology, like other polynomial approximations introduced previously, does not apply to random circuit sampling because the $k$th order approximation would simply result in the uniform distribution, in contrast to GBS.

Efficient approximation of experimental Gaussian boson sampling

We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_{\text{noisy}}$ of a generic noisy circuit instance and the output distribution $p_{\text{ideal}}$ of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as $F=\text{exp}(-2s\epsilon \pm O(s\epsilon^2))$, where $\epsilon$ is the probability of error per circuit location and $s$ is the number of two-qubit gates. Furthermore, if the noise is incoherent, the output distribution approaches the uniform distribution $p_{\text{unif}}$ at precisely the same rate and can be approximated as $p_{\text{noisy}} \approx Fp_{\text{ideal}} + (1-F)p_{\text{unif}}$, that is, local errors are scrambled by the random quantum circuit and contribute only white noise (uniform output). Importantly, we upper bound the total variation error (averaged over random circuit instance) in this approximation as $O(F\epsilon \sqrt{s})$, so the "white-noise approximation" is meaningful when $\epsilon \sqrt{s} \ll 1$, a quadratically weaker condition than the $\epsilon s\ll 1$ requirement to maintain high fidelity. The bound applies when the circuit size satisfies $s \geq \Omega(n\log(n))$ and the inverse error rate satisfies $\epsilon^{-1} \geq \tilde{\Omega}(n)$. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; it was an underlying assumption in complexity-theoretic arguments that low-fidelity random quantum circuits cannot be efficiently sampled classically. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

Random quantum circuits transform local noise into global white noise.

We study the properties of output distributions of noisy, random circuits. We obtain upper and lower bounds on the expected distance of the output distribution from the uniform distribution. These bounds are tight with respect to the dependence on circuit depth. Our proof techniques also allow us to make statements about the presence or absence of anticoncentration for both noisy and noiseless circuits. We uncover a number of interesting consequences for hardness proofs of sampling schemes that aim to show a quantum computational advantage over classical computation. Specifically, we discuss recent barrier results for depth-agnostic and/or noise-agnostic proof techniques. We show that in certain depth regimes, noise-agnostic proof techniques might still work in order to prove an often-conjectured claim in the literature on quantum computational advantage, contrary to what was thought prior to this work.

Tight bounds on the convergence of noisy random circuits to uniform.

A major challenge in the development of near-term quantum computers is to characterize the underlying quantum noise with minimal hardware requirements. We show that random circuit sampling (RCS) is a powerful benchmarking primitive that can be used to efficiently extract the total amount of quantum noise of a many qubit system by creating an exponential decay of fidelity. Compared with randomized benchmarking (RB) and its scalable variants, RCS benchmarking has the unique advantage of being flexible with the gate set, as the fidelity decay in RCS benchmarking comes from scrambling properties of random quantum circuits, which hold for generic gate sets and do not rely on any group structure. First, under a first order approximation in the noise rate, we rigorously prove the exponential decay of average fidelity of low-depth noisy random circuits, using a technique that maps random quantum circuits to a classical spin model. Second, we use the unbiased linear cross entropy as a sample efficient estimator of fidelity and numerically verify its correctness by simulating up to 20 qubits with different noise models. Third, we develop a theoretical model of the total variance of RCS benchmarking, which generalizes previous statistical analysis of cross entropy by considering the variance across different circuits. Finally, we experimentally demonstrate RCS benchmarking on IBM Quantum hardware with up to 20 superconducting qubits, which verifies our theoretical analysis. We expect RCS benchmarking to be widely applicable across different hardware platforms due to its flexibility with the gate set and architecture.

Roozbeh Bassirianjahromi

Papers

Benchmarking near-term quantum computers via random circuit sampling