Quantum computational advantage via high-dimensional Gaussian boson sampling

Abstract

Photonics is a promising platform for demonstrating a quantum computational advantage (QCA) by outperforming the most powerful classical supercomputers on a well-defined computational task. Despite this promise, existing proposals and demonstrations face challenges. Experimentally, current implementations of Gaussian boson sampling (GBS) lack programmability or have prohibitive loss rates. Theoretically, there is a comparative lack of rigorous evidence for the classical hardness of GBS. In this work, we make progress in improving both the theoretical evidence and experimental prospects. We provide evidence for the hardness of GBS, comparable to the strongest theoretical proposals for QCA. We also propose a QCA architecture we call high-dimensional GBS, which is programmable and can be implemented with low loss using few optical components. We show that particular algorithms for simulating GBS are outperformed by high-dimensional GBS experiments at modest system sizes. This work thus opens the path to demonstrating QCA with programmable photonic processors.

Fig. 1.. Different representations of a D = 2D optical delay GBS instance with lattice size a = 3.

(A) Circuit representation. The vertical lines with dots at the end represent beam splitters. (B) Bidimensional lattice representation. The vertices of the lattice represent the modes, while edges represent beam splitters. (C) Optical circuit representation. The modes are defined by time-bins traveling in a waveguide. The horizontal gray slabs at the bottom of the delays represent the beam splitters. The number of cycles C in a high-dimensional GBS instance corresponds to applying multiple times the gates contained in the green-dotted box in (A). This action physically maps to using concatenating C copies of the delays encircled in the green box in (C). Note that for simplicity, we have not shown the photon-number detectors used to probe the quantum state at the end of the circuit.

Fig. 2.. Absolute values of the entries of the unitary matrices associated with two high-dimensional GBS instances drawn from U.

On the left, we show an (a = 6, D = 3, C = 1) instance, and on the right, we show an (a = 15, D = 2, C = 2) instance. Note that we explicitly color the zero entries of the unitary white; thus, the color scale is discontinuous at this end.

Fig. 3.. Distribution of the total photon number for M = 216 single mode–squeezed states with squeezing parameter r = 0.8.

We assume a total transmission of η = 0.5 (corresponding to roughly 3 dB of loss) for the lossy distribution. Note that the lossless distribution has no support on odd numbers of photons, which explains why visually it looks as if it has more area under the curve.

Fig. 4.. The time cost of calculating a Hafnian of size n in double precision.

The stars indicate actual sizes computed in the Niagara supercomputer (60). The blue line is a fit to t_Niagara(n) = c_Niagaran³2^n/2 with the only fitting parameter c_Niagara = 5.42 × 10⁻¹⁵ s. The standard deviation of fitting parameter c_Niagara is 1.2 × 10⁻¹⁶ s, which would give error bands thinner than the width of the line. We find an equivalent expected time in Fugaku, among the most powerful supercomputers, by considering the ratio of their R_max scores (maximal LINPACK performance achieved) giving their performance in number of floating point operations per second. The conversion factor between the left scale for Niagara and the right scale for Fugaku is the ratio of R_max values of Fugaku and Niagara, or equivalently, c_Niagara/c_Fugaku = 122.8. Note that since the computation of Hafnians can be broken into the independent calculation of an exponential number of summands (known as an embarrassingly parallel computation), this scaling is expected to be quite accurate.