On the experimental verification of quantum complexity in linear optics

Abstract

The first quantum technologies to solve computational problems that are beyond the capabilities of classical computers are likely to be devices that exploit characteristics inherent to a particular physical system, to tackle a bespoke problem suited to those characteristics. Evidence implies that the detection of ensembles of photons, which have propagated through a linear optical circuit, is equivalent to sampling from a probability distribution that is intractable to classical simulation. However, it is probable that the complexity of this type of sampling problem means that its solution is classically unverifiable within a feasible number of trials, and the task of establishing correct operation becomes one of gathering sufficiently convincing circumstantial evidence. Here, we develop scalable methods to experimentally establish correct operation for this class of sampling algorithm, which we implement with two different types of optical circuits for 3, 4, and 5 photons, on Hilbert spaces of up to 50, 000 dimensions. With only a small number of trials, we establish a confidence > 99% that we are not sampling from a uniform distribution or a classical distribution, and we demonstrate a unitary specific witness that functions robustly for small amounts of data. Like the algorithmic operations they endorse, our methods exploit the characteristics native to the quantum system in question. Here we observe and make an application of a "bosonic clouding" phenomenon, interesting in its own right, where photons are found in local groups of modes superposed across two locations. Our broad approach is likely to be practical for all architectures for quantum technologies where formal verification methods for quantum algorithms are either intractable or unknown.The construction of a universal quantum computer, capable of implementing any quantum computation or quantum simulation, is a major long term experimental objective. However, it is expected that non-universal quantum machines, that exploit characteristics of their own physical system to solve specific problems, will outperform classical computers in the near-term [1]. Ensembles of single photons in linear optical circuits are a recently proposed example: despite being non-interacting particles, their detection statistics are described by functions that are intractable to classical computers -matrix permanents [2]. It is therefore believed that linear optics constitutes a platform for the efficient sampling of probability distributions that cannot be simulated by classical computers, with strong evidence provided in the case of circuits described by large random matrices [3][4][5][6][7].A universal quantum computer, running for example Shor's factoring algorithm [8], creates an exponentially large probability distribution with individual peaks at highly regular intervals that facilitate the solution to the factoring problem allowing efficient classical verification, as is the case for all problems in the NP complexity class [9]. Accordingly, correct operation of the...

Boson sampling for molecular vibronic spectra

“…3 require at most 3 photons per mode (see SI). Current photon counters are able to distinguish up to a few photon numbers (≤ 3) per mode [46]. The (single mode) squeezing parameters for formic acid are given as ln(Σ) [27], that is, ln(Σ) = diag (0.10, 0.07, 0.02, −0.06, −0.08, −0.11, −0.19) .…”

High-dimensional unitary transformations and boson sampling on temporal modes using dispersive optics

“…Unitary transformations on optical modes have been used to implement single-particle quantum gates [1,2], quantum simulations [3], and boson sampling [4][5][6][7][8][9][10][11]. Traditionally, these transformations are implemented on spatial modes using a system of beam splitters.…”

“…Traditionally, these transformations are implemented on spatial modes using a system of beam splitters. However, building a large interferometer implementing such a unitary transformation is experimentally challenging and the largest number of modes so far has been 21 [9].…”

“…, at which point photons in each mode are measured using single-photon detectors [4][5][6][7][8][9]. As we describe, boson sampling can analogously be performed in time by replacing spatial mode a j with temporal mode a tj and by replacing the beam-splitter array with dispersion ( Fig.…”

“…Furthermore, the experiment would require 900 detectors and if the photons came from heralded sources, 900 sources [15]. To date, experimental demonstrations of SMBS have been limited to 5 photons in 21 modes [9]. The number of modes in temporal mode boson sampling (TMBS) can be increased simply by increasing the dispersion or reducing the temporal width of input photons.…”