Exponential improvement in benchmarking multiphoton interference

Abstract

Several photonic quantum technologies rely on the ability to generate multiple indistinguishable photons. Benchmarking the level of indistinguishability of these photons is essential for scalability. The Hong-Ou-Mandel dip provides a benchmark for the indistinguishability between two photons, and extending this test to the multi-photon setting has so far resulted in a protocol that computes the genuine n-photon indistinguishability (GI). However, this protocol has a sample complexity that increases exponentially with the number of input photons for an estimation of GI up to a given additive error. To address this problem, we introduce new theorems that strengthen our understanding of the relationship between distinguishability and the suppression laws of the quantum Fourier transform interferometer (QFT). Building on this, we propose a protocol using the QFT for benchmarking GI that achieves constant sample complexity for the estimation of GI up to a given additive error for prime photon numbers, and sub-polynomial scaling otherwise, representing an exponential improvement over the state of the art. We prove the optimality of our protocol in many relevant scenarios and validate our approach experimentally on Quandela's reconfigurable photonic quantum processor, where we observe a clear advantage in runtime and precision over the state of the art. We therefore establish the first scalable method for computing multi-photon indistinguishability, which applies naturally to current and near-term photonic quantum hardware.

Figures (5)

• Figure 1: Illustration of the protocol. An $n$-photon state is input to an $n$-mode QFT interferometer, and certain output events are post-selected on using photon-number-resolving (PNR) detection. The estimated output probabilities from these events are then used to estimate $c_1$.
• Figure 2: Illustration of the sampling requirements to benchmark genuine $n$-photon indistinguishability (GI) ($c_1$) with fixed additive error and statistical confidence. The cyclic interferometer (CI) exhibits exponential overhead, whereas our QFT-based protocol achieves near-constant complexity for prime $n$ and sub-polynomial growth for non-prime $n$ in the worst case. The number of samples is established using concentration bounds derived from a Hoeffding inequality hoeffdingProbability1963. The upper and lower bound functions for the QFT (yellow lines) in the plot are proportional to $n^{\frac{1}{10}}$ and $(\frac{n}{n-1})^2$.
• Figure 3: Experimental implementation.a. Interference between three single photons using $QFT_3$ along with pseudo-number resolving (PPNR) detection up to 3-photon resolution. b. Interference between three single photons using a cyclic interferometer ($CI_3$) tuned using a single phase shifter implementing the phase $\phi$. The desired outcomes contain no bunching and so direct threshold detection suffices. Note that the first two modes and the last mode went unused in this experiment, since the interferometer was shifted so that the same three photons were measured for both methods. c. The interference fringe obtained from the output of $CI_3$ whose amplitude of oscillation quantifies the genuine 3-photon indistinguishability $c_1$ (vertical arrow). The horizontal line notated by $QFT_3$ indicates the value of $c_1$ obtained from the experiment illustrated in panel a.. d. A table summarizing the $n=3$ and $n=4$ genuine multi-photon indistinguishabilities measured using the reconfigurable chip for both methods.
• Figure 4: (left) The Perceval circuit to implement the three-photon quantum Fourier transform (QFT) measurement. Photons are input into modes 2, 4, and 6 while all other modes are vacuum. Routing is done to bring all three photons into the first QFT and then separate the outputs into three independent QFTs that implement the PPNR detection. (right) The Perceval circuit to implement the four-photon QFT measurement. The output modes of the interference layer are routed either to a four-photon PPNR QFT or sent directly to a threshold detector. The PERM operation is either an identity or a swap, giving rise to four circuits that are recombined to obtain the full 4-photon resolved distribution.
• Figure 5: (left) The Perceval circuit to implement the three-photon cyclic interferometer (CI) measurement. Photons are input into modes 2, 4, and 6 while all other modes are vacuum. (right) The Perceval circuit to implement the four-photon CI measurement. The photons are input into modes 2, 4, 6, and 8 while all other modes are vacuum. In both cases, the phase $\phi$ is varied to obtain the interference fringe needed to extract $c_1$.

Theorems & Definitions (27)

• Theorem 3.1: Q-marginals probability
• Theorem 3.2: GI success probability
• Definition A.1: t-periodic Fock state
• Definition A.2: Mode-assignment vector
• Definition A.3: Q-value
• Definition A.4: Partition state
• Definition A.5: OBB partition state
• Definition A.6: t-periodic partition state
• Proposition B.1
• proof