Hardware requirements for realizing a quantum advantage with deterministic single-photon sources

Boson sampling is a specialized algorithm native to the quantum photonic platform developed for near-term demonstrations of quantum advantage over classical computers. While clear useful applications for such near-term pre-fault-tolerance devices are not currently known, reaching a quantum advantage regime serves as a useful benchmark for the hardware. Here, we analyze and detail hardware requirements needed to reach quantum advantage with deterministic quantum emitters, a promising platform for photonic quantum computing. We elucidate key steps that can be taken in experiments to overcome practical constraints and establish quantitative hardware-level requirements. We find that quantum advantage is within reach using quantum emitters with an efficiency of $60\%\text{–}70\%$ and interferometers constructed according to a hybrid-mode-encoding architecture, constituted of Mach-Zehnder interferometers with an insertion loss of $0.0035\mathrm{dB}$ (a transmittance of $99.92\%$) per component.

Figure 1

(a) A general boson sampling setup, consisting of a source of multiphoton input states, a multiport interferometer, and detectors. The associated system losses $\rho$ are indicated for each subcomponent. (b) A boson sampling setup based on a deterministic single-photon source (SPS) and a spatially encoded multimode interferometer. The single-photon source emits a string of single photons in predetermined time bins, illustrated as filled red circles. A demultiplexer converts the photon stream into a $p$-photon source by deterministically switching each photon into a separate mode. The spatially encoded input state is sent into an interferometer constructed from a network of MZIs. Each MZI is illustrated as a cross, where each cross consists of two phase shifters (PSs) and two beam splitters (BSs) arranged according to the illustration at the bottom. (c) A boson sampling setup based on a deterministic single-photon source (SPS) and a time-bin interferometer. The SPS is identical to the case in (b) where $\tau$ denotes the time separation between subsequent photons. The interferometer consists of reconfigurable time-dependent MZIs, where time-dependent phase shifters are implemented using electro-optic modulators (EOMs), which are connected by delay lines illustrated as black lines, where one loop corresponds to a single time bin $\tau$ of delay. For both (b) and (c), it is assumed that up to $l$ photons may be lost throughout the setup, illustrated as a red circle with a dashed outline, due to residual optical loss associated with each component.

Figure 2

(a) Illustration of the Clements interferometer architecture, as detailed in Ref. [48]. Each cross corresponds to an MZI, which can be constructed from two 50:50 beam splitters (BSs) and two phase shifters (PSs). (b) Illustration of a rectangular interferometer with a larger number of output modes than input modes, the latter of which is equal to the number of input photons. The interferometer can be described by a rectangular matrix, hence the name. (c) Illustration of an interferometer with multiple-mode encodings, i.e., a hybrid-mode encoding. In this case, one mode encoding has spatial modes separated in the $x$ direction, while the other has modes separated in the $y$ direction.

Figure 3

Illustration of how time-bin interferometers can be constructed to implement specific features in interferometer architectures. The red bin symbols $\bigsqcup$ correspond to the modes of the interferometer, and $\tau$ is the temporal separation between photons, which is equal to the inverse of the SPS emission rate $\tau =r_{\text{single-photon}}^{-1}$. Numbers correspond to input modes, and numbers with primes correspond to output modes. All of the time-bin interferometer architectures make use of reconfigurable MZIs, illustrated as boxed crosses. As is shown in the bottom-left corner, these MZIs are constructed using time-dependent phase shifters $\varphi \left(\mathrm{t}\right)$ and $\theta \left(\mathrm{t}\right)$, which determine the unitary transformation U(t) effected by the interferometer. (a) Illustration of a rectangular interferometer architecture implemented with spatial modes on the left-hand side and with a time-bin interferometer on the right-hand side. The MZI columns are split into three categories ${\text{C}}_{\text{1}}\left(m\right)$, ${\text{C}}_{\text{2}}\left(m\right)$, and ${\text{C}}_{\text{3}}\left(m\right)$, where $m$ is the number of modes which is assumed to be an even number. (b) Upper: illustration of a specific MZI column for four input modes ${\text{C}}_{\text{1}}\left(4\right)$. Lower: operation protocol of the ${\text{C}}_{\text{1}}\left(m\right)$ MZI column for a general even number of input modes $m$. (c) Upper: illustration of a specific MZI column for four input modes ${\text{C}}_{\text{2}}\left(4\right)$. Lower: operation protocol of the ${\text{C}}_{\text{2}}\left(m\right)$ MZI column for a general even number of input modes $m$. This type of MZI column increases the total number of modes in the time-bin interferometer by two. (d) Upper: illustration of a specific MZI column for six input modes ${\text{C}}_{\text{3}}\left(6\right)$. Lower: operation protocol of the ${\text{C}}_{\text{3}}\left(m\right)$ MZI column for a general even number of input modes $m$. In the first and last time bins, the MZI must be configured to perform the operation ${\text{U}}_{\text{swap}}$, which swaps the two input modes. This ensures that the number of output modes from the column is equal to the number of input modes $m$.

Figure 4

Maximum per-photon loss for the full circuit to perform boson sampling in the QA regime versus degree of indistinguishability and number of detected photons and for different values $l$ of lost or colliding photons. The two upper plots are for the case where the number of modes scales quadratically with the number of photons $m={(p-l)}^2$, and the two bottom plots are for the case where the number of modes scales linearly with the number of photons $m=10(p-l)$. The two plots to the left are for spatial interferometers with a demultiplexed source, whereas the two plots on the right are for time-bin interferometers. White contours indicate the added number of lost photons $l$, which increases with the indistinguishability, and detected number of photons.

Figure 5

Plot of the requirements on MZI insertion loss ($x$ axis) and ${\rho }_{\text{sys}}-{\rho }_{\text{int}}$ ($y$ axis) with photon indistinguishability set to $x^2=0.96$. The upper plots show the requirements for Aaronson-Brod boson sampling, where the input state consists of 59 photons with the outputs postselected to contain 50 photon detection events. The lower plots show the requirements for Aaronson-Arkhipov boson sampling, where we send in 50 photons and detect 50 photons, not allowing for photon loss or collisions. The first, second, and third columns show hardware requirements for setups where the interferometers are constructed according to the Clements architecture, the rectangular architecture, and a set of hybrid architectures, respectively. For the Clements and rectangular architectures, the solid lines correspond to the requirements for spatially encoded architectures, whereas dashed lines correspond to the requirements for time-bin architectures. For the hybrid architectures, the solid lines correspond to requirements for an interferometer encoded over two spatial-mode encodings, whereas dashed lines correspond to requirements for an interferometer encoded over time bins and two spatial-mode encodings. The dotted vertical lines mark the estimated MZI loss for a state-of-the-art experimental realization with static, nonprogrammable MZIs [25]. The dashed-dotted vertical line in the plots for hybrid interferometers marks the estimated MZI loss for a state-of-the-art experimental realization with programmable MZIs [55].

Figure 6

Requirements on source efficiency for a given indistinguishability with a hybrid interferometer encoded in two mode encodings as per Fig. 1. Rectangular interferometers have been used within both mode encodings. The requirements are defined to allow for 100 samples to be obtained per day with a 1-GHz single-photon generation rate. For each value of indistinguishability, the number of photons detected and lost has been optimized to increase the loss tolerance while maintaining an error bound higher than 1% for the approximation algorithm.