Boundaries for quantum advantage with single photons and loop-based time-bin interferometers

TL;DR

This work defines concrete boundaries for quantum advantage in loop-based, time-bin Boson Sampling by introducing a progressive, causally-aware simulation strategy that decomposes complex interferometers into small, tractable components. Central to the approach is a lattice-path formalism that encodes the reachable state space as downsets bounded by maximal lattice paths, enabling precise memory-bound predictions. A key technical advance is extending the lattice-path framework to handle long loops, mode permutations, and mid-circuit measurements via permuted cumulative spaces, contraction, and merge operations. The authors validate a memory-prediction heuristic and use it to map out regimes where classical simulation remains viable versus regimes where quantum advantage may emerge, particularly for power-law loop architectures with base length $\ell \ge 5$. They also discuss extensions to lossy, distinguishable, and Gaussian Boson Sampling scenarios, providing a foundation for guiding experimental designs toward configurations more likely to exhibit quantum advantage.

Abstract

Loop-based boson samplers interfere photons in the time degree of freedom using a sequence of delay lines. Since they require few hardware components while also allowing for long-range entanglement, they are strong candidates for demonstrating quantum advantage beyond the reach of classical emulation. We propose a method to exploit this loop-based structure to more efficiently classically sample from such systems. Our algorithm exploits a causal-cone argument to decompose the circuit into smaller effective components that can each be simulated sequentially by calling a state vector simulator as a subroutine. To quantify the complexity of our approach, we develop a new lattice path formalism that allows us to efficiently characterize the state space that must be tracked during the simulation. In addition, we develop a heuristic method that allows us to predict the expected average and worst-case memory requirements of running these simulations. We use these methods to compare the simulation complexity of different families of loop-based interferometers, allowing us to quantify the potential for quantum advantage of single-photon Boson Sampling in loop-based architectures.

Figures (16)

• Figure 1: Overview of the results.(a) A time-bin boson sampler consisting of loops of different lengths composed in sequence. The input photons arrive in regular intervals, represented by time-bin modes with photon numbers $\left|n_0, n_1, \dots, n_{m-1}\right\rangle$. The loops couple earlier time-bin modes to later ones, and the lengths of the loops are given in time-bin units: thus a loop of length $\ell$ allows photons in modes $a$ and $a + \ell$ to interfere. This is shown below the physical schematic as a circuit. Note that by convention, the first $\ell$ time-bins are deterministically loaded into a loop of length $\ell$, and we omit drawing the initial vacuum modes inside the loop. (b)Progressive decomposition of the circuit, where each yellow box $P_a$ contains only beamsplitters in the causal cone of the measurement of mode $a$, except for those already captured by previous causal cones. This decomposition allows us to evolve the wavefunction only through components $P_a$ that all have a bounded effective number of modes, measure their top mode, and collapse the wavefunction into a subspace. (c) An example of our lattice path formalism that describes the reachable state space of a loop-based system throughout simulation. Here, we show the state space after evolving the input wavefunction $\left|1^5\right\rangle$ through component $P_0$ and measuring 1 photon in the top mode, thus collapsing the wavefunction into a subspace shown as the yellow lattice diagram. Each basis state $\left|n'_1, n'_2, n'_3, n'_4\right\rangle$ of the space that can be found with nonzero probability corresponds to a path within the yellow diagram that starts in the bottom left corner, ends at top right corner, and takes integer steps up or right. The numbers of photons correspond to vertical steps taken by a path, as shown for two example paths on the right-hand-side. Basis states of the full Fock space $\mathcal{F}^{4} \mathbb{C}^{4}$ not contained in the yellow diagram are not reachable.
• Figure 2: Example of a lattice path from $\mathcal{L}(3,2)$ in both the height and step representations. (red) Height representation $\lambda = (0,0,1,2)$ where the coordinates indicate the height of the path above the horizontal axis. This is the maximum $y$ coordinate of a point passed by the path at each $x$ coordinate. (blue) Step representation $\underline{n}^\lambda = \Delta\lambda = (0,0,1,1)$ with coordinates $\underline{n}^\lambda_a = \lambda_a - \lambda_{a-1}$ that represent the size of the vertical steps the path takes, shown by blue dashed arrows.
• Figure 3: An example showing how we obtain a maximal lattice path $\mu$ from a given circuit and input state, in this case $L_1$ and $\left|1^5\right\rangle$. On the output modes are sets of possible photon numbers measured there, and we show how these define the lattice path $\mu$ (red). The possible measurement outcomes are all the lattice paths lower than $\mu$ -- these are the paths constrained to the yellow-shaded region under $\mu$.
• Figure 4: A view of the input modes and beamsplitters influencing output mode 0 in a system $L_1 \fatsemi L_2 \fatsemi L_3$. We highlight in red the wires along which photons may reach output mode 0 on the right. The path reaching the maximum input mode is further emphasized with arrows ($\mathrel{}$). The relevant beamsplitters are drawn in red as well -- these are the beamsplitters that form the component $P_0$. Observe that the first loop $L_1$ is a staircase of nearest-neighbour beamsplitters, so if a mode $a$ is relevant, so are the modes $<a$; this fact is emphasized in the yellow area.
• Figure 5: Progressive simulation

Theorems & Definitions (39)

• definition 1: optical loop
• definition 2: power-law composites
• definition 3: lattice path
• definition 4: height representation
• definition 5: step representation
• lemma 1: bosonic states are lattice paths
• proof
• definition 6: cumulative space
• theorem 1: relevant modes
• definition 7: PCS