Measurement-Based Linear Optics

A major challenge in optical quantum processing is implementing large, stable interferometers. We offer a novel approach: virtual, measurement-based interferometers that are programed on the fly solely by the choice of homodyne measurement angles. The effects of finite squeezing are captured as uniform amplitude damping. We compare our proposal to existing (physical) interferometers and consider its performance for BosonSampling, which could demonstrate postclassical computational power in the near future. We prove its efficiency in time and squeezing (energy) in this setting.

Figure 1

Generating and measuring the temporal-mode quad-rail lattice [12]. Squeezed vacuum states (a) pass through six balanced beam splitters (b) and two delay loops (c) with delays $\mathrm{\Delta }t$ and $M\mathrm{\Delta }t$, with $M$ an odd integer. Cluster modes are generated and measured with homodyne detection (d) at the same rate. After $ML$ time steps of duration $\mathrm{\Delta }t$, the QRL is an ($M\times L$)-macronode lattice. Input (output) states can be inserted (removed) using a switching device (e) [14] on any of the four rails (one example shown).

Figure 2

Measurement pattern of the quad-rail lattice (in terms of distributed modes [18]) for measurement-based linear optics. The left-hand column of macronodes contains the inputs (green circles), and the right-hand column contains the outputs (pink squares), which can be swapped in and out, as shown in Fig. 1. Macronode columns are measured left to right, implementing measurement-based Mach-Zehnders interferometers [Eq. (2)] at each macronode (see legend).

Figure 3

The measurement pattern in Fig. 2 is equivalent to a tessellation of the two-mode gate in the red box, which consists of a variable beam splitter followed by a pair of independent phase delays.

Figure 4

(a) Homodyne detection on one macronode in the quad-rail lattice (with respect to the distributed modes) can be represented by the above circuit [18]. We have selected omodyne angles such that the transformation in Eq. (2) is implemented. (b) The macronode circuit is decomposed as a pair of CV teleportation circuits [23] (shown in the green boxes) sandwiched between linear optics. (c) Macronode measurement with finite-squeezing effects and gain-tuned displacements [25] is equivalent to a Mach-Zehnder interferometer with equal loss $\mathcal{L}$ on each arm. (d) Because the loss $\mathcal{L}$ is the same on both arms, it can be commuted to the beginning (see text).

Figure 5

Cluster-state squeezing levels currently required for MBLO-based BosonSampling of $n$ photons in $m=n^2$ modes using the temporal-mode implementation [12, 14, 16] and a simulated circuit of depth $k=m$ (see Fig. 3). The blue line corresponds to experiments that would take 1 day on average for a successful run, while the surrounding blue region spans 1 min (above) to 1 yr (below). Marked points: (a) 5 dB, demonstrated in a large-scale cluster state [14], (b) 15 dB, largest single-mode squeezing achieved in optics [33], (c) 20.1 dB, squeezing corresponding to a recently reported six-mode interferometer [11], (d) 36.3 dB, squeezing corresponding to $n=20$, a standard target [5] for BosonSampling to demonstrate an advantage over classical simulation.