Simulating arbitrary Gaussian circuits with linear optics

Linear canonical transformations of bosonic modes correspond to Gaussian unitaries, which comprise passive linear-optical transformations as effected by a multiport passive interferometer and active Bogoliubov transformations as effected by a nonlinear amplification medium. As a consequence of the Bloch-Messiah theorem, any Gaussian unitary can be decomposed into a passive interferometer followed by a layer of single-mode squeezers and another passive interferometer. Here, it is shown how to circumvent the need for active transformations. Namely, we provide a technique to simulate sampling from the joint input and output distributions of any Gaussian circuit with passive interferometry only, provided two-mode squeezed vacuum states are available as a prior resource. At the heart of the procedure, we exploit the fact that a beam splitter under partial time reversal simulates a two-mode squeezer, which gives access to an arbitrary Gaussian circuit without any nonlinear optical medium. This yields, in particular, a procedure for simulating with linear optics an extended boson sampling experiment, where photons jointly propagate through an arbitrary multimode Gaussian circuit followed by the detection of output photon patterns.

Figure 1

Beam splitter (red segment) of transmissivity $t$ is converted into a two-mode squeezer of gain $1/t$ under partial time reversal. The evolution of the first mode is described in the predictive picture (${\widehat{a}}_1\to {\widehat{a}}_1^{\prime }$), while that of the second mode is expressed in the retrodictive picture (${\widehat{a}}_2^{\prime }\to {\widehat{a}}_2^{\prime \prime }$). The output ${\widehat{a}}_2^{\prime \prime }$ is accessed via an EPR state (yellow star).

Figure 2

(a) Linear-optical circuit ${\mathcal{U}}_{\text{G}}$ simulates the Gaussian circuit ${\tilde{\mathcal{U}}}_{\text{G}}$ depicted in panel (b). The adjacent modes of $M$ two-mode squeezed vacuum states (yellow stars) are combined pairwise on beam splitters of transmissivity $t_j$ (red segments) and then sent onto two linear-optical circuits ${\mathcal{U}}_{\text{A}}$ and ${\mathcal{U}}_{\text{B}}$, preceded by a row of balanced beam splitters (green segments). The black lines refer to the physical time evolution from left to right, while the pink arrows show the information flow in the time-unfolded picture. (b) The Gaussian circuit ${\tilde{\mathcal{U}}}_{\text{G}}$ corresponds to the time-unfolded version of ${\mathcal{U}}_{\text{G}}$. The two-mode squeezers (TS) result from partial time reversal. The inset shows the equivalence of a two-mode squeezer sandwiched between two balanced beam splitters (green segments) and two single-mode squeezers (SS).

Figure 3

Beam splitter of transmissivity $t$ (red segment) is fed by two modes originating each from a two-mode squeezed vacuum state (yellow stars), and we focus on the probability of the photon-counting event $k_1,k_2,m_1,m_2$. By unfolding this linear optical circuit in time, the two-mode squeezed vacuum states can be viewed as “wires,” and we get a two-mode squeezer of gain $1/t$ with inputs $m_1,m_2$ and outputs $k_1,k_2$ (the pink arrows indicate the information flow in this time-unfolded picture). In order to account for the finite squeezing of the two-mode squeezed vacuum states, the two-mode squeezer of gain $1/t$ must actually be preceded and followed by a filtering operation in Fock basis.

Figure 4

(a) Depicted linear-optical circuit ${\mathcal{U}}_{\text{G}}^{\left(2\right)}$, which is a slight modification of the circuit ${\mathcal{U}}_{\text{G}}$ [cf. Fig. 2], simulates arbitrary Gaussian transformations. Here, the lines and arrows have the same meaning as in Fig. 2: the black lines illustrate the evolution of input TMSs, which propagate from left to right; the pink arrows show the information flow in our time-unfolded formalism. (b) The time unfolded version of the circuit ${\mathcal{U}}_{\text{G}}^{\left(2\right)}$ is equivalent to two disjoint arbitrary Gaussian transformations ${\tilde{\mathcal{U}}}_{\text{G}}^{\left(2\right)}$ and ${\widetilde{{\mathcal{U}}^{\prime }}}_{\text{G}}^{\left(2\right)}$, each achieved by means of the Bloch-Messiah decomposition.