Linear-nonlinear duality for circuit design on quantum computing platforms

TL;DR

The results demonstrate that key amplitude-level features of nonlinear optical processes can be simulated using only native single-qubit unitaries and measurement-based primitives on existing digital quantum hardware, and extends PDC-inspired entanglement-generation mechanisms beyond photonic architectures.

Abstract

Beam splitters (BSs) and optical parametric amplifiers (OPAs) can be described using Lie groups $SU(2)$ and $SU(1,1)$. Here, we show that the dynamical trajectories of these devices are connected via a Wick rotation on their respective group manifolds. This yields an exact amplitude-level duality between BSs of transmittance $η$ and OPAs of gain $g=1/η$. This geometric correspondence admits a compact tensor-network formulation, which we use to construct a circuit-model protocol that reproduces PDC transition amplitudes. This construction naturally leads to finite-dimensional, truncated PDC unitaries that exactly reproduce the first $q$ amplitudes of an ideal parametric amplifier. Our results demonstrate that key amplitude-level features of nonlinear optical processes can be simulated using only native single-qubit unitaries and measurement-based primitives on existing digital quantum hardware. This extends PDC-inspired entanglement-generation mechanisms beyond photonic architectures.

Figures (11)

• Figure 1: Schematics of a (a) two-mode beam splitter (BS) and an (b) optical parametric amplifier (PDC). The purple beam indicates the pump beam undergoing down-conversion. The bosonic operators of the input light beams are rotated by the action of each optical device as dictated by the unitaries in the equations \ref{['eqn:BS_rotation']} and \ref{['eqn:PDC_rotation']}, respectively. For a BS of transmittance $\eta$, the input modes are rotated by a $SU(2)$ unitary, while for a PDC of gain $g$, the input modes are rotated by a $SU(1,1)$ unitary that does not preserve the total number of photons entering the optical device.
• Figure 2: Beamsplitter as a Euclidean parametric amplifier. A PDC of gain $g$ is represented by a point on a one-parameter curve on the group manifold $SU(1,1)$. Locally, $SU(1,1) \cong SO(1,2)$, and the parametric amplifier acts transitively on the two-dimensional hyperbolic space (as represented in panel (a)). After the Wick rotation $g\to\eta$, the one-parameter curve in $SO(1,2)$ transforms into a curve in $SO(3)$ (locally, $SU(2)$), which acts transitively on the two-dimensional sphere (panel (b)). This identifies each beam splitter as a Euclidean parametric amplifier with "inverse" temperature $\eta = g^{-1}$.
• Figure 3: BS--PDC duality: (a) The probability amplitude of detecting $s$ and $l$ photons at the output of a parametric amplifier with gain $g$, given input photons $n$ and $m$, is dual to the probability amplitude of detecting $m,l$ photons, given $n,s$ entering a lossless beam splitter with transmittance $1/g$. (b) At the level of matrix elements, this equivalence requires swapping the photon numbers entering and leaving the lower mode. The equality in panel (b) must be interpreted modulo the global factor $1/\sqrt{g}$.
• Figure 4: Circuit-model scheme for the composite encoding in Fig. \ref{['fig:full_encoding']}. Different input states contribute different numbers of qubits to the binary encoding. The color code blue/red indicates the qubits associated with the photon numbers entering the second/first mode. To ensure that the composite encoding extends to a unitary map on all computational states, we introduce ancilla qubits that remain unaffected by the optical transformation. To exchange the number of occupants in the lower mode, two additional quantum registers of $\lceil\log m\rceil$ qubits each are required. These registers are prepared in $\lceil\log m\rceil$ EPR pairs, represented by cups at the start of the circuit, and subsequently measured in the Bell basis, represented by caps, following the standard categorical teleportation structure Biamonte/tensor_networks/2019.
• Figure 5: Composite encoding of occupation-number states into physical qubit states. Binary-encoded states are transformed into symmetric states by the composite map $E_{P}\circ E_{B}^{-1}: \mathcal{H}^{\mathcal{O}(\log_{2}(nm))}\to \mathcal{S}(\mathcal{H}^{\otimes (n+m)})$.