Fig. 1: Elementary operations.

Circuit representations illustrating the action on position-eigenstates \({\{\left|x\right\rangle \}}_{x\in {\mathbb{R}}}\) of bosonic modes (thick wires) and computational basis states \({\{\left|b\right\rangle \}}_{b\in \{0,1\}}\) of a qubit (thin gray wires). The two-mode bosonic addition gate \({e}^{-i{Q}_{1}{P}_{2}}\) can be decomposed into constantly many beam-splitters and single mode squeezing unitaries, see refs. ^17,18. A homodyne P-quadrature measurement applied to a state \(\left|\Psi \right\rangle \in {L}^{2}({\mathbb{R}})\) produces a sample \(p\in {\mathbb{R}}\) from the distribution with density function \(p \, \mapsto \, | \widehat{\Psi }(p){| }^{2}\), where \(\widehat{\Psi }(p):=\frac{1}{{(2\pi )}^{1/2}}\int \Psi (x){e}^{ipx}dx\) denotes the Fourier transform of Ψ. Homodyne Q-quadrature measurement is defined similarly. Controlled-phase space rotations are defined in terms of the number operator \(\widehat{N}=({Q}^{2}+{P}^{2}-I)/2\).