Figure 1

Simplified graphical formalism for the Gaussian pure states represented in this paper. (a) Full graph for a particular Gaussian pure state [30]. (b) Simplified graph for the same state, as used in this paper. Self-loops are not drawn; colored edges indicate the sign of the weights as in Eq. (15), with $\mathcal{C}=\frac{1}{\sqrt{2}}$ (see text) for all edges since each edge connects nodes having exactly two neighbors; colored nodes indicate the bipartite splitting, with white nodes receiving a Fourier transform to convert this to an approximate CV cluster state with the same graph, but now with weights given by Eq. (17).Reuse & Permissions

Figure 2

Temporal-mode GPEPS construction of a CV quantum wire using passive squeezing and linear optics. Two single-mode squeezers $S_1$ and $S_2$ generate vacuum $\mathrm{p̂}$- and $\mathrm{q̂}$-squeezed pulses of light (respectively, as shown) at regular intervals $\Delta t$. These pass through a simple 50:50 beam splitter $B_1$, resulting in a two-mode squeezed state. (Red arrows point from the first node to the second in Eq. (18) for each beam splitter.) The delay loop in the bottom line delays the bottom mode by $\Delta t$, allowing it to match up with the top mode of the subsequent pair emerging from $B_1$, resulting schematically in the graph shown in Eq. (21). The second 50:50 beam splitter $B_2$ implements sequentially each of the transformations indicated by the red arrows, resulting in the final graph of Eq. (22). These pulses head toward detectors $D_1$ and $D_2$, which implement the necessary $\mathrm{q̂}$ measurements (phase shifted as appropriate for the white nodes; see Sec. 2c), which are indicated by the arrows in Eq. (23), as well as the adaptive measurement-based quantum algorithm to be implemented on the one-dimensional CV quantum wire. The adaptiveness means that subsequent measurement bases generally must be chosen based on previous measurement outcomes. Most measurements will involve homodyne detection in a basis that must be calculated and updated before the arrival of the next pulse, but the ability to divert the beam to an efficient photon counter is also required for universal single-mode QC [6].Reuse & Permissions

Figure 3

Two equivalent representations of the GPEPS topology of the temporal-mode square-lattice CV cluster state illustrated in Eq. (25), adapted from Fig. 3 in Ref. [10]. Two-mode squeezed states are arranged as in Eq. (26), while red circles represent the four beam-splitter transformations indicated in Eqs. (26) and (27) for each macronode, as well as the projection using $\mathrm{q̂}$ measurements on three of the micronodes within each, eventually resulting in the graph of Eq. (28). The two-dimensional square-lattice graph (top) is formally infinite in one dimension but finite in the other. This graph can be redrawn as a multiply threaded infinite line graph (bottom). There are $M$ additional threadings of the line graph that pass through macronodes $M$ units apart, resulting in a square lattice with vertical dimension $M$. Note that $M$ must be odd to ensure that the graph is bipartite. (As shown, $M=5$.) The dotted links represent additional edges that would make the linear version translationally symmetric and equivalent to a square lattice on a cylinder with one unit of shear in the longitudinal direction. Such a family of graphs would still be universal for one-way QC because we can measure $\mathrm{q̂}$ on every $M$th node to delete it (and its links) and “unfold” the graph into an ordinary square lattice with a vertical dimension of $M-1$ [6].Reuse & Permissions

Figure 4

Temporal-mode GPEPS construction of a square-lattice CV cluster state using passive squeezing and linear optics. Two copies of the quantum-wire setup from Fig. 2 are used to generate the lattice. The upper one has the ordinary delay of $\Delta t$ and corresponds to the vertical links in Fig. 3 (top). The longer delay of $M\Delta t$ in the lower one gives the second threading of the wire and corresponds to the horizontal links in Fig. 3 (top). Beam splitters $B_3$ and $B_4$ implement the transformations indicated by red arrows in Eq. (26). [Red arrows point from the first node to the second in Eq. (18) for each beam splitter.] Following this, the 50:50 beam splitters $B_5$ and $B_6$ implement the transformations indicated by red arrows in Eq. (27), eventually resulting in the state with graph $\mathbf{Z}$ from Eq. (25). The outputs head to four detectors, which implement the $\mathrm{q̂}$ measurements (phase shifted as appropriate for the white nodes; see Sec. 2c) to project the state down to an ordinary square lattice, Eq. (28), as well as the adaptive measurement-based quantum algorithm to be implemented. The adaptiveness means that subsequent measurement bases generally must be chosen based on previous measurement outcomes. Most measurements will involve homodyne detection in a basis that must be calculated and updated before the arrival of the next pulse, but the ability to divert the beam to an efficient photon counter is also required for universal QC [6].Reuse & Permissions