Arbitrarily Large Continuous-Variable Cluster States from a Single Quantum Nondemolition Gate

We present a compact experimental design for producing an arbitrarily large optical continuous-variable cluster state using just one single-mode vacuum squeezer and one quantum nondemolition gate. Generating the cluster state and computing with it happen simultaneously: more entangled modes become available as previous modes are measured, thereby making finite the requirements for coherence and stability even as the computation length increases indefinitely.

Figure 1

Linear CV cluster state (quantum wire) made from squeezers and $C_Z$ gates. The graph for a quantum wire of arbitrary extent is shown at the bottom, with the three highlighted nodes corresponding to the partial experimental schematic above. Black lines represent optical paths (fiber or free space). Single-mode squeezers $S$ each produce a single pulse in a squeezed vacuum state (reduced variance in $\widehat{p}$), timed so that they intersect the previous and subsequent pulses within the appropriate $C_Z$ gates. The Heisenberg action of $C_Z$ on the input quadratures is shown diagramatically in the box at the bottom right (${\widehat{p}}_A\to {\widehat{p}}_A+{\widehat{q}}_B$ and ${\widehat{p}}_B\to {\widehat{p}}_B+{\widehat{q}}_A$, with ${\widehat{q}}_A$ and ${\widehat{q}}_B$ unchanged). Small numbered pulses are drawn next to the optical paths to illustrate their positions at a particular instant during the experiment. Dotted pulses show their locations (and their labels) at previous and subsequent times. Notice that pulses 3 and 4 are each shown twice, illustrating their flow through the experiment. Detectors $D$ implement the desired quantum algorithm through adaptive measurements that depend both on the algorithm and on previous measurement results [13]. We are ignoring boundary conditions (see text), so the quantum wire is effectively infinite, and the schematic shown is taken to be repeated indefinitely both to the left and to the right.

Figure 2

Linear CV cluster state made from just one squeezer and one $C_Z$ gate. The setup from Fig. 1 is “rolled up” into a compact design. The squeezer $S$ continually generates pulses of $\widehat{p}$-squeezed vacuum at regular intervals, with the length of the looping path chosen so that adjacent pulses are mode matched and phase locked within the $C_Z$ gate. The Heisenberg action of this gate is indicated in the box and is the same as in Fig. 1. The detector $D$ measures the pulses in sequence as they exit the $C_Z$ gate, with current measurement results affecting future measurement bases [13]. The numbered pulses next to the optical paths indicate the position of the independent modes in a single snapshot of the experiment, with pulse 1 having been emitted from $S$ before pulse 2, etc. Because the mode pulses are distinguished by the time of their emission from $S$, we say that this experiment generates a temporal-mode CV quantum wire.

Figure 3

Two-dimensional square-lattice graph (top) that is 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 that pass through nodes $M$ units apart, resulting in a square lattice with vertical dimension $M$. (In this case, $M=4$.) The dotted links represent additional $C_Z$ interactions 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 $\widehat{q}$ 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 ($M-1$) [13].

Figure 4

Two-dimensional square-lattice CV cluster state made from just one squeezer and one $C_Z$ gate. Beginning with the setup in Fig. 2, which is assumed to use horizontal polarization exclusively, polarizing beam splitters and a half wave plate (their action shown in the box at the bottom right) divert the output of the first pass through the $C_Z$ gate (inner loop) around for a second pass (outer loop) before heading to the detector $D$. This provides the additional “threadings” required for a square lattice (Fig. 3, dotted links included). The $C_Z$ gate must act in the same fashion when the input states are horizontally polarized as when they are vertically polarized (see box at the top right) [29]. The coil in the larger loop indicates that the length of this loop controls the vertical width of the lattice (the parameter $M$ from Fig. 3), with $M=4$ shown. The numbered pulses next to the optical paths indicate the position and polarization of the independent modes in a single snapshot of the experiment, with pulse 1 having been emitted from $S$ before pulse 2, etc. Because the mode pulses are distinguished by the time of their emission from $S$, we say that this experiment generates a temporal-mode CV cluster state.