Noise analysis of single-mode Gaussian operations using continuous-variable cluster states

We consider measurement-based quantum computation that uses scalable continuous-variable cluster states with a one-dimensional topology. The physical resource, known here as the dual-rail quantum wire, can be generated using temporally multiplexed offline squeezing and linear optics or by using a single optical parametric oscillator. We focus on an important class of quantum gates, specifically Gaussian unitaries that act on single quantum modes (qumodes), which gives universal quantum computation when supplemented with multi-qumode operations and photon-counting measurements. The dual-rail wire supports two routes for applying single-qumode Gaussian unitaries: The first is to use traditional one-dimensional quantum-wire cluster-state measurement protocols. The second takes advantage of the dual-rail quantum wire in order to apply unitaries by measuring pairs of qumodes called macronodes. We analyze and compare these methods in terms of the suitability for implementing single-qumode Gaussian measurement-based quantum computation.

Figure 1

Two ways of implementing measurement-based quantum computation on the dual-rail quantum wire (DRW). (a) Simplified graph [12] of the DRW. (b) The continuous-variable quantum-wire (CVW) protocol involves converting the DRW to a CVW by measurement of the position-quadrature ($\widehat{q}$) basis on the top qumodes [12], followed by single-qumode homodyne measurements in some quadrature bases ${\widehat{b}}_i$ to evolve and propagate the state to the right along the wire [6, 25, 26]. (c) The macronode protocol involves encoding the input state within the leftmost macronode (pair of qumodes). Each macronode is measured by homodyne detection of its constituent qumodes (${\widehat{b}}_{ia}$, ${\widehat{b}}_{ib}$). Graph weights [27] have been omitted for convenience. Color represents the sign and phase of each link. Blue and orange represents $\pm$ sign, respectively, and red is a complex sign of $i$. The magnitude of each red self-loop is ${\varepsilon }_{\text{D}}=sech\left(2\alpha \right)$ where $\alpha >0$ is the overall squeezing parameter [12]. The adjoining edges have magnitude $g_{\text{D}}=\frac{1}{2}tanh\left(2\alpha \right)$. The black edge and self-loops contained in the green ovals label the modes containing the encoded input state.

Figure 2

(a) Circuit diagram for an element of Gaussian measurement-based quantum computation. The input $|\psi \rangle$ is entangled with a momentum-squeezed vacuum state by a ${\widehat{C}}_Z\left(g\right)$ interaction [see Eq. (2.5)]. The dotted line encapsulates CVCS construction. After measuring the top qumode with outcome $m_1$, the following operations are applied to the output $|\psi \rangle$: logic gate ${\widehat{U}}_1$ [see Eq. (2.15)], a displacement $\widehat{X}\left(\frac{m_1}{g}\right)$ [see Eq. (2.16)], and noise $\widehat{N}(\varepsilon$) [see Eq. (2.19)]. (b) Graphical representation of a CVW with an input state on the leftmost qumode. When using the graphical representation, we must assume that the input state is Gaussian [27]. In Sec. 2c we use Wigner functions to generalize the description to mixed input states. The blue solid part of the wire is represented in the circuit diagram above.

Figure 3

Macronode-based computation applies a logical gate to an input encoded in a macronode by measuring the leftmost macronode. Here we assume that the input state is Gaussian (and represent it by the self-loop weight $z_{\psi }$) so that we can describe macronode measurement using the graphical calculus [27]. This is for illustrative purposes only, and the same statements can be made for general input states. The top part of the diagram shows a section of the DRW graph corresponding to the $(a,b)$ macronode decomposition—that is, each node represents a physical qumode. The pair of local homodyne measurements (${\widehat{b}}_{1a}$, ${\widehat{b}}_{1b}$) apply the state transformation $z_{\psi }\to z_{{\psi }^{\prime }}$ (this transformation is described in more detail in Fig. 4). On the bottom is the equivalent description using distributed modes [see Eq. (3.2)]. Notice that while the measurements in this decomposition are nonlocal (they are effectively Bell measurements), the input state $z_{\psi }$ and output state $z_{{\psi }^{\prime }}$ are localized on some graph node. Following either set of arrows from the bottom left to the bottom right then shows the graphical evolution of the input state. Also note that on the bottom pair of figures, the bottom right node is merely a spectator. Thus, we can describe this process using distributed modes by a single-qumode input, a two-qumode cluster state (the diagonally joined pair of nodes on the bottom left graph), and Bell measurements (which correspond to local measurements on the physical qumodes). This highlights the similarity between macronode computation and CV teleportation [15].

Figure 4

Circuit diagram showing a basic element of macronode-based quantum computation. The state immediately after the dotted line is equivalent to a small section of DRW using distributed modes, as shown in Fig. 3. In that figure, $|\psi \rangle$ is represented by $z_{\psi }$ (assumed Gaussian), and the pair of other modes is equivalent to the leftmost diagonally connected pair of nodes. The box labeled $B^{*}$ does not represent a physical gate. Instead, it is a change from distributed modes (blue solid circuit wires) to physical modes (red dashed circuit wires)— see Eq. (3.2). Computation proceeds via local physical homodyne measurements ${\widehat{b}}_{1(a,b)}$. The bottom qumode remains in the distributed-mode basis, which is why the circuit line remains blue. Alternatively, we can interpret this diagram in a different way: If $B^{*}$ is taken to be a physical 50:50 beamsplitter gate, and if the colors and subscript labels are ignored, this diagram shows how to construct a small section of the DRW, inject an input state, and use local macronode measurements to drive MBQC. This reveals that the transformation from physical to distributed modes, Eq. (3.2), is the same as that of the beamsplitter gate used in DRW construction [12]. The squeezing factor acting on the pair of vacuum modes is given by ${\varepsilon }_{\text{D}}^{-1/2}$, and the effective ${\widehat{C}}_Z$ interaction strength parameter is $t=tanh\left(2\alpha \right)$, where $\alpha$ is the overall squeezing parameter [12]. The total operation applied on the logical state is ${\widehat{N}}_{\text{m}}{\widehat{D}}_{\text{m}}{\widehat{V}}_1$ [see Eqs. (3.5), (3.7), and (3.11)].

Figure 5

Minimum scalar variance per rotation gate for three and four measurements on the CVW, $S_{\widehat{R}\left(\theta \right)}\left(3\right)$ (dashed blue) and ${\mathcal{S}}_{\widehat{R}\left(\theta \right)}\left(4\right)$ (solid red). Technically, only ${\mathcal{S}}_{\widehat{R}\left(\theta \right)}\left(4\right)$ has been minimized (represented by the calligraphic font) as $S_{\widehat{R}\left(\theta \right)}\left(3\right)$ is unique (it has no free measurement degree of freedom). Units on the vertical axis are such that the vacuum variance is $1/2$. Although the three-measurement protocol introduces the least noise at some particular $\theta$, for $\theta$ in the vicinity of $\frac{\pi }{2}$ or $\frac{3\pi }{2}$, which correspond to gates that cannot be implemented by three CVW measurements [26], the noise becomes arbitrarily large. On the other hand, the noise for the four-measurement protocol remains bounded for all $\theta$. In this plot, the squeezing parameter is $\alpha =0.5756$, corresponding to 5 dB of squeezing ($\#\text{dB}=10{log}_{10}{e}^{2\alpha }$), approximately the levels achieved in Ref. [15].

Figure 6

Minimum scalar variance per rotation gate using the dictionary protocol for three [$S_{\text{d},\widehat{R}\left(\theta \right)}\left(3\right)$, dashed blue] and four measurements [${\mathcal{S}}_{\text{d},\widehat{R}\left(\theta \right)}\left(4\right)$, solid red], as well as the general macronode protocol [$S_{\text{m},\widehat{R}\left(\theta \right)}^{\text{so}}\left(2\right)$, dot-dashed black] for two measurements. These are plotted as a function of $\theta$, which specifies the rotation applied to the state. Technically, only ${\mathcal{S}}_{\text{d},\widehat{R}\left(\theta \right)}\left(4\right)$ has been minimized (represented by calligraphic font) as $S_{\text{d},\widehat{R}\left(\theta \right)}\left(3\right)$ is unique (it has no free measurement degrees of freedom), and $S_{\text{m},\widehat{R}\left(\theta \right)}^{\text{so}}\left(2\right)$ is an upper bound on the minimum for the macronode protocol. Units on the vertical axis are such that the vacuum variance is $1/2$. The variance added by the general macronode protocol is upper bounded by the black line [corresponding to the suboptimal solution from Eq. (3.16)] that does not vary with angle. Although the three-macronode-measurement protocol introduces the least noise at some particular values of $\theta$, for $\theta$ in the vicinity of $\frac{\pi }{2}$ or $\frac{3\pi }{2}$, the noise diverges. These angles correspond to gates that cannot be implemented by three CVW measurements [26] (also expected here because the dictionary protocol is a node-for-node adaptation of the CVW protocol). The noise in the vicinity of these gates becomes arbitrarily large. On the other hand, the noise for the four-measurement protocol remains bounded for all $\theta$. In this plot, the squeezing parameter is $\alpha =0.5756$, corresponding to 5 dB of squeezing ($\#\text{dB}=10{log}_{10}{e}^{2\alpha }$), approximately the levels achieved in Ref. [15].