Quantum computing with multidimensional continuous-variable cluster states in a scalable photonic platform

Quantum computing is a disruptive paradigm widely believed to be capable of solving classically intractable problems. However, the route toward full-scale quantum computers is obstructed by immense challenges associated with the scalability of the platform, the connectivity of qubits, and the required fidelity of various components. One-way quantum computing is an appealing approach that shifts the burden from high-fidelity quantum gates and quantum memories to the generation of high-quality entangled resource states and high fidelity measurements. Cluster states are an important ingredient for one-way quantum computing, and a compact, portable, and mass producible platform for large-scale cluster states will be essential for the widespread deployment of one-way quantum computing. Here, we bridge two distinct fields—Kerr microcombs and continuous-variable (CV) quantum information—to formulate a one-way quantum computing architecture based on programmable large-scale CV cluster states. Our scheme can accommodate hundreds of simultaneously addressable entangled optical modes multiplexed in the frequency domain and an unlimited number of sequentially addressable entangled optical modes in the time domain. One-dimensional, two-dimensional, and three-dimensional CV cluster states can be deterministically produced. When combined with a source of non-Gaussian Gottesman-Kitaev-Preskill qubits, such cluster states enable universal quantum computation via homoyne detection and feedforward. We note cluster states of at least three dimensions are required for fault-tolerant one-way quantum computing with known error-correction strategies. This platform can be readily implemented with silicon photonics, opening a promising avenue for quantum computing on a large scale.

Figure 1

Universal quantum computation is possible by combining a source of non-Gaussian GKP qubits with multimode Gaussian unitaries and Gaussian measurements with feedforward [46, 47]. These elements are summarized in the top three bubbles embedded in the green box. Our scheme addresses the need for multimode Gaussian gates and measurements via the on-chip implementation of a CV cluster state. It achieves this by combining two elements: on-chip squeezed-light sources and linear optics, and homodyne measurements.

Figure 2

(Top) Classical frequency-comb generation. (Bottom) The spectra for the input pump (orange bar) and the output side-mode fields (blue bars).

Figure 3

Generation of the 0D cluster state. Middle panel illustrates the FWM process in the MR. The bottom panel is the graph representation of the output state. The edge-weight parameter is $C=1$.

Figure 4

Generation of the 1D cluster state. Top and bottom inserted panels depict the graph-state representations right after stage $\left(i\right)$ and $\left(ii\right)$. The upper rail is for spatial mode a, and the lower rail is for spatial mode b. $C=1$ for stage $\left(i\right)$ and $C=1/2$ for stage $\left(ii\right)$.

Figure 5

Generation of the 2D cluster state. Gray, green, and magenta shaded areas in the inserted figures denote the temporal modes $t_1,t_2,t_3\in T$, where $t_2=t_1+\delta t$ and $t_3=t_2+\delta t$, and $\delta t$ is the time delay due to the DL. For stages $\left(i\right), \left(ii\right), \left(iii\right)$, and $\left(iv\right), C=1, 1/2, 1/2$, and $1/2\sqrt{2}$, respectively. The labels $\mathrm{a}(4n+1), \mathrm{a}(4n-1), \mathrm{b}(4n+1)$, and $\mathrm{b}(4n-1)$ indicate the spatial mode indices (a and b) followed by the spectral mode index, where $n\in \mathbb{Z}$. Electrodes for the UMZI are shown in yellow.

Figure 6

Generation of programmable CV cluster states. The inserted figures show the graph states at stages $\left(i\right)$–$\left(v\right)$. The graphs at stages $\left(iv\right)$ and $\left(v\right)$ are coarse-grained so that each node contains four modes and each edge represents many connections between two particular macronodes (group of four modes). The magenta box indicates where single-mode input states, such as GKP qubit ancilla states, can be injected into the 3D lattice via the BMZI. For stages $\left(i\right)$–$\left(v\right), \mathcal{C}=1,1/2,1/4,1/4,$ and ${\left(4\sqrt{2}\right)}^{-1}$, respectively.

Figure 7

Introduction of the macronodes for Fig. 6. Each edge is labeled corresponding to the legend shown above. (a) The single sheet is equivalent to the graph state at stage $\left(iii\right)$, where multiple copies of the quad rail lattice cluster state multiplexed in time. $\mathcal{C}=1/4$ (b) 3D cluster state at stage $\left(v\right)$. (b1) Unit cell of the 3D cluster state. (b2) Single spacelike slice of the 3D cluster state. (b3) Single time-like slice of the 3D cluster state. $\mathcal{C}={\left(4\sqrt{2}\right)}^{-1}$.

Figure 8

(a) A (frequency)$\times$(time) array of two-mode CV cluster states. $\mathcal{C}=1$. (b) A train of frequency entangled dual-rail wire cluster states. $\mathcal{C}=1/2$. (c) A (frequency)$\times$(time) entangled 2D resource [33]. $\mathcal{C}=1/4$. (d) The bilayer square lattice cluster state [34]. $\mathcal{C}={\left(2\sqrt{2}\right)}^{-1}$. (e) A train of quad-rail lattice cluster states [55]. $\mathcal{C}=1/4$.

Figure 9

Shifting of cavity modes. $\delta {\omega }_X$ and $\delta {\omega }_S$ denote the frequency shifts from cross-phase modulation and self-phase modulation. $\delta {\omega }_l$ is the dispersion frequency shift of spectral mode $l$. (a) Zero-detuned ($\sigma =0$) cold cavity without considering dispersion effect (${\zeta }_2=0$). (b) Zero-detuned hot cavity without considering dispersion effect. (c) Zero-detuned cold cavity in the case of anomalous dispersion (${\zeta }_2>0$). (d) Zero-detuned cold cavity in the case of normal dispersion (${\zeta }_2<0$). (e) Balancing between dispersion and Kerr nonlinearity (e.g., $\sigma <0, {\zeta }_2>0$).

Figure 10

The spectrum of a stable Kerr-soliton frequency comb. Each frequency tooth can serve as a pump or a phase reference that addresses a corresponding spectral mode in the CV cluster state.

Figure 11

The squeezing and anti-squeezing spectra, normalized to the shot-noise limit, for (a) 0D, (b) 1D, (c) 2D, and (d) 3D CV cluster states. Blue solid (dashed) curves are the squeezing (anti-squeezing) spectra with the maximum squeezing levels in 10.18, 8.17, 6.25, and 4.03 dB in 0D, 1D, 2D, and 3D cases. Orange solid (dashed) curves are the squeezing (anti-squeezing) spectra at 3 dB squeezing.

Figure 12

Normalized squeezing levels for the 0D case versus the entangled pair index for four waveguide cross section configurations. Red dashed horizontal line is at the 3 dB squeezing level. The numbers of entangled pairs above the 3 dB squeezing level for the four cases are $l_{3\mathrm{dB}}=1340,500,331,\text{and}143$.

Figure 13

(a) Unit cell of the 3D cluster state with respect to physical modes. Every macronode is connected to eight other macronodes. Furthermore, each individual mode has $8\times 4=32$ neighbours. Edge conventions are as defined in Fig. 7. (b) Unit cell of the 3D cluster state graph in terms of distributed modes $\alpha , \beta , \gamma$, and $\delta$ [see Eq. (34)]. (c) Top view of (b). Constant time cross section showing a layer of grey and red macronodes and their connections to macronodes in the layers above and below. Solid (dotted) lines represent upwards (downwards) pointing edges.

Figure 14

(a) Local measurement of the physical modes via Eq. (35). (b) From the perspective of the distributed modes, such a measurement will in the general case project onto entangled states. (c) For some choices of angles, these projections will be onto partially or fully separable states. Choosing ${\theta }_{\text{a}}={\theta }_{\text{c}}$ and ${\theta }_{\text{b}}={\theta }_{\text{d}}$ makes the measurement separable with respect to the $\left(\alpha ,\beta |\gamma ,\delta \right)$ bipartition. This follows from two applications of Eq. (36). (d) Choosing ${\theta }_{\text{a}}={\theta }_{\text{b}}={\theta }_{\text{c}}={\theta }_{\text{d}}$ results in a local measurement with respect to the distributed modes. We define ${(n_{\text{a}},n_{\text{b}},n_{\text{c}},n_{\text{d}})}^{\text{T}}=\mathbf{A}{(m_{\text{a}},m_{\text{b}},m_{\text{c}},m_{\text{d}})}^{\text{T}}$. This follows from four applications of Eq. (36).

Figure 15

(a) In order to implement single-mode gates on all inputs, the grey macronodes in the (un-)shaded regions are measured in the $\widehat{p}(\pm \pi /4)$ basis, respectively. (b) The relative sign in the gate implemented in Eq. (40) depends on whether the input lies in a green shaded region, or not. Diagrammatic conventions are the same as in Figs. 13 and 13.

Figure 16

(a) The two square graphs from Fig. 13 have been redrawn side by side. Note that the time arrow is from left to right, rather than from top to bottom. The green modes on the left are the distributed modes that can contain an input. The red arrows denote the application of a 50:50 BSG, and the black arrows indicate the local basis measurement after all such BSGs. (b) Alternative representation of (a). The square graph cluster state is equivalent to two pairs stitched together by a 50:50 BSG [see Eq. (33)]. This picture more clearly shows that the measurement pattern implements sequential teleportation [see Eq. (38)].

Figure 17

(a) Subgraph of the full 3D cluster state represented using distributed modes. Some of the square graphs are shaded red or green to make the 3D layout clearer. (b) The top layer of the graph viewed from above. Macronodes A, B, C, and D contain input states encoded within $\alpha$ and $\gamma$ distributed modes. Solid lines are pointing out of the page, while dotted lines are pointing into the page. (c) Middle layer of the graph viewed from above. (d) Bottom layer of the graph viewed from above. These modes are not measured in this round, but will contain input states after the upper two layers are measured. (e) Macronode I is the only grey macronode where the homodyne angles are not all the same. The dotted arrows represent 50:50 BSGs that act before those represented by the solid arrows. After all 50:50 BSGs, the modes in this macronode are measured in the bases shown in black.

Figure 18

Beamsplitter gymnastics for ${\widehat{B}}_{\text{I}\alpha ,\text{I}\beta }$ and ${\widehat{B}}_{\text{I}\alpha ,\text{I}\gamma }$. For ${\widehat{B}}_{\text{I}\alpha ,\text{I}\beta }$, modes $\text{(1–10)}=(\text{D}\gamma ,\text{D}\delta ,\text{A}\gamma ,\text{A}\delta ,\text{M}\beta ,\text{I}\alpha ,\text{I}\beta ,\text{E}\alpha ,\text{Q}\gamma ,\text{N}\gamma )$. For ${\widehat{B}}_{\text{I}\alpha ,\text{I}\gamma }$, modes $\text{(1–10)}=(\text{D}\gamma ,\text{D}\delta ,\text{A}\gamma ,\text{A}\delta ,\text{M}\beta ,\text{I}\alpha ,\text{I}\gamma ,\text{G}\delta ,\text{Q}\gamma ,\text{O}\alpha )$. (a) The goal is to move the 50:50 BSG between the grey modes in the purple oval (6 and 7), replacing it with an operation that only acts on different modes. (b) Since modes 5 and 8 are measured in the same basis, we can use postprocessing to insert an extra 50:50 BSG [see Eq. (36)]. (c) Each square graph can be replaced with two entangled pairs and a pair of 50:50 BSG. The dotted lines indicate that these act before the other 50:50 BSGs [see Eq. (33)]. (d) By direct calculation using Eqs. (26), ${\widehat{B}}_{5,8}{\widehat{B}}_{6,7}{\widehat{B}}_{6,5}{\widehat{B}}_{8,7}={\widehat{B}}_{6,5}{\widehat{B}}_{8,7}{\widehat{B}}_{6,8}{\widehat{B}}_{7,5}$. (e) We can move the rightmost dotted beamsplitter using the second equality in Eq. (33). (f) Similarly, the leftmost dotted beamsplitter can be moved to the left and we can substitute the square graphs back in by four applications of Eq. (33). At this stage, the 50:50 BSG ${\widehat{B}}_{6,7}$ had been replaced with the two dotted 50:50 BSGs that act on either sides of the square graphs, thereby achieving our goal.

Figure 19

Beamsplitter gymnastics for ${\widehat{B}}_{\text{I}\gamma ,\text{I}\delta }$ and ${\widehat{B}}_{\text{I}\beta ,\text{I}\delta }$. For ${\widehat{B}}_{\text{I}\gamma ,\text{I}\delta }$, modes $\text{(1–10)}=(\text{B}\alpha ,\text{B}\beta ,\text{C}\alpha ,\text{C}\beta ,\text{G}\delta ,\text{I}\gamma ,\text{I}\delta ,\text{K}\gamma ,\text{O}\alpha ,\text{C}\alpha )$. For ${\widehat{B}}_{\text{Ib,}\text{Id}}$, modes $\text{(1–10)}=(\text{A}\gamma ,\text{A}\delta ,\text{C}\alpha ,\text{C}\beta ,\text{E}\alpha ,\text{I}\beta ,\text{I}\delta ,\text{K}\gamma ,\text{N}\gamma ,\text{P}\alpha )$. (a) The goal is to move the BSG between the grey modes in the purple oval (6 and 7), replacing it with an operation that only on other modes. (b) By following a similar sequence of steps as in Fig. 18 (or equivalently, using the proof shown in Ref. [34]) we arrive at this alternative description. (c) The leftmost dotted beamsplitter can be moved to the left and we can substitute the square graphs back in by four applications of Eq. (33). At this stage, the 50:50 BSG ${\widehat{B}}_{6,7}$ had been replaced with the two dotted 50:50 BSGs that act on either sides of the square graphs, thereby achieving our goal.

Figure 20

A full architecture of the integrated photonic one-way quantum computing.

Figure 21

On top we show the desription of the Gaussian pure state using the full graphical calculus. On the bottom we show the simplified notation described in the main text. (a) Two-mode continuous-variable cluster state. The rescaling parameter is $C=1$. (b) Four mode continuous-variable cluster state. The rescaling parameter is $C=1/\sqrt{2}$.