Deterministic generation of a two-dimensional cluster state

TL;DR

This paper presents a scalable, deterministic method for generating large two-dimensional photonic cluster states using temporal multiplexing, enabling universal measurement-based quantum computation with potential for fault tolerance.

Contribution

A novel scheme utilizing temporal multiplexing, delay loops, and beam-splitters to produce large 2D cluster states suitable for universal quantum computing.

Findings

Generated over 30,000 entangled modes in a cylindrical lattice

Achieved a 1250-mode-long cluster state for deep quantum circuits

Demonstrated potential for fault-tolerant quantum computation

Abstract

Measurement-based quantum computation offers exponential computational speed-up via simple measurements on a large entangled cluster state. We propose and demonstrate a scalable scheme for the generation of photonic cluster states suitable for universal measurement-based quantum computation. We exploit temporal multiplexing of squeezed light modes, delay loops, and beam-splitter transformations to deterministically generate a cylindrical cluster state with a two-dimensional (2D) topological structure as required for universal quantum information processing. The generated state consists of more than 30000 entangled modes arranged in a cylindrical lattice with 24 modes on the circumference, defining the input register, and a length of 1250 modes, defining the computation depth. Our demonstrated source of 2D cluster states can be combined with quantum error correction to enable fault-tolerant quantum computation.

Figures

Equations

$$\begin{array}{rl}{\hat{n}}_k^x=& {\hat{x}}_k^A+{\hat{x}}_k^B-{\hat{x}}_{k+1}^A-{\hat{x}}_{k+1}^B\\ & -{\hat{x}}_{k+N}^A+{\hat{x}}_{k+N}^B-{\hat{x}}_{k+N+1}^A+{\hat{x}}_{k+N+1}^B,\end{array}$$

$$\begin{array}{rl}{\hat{n}}_k^p=& {\hat{p}}_k^A+{\hat{p}}_k^B+{\hat{p}}_{k+1}^A+{\hat{p}}_{k+1}^B\\ & -{\hat{p}}_{k+N}^A+{\hat{p}}_{k+N}^B+{\hat{p}}_{k+N+1}^A-{\hat{p}}_{k+N+1}^B,\end{array}$$

Full text

Quantum computing represents a new paradigm for information processing that harnesses the inherent non-classical features of quantum physics to find solutions to problems that are computationally intractable on classical processors Ladd et al. (2010). In measurement-based, or cluster state, quantum computing (MBQC), the processing is performed via simple single-site measurements on a large entangled cluster state Raussendorf and Briegel (2001). This constitutes a simplification over the standard gate-based model of quantum computing, as it replaces complex coherent unitary dynamics with simple projective measurements. However, one of the outstanding challenges in realizing cluster state computation is the reliable, deterministic and scalable generation of non-classical entangled states suitable for universal information processing.

Several candidate platforms for scalable cluster state generation have been proposed and some experimentally realized, including solid state superconducting qubits Wang et al. (2018), trapped ion qubits Mandel et al. (2003); Lanyon et al. (2013) and photonic qubits or qumodes, in which qubits can be encoded, generated by parametric down-conversion Walther et al. (2005); Tokunaga et al. (2008); Yokoyama et al. (2013); Chen et al. (2014) or by quantum dots Schwartz et al. (2016). However, none of these implementations have demonstrated true scalability combined with computational universality. The largest cluster state generated to date is a temporally multiplexed photonic state comprising entangled modes in a long chain which however does not allow for universal computation due to its one-dimensional (1D) topological structure Yokoyama et al. (2013); Yoshikawa et al. (2016). To achieve universality, the dimension of the cluster state must be at least two. Several proposals for generating two-dimensional (2D) cluster states in different systems have been proposed Economou et al. (2010); Menicucci (2011); Wang et al. (2014); Alexander et al. (2016) but due to technical challenges, scalable and computationally universal cluster states have yet to be produced in any physical system.

We propose and demonstrate a highly scalable scheme for the generation of cluster states for universal quantum computation based on quantum continuous variables (CV) where information is encoded in the position or momentum quadratures of photonic harmonic oscillators Weedbrook et al. (2012). We use a temporally multiplexed source of optical Einstein-Podolsky-Rosen (EPR) states Reid et al. (2009) to generate a long string of entangled modes that is curled up and fused to form a 2D cylindrical array of entangled modes. Specifically, we generate a massive cluster state of more than $30\,000\text{\,}$ entangled modes comprising an input register of $2\times 12=24$ modes on which the input state may be encoded, and a length of $1250\text{\,}$ modes for encoding operations by projective measurements, only limited by the phase stability of our setup. In addition to being universal and deterministically generated, the source is operated under ambient conditions in optical fibers at the low-loss telecom wavelength of $1550\text{\,}\mathrm{n}\mathrm{m}$. These favorable operational conditions and specifications significantly facilitate further upscaling of the entangled state as well as its use in applications and fundamental studies.

The canonical approach to CV cluster state generation is to apply two-mode controlled-Z gates onto pairs of individually prepared eigenstates of the momentum (or phase quadrature) operators $\hat{p}_{i},\hat{p}_{j}$ in adjacent modes $i,j$. The gate is described by the unitary operation $\hat{C}_{Z}=e^{ig\hat{x}_{i}\hat{x}_{j}}$ where $\hat{x}_{i},\hat{x}_{j}$ are the position (amplitude quadrature) operators of mode $i$ and $j$, while $g$ is the interaction strength. Applying this gate to two modes leads to entanglement in the form of quantum correlations of the two modes’ quadratures. The operations and resulting state can be represented by a graph in which the nodes represent the momentum eigenstates while the edges (links) between the nodes represent the application of a controlled-Z operation where the interaction strength is given by the edge weight. In a practical implementation, the unphysical momentum eigenstates are replaced by highly squeezed states while the controlled-Z operations can be imitated by phase shifts and beam splitter transformations. To enable scalability, it has been suggested to use multiplexing of spatial modes Armstrong et al. (2012), frequency modes Menicucci et al. (2008); Cai et al. (2017), or temporal modes Menicucci (2011); Alexander et al. (2018). For example, Menicucci suggested using temporal multiplexing to form a 2D cluster state combining four squeezed state generators, five beam splitters, and two delay lines Menicucci (2011).

We propose a simpler approach to 2D cluster state generation lowering the experimental requirements (Fig. 1). The state is produced in four steps: i) Pairs of squeezed vacuum states are generated at $1550\text{\,}\mathrm{n}\mathrm{m}$ wavelength from two bow-tie shaped optical parametric oscillators (OPOs) by parametric down conversion Andersen et al. (2016). The states are defined in consecutive temporal modes of duration $\tau$ of the continuously generated OPO output. ii) The squeezed vacuum pairs in spatial modes $A$ and $B$ are interfered on a balanced beam splitter (denoted $\text{BS}_{1}$). This produces a train of pairwise EPR-entangled temporal modes exhibiting quantum correlation between the position and momentum quadratures. Each EPR pair can be represented by a simple graph of a single edge connecting two nodes. iii) A 1D cluster state is formed by delaying one arm of the interferometer by $\tau$ with respect to the other arm and interfering the resulting time-synchronized modes on another balanced beam splitter (denoted $\text{BS}_{2}$). The interference entangles EPR pairs along an indefinitely long chain creating a 1D graph. iv) In the final step, the 2D cluster state is produced by introducing another delay to one interferometer arm of duration $N\tau$ and interfering the resulting time-synchronized modes on a final beam splitter (denoted $\text{BS}_{3}$). This effectively curls up the graph and fuses the modes into an indefinitely long cylinder with $N$ nodes on the circumference as illustrated in Fig. 1 for $N=12$, leading to $2\times N=24$ input modes distributed on the two spatial modes $A$ and $B$. For detailed description of experimental implementations see Material and Methods See supplementary materials .

All states and operations involved are Gaussian, meaning they can be described by Gaussian distributions of the quadrature variables in phase space. In the formalism of graphical calculus for Gaussian states Menicucci et al. (2011), the generated graphs are so-called $\mathcal{H}$-graphs as they can be generated from vacuum by a single Hamiltonian, and have an edge weight of $g=i\sinh(2r)G$ where $r$ is the squeezing parameter of the two squeezing operations and $G=-1$ for the EPR-states, $\pm 1/2$ for the 1D graph and $\pm 1/4,1/2$ for the 2D graph. Due to the particular structure of the $\mathcal{H}$-graph generated here (it is self-inverse and bipartite—see Supplementary Text section 1.1 for details See supplementary materials ), it can be transformed into a cluster state by $\pi/2$ rotations in phase space leading to real edges of weight $g=\tanh(2r)G\rightarrow G$ for $r\rightarrow\infty$. Finally, as the $\pi/2$ phase space rotations can be absorbed into the measurement basis, or simply by appropriate re-definitions of quadratures on the rotated modes, the generated $\mathcal{H}$-graph state and its corresponding cluster state are completely equivalent. See Supplementary Text section 1.2 for details on the cluster state generation scheme See supplementary materials .

The produced cylindrical 2D cluster state can be shown to be a universal resource for quantum computing: In Fig. 2, the generated cylindrical cluster state is unfolded and projected into a square lattice by projective measurements in the position basis and $\pi/2$ phase-space rotations of different modes. Such a square lattice is a well-known universal resource for quantum computing Gu et al. (2009), and thus the initial cylindrical cluster state is itself universal. For computation it is not necessary to project the generated cluster state into a square lattice—rather, one would in general optimize the detector settings required for the gate to be implemented. For instance, with proper settings the cluster state can be projected into 1D dual-rail wires along the cylinder, an efficient resource for one-mode computation Yokoyama et al. (2013); Alexander et al. (2018) and with possible two-mode interactions between them—for details see Supplementary Text section 1.4 See supplementary materials . Doing so requires fast control of the measurement bases in between temporal modes, while in this work the cluster state is measured in fixed bases for state verification.

Multi-partite cluster state inseparability can be witnessed through the measurement of the uncertainties of the state nullifiers—linear combinations of position and momentum operators for which the cluster states are eigenstates with eigenvalue 0. E.g. for the ideal two-mode EPR state, the well-known nullifiers are $\hat{n}^{x}_{\text{EPR}}=\hat{x}_{A}-\hat{x}_{B}$ and $\hat{n}^{p}_{\text{EPR}}=\hat{p}_{A}+\hat{p}_{B}$ since $\hat{n}^{x}_{\text{EPR}}\ket{\rm{EPR}}=0$ and $\hat{n}^{p}_{\text{EPR}}\ket{\rm{EPR}}=0$. For our 2D cluster state, $|\text{2D}\rangle$, the nullifiers consist of 8 modes and are given by

[TABLE]

[TABLE]

as $\hat{n}_{k}^{x}|\text{2D}\rangle=0$ and $\hat{n}_{k}^{p}|\text{2D}\rangle=0$ (derived in the Supplementary Text section 1.3 See supplementary materials ), where the subscript indicates the temporal mode index with $N$ being the number of temporal modes in the cluster state circumference.

The practically realizable cluster state is never an exact eigenstate of the nullifiers since such a state is unphysical. The measurement outcomes of the nullifiers are therefore not exactly zero in every measurement but possess some uncertainties around zero. A condition for complete inseparability of the 2D cluster state (derived in the Supplementary Text section 2 See supplementary materials ) leads to a bound on the variances of all nullifiers of $3\text{\,}\mathrm{d}\mathrm{B}$ squeezing below the shot noise level. Therefore, to witness full inseparability, we must observe more than $3\text{\,}\mathrm{d}\mathrm{B}$ squeezing for all nullifiers. In Fig. 3, the measured nullifier variances are shown for a dataset of $1500\text{\,}$ nullifiers and they are all observed to be well below the $-3\text{\,}\mathrm{d}\mathrm{B}$ bound; we measure an averaged variance of $-4.7\text{\,}\mathrm{d}\mathrm{B}$ and $-4.3\text{\,}\mathrm{d}\mathrm{B}$ for $\hat{n}_{k}^{x}$ and $\hat{n}_{k}^{p}$, respectively. In the inset of Fig. 3, we present the measurement of a longer cluster state of $15\,000\text{\,}$ temporal modes corresponding to a measurement time of $4\text{\,}\mathrm{m}\mathrm{s}$. Although phase instabilities are clearly seen to affect the performance in terms of variations of the nullifier variances, all variances stay below the $-3\text{\,}\mathrm{d}\mathrm{B}$ bound. The 2D cluster of $2\times$15,000\text{,}$=$30,000\text{,}$$ modes is thus fully inseparable. Note that not all $30\,000\text{\,}$ modes of the cluster state need to exist simultaneously when performing projective measurements for computation. In fact, only a single temporal mode of the cluster state needs to exist while the remaining modes of the state are under construction. Hence, the cluster state can be immediately consumed for computation while being generated, with no additional state storage necessary—see Supplementary Text section 1.4 for a possible measurement scheme for computation on the cluster state.

With the deterministic generation of a universal 2D cluster state, we have for the first time (in parallel with Asavanant et al. Asavanant et al. (2019)) in any system constructed a platform for universal MBQC. Its scalability was demonstrated by entangling $30\,000\text{\,}$ optical modes in a 2D lattice that includes 24 input modes and allows for a computation depth of $1250\text{\,}$ modes. Since only a few modes exist simultaneously, we are not limited by the coherence time of the light source, and thus the number of operations depends only on the phase stability of the system. The computational depth can therefore be unlimited by implementing continuous feedback control of the system for phase stabilization as demonstrated for the 1D photonic cluster state in Yoshikawa et al. (2016). The results presented here and in Asavanant et al. (2019) are similar: Both 2D cluster states are generated deterministically in the CV regime with comparable size and amount of squeezing in the nullifier variance. However, with only two squeezing sources, three interference points, and operation in fiber, the experimental setup demonstrated here is simpler, while in Asavanant et al. (2019) larger bandwidth OPOs are demonstrated resulting in shorter delay lines. In both systems, the number of input modes can be readily increased by using OPOs with larger bandwidths, possibly combined with a longer time delay of the second interferometer. E.g. using OPOs with a $1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ bandwidth (65 times wider) and a twice as long interferometer delay, a state with $\sim$1500\text{,}$$ input modes can be generated. Large bandwidth OPOs have been demonstrated, but phase stability and losses in the delay lines are more challenging. While phase fluctuation is only a matter of experimental control on which we expect to improve with continuous phase stabilization, delay losses are unavoidable and increasing the OPO bandwidth may be a better solution than increasing the delay lengths.

CV cluster states are described by Gaussian statistics, but it is known that an element (state, operation, or measurement) of non-Gaussian quadrature statistics is required for universal quantum computing Bartlett and Sanders (2002). Such an element could be a photon number resolving detector (PNRD) or an ancillary cubic-phase state Alexander et al. (2018); Gottesman et al. (2001). Despite recent experimental efforts in developing high-efficiency PNRD Thekkadath et al. (2019) and deterministically generating optical states with non-Gaussian statistic Hacker et al. (2019), the formation of the required non-Gaussianity of the cluster state still constitutes an important challenge to be tackled in the future. Another currently limiting factor towards quantum computation is the existence of finite squeezing in the cluster leading to excess quantum noise and thus computational errors. However, these errors can be circumvented using Gottesman-Kitaev-Preskill (GKP) state encoding Gottesman et al. (2001) concatenated with traditional qubit error correction schemes leading to fault-tolerant computation with a 15–$17\text{\,}\mathrm{d}\mathrm{B}$ squeezing threshold Walshe et al. (2019). Another recently discovered advantage of the GKP encoding is that in addition to fault-tolerance, it also allows for universality without adding extra non-Gaussian states or operations Baragiola et al. (2019). While GKP states have recently been produced in the microwave regime Campagne-Ibarcq et al. (2019) and in trapped-ion mechanical oscillators Flühmann et al. (2019), their production in the optical regime remains a task for future work. For further discussion on quantum computation using the generated cluster state, see Supplementary Text section 1.4 See supplementary materials . Although a path towards fault-tolerant universal quantum computing using CV cluster states has been established, it is highly likely that the first demonstrations of CV quantum computation will be non-universal algorithmic sub-routines such as boson sampling and instantaneous quantum computing Su et al. (2018). With the large, but noisy cluster state demonstrated here, interesting future work will be to implement basic Gaussian circuits and investigate e.g. the attainable circuit depth. Furthermore, the technique of folding a 1D cluster state into a 2D structure could be extended, using an additional interferometer, to form 3D cluster states which might be suitable for topologically protected MBQC.

Acknowledgements

We thank R. N. Alexander for useful discussion on the final manuscript and J. B. Brask for proofreading. The work was supported by the Danish National Research Foundation through the Center for Macroscopic Quantum States (bigQ, DNRF142), and the VILLUM FOUNDATION Young Investigator Programme (grant no. 10119).

Author contributions: M.V.L. and U.L.A. conceived the project. J.S.N., X.G. and C.R.B. designed and built the squeezing sources, while M.V.L. developed the theoretical background, designed the experiment and built the setup. M.V.L. performed the experiments and data analysis. The project was supervised by U.L.A. and J.S.N. The manuscript was written by U.L.A., M.V.L. and J.S.N. with feedback from all authors.

Data availability: Experimental data and analysis code is available at figshare Larsen et al. (2019a).