Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture

Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction. However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction. In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation. To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources. Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation. To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state. We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.

Popular Summary

Quantum computation has great potential to efficiently solve some problems that are considered hard for conventional computers. To get to that point, quantum computers significantly larger than the current state of the art are needed. However, scaling up quantum processors with current technologies while keeping the noise level down is a serious challenge, largely due to the delicate coherent dynamics required to implement quantum gates.

In this work, we propose an optical architecture for computation based on projective measurements, the so-called measurement-based quantum computation, which avoids the need for coherent dynamics and is inherently scalable. The scalability of our architecture is derived from the deterministic nature of linear optics and homodyne detection in combination with the use of a temporal encoding that allows just a few optical devices to be reused over and over. This leads to a surprisingly simple optical setup, which is even directly compatible with an optical network. Furthermore, our architecture is designed to support state-of-the-art error-correction methods to combat the unavoidable noise in quantum computation, and we show by careful simulation that fault-tolerant computation is possible with feasible resource requirements.

The architecture we propose is experimentally simple, scalable, and fault-tolerant, and is therefore promising for near-future implementations of photonic quantum computation.

Figure 1

Conceptual illustration of the computation scheme. $n\times m$ wires of two-mode entangled states are prepared in $n\times m$ resource-preparation gadgets (“Res. prep.”). States with input information for computation, ${|{\psi }_{\mathrm{in}}⟩}_{\mathrm{GKP}}$, are switched into each wire for computation using an optical switch. With a beam splitter (marked with a red arrow) and two detectors for each wire, single-mode gates are implemented in each wire when the resource-preparation gadget is fed with squeezed states, ${|0⟩}_{\mathrm{sq}}$ [9, 30]. Feeding the resource-preparation gadget with special GKP states, namely qunaught states, ${|\varnothing ⟩}_{\mathrm{GKP}}$, allows GKP quadrature correction of encoded GKP qubits [28]. Neighboring wire setups are connected using variable beam splitters (VBSs), allowing tunable coupling for implementing two-mode gates and thereby enabling multimode computation.

Figure 2

(a) Temporally encoded computational setup for our scheme. The scheme consists of three parts: the resource-preparation gadget; the computational level, where the computation takes place; and the temporally delocalized measurement device (TDMD) for gate implementation by projective measurements. This scheme utilizes temporal multiplexing of two spatial modes, $A$ and $B$, marked in the computational level. (b) Wires of two-mode entanglement at the computational level shown in the time domain, here for the simple case of $nm=9$. The bold lines represent two-mode entanglement, while the thin lines indicate the temporal overlap of $A$ and $B$. The wires begin with $nm$ input states in temporal modes 0 to 8, switched in using an optical switch in $A$ in the computational level. The colors of the wires have no physical meaning and are used merely to indicate different wires. (c) Wires rearranged into a 3D time lattice, where the input is encoded onto the $n\times m$ end surface, while gates are implemented by teleportation along the wires in the third dimension. Here, the red arrows represent the first beam splitter of the TDMD, while the dotted blue and green arrows represent the variable beam splitters of the TDMD. The first ten temporal modes in (b) from 0 to 9 are labeled in (c).

Figure 3

Implementation of GKP quadrature correction by qubit teleportation. Qunaught states, ${|\varnothing ⟩}_{\mathrm{GKP}}$, are injected into the resource-preparation gadget, thereby preparing a two-mode GKP-qubit Bell state, shown as two connected rectangles before the $nm$ delay. In the computational level, after the $nm$ delay, one part of the Bell state overlaps in time with the GKP-qubit state to be corrected, ${|\psi ⟩}_{\mathrm{GKP}}$, shown as a circle. When the TDMD is set to perform a Bell measurement (the two VBSs are open and left out in the figure), ${|\psi ⟩}_{\mathrm{GKP}}$ is teleported through the Bell state and projected into a purified GKP qubit state by the Kraus operator in Eq. (6). At the bottom is the corresponding graph as it would appear in the 3D time lattice in Fig. 2, as well as the corresponding circuit diagram.

Figure 4

(a) Illustration of a logical qubit for the $d=5$ rotated surface code. The white and gray circles represent odd and even data qubits, respectively. The green and red circles represent $Z$- and $X$-measure qubits, respectively. The two-mode gate operations are listed in Eq. (9), and the labels 1 to 4 indicate the time steps in which those gates are implemented. (b),(c) Illustration of one round of syndrome measurements, including the initialization of the encoded GKP ancilla qubits, four time steps for the coupling between the data and ancilla qubits by four measurement-induced two-mode gates, and a measurement using the TDMD. For the surface-GKP code, GKP quadrature corrections are performed on data qubits at the beginning of a surface-code syndrome measurement cycle. For the surface-4-GKP code, GKP quadrature corrections are performed on all data and ancilla qubits after each gate. (d) Orientation of the surface code in the 3D time lattice of the computation scheme shown in Fig. 2. Here, the surface code is encoded in two dimensions of the time lattice with vertices corresponding to encoded GKP qubits, while gates are encoded in each step along the third dimension of the time lattice. (e,f) Example of qubit errors induced by GKP quadrature correction in the surface-4-GKP code. In (e), an $\hat{X}$ Pauli error occurs in a measure-$Z$ ancilla after the first two-mode gate of a surface-code syndrome measurement round. The error then propagates to neighboring data and measure-$X$ ancillas via the subsequent two-mode gates used to measure the surface-code stabilizers. In (f), a $\hat{Z}$ Pauli error occurs in a data qubit after the first two-mode gate and propagates to two measure-$X$ ancillas. See Appendix pp3 for an examination of all possible Pauli errors, how they propagate, and the corresponding edges in the matching graphs for decoding.

Figure 5

(a) Simulated logical $\hat{Z}$ and $\hat{X}$ error probabilities for the surface-4-GKP code as a function of the (identical) squeezing of the ${|0⟩}_{\mathrm{sq}}$ states used for gate implementation, the GKP qubits encoding the surface code, and the ${|\varnothing ⟩}_{\mathrm{GKP}}$ states used for quadrature correction. The logical error probability is shown for different code distances $d$, and the fault-tolerant threshold where the logical error rate decreases with increasing code distance is seen to be at 12.7 dB of squeezing. The error bars of standard deviation are estimated by bootstrapping. (b) Squeezing threshold of the surface-4-GKP code as a function of the probability with which a ${|\varnothing ⟩}_{\mathrm{GKP}}$ state is replaced with a ${|0⟩}_{\mathrm{sq}}$ state in the resource preparation. Here, the threshold is estimated as the crossing point of the $d=7$ and $d=9$ logical error rates. For zero replacement probability, the threshold is that in (a).

Figure 6

Simulation results for all four simulated cases. Here, the case of the surface-4-GKP code using $p_{\sigma }(z)$ gives the results shown in the main text in Fig. 5. The error bars of standard deviation are estimated by bootstrapping.

Figure 7

Sections of the $Z$ (a) and $X$ (b) matching graphs, here with labels corresponding to the code distance $d=5$ similarly to Fig. 4. Each two-mode gate is labeled with a number corresponding to the step in the $i\mathrm{th}$ round of syndrome measurements.