Robust fault tolerance for continuous-variable cluster states with excess antisqueezing

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Robust fault tolerance for continuous-variable cluster states with excess antisqueezing

Blayney W. Walshe, Lucas J. Mensen, Ben Q. Baragiola, and Nicolas C. Menicucci

Phys. Rev. A 100, 010301(R) – Published 22 July 2019

Abstract

The immense scalability of continuous-variable cluster states motivates their study as a platform for quantum computing, with fault tolerance possible given sufficient squeezing and appropriately encoded qubits [N. C. Menicucci, Phys. Rev. Lett. 112, 120504 (2014)]. Here, we expand the scope of that result by showing that additional antisqueezing has no effect on the fault-tolerance threshold, removing the purity requirement for experimental continuous-variable cluster-state quantum computing. We emphasize that the appropriate experimental target for fault-tolerant applications is to directly measure 15–17 dB of squeezing in the cluster state rather than the more conservative upper bound of 20.5 dB.

Received 15 March 2019

DOI:https://doi.org/10.1103/PhysRevA.100.010301

Physics Subject Headings (PhySH)

Optical quantum information processing Quantum computation Quantum error correction Quantum information processing with continuous variables

Quantum Information, Science & Technology

Authors & Affiliations

Blayney W. Walshe ^*, Lucas J. Mensen, Ben Q. Baragiola, and Nicolas C. Menicucci

Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia

^*blayneyw@gmail.com

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Issue

Vol. 100, Iss. 1 — July 2019

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Images

Figure 1
CV resource state for fault-tolerant quantum computation. In the original proposal, Ref. [19], the base layer consists of nodes that are momentum-squeezed vacuum states, with the lines connecting them representing ${\hat{C}}_{Z} [1] = e^{i \hat{q} \otimes \hat{q}}$ gates. Here, we consider each node to be an impure momentum-squeezed thermal state, which has additional noise in the position quadrature. The cyan nodes are GKP-encoded $| 0_{L} 〉$ states [18], attached via ${\hat{C}}_{Z} [1]$ gates at regular intervals like flowers in a flowerbed. These are used for error correction, while the gray and white nodes implement the computation (see text). The yellow highlighted nodes represent two-mode (left) and one-mode (right) error-corrected gates, described in more detail in Fig. 2.
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Figure 2
The subgraphs sufficient to enact any (a) single-qubit and (b) two-qubit logical-Clifford gate, such as the $C_{Z} [- 1]$ gate, followed by GKP error correction [19]. These have been cut from the flowerbed using $\hat{q}$ measurements on the adjacent nodes. The cyan nodes are GKP $| 0_{L} 〉$ for error correction [18], the blank nodes are squeezed thermal states [Eq. (1)], in is the output node of the previous gate, and the arrows indicate the direction of the measurement sequence.
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Figure 3
Phase-space comparison of the vacuum ( $s = 1$ ) and a 5-dB $p$ -squeezed vacuum state ( $s = 1.78$ ). Additional antisqueezing in the $q$ quadrature models a squeezed thermal state (dashed). Shown are 1- $σ$ error ellipses for the thermal-state Wigner functions, Eq. (1).
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Figure 4
The quantum circuit (read right to left) describing one measurement and outcome-dependent displacement as part of the single-mode gate shown in Fig. 2. $W_{in}$ is the Wigner function for an arbitrary quantum state, and $W_{ɛ, κ}$ is that for the squeezed thermal state [Eq. (1)] at the next node in the cluster. $W_{a}$ is the input state after the shear gate, and $W_{b}$ is the two-mode Wigner function after the ${\hat{C}}_{Z} [1]$ gate. The unnormalized, outcome-dependent output state is represented by ${\tilde{W}}_{out}^{(t)}$ .
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Figure 5
The quantum circuit (read right to left) representing the action of $\hat{p}$ measurements on the two nodes connecting the input states in Fig. 2. The measurement-dependent correction step is included. $W_{in}$ is the Wigner function for an arbitrary two-mode input state, and $W_{ɛ, κ}$ represents the squeezed thermal state [Eq. (1)] at each of the two connecting nodes in the cluster. $W_{a}$ is the total Wigner function after the ${\hat{C}}_{Z} [1]$ gates. The unnormalized, outcome-dependent output state is represented by ${\tilde{W}}_{out}^{(r, t)}$ .
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