Downloading many-qubit entanglement from continuous-variable cluster states

TL;DR

This work presents a top-down protocol to harvest many-qubit entanglement from efficiently generated continuous-variable cluster states by teleporting the CV entanglement into auxiliary qubits via conditional displacement and one-bit teleportation in a displaced-GKP basis. An equivalent circuit maps dominant CV imperfections into single-qubit preparation errors, enabling standard qubit quantum error correction to mitigate the downloaded entanglement. The analysis yields practical squeezing thresholds, notably 11.9 dB for fault-tolerant quantum computation and 5.4 dB for robust quantum memory or non-FTQC, and discusses hardware platforms where the protocol can be realized. The framework integrates CV and qubit technologies, offering a scalable route to versatile qubit cluster states with potential impact on quantum computation, sensing, and communication in hybrid systems.

Abstract

Many-body entanglement is an essential resource for many quantum technologies, but its scalable generation has been challenging on qubit platforms. However, the generation of continuous-variable (CV) entanglement can be extremely efficient, but its utility is rather limited. In this work, we propose a scheme to combine the best of both qubit and CV approaches: a systematic method to download useful many-qubit entanglement from the efficiently generated CV cluster states. Our protocol is based on one-bit teleportation of the qubit correlation encoded in the displaced Gottesman-Kitaev-Preskill basis. To characterize the practical performance of our scheme, we develop an equivalent circuit to map dominant CV errors to single-qubit preparation errors. Particularly, we relate finite squeezing error to qubit erasure, and show that only 5.4 dB squeezing is sufficient to implement robust qubit memory or quantum computation (QC), and 11.9 dB for fault-tolerant QC. Our protocol can be implemented with the operations that are common in many bosonic platforms.

Figures (8)

• Figure 1: a) Three-step protocol to download a qubit cluster state from a CV cluster state. b) Flat wavefunction of $\ket{0}_p$ can be viewed as an equal superposition of displaced GKP basis states. c) Circuit corresponds to Steps 2 and 3 of the protocol. The initial qumode state is assumed a general displaced GKP qubit $\ket{\psi^\textrm{gkp}_{\mu,\nu}}$. d) Standard one-bit teleportation circuit.
• Figure 2: a) Circuit of our protocol when qumodes are prepared as thermal CVCS. b) Equivalent circuit, which each qubit is introduced errors through interacting with squeezed thermal states before being entangled by ideal CZ gates. c) Illustration of qubit amplitude imbalance after interacting with finitely squeezed vacuum. d) Cluster states after weak measurement correction. Qubits are probabilistically restored to balanced amplitudes or deleted from the cluster.
• Figure 3: a) Detector inefficiency introduces both $q$ and $p$ quadrature noises. $p$-noise is irrelevant after $q$-measurement, while $q$-noise can be lumped into CVCS imperfection. b) Correlation induced by apparatus artifacts can be canceled by introducing mode-dependent thermalization and squeezing, additional beam-splitter array, and modifying strength of CPHASE, i.e. $\hat{\bm{C}}^{\mathrm{CV}}_Z(g') \equiv \prod_{(i,j) \in E} e^{ig'\hat{q}_i \hat{q}_j}$. c) Weak CD strength can be compensated by interacting the qubits with multiple copies of CVCS.
• Figure S1: Left: One-bit teleportation circuit. Right: One-bit teleportation circuit with extra $\hat{R}_Z$ gates added. $l \in \{0, 1\}$ is the logical value measured from the qubit. We have defined $\ket*{\psi} \equiv \alpha \ket*{0} + \beta \ket*{1}$ and added the identity gate as a combination of $\hat{R}_Z$ gates for an easier comparison with the one-bit teleportation in displaced GKP states.
• Figure S2: Left: Conditional displacement is a CNOT gate in the displaced GKP basis plus a phase shift on the physical qubit. Since $q = (2m+l)\sqrt{\pi} + \mu$, we can express the qumode measurement outcome and post-processing in terms of $l, \nu$. Right: One-bit teleportation without post-processing, so the teleported state will contain the teleportation by-product.