[Paper Review] Hardware efficient autonomous error correction with linear couplers in superconducting circuits

Ziqian Li, Tanay Roy+3 more|arXiv (Cornell University)|Mar 2, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes the Star code, a hardware-efficient autonomous quantum error correction scheme for superconducting circuits that uses linear couplers to enable two-photon transitions instead of challenging four-photon processes. It achieves quadratic improvement in logical qubit lifetime by engineering dissipative pathways through tunable couplers and lossy resonators, enabling robust error suppression with minimal experimental overhead.

ABSTRACT

Large-scale quantum computers will inevitably need quantum error correction (QEC) to protect information against decoherence. Given that the overhead of such error correction is often formidable, autonomous quantum error correction (AQEC) proposals offer a promising near-term alternative. AQEC schemes work by transforming error states into excitations that can be efficiently removed through engineered dissipation. The recently proposed AQEC scheme by Li et al., called the Star code, can autonomously correct or suppress all single qubit error channels using two transmons as encoders with a tunable coupler and two lossy resonators as a cooling source. The Star code requires only two-photon interactions and can be realized with linear coupling elements, avoiding experimentally challenging higher-order terms needed in many other AQEC proposals, but needs carefully selected parameters to achieve quadratic improvements in logical states' lifetimes. Here, we theoretically and numerically demonstrate the optimal parameter choices in the Star Code. We further discuss adapting the Star code to other planar superconducting circuits, which offers a scalable alternative to single qubits for incorporation in larger quantum computers or error correction codes.

Motivation & Objective

• To develop a scalable, hardware-efficient autonomous quantum error correction (AQEC) scheme for superconducting qubits that avoids high-order nonlinear interactions.
• To enable robust suppression of single-photon loss and dephasing errors using only two-photon processes and linear coupling elements.
• To identify optimal parameter regimes that maximize logical qubit lifetime improvement through analytical and numerical analysis.
• To demonstrate adaptability of the Star code to alternative qubit types such as fluxons, enabling broader integration into planar superconducting architectures.

Proposed method

• The Star code encodes a logical qubit in the degenerate dark states of a two-transmon system coupled to two lossy resonators via tunable linear couplers.
• It leverages engineered dissipation through XX-type couplings between transmons and resonators, enabling autonomous correction of photon loss errors via two-photon transitions.
• The Hamiltonian is transformed into a rotating frame where the logical states emerge as dark states, minimizing decay to the environment.
• Analytical derivation of logical lifetime scaling is performed under the rotating wave approximation, with numerical simulations validating the results under non-ideal conditions.
• Parameter optimization is conducted by analyzing the impact of detunings, Rabi drives, and qubit-resonator coupling strengths on error suppression and lifetime enhancement.
• The scheme is extended to other qubit platforms, such as fluxons, by preserving the essential symmetry and coupling structure of the original design.

Figure 1: Star code protocol. (a) An example of hardware layout. Two transmons are individually coupled to two resonators dispersively. The dashed box between the two transmons represents any tunable coupling element that can provide sufficient QQ red and blue sideband interactions. (b) Four QQ side

Experimental results

Research questions

• RQ1 Can a two-photon-based AQEC scheme achieve quadratic improvement in logical qubit lifetime without requiring four-photon drives or complex nonlinear elements?
• RQ2 How do variations in drive Rabi frequency, detuning, and coupling strength affect the performance and robustness of the Star code?
• RQ3 To what extent is the Star code resilient to non-ideal parameters such as residual ZZ interactions and qubit-resonator coupling?
• RQ4 Can the Star code be generalized to other superconducting qubit architectures, such as fluxons, while maintaining its error correction efficiency?

Key findings

• The Star code achieves a quadratic improvement in logical qubit lifetime by leveraging two-photon transitions and engineered dissipation, outperforming conventional schemes with similar resource overhead.
• Optimal parameter regimes exist where the logical state lifetime scales quadratically with the error correction rate, confirmed by both analytical modeling and numerical simulations.
• Residual qubit-resonator ZZ coupling has a weak effect on logical lifetime when the sideband rate exceeds the coupling strength, indicating robustness to moderate experimental imperfections.
• Photon excitation in readout resonators poses a risk of logical error but can be mitigated by fast error correction, as long as the correction rate is sufficiently high.
• The scheme is insensitive to asymmetries in drive Rabi frequencies and maintains performance as long as the energy gap between logical states remains large relative to parameter fluctuations.
• The Star code can be adapted to other qubit platforms, such as fluxons, by preserving the essential coupling topology and dark state structure, enabling scalable integration in planar circuits.

Figure 2: Error correction cycle for $\left|L_{0}\right\rangle$ . The effective $\left|L_{0}\right\rangle$ refilling rate $\Gamma_{R}$ is shown in the purple arrow. A second photon loss can happen at rate $\gamma$ before the completion of the refilling cycle. Population transfer to the grey dash box