Hardware-efficient autonomous error correction with linear couplers in superconducting circuits

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Hardware-efficient autonomous error correction with linear couplers in superconducting circuits

Ziqian Li, Tanay Roy, David Rodríguez Pérez, David I. Schuster, and Eliot Kapit

Phys. Rev. Research 6, 013171 – Published 15 February 2024

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.

Received 31 March 2023
Revised 14 November 2023
Accepted 23 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013171

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Open quantum systems & decoherence Quantum coherence & coherence measures Quantum error correction

Quantum Information, Science & Technology

Authors & Affiliations

Ziqian Li ^1,2,3,*, Tanay Roy ^1,2,*,†, David Rodríguez Pérez ⁴, David I. Schuster^1,2,5,3, and Eliot Kapit⁴

¹James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
²Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
³Department of Applied Physics, Stanford University, Stanford, California 94305, USA
⁴Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA
⁵Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA

^*These authors contributed equally to this work.
^†Present address: Superconducting Quantum Materials and Systems Center, Fermi National Accelerator Laboratory (FNAL), Batavia, Illinois 60510, USA.

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Vol. 6, Iss. 1 — February - April 2024

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Images

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 sufficiently strong QQ red and blue sideband interactions. (b) Four QQ sideband mixing configurations in the logical static frame. All sidebands are applied with an equal rate $W$ and specific detuning choices ( $\pm ν_{0 / 1}$ ) to construct the logical manifold. (c) Energy diagram in the rotating frame. The green dashed box covers the logical states and error states involved in the AQEC protocol, and the grey dashed box includes the other stray eigenstates that have suppressed population transfer by the energy gap. Error states from single-photon loss are restored to the parent logical states through individual correction paths. The QR sidebands (rate $Ω / 2$ ), the resonators' photon decay (rate $κ$ ), and the transmon $T_{1}$ decay (rate $γ$ ) are shown in the blue, brown, and black arrows, respectively.
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Figure 2
Error correction cycle for $| L_{0} 〉$ . The effective $| L_{0} 〉$ refilling rate $Γ_{R}$ is shown in the purple arrow. A second photon loss can happen at rate $γ$ before the completion of the refilling cycle. Population transfer to the grey dashed box is marked with blue dashed arrows, and the population transfer is suppressed by the energy difference $O (ν)$ .
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Figure 3
Logical lifetime ( $T_{L}$ ) as a function of detunings and sideband rates. Simulations are performed up to 200 $μ s$ with $T_{1} = 20 μ s$ for both transmons. The logical $T_{L}$ are extracted by fitting the last 180 $μ s$ to an exponential decay profile. (a) Two-dimensional (2D) scan of QQ sideband detunings $ν_{0}$ and $ν_{1}$ . Other parameters used in the simulation: ${α_{1}, α_{2}, W, Ω, κ} = {- 160, - 260, 5, 1, 0.5} MHz$ . Optimal performance is obtained around $ν_{0} = - ν_{1} = \pm W / \sqrt{3}$ . (b) 2D scan of QQ and QR sideband rates $W$ and $Ω$ . Parameters are set to be $ν_{0} = - ν_{1} = W / \sqrt{3}$ , and $Ω = κ$ for best AQEC performance. Simulations show significantly improved performance around $Ω = W / 10$ .
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Figure 4
Logical lifetime improvement as a function of transmon $T_{1}$ (considered identical for both transmons). Quadratic lifetime improvement (roughly linear improvement in the lifetime ratio) under AQEC is clearly seen in the plot. Logical $T_{L}$ are extracted by fitting traces to the exponential decay curve $A exp (- t / T_{L}) + C$ (with $A$ and $C$ being free parameters), and the improvement ratio is $T_{L} / T_{1}$ . Error bars (one standard deviation) for $T_{L}$ are smaller than the marker size. Each simulation is run up to 800 $μ s$ , and the short period is not included in the fitting. Other parameters used in the simulation are ${α_{1}, α_{2}, W, ν_{0}, ν_{1}, Ω, κ} = {- 160, - 260, 10, 5.77, - 5.77, 0.71, 0.5} MHz$ . The analytic expression (solid lines) matches the simulation result. The depolarization lifetime of $| L_{1} 〉$ is almost the same as $| L_{0} 〉$ in simulation. All simulated logical lifetimes here are above the break-even point.
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Figure 5
Simulated logical states' lifetime in the presence of qubit-resonator dispersive coupling $χ$ . Parameter used in the simulation: ${α_{1}, α_{2}, W, ν_{0}, ν_{1}, Ω, κ} = {- 160, - 260, 10, 5.77, - 5.77, 0.71, 0.5} MHz, T_{1} = 60 μ s$ .
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Figure 6
Simulated logical states' lifetime and physical qubit lifetime. Parameter used in the simulation: ${α_{1}, α_{2}, W, ν_{0}, ν_{1}, Ω, κ} = {- 160, - 260, 10, 5.77, - 5.77, 0.71, 0.5} MHz, T_{1} = 60 μ s$ .
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