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Paper 1

Q-Pandora Unboxed: Characterizing Noise Resilience of Quantum Error Correction Codes

Avimita Chatterjee, Subrata Das, Swaroop Ghosh

Year

2023

Journal

arXiv preprint

DOI

arXiv:2308.02769

arXiv

2308.02769

Quantum error correction codes (QECCs) are critical for realizing reliable quantum computing by protecting fragile quantum states against noise and errors. However, limited research has analyzed the noise resilience of QECCs to help select optimal codes. This paper conducts a comprehensive study analyzing two QECCs - rotated and unrotated surface codes - under different error types and noise models using simulations. Among them, rotated surface codes perform best with higher thresholds attributed to simplicity and lower qubit overhead. The noise threshold, or the point at which QECCs become ineffective, surpasses the error rate found in contemporary quantum processors. When confronting quantum hardware where a specific error or noise model is dominant, a discernible hierarchy emerges for surface code implementation in terms of resource demand. This ordering is consistently observed across unrotated, and rotated surface codes. Our noise model analysis ranks the code-capacity model as the most pessimistic and circuit-level model as the most realistic. The study maps error thresholds, revealing surface code's advantage over modern quantum processors. It also shows higher code distances and rounds consistently improve performance. However, excessive distances needlessly increase qubit overhead. By matching target logical error rates and feasible number of qubits to optimal surface code parameters, our study demonstrates the necessity of tailoring these codes to balance reliability and qubit resources. Conclusively, we underscore the significance of addressing the notable challenges associated with surface code overheads and qubit improvements.

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Paper 2

Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization

Ignacio García-Mata, John Martin, Olivier Giraud, Bertrand Georgeot, Rémy Dubertrand, Gabriel Lemarié

Year

2022

Journal

arXiv preprint

DOI

arXiv:2209.04337

arXiv

2209.04337

The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $ξ_\parallel$ than the one in the transverse direction, $ξ_\perp$. Importantly, these two lengths have different critical behaviors: $ξ_\parallel$ diverges with a critical exponent $ν_\parallel=1$, while $ξ_\perp$ reaches a finite universal value ${ξ_\perp^c}$ at the transition point $W_c$. Indeed, $ξ_\perp^{-1} \approx {ξ_\perp^c}^{-1} + ξ^{-1}$, with $ξ\sim (W-W_c)^{-ν_\perp}$ associated with a new critical exponent $ν_\perp = 1/2$, where $\exp( ξ)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $ν=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $ξ_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

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