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arXiv:1711.03968 · Physical review. B./Physical review. B

Dephasing of Majorana-based qubits

Christina Knapp, Torsten Karzig, Roman M. Lutchyn, Chetan Nayak

Published Mar 7, 2018• 84 citations
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Abstract

We analyze charging-energy-protected Majorana-based qubits, focusing on the residual dephasing that is present when the distance between Majorana zero modes (MZMs) is insufficient for full topological protection. We argue that the leading source of dephasing is $1/f$ charge noise. This noise affects the qubit as a result of the hybridization energy and charge distribution associated with weakly overlapping MZMs, which we calculate using a charge-conserving formalism. We estimate the coherence time to be hundreds of nanoseconds for Majorana-based qubits whose MZM separation is $L\ensuremath{\sim}5\ensuremath{\xi}$ (with $\ensuremath{\xi}$ being the coherence length). The coherence time grows exponentially with MZM separation and eventually becomes temperature limited for $L/\ensuremath{\xi}\ensuremath{\sim}30$.

DOI
10.1103/physrevb.97.125404

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We analyze charging-energy-protected Majorana-based qubits, focusing on the residual dephasing that is present when the distance between Majorana zero modes (MZMs) is insufficient for full topological protection. We argue that the leading source of dephasing is 1/f1/f1/f charge noise. This noise affects the qubit as a result of the hybridization energy and charge distribution associated with weakly overlapping MZMs, which we calculate using a charge-conserving formalism. We estimate the coherence time to be hundreds of nanoseconds for Majorana-based qubits whose MZM separation is L\ensuremath∼5\ensuremathξL\ensuremath{\sim}5\ensuremath{\xi}L\ensuremath∼5\ensuremathξ (with \ensuremathξ\ensuremath{\xi}\ensuremathξ being the coherence length). The coherence time grows exponentially with MZM separation and eventually becomes temperature limited for L/\ensuremathξ\ensuremath∼30L/\ensuremath{\xi}\ensuremath{\sim}30L/\ensuremathξ\ensuremath∼30.
Topological quantum computation encodes quantum information nonlocally by nucleating non-Abelian anyons separated by distances LLL, typically spanning the qubit device size. This nonlocality renders topological qubits exponentially immune to dephasing from all sources of classical noise with operator support local on the scale of LLL. We perform detailed analytical and numerical analyses of a time-domain Ramsey-type protocol for noisy Majorana-based qubits that is designed to validate this coveted topological protection in near-term devices such as the so-called ``tetron'' design. By assessing dependence of dephasing times on tunable parameters, e.g., magnetic field, our proposed protocol can clearly distinguish a bona fide Majorana qubit from one constructed from semilocal Andreev bound states, which can otherwise closely mimic the true topological scenario in local probes. In addition, we analyze leakage of the qubit out of its low-energy manifold due to classical-noise-induced generation of quasiparticle excitations; leakage limits the qubit lifetime when the bulk gap collapses, and hence our protocol further reveals the onset of a topological phase transition. This experiment requires measurement of two nearby Majorana modes for both initialization and readout---achievable, for example, by tunnel coupling to a nearby quantum dot---but no further Majorana manipulations, and thus constitutes an enticing prebraiding experiment. Along the way, we address conceptual subtleties encountered when discussing dephasing and leakage in the context of Majorana qubits.
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A recent experiment reported coherent parity oscillations between minimal Kitaev chains, demonstrating a Majorana parity qubit (arXiv:2607.09511). Operations on this qubit are controlled by coupling two Kitaev chains and are intrinsically unprotected against noise in the coupling strength. Here we show how the exponential protection in the underlying chains can nevertheless be revealed through unprotected qubit operations. To this end, we develop an analytical theory of quasistatic dephasing for two coupled NNN-site Kitaev chains at the sweet spot. In Rabi experiments, protection can be resolved by adapting the operating point with chain length: depending on the coupling architecture, it appears as an exponentially increasing optimized number of coherent oscillations or as a longer coherence time. Ramsey experiments show an exponentially increasing coherence time, provided the chains can be sufficiently decoupled. Our analytical theory takes into account the non-Gaussian splitting statistics of the Kitaev chains, allowing us to quantitatively predict the scaling with chain length. Moreover, we determine how residual noise and imperfections limit the range of chain lengths over which this scaling can be observed. Our results show how qubit dynamics can be used to probe Majorana protection in experimentally realistic devices.
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We study the decoherence dynamics of Majorana qubit braiding operations in a T-junction topological superconducting chain (TSC) system, in which the braiding is performed by controlling the electron-chemical potentials of the TSCs and the couplings between them. By solving rigorously the Majorana qubit dynamics, we show how the Majorana qubit coherence is generated through Bogoliubon correlations formed by exchanging Majorana zero modes (MZMs) between two TSCs in braiding operations. We also demonstrate with the exact master equation how MZMs and also the Bogoliubon correlations dissipate due to charge fluctuations of the controlling gates at both the zero and finite temperatures. As a result, Majorana qubit coherence and also the fermion parity conservation cannot be immune from local perturbations during braiding operations.