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arXiv:1911.02582 · Physical review. B./Physical review. B

Dephasing and leakage dynamics of noisy Majorana-based qubits: Topological versus Andreev

Ryan V. Mishmash, Bela Bauer, Felix von Oppen, Jason Alicea

Published Feb 3, 2020• 41 citations
arXivHTMLPDF

Abstract

Topological quantum computation encodes quantum information nonlocally by nucleating non-Abelian anyons separated by distances $L$, 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 $L$. 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.

DOI
10.1103/physrevb.101.075404

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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.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f1/f1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2C=2C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Robustness of edge states and non-Abelian excitations of topological states of matter promises quantum memory and quantum processing, which are naturally immune to microscopic imperfections such as static disorder. However, topological properties will not in general protect quantum systems from time-dependent disorder or noise. Here we take the example of a network of Kitaev wires with Majorana edge modes storing qubits to investigate the effects of classical noise in the crossover from the quasistatic to the fast fluctuation regime. We present detailed results for the Majorana edge correlations, and fidelity of braiding operations for both global and local noise sources preserving parity symmetry, such as random chemical potentials and phase fluctuations. While in general noise will induce heating and dephasing, we identify examples of long-lived quantum correlations in the presence of fast noise due to motional narrowing, where external noise drives the system rapidly between the topological and nontopological phases.
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 $L\ensuremath{\sim}5\ensuremath{\xi}$ (with $\ensuremath{\xi}$ being the coherence length). The coherence time grows exponentially with MZM separation and eventually becomes temperature limited for $L/\ensuremath{\xi}\ensuremath{\sim}30$.
We study a series of dynamical protocols which involve periodically driving a quantum dot coupled to a putative nanowire hosting Majorana zero modes (MZMs) to (i) reduce the hybridization between MZMs, (ii) improve the coherence of the Majorana qubit with respect to 1/f1/f1/f dephasing noise and quasiparticle poisoning, and (iii) provide a definitive test to differentiate Andreev bound states (ABSs) from MZMs. The protocols are based on the notion of draiding, exchanging a pair of Majoranas twice, repeatedly, at high frequency [I. Martin and K. Agarwal, PRX Quantum 1, 020324 (2020)]. In this process, the exchanged Majorana operators acquire a robust minus sign such that terms in the Hamiltonian, linear in either operator, vanish on average. The four protocols proposed implement draiding by coupling quantum dot(s) to the end(s) of the nanowire. They are treated using Floquet theory and numerical simulations. The hybridization energy and decoherence rate are shown to be reduced by several orders of magnitude, in accordance with theoretical expectations, when the protocols are implemented on nanowires described by experimentally relevant parameters. The tunneling conductance computed in this Floquet setting reveals zero-bias peaks (ZBPs) that become more centered at zero voltage bias. When these protocols are implemented on nanowires supporting ABSs that mimic MZMs in ZBP measurements, the qubit coherence deteriorates, in stark contrast to the case where the nanowire supports MZMs and the coherence drastically improves, thus serving as a dynamical test to distinguish MZMs from trivial bound states.
Topology-related ideas might lead to noise-resilient quantum computing. For example, it is expected that the slow spatial exchange (``braiding'') of Majorana zero modes in superconductors yields quantum gates that are robust against disorder. Here, we report our numerical experiments, which describe the dynamics of a Majorana qubit built from quantum dots controlled by time-dependent gate voltages. Our protocol incorporates nonprotected control, braiding-based protected control, and readout of the Majorana qubit. We use the Kitaev chain model for the simulations, and we focus on the case when the main source of errors is quasistatic charge noise affecting the hybridization energy splitting of the Majorana modes. We provide quantitative guidelines to suppress both diabatic errors and disorder-induced qubit dephasing, such that a fidelity plateau is observed as the hallmark of the topological quantum gate. Our simulations predict realistic features that are expected to be seen in future braiding experiments with Majorana zero modes and other topological qubit architectures.
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.