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arXiv:2112.08392 · Physical review. B./Physical review. B

Dynamical approach to improving Majorana qubits and distinguishing them from trivial bound states

Brett Min, Bastien Fajardo, T. Pereg-Barnea, Kartiek Agarwal

Published Apr 15, 2022• 7 citations
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Abstract

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 $1/f$ 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.

DOI
10.1103/physrevb.105.155412

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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.
We propose and analyze a family of periodic braiding protocols in systems with multiple localized Majorana modes (majoranas) for the purposes of Hamiltonian engineering.The protocols rely on double braids-draids-which flip the signs of both majoranas, as one is taken all the way around the other.Rapid draiding can be used to dynamically suppress some or all intermajorana couplings.Suppressing all couplings can drastically reduce residual majorana dynamics, producing a more robust computational subspace.Nontrivial topological models can be obtained by selectively applying draids to some of the overlapping (imperfect) majoranas.Remarkably, draids can be implemented without having to physically braid majoranas or performing projective measurements.For instance, we show that draids can be performed by periodically modulating the coupling between a quantum dot and a topological superconducting wire to dynamically suppress the hybridization of majoranas in the quantum wire.In current experimental setups, this could lead to suppression of this coupling by a few orders of magnitude.The robustness of this protocol can be shown to parallel the topological robustness of physically braided majoranas.We propose an architecture that implements draids between distant majorana modes within a quantum register using a setup with multiple quantum dots and also discuss measurement-based ways of implementing the same.
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.
Braiding Majorana zero modes (MZMs) is the key procedure toward topological quantum computation. We show such braiding can be well performed in a parallel semiconductor-superconductor nanowire structure. Considering the fact that the low-energy Andreev bound states (ABSs) usually mix with the MZMs in the present setup, we further investigate the braiding properties of MZMs when an ABS is presented. Our numerical simulation suggests that ABS can be regarded as a pair of weakly coupled MZMs. The dynamical hybridization of MZMs plus the non-Abelian braiding of MZMs would induce an arbitrary rotation on the Bloch sphere of a single qubit. Remarkably, such rotation is manipulable since the rotation parameters could be individually modulated. Thus, the dynamic evolution can be eliminated and the non-Abelian braiding statistics, independent of the braiding time, retrieves.
Majorana zero modes, characterized by non-Abelian exchange statistics, have been widely viewed as prime candidates for implementing topological quantum computing. The associated braiding and fusion operations realize topologically protected quantum gates and qubit measurements, thus providing a feasible pathway toward intrinsically fault-tolerant quantum computation. However, the experimental realization of controllable braiding and fusion of Majorana zero modes remains one of the most formidable challenges in the field.In this paper, building upon recent theoretical advances in quantum-dot-assisted schemes for realizing the braiding and fusion of Majorana zero modes, we provide a systematic introduction to the underlying theoretical foundations and detailed operational procedures, and further elucidate the experimental feasibility of the approach.Furthermore, this paper presents two lines of investigation developed within the proposed framework. Firstly, based on the quantum-dot-assisted braiding scheme, the dynamical evolution of Andreev bound states in Majorana nanowires is revealed. Secondly, employing the quantum-dot-assisted fusion scheme, numerical simulations of charge transfer along different fusion paths clarify the nontrivial fusion rules of Majorana zero modes and the corresponding operational procedures.
Majorana zero modes are emergent zero-energy quasiparticle excitations in certain superconducting systems that can be viewed as fractionalized or “half” electrons. These quasiparticles obey non-Abelian braiding statistics which is one manifestation of such half-electron character. Due to such non-Abelian braiding property, Majorana zero mode pairs hold promise as potential qubits for topological quantum computation. It is somewhat surprising that, at least theoretically, ordinary one-dimensional semiconductor systems can be induced to host such esoteric Majorana modes as edge states if some precise experimental conditions are satisfied. Because of the relative simplicity of material and experimental requirements to host Majorana modes, there has been a flurry of experimental effort to realize them in semiconductor nanowire systems. While experimental efforts have produced preliminary evidence for the presence of Majorana zero modes in these systems, a thorough confirmation is lacking. The experimental signature in question is the presence of a zero-bias conductance-peak that, while necessary, is not a sufficient criterion to establish the presence of underlying Majorana modes. Given the importance of Majorana braiding for topological quantum computation and skepticism over the presence of Majorana modes in these experimental systems, it would seem natural to attempt braiding these putative Majorana modes in the near future. In that case, an observation of non-Abelian statistics would provide the necessary and sufficient condition in favor of Majorana presence in the studied experimental systems. This thesis has three distinct parts. First, we assume perfect Majorana modes as given that can be successfully braided. In this case, we calculate the diabatic error due to the finite speed of braiding when the system is coupled to a Bosonic bath. Next, we grant that the mechanism for zero-bias conductance-peak is indeed topological, albeit the underlying Majorana modes may be imperfect (the modes are not precisely at zero energy). We study the interplay of dissipation and finite energy splitting of the Majorana modes and study its consequence regarding the probability of successful braiding. Lastly, we propose studying the correlation between independent left and right conductance measurements as a means to distinguish between a topological versus a non-topological the mechanism underlying the observed zero-bias conductance-peak.
Quasiparticle poisoning has remained one of the main challenges in the implementation of Majorana-based quantum computing. It inevitably occurs when the system hosting Majorana qubits is not completely isolated from its surrounding, thus considerably limiting its computational time. We propose the use of periodic driving to generate multiple Majorana fermions at each end of a single quantum wire, which naturally provides the necessary resources to implement active quantum error corrections with minimal space overhead. In particular, we present a stabilizer code protocol that can specifically detect and correct any single quasiparticle poisoning event. Such a protocol is implementable via existing proximitized semiconducting nanowire proposals, where all of its stabilizer operators can be measured from an appropriate Majorana parity dependent four-terminal conductance.