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arXiv:1303.4379 · Physical Review B

Flux-controlled quantum computation with Majorana fermions

Timo Hyart, Bernard van Heck, Ion Cosma Fulga, Michele Burrello, Anton Akhmerov, C. W. J. Beenakker

Published Jul 17, 2013• 344 citations
arXivHTMLPDF

Abstract

Majorana fermions hold promise for quantum computation, because their non-Abelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of well-separated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully flux controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initialization-braiding-measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory (RAMM), a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multiqubit measurements allow for the efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.

DOI
10.1103/physrevb.88.035121

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•Jan 1, 2016•0 citations
Majorana fermions hold promise for quantum computation, because their non-Abelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of well-separated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully flux controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initialization-braiding-measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory (RAMM), a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multiqubit measurements allow for the efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.
Motivated by recent experimental progress towards the measurement and manipulation of Majorana fermions with superconducting circuits, we propose a device interfacing Majorana fermions with circuit quantum electrodynamics. The proposed circuit acts as a charge parity detector changing the resonance frequency of a superconducting $\ensuremath{\lambda}/4$ resonator conditioned on the parity of charges on nearby gates. Operating at both charge and flux sweet spots, this device is highly insensitive to environmental noise. It enables high-fidelity single-shot quantum nondemolition readout of the state of a pair of Majorana fermions encoding a topologically protected qubit. Additionally, the interaction permits the realization of an arbitrary phase gate on the topological qubit, closing the loop for computational completeness. Away from the charge sweet spot, this device can be used as a highly sensitive charge detector with a sensitivity better than ${10}^{\ensuremath{-}4}\text{e}/\sqrt{\text{Hz}}$ and bandwidth larger than 1 MHz.
We propose a topological qubit in which braiding and readout are mediated by the $4\ensuremath{\pi}$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a trijunction. In order to control the superconducting phase differences at the trijunction, the nanowires are enclosed in a ring made of a conventional superconductor. In order to perform initialization and readout, one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.
In this paper, we present a hybrid scheme for topological quantum computation in a system of cold atoms trapped in an atomic lattice. A topological qubit subspace is defined using Majorana fermions which emerge in a network of atomic Kitaev one-dimensional wires. We show how braiding can be efficiently implemented in this setup and propose a direct way to demonstrate the non-Abelian nature of Majorana fermions via a single-parity measurement. We then introduce a proposal for the efficient, robust, and reversible mapping of the topological qubits to a conventional qubit stored in a single atom. There, well-controlled standard techniques can be used to implement the missing gates required for universal computation. Our setup is complemented with an efficient nondestructive protocol to check for errors in the mapping.
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.
Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parity-protected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system --- irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parity-protected rotations and read-out of a topological (top) qubit. The advantage over an earlier proposal using a flux qubit is that the coupling can be switched on and off with exponential accuracy, promising a reduced sensitivity to charge noise.
Quantum computation is a multifaceted field of research aiming for the physical realization of quantum systems and their manipulation. This thesis discusses the combination of two notable approaches in the pursuit of a fully operational quantum computer---circuit quantum electrodynamics and topological quantum computation based on Majorana quasiparticles. In circuit quantum electrodynamics quantum information is stored into small superconducting circuit elements whose interaction with electromagnetic radiation in the range of microwaves allows to process quantum information very efficiently. This approach has proven extremely useful for control and readout of superconducting qubits, i.e., small circuit elements that carry quantum information. Because of remarkably strong light-matter couplings that can be achieved for superconducting qubits in microwave resonators, the circuit quantum electrodynamics architecture is particularly useful to perform highly sensitive quantum measurements.Superconductivity by itself is an intriguing state of matter that shows a great variety of different phenomena. In particular, the discovery of topological phases in superconductors opened new horizons for quantum computation. One notable system that admits topological superconductivity is a semiconductor-superconductor nanowire with special zero modes occurring at its ends. These so-called Majorana zero modes are remarkably robust against decoherence and therefore well-suited for fault-tolerant quantum computation.The first part of this thesis examines the coupling of Majorana zero modes to electromagnetic radiation with microwave frequencies. The light-matter coupling mechanism that is considered here arises for Majorana zero modes located at a voltage-biased superconducting tunneling junction. The emission of microwave radiation in presence of Majorana zero modes gives rise to coherent radiation which is emitted at half of the usual Josephson frequency. On the basis of this fractional Josephson radiation, we propose a microwave readout scheme for Majorana qubits. As usual for typical measurements in circuit quantum electrodynamics, the proposed readout implements a quantum non-demolition measurement for Majorana qubit.In the last part of the thesis we propose a novel scheme for the implementation of measurement-induced entanglement between remote superconducting qubits as required for quantum communication. By detecting a single photon, that passes a Mach-Zehnder interferometric setup, deterministic entanglement with single-shot efficiency is achieved. This scheme essentially relies on the strong coupling between the qubits and the photon.