Motivated by the need of quantum measurement of Majorana qubits and surface-code stabilizers, we analyze the performance of a double-dot interferometer under the influence of environment noise. The double-dot setup design allows accounting for the full multiple tunneling process between the dots through the Majorana island, within a master equation approach. In the co-tunneling regime, which results in a Majorana-mediated effective coupling between the dots, the master equation approach allows us to obtain analytic solutions for the measurement currents. The measurement quality, characterized by figures of merit such as the visibility of measurement signals, is carried out in regard to the unusual decoherence effect rather than `which-path' dephasing. The results obtained in this work are expected to be useful for future experiments of Majorana qubit and stabilizer measurements.
We propose an Aharonov-Bohm interferometer consisted of two topological superconducting chains (TSCs) to generate coherence of Majorana qubits, each qubit is made of two Majorana zero modes (MZMs) with the definite fermion parity. We obtain the generalized exact master equation as well as its solution and study the real-time dynamics of the MZM qubit states under various operations. We demonstrate that by tuning the magnetic flux, the decoherence rates can be modified significantly, and dissipationless MZMs can be generated. By applying the bias voltage to the leads, one can manipulate MZM qub…
We study and generalize the class of qubit topological stabilizer codes that arise in the Abelian phase of the honeycomb lattice model. The resulting family of codes, which we call `matching codes' realize the same anyon model as the surface codes, and so may be similarly used in proposals for quantum computation. We show that these codes are particularly well suited to engineering twist defects that behave as Majorana modes. A proof of principle system that demonstrates the braiding properties of the Majoranas is discussed that requires only three qubits.
We propose a physical realization of a commuting Hamiltonian of interacting Majorana fermions realizing $Z_{2}$ topological order, using an array of Josephson-coupled topological superconductor islands. The required multi-body interaction Hamiltonian is naturally generated by a combination of charging energy induced quantum phase-slips on the superconducting islands and electron tunneling. Our setup improves on a recent proposal for implementing a Majorana fermion surface code [1], a 'hybrid' approach to fault-tolerant quantum computation that combines (1) the engineering of a stabilizer Hamil…
DC Conductance of X-shaped Majorana Interferometer reveals Non-Abelian Anyon Statistics
Heinrich-Heine-Universität（海因里希·海涅大学）；I.N.F.N., Gruppo collegato di Cosenza（意大利国家核物理研究所，科森扎协作组）；Università della Calabria（卡塔尼亚大学）；RWTH Aachen University（亚琛工业大学）
该研究提出四端 X 形手性马约拉纳干涉仪，利用线性响应直流电导张量检测伊辛任意子非阿贝尔统计。通过手征玻色化理论发现其电导张量各向同性，能产生非零电导，可直接证明非阿贝尔统计，补充了双臂干涉仪研究。
Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric non-local parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common +1 eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several know…
We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance $4$ codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance $4$ and $6$ codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.
Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学陈省身理论物理研究所）
We introduce an exactly solvable model of interacting Majorana fermions realizing $Z_{2}$ topological order with a $Z_{2}$ fermion parity grading and lattice symmetries permuting the three fundamental anyon types. We propose a concrete physical realization by utilizing quantum phase slips in an array of Josephson-coupled mesoscopic topological superconductors, which can be implemented in a wide range of solid state systems, including topological insulators, nanowires or two-dimensional electron gases, proximitized by $s$-wave superconductors. Our model finds a natural application as a Majorana…
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 MFs 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 corr…
We refine recent local unitary entanglement classification for symmetric pure states of $n$ qubits (that is, states invariant under permutations of qubits) using local unitary stabilizer subgroups and Majorana configurations. Stabilizer subgroups carry more entanglement distinguishing power than do the stabilizer subalgebras used in our previous work. We extend to mixed states recent results about local operations on pure symmetric states by showing that if two symmetric density operators are equivalent by a local unitary operation, then they are equivalent via a local unitary operation that i…
We initiate the study of Majorana fermion codes. These codes can be viewed as extensions of Kitaev's 1D model of unpaired Majorana fermions in quantum wires to higher spatial dimensions and interacting fermions. The purpose of Majorana fermion codes (MFCs) is to protect quantum information against low-weight fermionic errors, that is, operators acting on sufficiently small subsets of fermionic modes. We examine to what extent MFCs can surpass qubit stabilizer codes in terms of their stability properties. A general construction of 2D MFCs is proposed which combines topological protection based …
We establish a unified framework for Majorana-based fault-tolerant quantum computation with Majorana surface codes and Majorana color codes. All logical Clifford gates are implemented with zero time overhead. This is done by introducing a protocol for Pauli product measurements with tetrons and hexons which only requires local 4-Majorana parity measurements. An analogous protocol is used in the fault-tolerant setting, where tetrons and hexons are replaced by Majorana surface code patches, and parity measurements are replaced by lattice surgery, still only requiring local few-Majorana parity me…
We present a new measurement-based scheme for performing braiding operations on Majorana zero modes and for detecting their non-Abelian statistics without moving or hybridizing them. In our scheme, the topological qubit encoded in any pair of well-separated Majorana zero modes in a mesoscopic superconductor island is read out from the transmission phase shift in electron teleportation through the island in the Coulomb blockade regime. We propose experimental setups to measure the teleportation phase shift via conductance in an electron interferometer or persistent current in a closed loop.
Surface codes have emerged as promising candidates for quantum information processing. Building on the previous idea to realize the physical qubits of such systems in terms of Majorana bound states supported by topological semiconductor nanowires, we show that the basic code operations, namely projective stabilizer measurements and qubit manipulations, can be implemented by conventional tunnel conductance probes and charge pumping via single-electron transistors, respectively. The simplicity of the access scheme suggests that a functional code might be in close experimental reach.
Color-code quantum computation seamlessly combines Majorana-based hardware with topological error correction. Specifically, as Clifford gates are transversal in two-dimensional color codes, they enable the use of the Majoranas' nonabelian statistics for gate operations at the code level. Here, we discuss the implementation of color codes in arrays of Majorana nanowires that avoid branched networks such as T-junctions, thereby simplifying their realization. We show that, in such implementations, nonabelian statistics can be exploited without ever performing physical braiding operations. Physica…
Pauli spectrum and nonstabilizerness of random fermionic Gaussian states
Universität zu Köln（科隆大学）；Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
Fracton topological phases host fractionalized topological quasiparticles with restricted mobility, with promising applications to fault-tolerant quantum computation. While a variety of exactly solvable fracton models have been proposed, there is a need for platforms to realize them experimentally. We show that a rich set of fracton phases emerges in interacting Majorana band models whose building blocks are within experimental reach. Specifically, our Majorana constructions overcome a principal obstacle, namely the implementation of the complicated spin cluster interactions underlying fracton…
The Majorana code is an example of a stabilizer code where the quantum information is stored in a system supporting well-separated Majorana Bound States (MBSs). We focus on one-dimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a parity-preserving thermal environment. We apply the Davies prescription, a standard method that describes the basic aspects of a thermal environment, and derive a master equation in the Born-Markov limit. We first focus on a single wire with immobile MBSs and perform error correction to…
Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models…
Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the AdS/CFT correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC). Relating its logical code basis to Majorana dimers, we ef…
Quantum error correction is crucial for any quantum computing platform to achieve truly scalable quantum computation. The surface code and its variants have been considered the most promising quantum error correction scheme due to their high threshold, low overhead, and relatively simple structure that can naturally be implemented in many existing qubit architectures, such as superconducting qubits. The recent development of Floquet codes offers another promising approach. By going beyond the usual paradigm of stabilizer codes, Floquet codes achieve similar performance while being constructed …
One of the main challenges for quantum computation is that while the number of gates required to perform a non-trivial quantum computation may be very large, decoherence and errors in realistic quantum architectures limit the number of physical gate operations that can be performed coherently. Therefore, an optimal mapping of the quantum algorithm into the physically available set of operations is of crucial importance. We examine this problem for a measurement-only topological quantum computer based on Majorana zero modes, where gates are performed through sequences of measurements. Such a sc…
In this paper we study the Hamming-like fermionic code encoding three-qubits into sixteen Majorana modes recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure however, it can be obtained from two, qubit stabilizer ones via a glueing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group $E_8$. In arriving at these results we develop a general formalism f…
Stabilizer-Rényi Microscopy of Critical Correlations in Interacting Fermions
School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Hefei National Laboratory（合肥国家实验室）
A remarkable property of quantum mechanics in two-dimensional (2D) space is its ability to support "anyons," particles that are neither fermions nor bosons. Theory predicts that these exotic excitations can be realized as bound states confined near topological defects, like Majorana zero modes trapped in vortices in topological superconductors. Intriguingly, in the simplest cases the nontrivial phase that arises when such defects are "braided" around one another is not intrinsically quantum mechanical; rather, it can be viewed as a manifestation of the geometric (Pancharatnam-Berry) phase in w…
Freie Universität Berlin（柏林自由大学）；Technology Innovation Institute (TII)（技术创新研究所）；Tsinghua University（清华大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料能源中心）
