Quantum computation with Turaev-Viro codes

For a 3-manifold with triangulated boundary, the Turaev-Viro topological invariant can be interpreted as a quantum error-correcting code. The code has local stabilizers, identified by Levin and Wen, on a qudit lattice. Kitaev’s toric code arises as a special case. The toric code corresponds to an abelian anyon model, and therefore requires out-of-code operations to obtain universal quantum computation. In contrast, for many categories, such as the Fibonacci category, the Turaev-Viro code realizes a non-abelian anyon model. A universal set of fault-tolerant operations can be implemented by deforming the code with local gates, in order to implement anyon braiding. We identify the anyons in the code space, and present schemes for initialization, computation and measurement. This provides a family of constructions for fault-tolerant quantum computation that are closely related to topological quantum computation, but for which the fault tolerance is implemented in software rather than coming from a physical medium.     

From entanglement renormalisation to the disentanglement of quantum double models

We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models in which the Hamiltonian is gradually simplified along a parallel simplification of the connectivity of the lattice. We consider the case of Kitaev’s quantum double models, both Abelian and non-Abelian, and we obtain a rederivation of the known map of the toric code to two Ising chains; we pay particular attention to the non-Abelian models and discuss their space of states on the torus. Ultimately, the construction is universal for such models and its essential feature, the lattice simplification, may point towards a renormalisation of the metric in continuum theories.     

Topological order in an exactly solvable 3D spin model

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest-neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on Ω(R²) qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.     

On the stability of topological phases on a lattice

We study the stability of anyonic models on lattices to perturbations. We establish a cluster expansion for the energy of the perturbed models and use it to study the stability of the models to local perturbations. We show that the spectral gap is stable when the model is defined on a sphere, so that there is no ground state degeneracy. We then consider the toric code Hamiltonian on a torus with a class of abelian perturbations and show that it is stable when the torus directions are taken to infinity simultaneously, and is unstable when the thin torus limit is taken.     

A symmetry principle for topological quantum order

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev’s Toric code model and Wen’s plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.     

The real topological vertex at work

We develop the real vertex formalism for the computation of the topological string partition function with D-branes and O-planes at the fixed point locus of an anti-holomorphic involution acting non-trivially on the toric diagram of any local toric Calabi–Yau manifold. Our results cover in particular the real vertex with non-trivial fixed leg. We give a careful derivation of the relevant ingredients using duality with Chern–Simons theory on orbifolds. We show that the real vertex can also be interpreted in terms of a statistical model of symmetric crystal melting. Using this latter connection, we also assess the constant map contribution in Calabi–Yau orientifold models. We find that there are no perturbative contributions beyond one-loop, but a non-trivial sum over non-perturbative sectors, which we compare with the non-perturbative contribution to the closed string expansion.     

Structure of 2D Topological Stabilizer Codes

We provide a detailed study of the general structure of translationally invariant two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. We show that they can be understood in terms of the homology of string operators that carry a certain topological charge. In subsystem codes, two dual kinds of charges appear. We prove that two non-chiral codes are equivalent under local transformations iff they have isomorphic topological charges. Our approach emphasizes local properties over global ones.     

Knot invariants and new weight systems from general 3D TFTs

We introduce and study the Wilson loops in general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in the Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich’s graph complex to accommodate the Wilson loop). We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKähler manifolds.     

Trivial and non-trivial superconductivity in dsDNA

A double-stranded DNA (dsDNA) is modeled by two coupled one-dimensional Kitaev's chain and the topological superconductivity is studied. It is shown that the zero energy mode exists under some specific conditions. The wave function of zero mode is calculated and it is shown that the Majorana quasi-particles exist on the ends of each strand. By calculating the winding number, we show that the topological phase transition can happen if the hopping integral between two strands is very smaller than the pairing potential between the Cooper pairs. It means that the dsDNA behaves as a trivial superconductor, commonly, but single-stranded DNA (or two coupled ssDNA with very small hopping between them) may behave as a non-trivial superconductor. Finally, we suggest an experimental setup for probable detection of Majorana quasi-particle in DNA.     

Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion,a Proof of the BKMP Conjecture

The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for \({\mathbb{C}^3}\). In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion.One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to the special geometry property implied by the topological recursion.     

Standard form of qudit stabilizer groups

We investigate stabilizer codes with carrier qudits of equal dimension D, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding stabilizer, and this implies that the code and its stabilizer are dual to each other. We also show that any qudit stabilizer can be put in a canonical, or standard, form using a series of Clifford gates, and we provide an explicit efficient algorithm for doing this. Our work generalizes known results that were valid only for prime dimensional systems and may be useful in constructing efficient encoding/decoding quantum circuits for qudit stabilizer codes and better qudit quantum error correcting codes.     

A Family of Degenerate Codes for Depolarizing Channels

Quantum degenerate code may improve the hashing bound of quantum capacity. We propose a family of quantum degenerate codes derived from two-colorable graphs. The coherent information of the codes is analytically obtained as a function of the channel noise for the depolarizing channel. We find a new code which has a higher noise threshold than that of the repetition code.     

Jointly-check iterative decoding algorithm for quantum sparse graph codes

<正>For quantum sparse graph codes with stabilizer formalism,the unavoidable girth-four cycles in their Tanner graphs greatly degrade the iterative decoding performance with a standard belief-propagation(BP) algorithm.In this paper, we present a jointly-check iterative algorithm suitable for decoding quantum sparse graph codes efficiently.Numerical simulations show that this modified method outperforms the standard BP algorithm with an obvious performance improvement.     

Topological phases: An expedition off lattice

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of ‘baby universe’, here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger’s theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.     

Implementation of single-qubit and CNOT gates by anyonic excitations of two-body topological color code

The anyonic excitations of topological two-body color code model are used to implement a set of gates. Because of two-body interactions, the model can be simulated in optical lattices. The excitations have nontrivial mutual statistics, and are coupled to nontrivial gauge fields. The underlying lattice structure provides various opportunities for encoding the states of a logical qubit in anyonic states. The interactions make the transition between different anyonic states, so being logical operation in the computational bases of the encoded qubit. Two-qubit gates can be performed in a topological way using the braiding of anyons around each other.     

ZN Gauge theories on a lattice and quantum memory

In the present paper we shall study (2+1)-dimensional Z_N gauge theories on a lattice. It is shown that the gauge theories have two phases, one is a Higgs phase and the other is a confinement phase. We investigate low-energy excitation modes in the Higgs phase and clarify relationship between the Z_N gauge theories and Kitaev’s model for quantum memory and quantum computations. Then we study effects of random gauge couplings (RGC) which are identified with noise and errors in quantum computations by Kitaev’s model. By using a duality transformation, it is shown that time-independent RGC give no significant effects on the phase structure and the stability of quantum memory and computations. Then by using the replica methods, we study Z_N gauge theories with time-dependent RGC and show that nontrivial phase transitions occur by the RGC.     

A network analysis of the Chinese stock market

In many practical important cases, a massive dataset can be represented as a very large network with certain attributes associated with its vertices and edges. Stock markets generate huge amounts of data, which can be use for constructing the network reflecting the market’s behavior. In this paper, we use a threshold method to construct China’s stock correlation network and then study the network’s structural properties and topological stability. We conduct a statistical analysis of this network and show that it follows a power-law model. We also detect components, cliques and independent sets in this network. These analyses allows one to apply a new data mining technique of classifying financial instruments based on stock price data, which provides a deeper insight into the internal structure of the stock market. Moreover, we test the topological stability of this network and find that it displays a topological robustness against random vertex failures, but it is also fragile to intentional attacks. Such a network stability property would be also useful for portfolio investment and risk management.     

量子稳定子码的差错纠正与译码网络构建

寻找差错症状与差错算子之间映射关系是量子译码网络的核心内容,也是量子译码网络实现纠错功能的关键.给出了比特翻转差错症状矩阵和相位翻转差错症状矩阵的定义,将任意Pauli差错算子的差错症状表示为比特翻转差错症状矩阵和相位翻转差错症状矩阵的线性组合.研究发现,量子稳定子码的差错症状矩阵由其校验矩阵所决定,从而可将差错症状矩阵与差错算子之间的映射关系转化为校验矩阵与差错算子之间的映射关系,使得所有关于差错症状的分析都可以通过分析其校验矩阵来实现.这与经典线性码的差错症状与奇偶校验矩阵之间的关系类似,因此可以将经 关键词： 稳定子码 校验矩阵 差错症状 Pauli算子     

An Anyon Model in a Toric Honeycomb Lattice

We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model is helpful to understand the toric code of anyons in a more symmetric way. On the other hand, there is a direct relation between this toric honeycomb model and a boundary coupled Ising chain array in a square lattice via Jordan-Wignertransformation. We discuss the equivalence between these two modelsin the low-lying sector and realize these anyon excitations in a conventional fermion system. The analysis for the ground state degeneracy in the last section can also be thought of as a complementarity of our previous work [Phys. A: Math. Theor. 43 (2010) 105306].