Although it is well-known that all quantum states can be produced by single-qubit measurements on the cluster states, it is not a simple task to explicitly find which measurement patterns on the cluster states can generate different quantum states. In this paper, we introduce a general scheme to find measurement patterns corresponding to Calderbank--Shor--Steane (CSS) topological states containing Kitaev's toric code states and color code states on different lattices and in different dimensions. Furthermore, we find a measurement pattern for generating non-Abelian anyons where measurement-induced defects on a toric code state play the role of Ising anyons. We also support our scheme by a graphical notation where, by following a few simple graphical transformations, one will be able to convert a CSS topological state to a cluster state. Our scheme can also be used for experimental realization of anyons on cluster states.
Calderbank-Shor-Steane (CSS) codes are a class of quantum error correction codes that contains the toric code and fracton models. A procedure called foliation defines a cluster state for a given CSS code. We use the CSS chain complex and its tensor product with other chain complexes to describe the topological structure in the foliated cluster state, and argue that it has a symmetry-protected topological order protected by generalized global symmetries supported on cycles in the foliated CSS chain complex. We demonstrate the so-called anomaly inflow between CSS codes and corresponding foliated cluster states by explicitly showing the equality of the gauge transformations of the bulk and boundary partition functions defined as functionals of defect world-volumes. We show that the bulk and boundary defects are related via measurement of the bulk system. Further, we provide a procedure to obtain statistical models associated with general CSS codes via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models with insertion of twist defects. We also study the measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models based on recent proposals in the literature, and demonstrate a non-invertible fusion of duality operators. Using the cluster-state entanglers, we construct the so-called strange correlator for general CSS/fracton models. Finally, we introduce a new family of subsystem-symmetric quantum models each of which is self-dual under the generalized Kramers-Wannier-Wegner duality transformation, which becomes a non-invertible symmetry.
The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local
C∗-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by
Rep(D(Z2×Z2))≃Rep(D(Z2))⊠Rep(D(Z2)), which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting
X- and
Z-type checks. We introduce CSS-like codes on
G-valued qudits for any finite group
G that reduce to qubit CSS codes for
G=Z2 yet generalize the Kitaev quantum double model for general groups. The
X-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while
Z-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when
G is non-Abelian and simple, every
G-covariant group-CSS code with suitably upper-bounded
Z-check weight and lower-bounded
Z-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.
In this thesis we explore the possibility of creating continuousvariable \nquantum systems that are capable of supporting universal \nquantum computation. We begin by examining the measurement-based \nmodel, which employs sequences of measurements on highly entangled \nresource states, known as a cluster states. We suggest a method for \nthe construction of Gaussian cluster states based on ensembles of atoms \nand quantum non-demolition interactions. We then go on to expand \nour model to allow for the inclusion of light modes as part of the cluster. \nThis yields a new class of states, the composite cluster states. \nThis leads us to propose a new architecture for the measurement-based \nmodel that uses these composite clusters to increase resource e ciency \nand reduce computational errors. \nThe second part of this thesis concerns topological quantum computation. \nIn states exhibiting topological degrees of freedom, quantum \ninformation can be stored as a non-local property of the physical system \nand manipulated by braiding quasiparticles known as anyons. Here \nwe show how these ideas can be extended to continuous variables. We \nestablish a continuous variable analogue of the Kitaev toric code, show \nthat excitations correspond to continuous versions of Abelian anyons \nand investigate their behaviour under the condition of nite squeezing \nof the resource state. \nFinally, we expand our continuous variable topological model to \ninclude non-abelian excitations by constructing superpositions of CV \ntoric code anyons. We derive the fusion and braiding behaviour of \nthese non-abelian excitations and nd that they correspond to a CV \nanalog of Ising anyons. Using these resources, we go on to suggest \na computational scheme that encodes qubits within the fusion spaces \nof the CV Ising anyons and derive one- and two-qubit quantum gates \noperations that are implemented in a topological manner.
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
The Measurement-based quantum computation provides an alternate model for quantum computation compared to the well-known gate-based model. It uses qubits prepared in a specific entangled state followed by single-qubit measurements. The stabilizers of cluster states are well defined because of their graph structure. We exploit this graph structure extensively to design non-CSS codes using measurement in a specific basis on the cluster state. % We aim to construct
[[n,1]] non-CSS code from a
(n+1) qubit cluster state. The procedure is general and can be used specifically as an encoding technique to design any non-CSS codes with one logical qubit. We show there exists a
(n+1) qubit cluster state which upon measurement gives the desired
[[n,1]] code.