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arXiv:2104.03669 · Physical Review A

Color codes with twists: Construction and universal-gate-set implementation

Manoj G. Gowda, Pradeep Kiran Sarvepalli

Published Jul 13, 2021• 2 citations
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Abstract

Twists are defects in a lattice that can be used to perform encoded computations. Three basic types of twists can be introduced in color codes: twists that permute color, charge of anyons, and domino twists that permute the charge label of an anyon with a color label. In this paper, we study a subset these twists from a coding theoretic viewpoint. Specifically, we discuss systematic construction of charge permuting and color permuting twists in color codes. We show that by braiding alone, Clifford gates can be realized in color codes with charge permuting twists. We also discuss implementing single-qubit Clifford gates by a Pauli frame update and CNOT gate by braiding holes around twists in color codes with color permuting twists. Finally, we also discuss implementing a non-Clifford gate by state injection, thus completing the realization of a universal gate set.

DOI
10.1103/physreva.104.012603

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Twists are defects in a lattice that can be used to perform encoded computations. Three basic types of twists can be introduced in color codes: twists that permute color, charge of anyons, and domino twists that permute the charge label of an anyon with a color label. In this paper, we study a subset these twists from a coding theoretic viewpoint. Specifically, we discuss systematic construction of charge permuting and color permuting twists in color codes. We show that by braiding alone, Clifford gates can be realized in color codes with charge permuting twists. We also discuss implementing single-qubit Clifford gates by a Pauli frame update and CNOT gate by braiding holes around twists in color codes with color permuting twists. Finally, we also discuss implementing a non-Clifford gate by state injection, thus completing the realization of a universal gate set.
twists are defects in the lattice which can be utilized to perform computations on encoded data. twists have been studied in various classes of topological codes like qubit and qudit surface codes, qubit color codes and qubit subsystem color codes. they are known to exhibit projective non-abelian statistics which is exploited to perform encoded gates. in this paper, we initiate the study of twists in qudit color codes over odd prime alphabet. to the best of our knowledge, this is the first study of twists in qudit color codes. specifically, we present a systematic construction of twists in qudit color codes that permute both charge and color of the excitations. we also present a mapping between generalized pauli operators and strings in the lattice. making use of the construction, we give protocols to implement generalized clifford gates using charge-color permuting twists.
Twists are defects in the lattice which can be utilized to perform computations on encoded data. Twists have been studied in various classes of topological codes like qubit and qudit surface codes, qubit color codes, and qubit subsystem color codes. They are known to exhibit projective non-Abelian statistics, which is exploited to perform encoded gates. In this paper, we initiate the study of twists in qudit color codes over the odd prime alphabet. Specifically, we present a systematic construction of twists in qudit color codes that permute both charge and color of the excitations. We also present a mapping between generalized Pauli operators and strings in the lattice. Making use of the construction, we give protocols to implement generalized Clifford gates using charge-and-color-permuting twists.
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.
We describe in detail how to perform universal fault-tolerant quantum computation on a two-dimensional color code, making use of only nearest neighbor interactions. Three defects (holes) in the code are used to represent logical qubits. Triple-defect logical qubits are deformed into isolated triangular sections of color code to enable transversal implementation of all single logical qubit Clifford group gates. Controlled-NOT (CNOT) is implemented between pairs of triple-defect logical qubits via braiding.
Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.