Topological subsystem color codes add an important feature to the advantages of topological codes: error tracking only involves measuring two local operators in a two-dimensional setting. Unfortunately, currently known methods for computing with them are highly unpractical. We give a mechanism for implementing all Clifford gates by code deformation in a planar setting. In particular, we use twist braiding and express its effects in terms of certain colored Majorana operators.
Topological subsystem color codes add an important feature to the advantages of topological codes: error tracking only involves measuring two local operators in a two-dimensional setting. Unfortunately, currently known methods for computing with them are highly unpractical. We give a mechanism for implementing all Clifford gates by code deformation in a planar setting. In particular, we use twist braiding and express its effects in terms of certain colored Majorana operators.
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.
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' non-Abelian 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, non-Abelian statistics can be exploited without ever performing physical braiding operations. Physical braiding operations are replaced by Majorana tracking, an entirely software-based protocol which appropriately updates the Majoranas involved in the color-code stabilizer measurements. This approach minimizes the required hardware operations for single-qubit Clifford gates. For Clifford completeness, we combine color codes with surface codes, and use color-to-surface-code lattice surgery for long-range multitarget CNOT gates which have a time overhead that grows only logarithmically with the physical distance separating control and target qubits. With the addition of magic state distillation, our architecture describes a fault-tolerant universal quantum computer in systems such as networks of tetrons, hexons, or Majorana box qubits, but can also be applied to nontopological qubit platforms.
In fault-tolerant quantum computation [1], quantum information is protected from noise by encoding it in somewhat non-local degrees of freedom, thus distributing it among many smaller subsystems, typically qubits. This makes sense under the physically relevant assumption that interactions with the environment have a local nature. The implementation of gates, consequently, must be also as local as possible to preserve the structure of noise. This is naturally achieved with transversal gates, i.e. unitary operators that transform encoded states by acting separately on suitable subsystems, in the simplest case independently on each qubit. Unfortunately, no code admits a universal transversal set of gates [2]. Topological quantum error correcting codes [3] introduce a richer notion of locality by considering the spatial location of the physical qubits, which are assumed to be arranged on a lattice. They come in families parametrized with a lattice size, for a fixed spatial dimension. Their defining features are (i) that the measurements needed to recover information about errors only involve a few neighbouring qubits and (ii) that no encoded information can be recovered without access to a number of physical qubits comparable to the system size. Rather than sticking to the above definition of transversality, for topological codes it is natural to consider instead quantum circuits of fixed depth with geometrically local gates [4, 5]. Remarkably, it has been recently shown that spatial dimension imposes constraints on the transversal gates that are allowed [5]. In particular, this is true in the case of topological stabilizer codes, where it has been found that transversal gates on D-dimensional codes have to belong to certain sets PD. A first main result of this work [6] is that codes that saturate these bounds can be constructed for every D, in the sense that they admit at least a transversal gate in PD (not belonging to PD−1). In particular, the result applies to color codes [7, 8, 9, 10], where the single-qubit gate RD := ( 1 0 0 e2πi/2 D ) is transversal (in the more strict non-topological sense).
We describe a method to use measurements and correction operations in order to implement the Clifford group in a stabilizer code, generalising a result from [Bombin,2011] for topological subsystem colour codes. In subsystem stabilizer codes of distance at least 3 the process can be implemented fault-tolerantly. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only three observables. This is an alternative to the method proposed by [Paetznick and Reichardt, 2013] to generate a set of gates which is universal for quantum computing for this code. The construction is inspired by the description of code rewiring from [Colladay and Mueller, 2018]. Inspired by the code rewiring strategy of [Colladay and Mueller, 2018], we describe a method to use measurements and correction operations in order to implement the Clifford group in the code space of any stabilizer code, and we specify a sufficient set of conditions under which the distance of the code is preserved throughout. In particular this provides a method to implement a logical Hadamard-type gate within the 15-qubit Reed-Muller quantum code by measuring and correcting only two observables, providing the only non-transversal gate required for universality. Furthermore this approach is applicable to the toric code and quantum LDPC code
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
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.