Unfolding the color code

Abstract

The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d-dimensional closed manifold is equivalent to multiple decoupled copies of the d-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d-dimensional color code with d 1 + boundaries of d 1 + distinct colors, we find that the code is equivalent to multiple copies of the d-dimensional toric code which are attached along a d( 1) − -dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d-dimensional toric code admits logical non-Pauli gates from the dth level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d-qubit control-Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.

Topological color code and symmetry-protected topological phases

“…In this setting, a looplike SPT excitation involves electric charges from two copies of the toric code, each possessing one copy of the Z 2 symmetry. This conclusion also follows from the unitary equivalence of the color code and two decoupled copies of the toric code on a closed manifold [23].…”

“…First, anyonic excitations with three different color labels are not independent from each other since applications of Pauli X and Z operators on a single qubit create composites of anyons m A m B m C ,e A e B e C , respectively. In other words, the following fusion channels exist: In fact, it is known that, on a closed manifold, the twodimensional color code is equivalent to two decoupled copies of the toric code under a local unitary transformation [21][22][23].…”

“…It is known that, on a closed manifold, the three-dimensional color code is equivalent to three decoupled copies of the toric code under a local unitary transformation [23,38]. Note that the three-dimensional color code has volume and membrane phase operators.…”

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“…[23,25,26]. Thus, we see a very concrete realization of this mapping using code concatenation: the code concatenation perspective directly suggests ways of doing noisy syndrome measurements and implementing a decoder.…”

“…The triangular code encodes a single qubit and the oddness of the number of physical qubits implies that X = X all and Z = Z all form a pair of anti-commuting logical operators (X all acts as Pauli X on all qubits in the lattice). A multi-qubit color code can be obtained using a polygon of higher degree [23]. By multiplying these logical operators with check operators, one can deform the logical operators to operators on a boundary of the triangular lattice.…”