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Color codes with domino twists: construction, logical measurements, and computation

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Abstract

Twists are defects that are used to encode and process quantum information in topological codes like surface and color codes. Color codes can host three basic types of twists, namely charge-permuting, color-permuting, and domino twists. In this paper, we study domino twists from the viewpoint of computation. Specifically, we give a systematic construction for domino twists in qubit color codes. We also present protocols for measurement of logical qubits. Finally, we show that all Clifford gates can be implemented by braiding twists.

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Acknowledgements

The author would like to thank Prof. Pradeep Sarvepalli and Guillaume Dauphinias for helpful discussions.

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  1. Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India

    Manoj G. Gowda

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Appendices

Stabilizer generators as strings

In this section, we present strings corresponding to a subset of stabilizer generators. Note that stabilizers do not produce syndromes. As a result, the strings corresponding to stabilizers are closed strings.

Fig. 14
Fig. 14
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String description for stabilizers: a normal faces b twists

String description for normal stabilizer generators and twists is shown in Fig. 14a and b, respectively.

Proof of Lemma 1

Let \(\tau = t / 2\) be the number of twist pairs and let \(\ell \) be the length of the virtual path (i.e., the path formed by common edges to the half-bricks). For each twist pair, there are \(\ell - 1\) two-valent vertices and the total number of two-valent vertices is \(\tau (\ell - 1 )\). Since the graph is embedded on two-dimensional plane, we have \(n + f - e = 2\), where n and f are number of vertices and faces, respectively (note that f also includes the domain wall as one face and the unbounded green face). Counting the degree of each vertex, we get \(2e = 3[n - \tau (\ell - 1 )] + 2\tau (\ell - 1 ) = 3n + \tau - \tau \ell \), where e is the number of edges in the lattice. Using the above result, we get, \(f = (n/2) - (\tau / 2) (\ell - 1) + 2 \). Removing the face counted for the domain wall for every twist pair (the number of such faces is \(\tau \)), we get \(f = (n/ 2) - (\tau / 2) (\ell + 1) + 2\). The number of stabilizers defined on normal faces and twists is 2f. Including the brick stabilizers (which are \(\tau \ell \) in number) and removing dependencies, we the number of independent stabilizers as

$$\begin{aligned} s = 2f + \tau \ell - 3 = n - \tau + 1 = n - \frac{t}{2} + 1. \end{aligned}$$
(B1)

Therefore, the number of encoded qubits is \(k = \frac{t}{2} - 1\), where t is the number of twists in the lattice.

Fig. 15
Fig. 15
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Combining two Majorana operators to get a logical operator

Combining two Majorana operators to get a logical operator

The operators \(c_m\) are open string operators and therefore do not belong to \( \mathcal {C}( \mathcal {S})\), the centralizer of stabilizer group. However, the operator \(-ic_mc_{m+1}\) is closed loop operator and is in \(\mathcal {C}(\mathcal {S})\), see Fig. 15. The operator \(-ic_mc_{m+1}\) is called Fermionic string operator (as its measurement outcome indicates the presence or absence of Fermions). Measurement outcome of this operator gives the total charge of the twists (fusion outcome). If the outcome is \(+1\), then the twists fuse to vacuum and to fermion \(\psi \) if the outcome is \(-1\).

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Gowda, M.G. Color codes with domino twists: construction, logical measurements, and computation. Quantum Inf Process 24, 87 (2025). https://doi.org/10.1007/s11128-025-04703-y

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