Majorana braiding simulations with projective measurements

Abstract

We summarize the key ingredients required for universal topological quantum computation using Majorana zero modes in networks of topological superconductor nanowires. Particular emphasis is placed on the use of both sparse and dense logical qubit encodings, and on the transitions between them via projective parity measurements. Combined with hybridization, these operations extend the computational capabilities beyond braiding alone and enable universal gate sets. In addition to outlining the theoretical foundations-including the algebra of Majorana operators, along with the stabilizer formalism-we introduce an efficient numerical method for simulating the time-dependent dynamics of such systems. This method, based on the time dependent Pfaffian formalism, allows for the classical simulation of realistic device architectures that incorporate braiding, projective measurements, and disorder. The result is a semi-pedagogical overview and computational toolbox designed to support further exploration of topological quantum computing platforms.

Figures (5)

• Figure 1: Schematic of the sparse encoding for a single qubit, with the logical qubit labelled along with the parity conserving ancilla. Hybridization between pairs of Majoranas results in continuous $X$- or $Z$-rotation around the Bloch sphere as indicated.
• Figure 2: Schematic of the dense encoding for two qubits, with both logical qubits labelled along with the parity conserving ancilla. Hybridization between pairs of Majoranas causes rotations in the logical code space as indicated.
• Figure 3: Schematic of the sparse to dense projection process. (a) Sparse encoding for two logical qubits each highlighted in yellow. (b) Illustration of the $\Pi^-_{45}$ projection, with the measurement conducted via pairwise fusion, mapping the qubit to the dense encoding. (c) Resulting dense encoding of the two logical qubits in terms of six MZMs highlighted in orange. Note that $\gamma_4$, $\gamma_5$ are absent from the encoding. (d) Here we highlight the MZMs involved in the four-point $\Pi^-_{1234}$ projection, which maps the qubit back to the sparse encoding.
• Figure 4: Glossary of braids in the even parity subspace, all implemented in the $-i\gamma_{2i-1}\gamma_{2i}$ basis as utilized in the text. The first three entries, correspond to a $\sqrt{Z}$, $\sqrt{X}$ and Hadamard gate respectively. The next two entries correspond to controlled NOT gates, with braids enacted in the dense encoding and dashed lines, with the target being the first and second qubit respectively. Subsequently, the next entry is a controlled-$Z$ gate in the dense encoding. The next two entries being phase gates in the $Z$ and $X$ direction respectively, on a single sparse qubit, with the last entry being the measurement protocol, utilizing MZM fusion.
• Figure 5: Glossary of braids to two-qubit entangling gates in the odd parity sector of the dense encoding. First two entries correspond to CNOT gates, with braids enacted in the dense encoding. Last entry corresponds to the controlled-$Z$ gate. Braids for single-qubit gates are the same as in the even parity sector, shown in Fig. \ref{['fig:Glossary']}.