One-dimensional error-correcting code for Majorana qubits

Abstract

Although Majorana platforms are promising avenues to realizing topological quantum computing, they are still susceptible to errors from thermal noise and other sources. We show that the error rate of Majorana qubits can be drastically reduced using a 1D repetition code. The success of the code is due the imbalance between the phase error rate and the flip error rate. We demonstrate how a repetition code can be naturally constructed from segments of Majorana nanowires. We find the optimal lifetime may be extended from a millisecond to over one second.The main road block in achieving quantum computation [1] is dealing with quantum error. Isolating a bit of quantum information from its environment is challenging enough, however, in order to realize a useful quantum computation machine it is necessary to maintain coherence for thousands of entangled qubits. Topological qubits are useful in that they have built-in fault tolerance due to the spatial separations between the anyons and the boundary modes [2]. Majorana zero modes [3][4][5], which appear as end modes of p-wave superconducting nanowires, are one the most promising directions in topological quantum computing [4,[6][7][8][9][10][11][12][13][14]. These Majorana end modes can store information non-locally and can be braided to perform topologically protected logic gates [15][16][17][18][19][20][21][22].Although topological qubits have some level of protection from error they will still require error correction in order to be fully implemented as computational qubits. A perfect Majorana qubit would be infinitely long and held at zero temperature. Nonzero temperatures lead to a finite quasiparticle density, which will cause errors in the qubit. There exist error correction codes such as the toric code [2], surface codes [23][24][25][26], and color codes [27][28][29], which can be implemented on Majorana qubits [30][31][32][33][34][35][36][37] or in other schemes such as planar codes [38,39]. However, these error correction schemes require a great deal of overhead, having a large number of redundant qubits in order to catch and correct error. As Kitaev pointed out [2], any topological phase of matter can be identified as an error correcting code. In this vain, we ask if the 1D fermionic topological phase [40,41] built from a chain of Majorana nanowires can be identified with a "fermion-parity protected error correcting code". Such a chain would require only a line of physical qubits instead of a surface.In this paper, we show how a chain of Majorana nanowires can be used to significantly improve the qubit lifetime, because of a hierarchy of different error types in Majorana qubits. Due to an unexpectedly high observed density of quasiparticles [42][43][44][45][46][47], we argue that phase errors in Majorana qubits are orders of magnitude greater than bit flip errors. This phase error can be corrected at the expense of the much smaller bit flip error using the repetition code [48][49][50]. We describe the repetition code in the language of Majorana qubits. The code...

Nanowires: Exponential speedup in quantum computing

“…Because of the spatial separations between the boundary modes and anyons, topological qubits have inherent fault tolerance, which makes them valuable in this context. One of the most promising approaches in topological quantum computing is the study of Majorana zero modes, which arise as end modes of p-wave superconducting nanowires [ 151 ]. Topological qubits may become incoherent as a result of quasiparticle poisoning (QP), a solely fermionic mistake that modifies the fermion parity connected to Majorana degrees of freedom [ 152 ].…”