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Quantum Information and Computation

Small Majorana fermion codes

Mathew B. Hastings

Published Nov 1, 2017• 4 citations
Publisher

Abstract

We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 4 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 4 and 6 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.

DOI
10.26421/qic17.13-14-7

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We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 4 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 4 and 6 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.
We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 444 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 444 and 666 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.
We initiate the study of Majorana fermion codes (MFCs). These codes can be viewed as extensions of Kitaev's one-dimensional (1D) model of unpaired Majorana fermions in quantum wires to higher spatial dimensions and interacting fermions. The purpose of MFCs is to protect quantum information against low-weight fermionic errors, that is, operators acting on sufficiently small subsets of fermionic modes. We examine to what extent MFCs can surpass qubit stabilizer codes in terms of their stability properties. A general construction of 2D MFCs is proposed that combines topological protection based on a macroscopic code distance with protection based on fermionic parity conservation. Finally, we use MFCs to show how to transform any qubit stabilizer code to a weakly self-dual CSS code.
Quantum error correction protocols protect against local errors by storing logical information non-locally. This poses a challenge: how to design efficient logical gates on the non-local ``hidden'' logical information, and how to implement these gates using the local physical operations. We develop a general description of planar Pauli stabilizer codes and protocols for logical operations in terms of point-like particles called Majorana fermions. Information is stored in the pairwise fermion parities of spatially separated Majorana fermions. The description in terms of Majorana fermions captures not only large distance asymptotics, but also all scales down to the lattice constants. We exploit this locality to densely pack logical information in spacetime. The simplest application is to a static case: dense memory. More importantly, we implement fault-tolerant Majorana motion and leverage this primitive to design braiding-based logical gates. This approach reduces space overhead of logical operations resulting in an improved logical error rate given fixed number of physical qubits. We illustrate a practical use of our approach by designing and benchmarking of 2-qubit Clifford gates. We find numerically that our protocol outperforms lattice surgery in this setting for near-term error rates and realistic device constraints. More generally, introduction of compact motion of Majorana fermions as an efficient computational primitive opens a promising new route for the design of low overhead error correction protocols.
Abstract — Optimal quantum stabilizer codes of distance 3 are explicitly constructed for all lengths except for the following four families of lengths 8fm − {1, 2} and fm+2 − {2, 3} with fm = 4 m −1 3 and m ≥ 2 being integer, for which our codes are of the best parameters known and are only one logical qubit less than the quantum Hamming bound. The optimality of our codes is ensured by saturating either the quantum Hamming bound or a stronger bound for three families of lengths 8fm + {1, 2} and fm+2 − 1 with m ≥ 1 derived from the linear programming bound. For the lengths less than 128 three previously unknown codes [[36, 29, 3]], [[37, 30, 3]] and [[81, 73, 3]] have been found. Index Terms — quantum error correction, 1-error correcting stabilizer codes, quantum Hamming bound, linear programming bound, optimal codes I.
The tetron architecture is a promising candidate for topological quantum computation. Each tetron Majorana island has four Majorana zero modes, and possible measurements are constrained to span zero or two Majoranas per tetron. Such measurements are known to be sufficient for correcting so-called "bosonic errors," which affect an even number of Majoranas per tetron. We demonstrate that such measurements are also sufficient for correcting "fermionic errors," which affect an odd number of Majoranas per tetron. In contrast, previous proposals for "fermionic error correction" on tetrons introduce more experimental challenges. We show that "fermionic codes" can be derived from traditional "bosonic codes" by inclusion of tetrons in the stabilizer group.
We introduce error-correcting codes that can correct for fermion parity-violating (quasiparticle poisoning) and parity-conserving errors in systems of complex fermions and of Majorana fermions. After establishing properties of fermion codes, we introduce a generic construction of fermion codes from weakly self-dual classical, binary error-correcting codes. We use this method to construct (i) the shortest fermion code to correct for quasiparticle poisoning errors, (ii) translationally-invariant fermion codes and (iii) other codes that correct higher-weight errors. We conclude by discussing a possible physical realizations of one such code in mesoscopic superconductors hosting Majorana zero modes.