Optimal quantum subsystem codes in 2-dimensions

Abstract

Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1{n}_2$, $K=k$, and $D=\min (d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of 2-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge-fixed to certain hypergraph product codes that do.

18 pages, 6 figures

References in corpus (22)

• Fault-tolerant quantum computation by anyons
• Good Quantum Error-Correcting Codes Exist
• Stabilizer Codes and Quantum Error Correction
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Sparse Graph Codes for Quantum Error-Correction
• Quantum codes on a lattice with boundary
• Stabilizer Formalism for Operator Quantum Error Correction
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Subsystem fault tolerance with the Bacon-Shor code
• Tradeoffs for reliable quantum information storage in 2D systems
• Topological Subsystem Codes
• Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
• Asymptotically Good Quantum Codes
• A theory of single-shot error correction for adversarial noise
• Quantum Expander Codes
• Subsystem codes with spatially local generators
• Constructions and Noise Threshold of Topological Subsystem Codes
• On the logical operators of quantum codes
• Optimal error correction in topological subsystem codes
• Numerical study of hypergraph product codes
• Topological Subsystem Codes From Graphs and Hypergraphs

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• Robust universal Hamiltonian quantum computing using two-body interactions