Abstract
In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of O(nα(n)), where n is the number of physical qubits and α is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, α(n)≤3. We prove that our algorithm performs optimally for errors of weight up to (d−1)/2 and for loss of up to d−1 qubits, where d is the minimum distance of the code. Numerically, we obtain a threshold of 9.9% for the 2d-toric code with perfect syndrome measurements and 2.6% with faulty measurements.
Highlights
The main obstacle to the construction of a quantum computer is the unavoidable presence of errors, which left unchecked quickly destroy quantum information
We design a decoding algorithm for topological codes that runs in the worst case in almost-linear time in the number of physical qubits n, with a high threshold (See Table 1)
We focus on the worst case complexity and on the average case complexity since it is the maximum running time of the decoder that will determine the clock-time of the quantum computer
Summary
The main obstacle to the construction of a quantum computer is the unavoidable presence of errors, which left unchecked quickly destroy quantum information. Topological codes, in particular Kitaev’s surface code [42], are currently expected to form the core architecture of this first generation of quantum computers, due to their high thresholds and their locality. To use these codes, we require a classical decoding algorithm, which must process measurement information fast enough to keep pace with the clock-speed of the quantum device. We design a decoding algorithm for topological codes that runs in the worst case in almost-linear time in the number of physical qubits n, with a high threshold (See Table 1). Further discussion of the complexity scaling, and numerical simulations is given in the Appendices
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