Let be linear codes on

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch Quantum error correction

Problem

Let C2⊂C1C_2\subset C_1 be linear codes on {0,1}n\{0,1\}^n, of dimensions k2<k1k_2<k_1, and form the coset states ∣x+C2⟩=∣C2∣−1/2∑y∈C2∣x+y⟩|x+C_2\rangle=|C_2|^{-1/2}\sum_{y\in C_2}|x+y\rangle for x∈C1x\in C_1 (addition mod 2). ((a)) Show that ∣x+C2⟩=∣x′+C2⟩|x+C_2\rangle=|x'+C_2\rangle (as sets of computational basis states, hence as quantum states) if and only if x+x′∈C2x+x'\in C_2, so distinct coset states correspond to distinct cosets of C2C_2 inside C1C_1. ((b)) Show that C1C_1 splits into exactly 2k1−k22^{k_1-k_2} disjoint cosets of C2C_2, each of size 2k22^{k_2}. ((c)) Show that coset states built from different cosets are orthonormal, and hence that the resulting CSS code encodes exactly k1−k2k_1-k_2 logical qubits into nn physical qubits. ((d)) Check the count for the Steane code, where C1C_1 is the Hamming [7,4,3][7,4,3] code and C2C_2 is its [7,3,4][7,3,4] dual.

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