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Magic-state distillation is a protocol to purify noisy quantum states using quantum stabilizer codes. In this paper, we provide conditions for identifying magic states along with analytical expressions for computing the noise threshold. Through quantum entropic results, we prove the convergence of the magic-state distillation procedure. We derive analytical results to show how the noise threshold scales as a function of the minimum distance and the number of physical qudits of the underlying quantum stabilizer code. We distill |T and |S states with thresholds 0. 498 (theoretical limit is 0. 5) and 0. 71 (theoretical limit is 0. 75) over the depolarizing channel using [31, 1, 16]₂ quantum Bose-Chaudhuri-Hocquenghem code and [29, 1, 15]₃ quantum qutrit code, respectively. Finally, encoding algorithms based on quantum Calderbank-Shor-Steane (CSS) and non-CSS qudit stabilizer codes along with circuits are proposed for magic-state distillation.
Sharma et al. (Tue,) studied this question.
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