Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder

<jats:p>Recent work [M. J. Gullans , ] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a nonzero encoding rate in correcting errors even if the random circuits on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>n</a:mi></a:math> qubits, embedded in one spatial dimension (1D), have a logarithmic depth <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>d</b:mi><b:mo>=</b:mo><b:mi>O</b:mi><b:mo>(</b:mo><b:mo form="prefix">log</b:mo><b:mi>n</b:mi><b:mo>)</b:mo></b:mrow></b:math>. However, this was demonstrated only for a simple erasure noise model. In this work, we discover that, for the same class of codes, this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>d</d:mi><d:mo>=</d:mo><d:mi>O</d:mi><d:mo>(</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow></d:math>-depth random encoding circuits, can be attained for the above codes even when the circuit depth is restricted to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>d</e:mi><e:mo>=</e:mo><e:mi>O</e:mi><e:mo>(</e:mo><e:mo form="prefix">log</e:mo><e:mi>n</e:mi><e:mo>)</e:mo></e:mrow></e:math> in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mo form="prefix">log</g:mo></g:math>-depth encoding circuits in 1D.</jats:p> <jats:sec> <jats:title/> <jats:supplementary-material> <jats:permissions> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2024</jats:copyright-year> </jats:permissions> </jats:supplementary-material> </jats:sec>