Towards local testability for quantum coding

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-faces of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-cube (for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mi>p</mml:mi></mml:math>) and stabilizer constraints with faces of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x00B1;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mstyle displaystyle="false" scriptlevel="0"><mml:mrow><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mfrac linethickness="0"><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mfrac><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math> physical qubits and displays local testability with a soundness of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&amp;#x03A9;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> beating the current state-of-the-art of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>log</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-cube by arbitrary linear classical codes of length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.