Universal Adapters between Quantum Low-Density Parity Check Codes

We propose the repetition code adapter as a way to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks, thus providing a flexible tool for fault-tolerant computation with quantum LDPC codes. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The construction achieves joint logical Pauli measurement of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>t</a:mi> </a:math> weight <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>O</c:mi> <c:mo stretchy="false">(</c:mo> <c:mi>d</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> operators using <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>O</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>d</g:mi> <g:mo stretchy="false">)</g:mo> </g:math> time and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mover> <k:mi>O</k:mi> <k:mo stretchy="false">~</k:mo> </k:mover> </k:mrow> <k:mo stretchy="false">(</k:mo> <k:mi>t</k:mi> <k:mi>d</k:mi> <k:mo stretchy="false">)</k:mo> </k:math> additional qubits and checks, up to a factor polylogarithmic in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>d</p:mi> </p:math> . As a special case, for some geometrically local codes in fixed <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>D</r:mi> <r:mo>≥</r:mo> <r:mn>2</r:mn> </r:math> dimensions, only <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>O</t:mi> <t:mo stretchy="false">(</t:mo> <t:mi>t</t:mi> <t:mi>d</t:mi> <t:mo stretchy="false">)</t:mo> </t:math> additional qubits and checks are required instead. By extending the adapter in the case <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>t</x:mi> <x:mo>=</x:mo> <x:mn>2</x:mn> </x:math> , we also construct a toric code adapter that uses <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>O</z:mi> <z:mo stretchy="false">(</z:mo> <z:msup> <z:mi>d</z:mi> <z:mn>2</z:mn> </z:msup> <z:mo stretchy="false">)</z:mo> </z:math> additional qubits and checks to perform addressable logical gates on arbitrary LDPC codes via Dehn twists. To obtain these results, we develop a more general form of graph edge expansion and the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mrow> <db:mi mathvariant="sans-serif">S</db:mi> <db:mi mathvariant="sans-serif">k</db:mi> <db:mi mathvariant="sans-serif">i</db:mi> <db:mi mathvariant="sans-serif">p</db:mi> <db:mi mathvariant="sans-serif">T</db:mi> <db:mi mathvariant="sans-serif">r</db:mi> <db:mi mathvariant="sans-serif">e</db:mi> <db:mi mathvariant="sans-serif">e</db:mi> </db:mrow> </db:math> algorithm, which ensures a sparse transformation between different weight-2 check bases for the classical repetition code.