Feasibility of logical Bell state generation in memory assisted quantum networks

This study explores the feasibility of utilizing quantum error correction to generate and store logical Bell states in heralded quantum entanglement protocols, crucial for quantum repeater networks. Two lattice surgery-based protocols (local and nonlocal) are introduced to establish logical Bell states between distant nodes using an intermediary node. We simulate the protocols using realistic experimental parameters, including ion trap memories, noisy optical channels, frequency conversion, and nondestructive detection of photonic qubits. The study evaluates rotated and planar surface codes alongside Bacon-Shor codes for small code distances (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>3</a:mn><a:mo>,</a:mo><a:mn>5</a:mn></a:mrow></a:math>) under depolarizing and physical noise models. Pseudothresholds are identified, with physical error rates above <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>p</b:mi><b:mi>err</b:mi></b:msub><b:mo>∼</b:mo><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>3</b:mn></b:mrow></b:msup></b:mrow></b:math> offering no advantage over unencoded Bell states under depolarizing noise. Pseudo-thresholds are also reevaluated in terms of gate error rates <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>p</c:mi><c:msub><c:mi>err</c:mi><c:mi>H</c:mi></c:msub></c:msub></c:math>, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>p</d:mi><d:msub><d:mi>err</d:mi><d:mrow><d:mi>C</d:mi><d:mi>X</d:mi></d:mrow></d:msub></d:msub></d:math>, and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>p</e:mi><e:msub><e:mi>err</e:mi><e:mi>M</e:mi></e:msub></e:msub></e:math>. For a distance of 1 km between the end node and the intermediary, an advantage over unencoded Bell-state heralded protocols requires reducing gate error rates by an order of magnitude (<f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mn>0.1</f:mn><f:msub><f:mi>p</f:mi><f:msub><f:mi>err</f:mi><f:mi>H</f:mi></f:msub></f:msub></f:mrow></f:math>, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mn>0.1</g:mn><g:msub><g:mi>p</g:mi><g:msub><g:mi>err</g:mi><g:mrow><g:mi>C</g:mi><g:mi>X</g:mi></g:mrow></g:msub></g:msub></g:mrow></g:math>, and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mn>0.1</h:mn><h:msub><h:mi>p</h:mi><h:msub><h:mi>err</h:mi><h:mi>M</h:mi></h:msub></h:msub></h:mrow></h:math>). These results highlight the need for significant hardware improvements to implement logical Bell state protocols with quantum memories. Additionally, the nonlocal protocol rate was analyzed achieving rates up to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mo>(</i:mo><i:mn>32.53</i:mn><i:mo>±</i:mo><i:mn>1.53</i:mn><i:mo>)</i:mo><i:mspace width="0.16em"/><i:mi>Hz</i:mi></i:mrow></i:math> over distances of 1 to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mn>80</k:mn><k:mspace width="4pt"/><k:mi>km</k:mi></k:mrow></k:math> between the end node and the intermediary node.