Fusion for high-dimensional linear-optical quantum computing with improved success probability

Type-II fusion is a probabilistic entangling measurement that is essential to measurement-based linear-optical quantum computing and can be used for quantum teleportation more broadly. However, it remains under-explored for high-dimensional qudits. Our main result gives a Type-II fusion protocol with proven success probability approximately <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>2</a:mn><a:mo>/</a:mo><a:msup><a:mi>d</a:mi><a:mn>2</a:mn></a:msup></a:math> for qudits of arbitrary dimension <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi></c:math>. This generalizes a previous method that only applied to even-dimensional qudits. We believe this protocol to be the most efficient known protocol for Type-II fusion, with the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>5</e:mn></e:math> case beating the previous record by a factor of approximately <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>723</g:mn></g:math>. We discuss the construction of the required <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">(</i:mo><i:mi>d</i:mi><i:mo>−</i:mo><i:mn>2</i:mn><i:mo stretchy="false">)</i:mo></i:math>-qudit ancillary state using a silicon spin-qudit ancilla coupled to a microwave cavity through time-bin multiplexing. We then introduce a general framework of extra-dimensional corrections, a natural technique in linear optics that can be used to nondeterministically correct nonmaximally entangled projections into Bell measurements. We use this method to analyze and improve several different circuits for high-dimensional Type-II fusion and compare their benefits and drawbacks.