| Summary: | A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere <inline-formula> <math display="inline"> <semantics> <msup> <mi>S</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold <inline-formula> <math display="inline"> <semantics> <msup> <mi>M</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>. More precisely, the <i>d</i>-dimensional POVMs defined from subgroups of finite index of the modular group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mi>S</mi> <mi>L</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> correspond to <i>d</i>-fold <inline-formula> <math display="inline"> <semantics> <msup> <mi>M</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>- coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and <inline-formula> <math display="inline"> <semantics> <msup> <mi>M</mi> <mn>3</mn> </msup> </semantics> </math> </inline-formula>’s obtained from Dehn fillings are explored.
|
|---|
