Abstract
We give a scheme for interpreting shaded tangles as quantum circuits, with the property that if two shaded tangles are ambient isotopic, their corresponding computational effects are identical. We analyse 11 known quantum procedures in this way—including entanglement manipulation, error correction and teleportation—and in each case present a fully topological formal verification, yielding generalized procedures in some cases. We also use our methods to identify two new procedures, for topological state transfer and quantum error correction. Our formalism yields in some cases significant new insight into how the procedures work, including a description of quantum entanglement arising from topological entanglement of strands, and a description of quantum error correction where errors are ‘trapped by bubbles’ and removed from the shaded tangle.
Highlights
In this paper, we introduce a new knot-based language for designing and verifying quantum circuits
We read our shaded tangles as quantum circuits, with time flowing from bottom to top, and with individual geometrical features of the diagrams—such as shaded regions, cups and caps and crossings—interpreted as distinct quantum circuit components, such as qudits,1 qudit preparations, and certain 1- and 2-qudit gates
We show that our semantics is sound with respect to this isotopy relation: that is, if two shaded tangles are isotopic, they are equivalent as quantum circuits, in the sense that the quantum computations they describe have identical underlying linear maps
Summary
We introduce a new knot-based language for designing and verifying quantum circuits. We show that our semantics is sound with respect to this isotopy relation: that is, if two shaded tangles are isotopic, they are equivalent as quantum circuits, in the sense that the quantum computations they describe have identical underlying linear maps This yields a method for the design and verification of quantum procedures. Carmi and Moskovic develop a theory of tangle machines [55], where tangle diagrams ( not 2-shaded as we use them here) represent networks which can process information; these authors show their ideas apply to adiabatic quantum computation, which is quite different to the circuit model of quantum computation to which our work is most closely connected. This Hadamard has been used in the cluster state literature for neighbourhood inversion on a cluster graph [4, proposition 5], an operation we verify in §4e for a linear graph
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