Shaded tangles for the design and verification of quantum circuits

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 analyze 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 2 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.