Constant-overhead compilation without full adaptivity

Determine whether constant-overhead compilation of quantum circuits over arbitrary one- and two-qubit gates into the discrete gate set {H, T, CNOT} can be achieved without full adaptivity, for example using a mixed-unitary channel formed by randomly selecting a circuit from a distribution and then implementing it on a quantum computer.

Background

The paper establishes that any circuit of G arbitrary one- and two-qubit gates can be implemented to inverse-polynomial diamond-distance error using O(G) gates from {H, T, CNOT}, provided the compiled circuit may use adaptivity, including intermediate measurements and classically controlled future operations. It also proves an O(G + log(1/epsilon)) bound for circuits over the dyadic gate set.

The authors identify the role of full adaptivity as an unresolved aspect of their construction. In particular, they ask whether the same constant-overhead scaling can be obtained by a non-fully-adaptive implementation based on a mixed-unitary channel: randomly choose a circuit from a distribution and then execute the selected circuit on a quantum computer. The paper does not resolve this question.