Solovay-Kitaev Decomposition Strategy for Single-Qubit Channels

Abstract

Inspired by the Solovay-Kitaev decomposition for approximating unitary operations as a sequence of operations selected from a universal quantum computing gate set, we introduce a method for approximating any single-qubit channel using single-qubit gates and the controlled-not (cnot). Our approach uses the decomposition of the single-qubit channel into a convex combination of "quasiextreme" channels. Previous techniques for simulating general single-qubit channels would require as many as 20 cnot gates, whereas ours only needs one, bringing it within the range of current experiments.

Simulation of single-qubit open quantum systems

“…Due to our restriction to the single qubit case our notion of efficiency has no dependence on the system size, which remains a constant. However, as in [44], we restrict ourselves to quantum circuits requiring only a single ancilla qubit, the smallest possible minimal dilation for a non-unitary single-qubit channel.…”

“…It is important to note that each member T t of an arbitrary semigroup of single-qubit channels {T t } is itself a single-qubit channel, and therefore in principle, using the methods of Wang et al [44], can be simulated within 1-norm distance using O(log 3.97 (1/ )) gates from any specified single qubit set S and one CNOT, acting on only the system qubit and a single ancilla. However in order to utilise this method, which may even be improved [50,51] to require only O(log(1/ )) such gates, it is necessary to first obtain a decomposition of the channel T t into a convex sum of quasi-extreme channels, which in order to do explicitly requires specification of the generator.…”

“…We proceed by following the strategy, introduced in [44], of decomposing the channels T (θ k ) t into the convex sum of quasi-extreme channels. These quasi-extreme channels require only two Kraus operators for implementation, and hence can be simulated using a unitary circuit acting on only a single ancilla qubit.…”

“…These results allow us to efficiently implement the recombination methods of Bacon et al [20], such that in order to construct efficient quantum circuits for the simulation of arbitrary Markovian dynamics of a qubit it is only necessary to construct efficient circuits for the simulation of semigroups corresponding to the continuous one parameter family of generators defined by Bacon et al [20]. Furthermore, recently Wang et al [44] have shown how to utilise convex properties of the set of single-qubit quantum channels [45] to simulate any such channel via unitary circuits requiring only a single ancilla qubit, as opposed to the two-ancilla qubits required by straightforward implementations of the Stinespring dilation. We utilise these results for the construction of circuits for the simulation of the semigroups required by Bacon et al [20], such that after recombination we obtain an explicit unitary circuit, with size scaling polynomially with respect to time, consisting only of CNOT gates and single qubit gates and requiring only a single ancilla qubit, for the simulation up to any desired accuracy of an arbitrary single-qubit quantum dynamical semigroup.…”

“…In Section IV we generalise results from [10] into the setting applicable for the problem addressed here, effectively demonstrating a method for the efficient recombination of the generators decomposed in Section III. Finally, in Section V we exploit the methods introduced in [44] in order to provide explicit efficient unitary circuits for the semigroups corresponding to the generators resulting from the decomposition in Section III.…”