We consider the problem of Clifford testing, which asks whether a black-box n-qubit unitary is a Clifford unitary or at least ε-far from every Clifford unitary. We give the first 4-query Clifford tester, which decides this problem with probability poly(ε). This contrasts with the minimum of 6 copies required for the closely-related task of stabilizer testing. We show that our tester is tolerant, by adapting techniques from tolerant stabilizer testing to our setting. In doing so, we settle in the positive a conjecture of Bu, Gu and Jaffe, by proving a polynomial inverse theorem for a non-commutative Gowers 3-uniformity norm. We also consider the restricted setting of single-copy access, where we give an O(n)-query Clifford tester that requires no auxiliary memory qubits or adaptivity. We complement this with a lower bound, proving that any such, potentially adaptive, single-copy algorithm needs at least Ω(n1/4) queries. To obtain our results, we leverage the structure of the commutant of the Clifford group, obtaining several technical statements that may be of independent interest.

We prove a conjecture of Arunachalam & Dutt on the existence of a tolerant stabilizer testing algorithm, and achieve an exponential improvement in the parameters of the tester. Key to our argument is a generalized uncertainty relation for sets of Pauli operators, based on the Lovász theta function.

We consider the problem of testing whether an unknown $n$-qubit quantum state $|ψ\rangle$ is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using $O(n)$ copies, and conversely prove that $Ω(\sqrt{n})$ copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.

A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades and have found applications in other fields such as quantum information theory. Therefore it is natural to consider the computational complexity of deciding whether a given graph G has a vertex-minor isomorphic to another graph H. Here we prove that this decision problem is NP-complete, even when restricting H and G to be circle graphs, a class of graphs that has a natural relation to vertex-minors.