A complete theory of the Clifford commutant

Understand

The Clifford group plays a central role in quantum information science.

• It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation.
• At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with $k$-fold tensor powers of Clifford unitaries.
• Previous understanding of this commutant has been limited to relatively small values of $k$, constrained by the number of qubits $n$.