Unifying the Clifford hierarchy via symmetric matrices over rings

The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a $2^k\mathrm{th}$ root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of $k$ and the degree of the polynomial. For qubit systems, we consider $k\mathrm{th}$-level diagonal unitaries that can be described just by quadratic forms of the state over the ring ${\mathbb{Z}}_{2^k}$ of integers modulo $2^k$. The quadratic forms involve symmetric matrices over ${\mathbb{Z}}_{2^k}$ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over ${\mathbb{Z}}_{2^k}$ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the $\mathrm{Clifford}+T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.