Sarah Meng Li

We present a complete set of rewrite rules for multi-qudit Clifford circuits, where \emph{qudit} denotes a d-level quantum system with d an odd prime. Completeness means that any two Clifford circuits representing the same linear map can be transformed into each other using these rules. In total, there are 19 \emph{Clifford relations}, each involving at most three qudits and admitting an intuitive interpretation. Our approach leverages the isomorphism between the symplectic group $\mathrm{Sp}(2n, \mathbb{Z}_d)$ and the quotient of the Clifford group by the Pauli group. We first derive a complete set of \emph{symplectic relations} for $\mathrm{Sp}(2n, \mathbb{Z}_d)$, and then lift them to Clifford relations by incorporating Pauli corrections. To do this, we introduce a \emph{symplectic normal form} that captures the stabiliser tableau of a Clifford operator and is unique up to Pauli correction. This simplification enables a streamlined derivation of a complete set of 66 relations, which we further compress to 18 symplectic relations. Our computations in $\mathrm{Sp}(2n, \mathbb{Z}_d)$ are formalised in the Agda proof assistant, providing a machine-verified proof of correctness.

The ability to fault-tolerantly prepare cat states, also known as multi-qubit GHZ states, is an important primitive for quantum error correction. It is required for Shor-style syndrome extraction, and can also be used as a subroutine for doing fault-tolerant state preparation of CSS codewords. Existing approaches to fault-tolerant cat state preparations have been found using computationally expensive heuristics involving SAT solving, reinforcement learning or exhaustive analysis. In this paper we constructively find optimal circuits for cat states in a scalable way. In particular, we derive formal lower bounds on the number of CNOT gates required for circuits implementing n-qubit cat-states that do not spread errors of weight at most t for values t = 1, ..., 5. We do this by using fault-equivalent rewrites of ZX-diagrams to reduce it to a problem of characterising certain 3-regular simple graphs. We provide explicit constructions for circuits that match this lower bound for all n and t <= 5. Furthermore, we use SAT solvers to construct circuits for all n <= 50 and t <= 7. We additionally show how to trade CNOT count against depth, in particular allowing us to construct constant-depth fault-tolerant implementations using O(n) ancilla and O(n) CNOT gates.