SyQMA: A memory-efficient, symbolic and exact universal simulator for quantum error correction

The classical simulation of universal quantum circuits is crucial both fundamentally and practically for quantum computation. We propose SyQMA, a simulator with several convenient features, particularly suited for quantum error correction (QEC). SyQMA simulates universal quantum circuits with incoherent Pauli noise and computes exact expectation values and measurement probabilities as symbolic functions of circuit parameters: rotation angles, measurement outcomes, and noise rates. This simulator can sample measurement outcomes, enabling the simulation of dynamic quantum programs where circuit composition depends on prior measurement outputs. For QEC, it performs circuit-level maximum-likelihood decoding, provides exact symbolic expressions for logical error rates, and verifies the fault distance of fault-tolerant (FT) stabiliser and magic state preparation protocols. These features are enabled by an intuitive extension of stabiliser simulators, where each non-Clifford Pauli rotation and incoherent Pauli channel is compactly represented via auxiliary qubits and a modified trace. Representing the state requires only polynomial memory and time, while computing expectation values and measurement probabilities takes exponential time in the number of non-Clifford rotations and deterministic measurements, but only polynomial memory. The FT preparation of stabiliser and magic states, including the first stage of magic state cultivation, is analysed without approximations. We also exactly convert the disjoint error probabilities of a general multi-qubit Pauli channel to independent ones, a key step for creating and sampling from detector error models. The code is publicly available and open-source.