Fig. 1

Circuit model and algorithm flow. a Schematic of the parameterised quantum circuits that can be simulated by lowesa. The light boxes are arbitrary (noisy) Clifford gates, the blue boxes are parameterised Pauli rotations and the red kites represent Pauli noise channels. b Diagrammatic sketch of lowesa as described in Algorithm 1 applied to circuits given by Eq. (1). The Pauli operator P is propagated backwards through the circuit where every Clifford gate transforms it into another Pauli, and the decomposition of the parameterised Pauli rotations into process modes D₀, D₁, D₋₁ splits the propagation up into paths that may annihilate. A cutoff of ℓ = 2 is chosen which artificially annihilates paths that branch into D₁, D₋₁ more than 2 times.