Simulation modes

Every simulation computes the same expectation value,

$$f_L(\theta) = \mathrm{Tr}\!\left[\varrho\, C_L(\theta)^\dagger H\, C_L(\theta)\right],$$

for an observable $H$, a reference state $\varrho$, and a parameterised circuit $C_L(\theta)$ (see Propagation Algorithm). What differs between the two simulation modes is which object is expanded as a sum of Majorana monomials and pushed through the circuit: the observable (Heisenberg picture) or the state (Schrödinger picture). The mode is chosen at construction time — see Initialisation and updates.

Heisenberg picture (default)

The observable is expanded as $H = \sum_\nu c_\nu M_\nu$ and propagated backwards through the circuit, applying the gates in reverse order while the reference state is held fixed,

$$H_{j+1} = U_{L-j}^\dagger\, H_j\, U_{L-j}, \qquad H_0 = H,$$

and the expectation value $\mathrm{Tr}[\varrho\, H_L]$ is read off at the end (see Propagation Algorithm).

Schrödinger picture

The roles are swapped: the state $\varrho$ is expanded in the Majorana basis and evolved forwards through the circuit, with the observable held fixed. Under common assumptions for fermionic circuits, length truncation is equivalent in both pictures (Chakraborty et al., 2026), though matching the Heisenberg-picture result requires a slightly looser cutoff on the state (the schrodinger_cutoff setting; see Initialisation and updates). The Schrödinger picture is less common, but useful for evaluating many observables against the same state.

See also

• Propagation Algorithm — the back-propagation loop, gate application, and truncation, written for the Heisenberg picture.
• Initialisation and updates — how to select the picture at construction time.