  The Wigner function formalism serves as a crucial tool for simulating
continuous-variable and odd-prime dimensional quantum circuits, as well as
assessing their classical hardness. However, applying such a formalism to qubit
systems is limited due to the negativity in the Wigner function induced by
Clifford operations. In this work, we introduce a novel classical simulation
method for non-adaptive Clifford circuits based on the framed Wigner function,
an extended form of the qubit Wigner function characterized by a binary-valued
frame function. Our approach allows for updating phase space points under
Clifford circuits without inducing negativity in the Wigner function by
switching to a suitable frame when applying each Clifford gate. By leveraging
this technique, we establish a sufficient condition for efficient classical
simulation of Clifford circuits even with non-stabilizer inputs, where direct
application of the Gottesmann-Knill tableau method is not feasible. We further
develop a graph-theoretical approach to identify classically simulatable
marginal outcomes of Clifford circuits and explore the number of simulatable
qubits of log-depth circuits. We also present the Born probability estimation
scheme using the framed Wigner function and discuss its precision. Our approach
opens new avenues for quasi-probability simulation of quantum circuits, thereby
expanding the boundary of classically simulatable circuits.
