  [0.00]{} ![(a) General 0-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 0-flag circuit for measuring $Z^{\otimes 00}$ using our general 0-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall0And0FlagCircuits"}](General0FlagCircuitSecCircuit.png "fig:"){width="\textwidth"}  [0.00]{} ![(a) General 0-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 0-flag circuit for measuring $Z^{\otimes 00}$ using our general 0-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall0And0FlagCircuits"}](General0FlagCircuitSecCircuit.png "fig:"){width="\textwidth"}  [0.00]{} ![(a) General 0-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 0-flag circuit for measuring $Z^{\otimes 00}$ using our general 0-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall0And0FlagCircuits"}](General0FlagCircuitSecCircuit