  [0.00]{} ![ Illustration of 0-flag circuits for measuring (a) $Z^{\otimes 0}$ requiring only two flag qubits and (b) $Z^{\otimes 0}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag0CircuitExamples"}](Weight0Generator.png "fig:"){width="\textwidth"}  [0.00]{} ![ Illustration of 0-flag circuits for measuring (a) $Z^{\otimes 0}$ requiring only two flag qubits and (b) $Z^{\otimes 0}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag0CircuitExamples"}](Weight0Generator.png "fig:"){width="\textwidth"}  