We here use 𝑘 to index our qubits, where each qubit 𝑘 maps to a unique spin orbital, 𝜓 12 . 𝑋 and 𝑌 represent the corresponding Pauli gates, whereas 𝑍 B→ is the application of Pauli 𝑍 gates to all qubits 𝑘 -1, 𝑘 -2, … , 1. Note that Eqs. ( ) and ( ) are identical except for the sign (phase) of the Y-gate. In practice, this similarity means that the measurements of ⟨𝑎