Exact & formalControlled gate
Controlled-Y gate
A controlled version of the Pauli-Y gate that flips the target bit and attaches a ±i phase exactly when the control qubit is |1⟩, completing the CX/CY/CZ family of controlled Paulis.
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The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
The design was verified by construction plus measured evidence: statistical re-execution, small-instance analytic agreement, sub-block, echo, or invariant checks. Scale-specific bugs can still survive.
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A controlled version of the Pauli-Y gate that flips the target bit and attaches a ±i phase exactly when the control qubit is |1⟩, completing the CX/CY/CZ family of controlled Paulis.
A diagonal two-qubit entangling gate that applies a -1 phase exactly when both qubits are |1⟩, and is symmetric between the two qubits it acts on.
A two-qubit permutation gate defined as two CNOTs with reversed control and target applied back to back — a genuinely different two-CNOT gate from SWAP (which uses three), and notable for not being self-inverse.
The single-qubit no-op gate that leaves every state exactly unchanged, used as a placeholder in circuit diagrams, a timing/idle slot on real hardware, and the base case for gate-composition identities.
The Hermitian conjugate (and inverse) of the S gate, applying a -π/2 phase to |1⟩ and undoing whatever an S gate did earlier in a circuit.
A quarter-turn phase gate that exposes the difference between global and relative phase.
A Clifford square root of the Pauli-X gate that is the native physical single-qubit gate (an X_{π/2} pulse) on most superconducting hardware.
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