Quantum Gates Learning Gallery

Explore the I, X, Y, Z, H, S, and T single-qubit gates.

These supplementary illustrations explain quantum gates discussed in the Quantum AI Systems volumes. They were developed as companion learning resources and do not appear as numbered figures in the published books.

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Copyright © QuSciTech Press and Dr. Joe Wilson. Reproduction, redistribution, automated extraction, and commercial reuse are prohibited without written permission.

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7 gate visuals
Identity Gate (I)

Identity Gate (I)

Leaves the qubit unchanged.

a|0⟩ + b|1⟩ → a|0⟩ + b|1⟩

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Pauli-X Gate (Quantum NOT)

Pauli-X Gate (Quantum NOT)

Swaps the amplitudes of |0⟩ and |1⟩.

a|0⟩ + b|1⟩ → b|0⟩ + a|1⟩

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Pauli-Y Gate

Pauli-Y Gate

Swaps basis states and changes their phases.

a|0⟩ + b|1⟩ → −ib|0⟩ + ia|1⟩

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Pauli-Z Gate (Phase Flip)

Pauli-Z Gate (Phase Flip)

Changes relative phase; immediate 0/1 measurement probabilities stay unchanged.

a|0⟩ + b|1⟩ → a|0⟩ − b|1⟩

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Hadamard Gate

Hadamard Gate

Converts basis states into equal superpositions and can reveal phase differences.

H|0⟩ = |+⟩; H|1⟩ = |−⟩

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S Gate (90° Phase Gate)

S Gate (90° Phase Gate)

Adds 90° of relative phase to the |1⟩ component; sends |+⟩ to |+i⟩.

a|0⟩ + b|1⟩ → a|0⟩ + ib|1⟩

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T Gate (45° Phase Gate)

T Gate (45° Phase Gate)

Adds 45° of relative phase to the |1⟩ component.

a|0⟩ + b|1⟩ → a|0⟩ + b e^(iπ/4)|1⟩

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