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Single-Qubit Gates

Single-qubit gates are unitary transformations that operate on individual qubits in the quantum register. Each gate applies a 2×2 unitary matrix to the target qubit while preserving the joint Hilbert space structure.

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Hadamard Gate (H)

The Hadamard gate creates an equal superposition between |0⟩ and |1⟩ basis states.

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Matrix Representation

H = 1/√2 × [[1,  1],
            [1, -1]]

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Implementation

From quantum_computer.py:861-873: 
class HadamardGate(IQuantumGate):
    """H = [[1,1],[1,-1]] / sqrt(2)."""
    
    def apply(self, state, backend, targets, params):
        s = 1.0 / math.sqrt(2.0)
        u = torch.tensor([[s, s], [s, -s]], dtype=torch.complex64)
        for t in targets:
            state = _single_qubit_unitary(state, t, u, backend)
        return state

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Usage Example

from quantum_computer import QuantumCircuit, QuantumComputer

circuit = QuantumCircuit(n_qubits=1)
circuit.h(0)  # Apply Hadamard to qubit 0

qc = QuantumComputer()
result = qc.run(circuit)
# Result: |0⟩ → (|0⟩ + |1⟩)/√2

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Pauli-X Gate (X)

The Pauli-X gate is the quantum NOT gate, flipping |0⟩ ↔ |1⟩.

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Matrix Representation

X = [[0, 1],
     [1, 0]]

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Implementation

From quantum_computer.py:876-888: 
class PauliXGate(IQuantumGate):
    """X = [[0,1],[1,0]]."""
    
    def apply(self, state, backend, targets, params):
        u = torch.tensor([[0, 1], [1, 0]], dtype=torch.complex64)
        for t in targets:
            state = _single_qubit_unitary(state, t, u, backend)
        return state

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Usage Example

circuit = QuantumCircuit(n_qubits=1)
circuit.x(0)  # Apply X gate to qubit 0

result = qc.run(circuit)
# Result: |0⟩ → |1⟩

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Pauli-Y Gate (Y)

The Pauli-Y gate applies a π rotation around the Y-axis of the Bloch sphere.

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Matrix Representation

Y = [[0, -i],
     [i,  0]]

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Implementation

From quantum_computer.py:890-902: 
class PauliYGate(IQuantumGate):
    """Y = [[0,-i],[i,0]]."""
    
    def apply(self, state, backend, targets, params):
        u = torch.tensor([[0, -1j], [1j, 0]], dtype=torch.complex64)
        for t in targets:
            state = _single_qubit_unitary(state, t, u, backend)
        return state

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Usage Example

circuit = QuantumCircuit(n_qubits=1)
circuit.y(0)  # Apply Y gate to qubit 0

result = qc.run(circuit)
# Result: |0⟩ → i|1⟩

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Pauli-Z Gate (Z)

The Pauli-Z gate applies a phase flip: |0⟩ → |0⟩, |1⟩ → -|1⟩.

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Matrix Representation

Z = [[1,  0],
     [0, -1]]

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Implementation

From quantum_computer.py:904-916: 
class PauliZGate(IQuantumGate):
    """Z = [[1,0],[0,-1]]."""
    
    def apply(self, state, backend, targets, params):
        u = torch.tensor([[1, 0], [0, -1]], dtype=torch.complex64)
        for t in targets:
            state = _single_qubit_unitary(state, t, u, backend)
        return state

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Usage Example

circuit = QuantumCircuit(n_qubits=1)
circuit.z(0)  # Apply Z gate to qubit 0

result = qc.run(circuit)
# Result: |1⟩ → -|1⟩

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Identity Gate (I)

The identity gate leaves the qubit state unchanged. While not explicitly implemented as a separate gate class, it corresponds to applying no operation or can be represented using the gate registry.

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Matrix Representation

I = [[1, 0],
     [0, 1]]

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Usage

The identity operation is implicitly applied when no gate is specified for a qubit in a circuit layer.

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Gate Registry

All single-qubit gates are registered in the _GATE_REGISTRY dictionary (quantum_computer.py:1157-1174): 
_GATE_REGISTRY: Dict[str, IQuantumGate] = {
    "H":      HadamardGate(),
    "X":      PauliXGate(),
    "Y":      PauliYGate(),
    "Z":      PauliZGate(),
    "S":      SGate(),
    "T":      TGate(),
    # ... other gates
}

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Technical Details

All single-qubit gates operate via the _single_qubit_unitary function (quantum_computer.py:737-784), which:
  1. Identifies basis state pairs (k0, k1) that differ only in the target qubit bit
  2. Applies the 2×2 unitary matrix to the amplitude pair:
    • α’ = u[0,0]×α + u[0,1]×α_
    • α’ = u[1,0]×α + u[1,1]×α_
  3. Handles complex multiplication of matrix elements with spatial wavefunctions
  4. Preserves the joint Hilbert space structure for entanglement

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See Also

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