Single-Qubit States and Gates

Single-Qubit States and Gates#

Pauli matrices, Hadamard, phase gates, rotations on the Bloch sphere. The universal single-qubit gate set.

Single-qubit gates are the building blocks of quantum circuits. Each gate is a \(2 \times 2\) unitary matrix that maps one qubit state to another. On the Bloch sphere, every single-qubit gate corresponds to a rotation. This section catalogs the essential gates and shows how to combine them.

The Pauli Matrices#

The three Pauli matrices, together with the identity, form a basis for all \(2 \times 2\) Hermitian matrices. They are the most important operators in quantum computing.

\[\begin{split}I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\end{split}\]

Properties#

\(X^2 = Y^2 = Z^2 = I\) (each is its own inverse)
\(XY = iZ\), \(YZ = iX\), \(ZX = iY\) (cyclic commutation)
Eigenvalues: \(\pm 1\) for all three
Trace: \(\text{Tr}(X) = \text{Tr}(Y) = \text{Tr}(Z) = 0\)

Actions#

Gate	Action	Bloch sphere
\(X\)	\(\|0\rangle \leftrightarrow \|1\rangle\) (bit flip)	\(180°\) rotation about \(x\)-axis
\(Y\)	\(\|0\rangle \mapsto i\|1\rangle\), \(\|1\rangle \mapsto -i\|0\rangle\)	\(180°\) rotation about \(y\)-axis
\(Z\)	\(\|0\rangle \mapsto \|0\rangle\), \(\|1\rangle \mapsto -\|1\rangle\) (phase flip)	\(180°\) rotation about \(z\)-axis

The Hadamard Gate#

The Hadamard gate creates and destroys superpositions:

\[\begin{split}H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\end{split}\]

\[H|0\rangle = |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad H|1\rangle = |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}\]

Properties:

\(H^2 = I\) (self-inverse)
\(H = \frac{X + Z}{\sqrt{2}}\) (midway between \(X\) and \(Z\))
On the Bloch sphere: \(180°\) rotation about the axis \((\hat{x} + \hat{z})/\sqrt{2}\)

The Hadamard gate changes the measurement basis: measuring \(H|\psi\rangle\) in the \(Z\)-basis is equivalent to measuring \(|\psi\rangle\) in the \(X\)-basis.

Phase Gates#

Phase gates modify the relative phase between \(|0\rangle\) and \(|1\rangle\):

\[\begin{split}R_\phi = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{pmatrix}\end{split}\]

Important special cases:

Gate	\(\phi\)	Matrix	Effect
\(S\)	\(\pi/2\)	\(\begin{pmatrix}1&0\\0&i\end{pmatrix}\)	Quarter turn (\(Z\)-axis)
\(T\)	\(\pi/4\)	\(\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}\)	Eighth turn (\(Z\)-axis)
\(Z\)	\(\pi\)	\(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\)	Half turn (\(Z\)-axis)

The \(T\) gate is especially important: it cannot be efficiently simulated classically when combined with Clifford gates, making it the resource for universal quantum computation.

The Clifford + T Gate Set

The Clifford group (generated by \(H\), \(S\), and CNOT) can be efficiently simulated classically by the Gottesman-Knill theorem. Adding the \(T\) gate makes the set universal: any unitary can be approximated to arbitrary precision. The Solovay-Kitaev theorem guarantees that \(O(\log^c(1/\epsilon))\) gates suffice for precision \(\epsilon\).

Rotation Gates#

General rotations about the Bloch sphere axes:

\[\begin{split}R_x(\theta) = e^{-i\theta X/2} = \begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\end{split}\]

\[\begin{split}R_y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\end{split}\]

\[\begin{split}R_z(\theta) = e^{-i\theta Z/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\end{split}\]

Any single-qubit unitary \(U\) can be decomposed as:

\[U = e^{i\gamma} R_z(\alpha) R_y(\beta) R_z(\delta)\]

for suitable angles \(\gamma, \alpha, \beta, \delta\). This is the \(ZYZ\) decomposition.

Universal Single-Qubit Gate Sets#

A set of gates is universal for single-qubit operations if any single-qubit unitary can be expressed (or approximated) as a finite product of gates from the set.

Gate set	Universal?	Notes
\(\{R_y(\theta), R_z(\phi)\}\) for all \(\theta, \phi\)	Exact	Continuous parameters
\(\{H, T\}\)	Approximate	Dense in SU(2), Solovay-Kitaev applies
\(\{H, S\}\)	No	Generates finite Clifford group only
\(\{X, H\}\)	No	Cannot generate arbitrary phases

Code Examples#

Qiskit

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
import numpy as np

qc = QuantumCircuit(1)

# Apply gates
qc.h(0)          # Hadamard: |0> -> |+>
qc.t(0)          # T gate: adds pi/4 phase to |1>
qc.rx(np.pi/3, 0)  # Rx rotation by pi/3
qc.rz(np.pi/6, 0)  # Rz rotation by pi/6

# Get the statevector
sv = Statevector.from_instruction(qc)
print("State:", sv)
print("Probabilities:", sv.probabilities())

Cirq

import cirq
import numpy as np

qubit = cirq.LineQubit(0)

circuit = cirq.Circuit([
    cirq.H(qubit),                     # Hadamard
    cirq.T(qubit),                     # T gate
    cirq.rx(rads=np.pi/3)(qubit),      # Rx rotation
    cirq.rz(rads=np.pi/6)(qubit),      # Rz rotation
])

print(circuit)

# Simulate
sim = cirq.Simulator()
result = sim.simulate(circuit)
print("State vector:", result.final_state_vector)

PennyLane

import pennylane as qml
import numpy as np

dev = qml.device("default.qubit", wires=1)

@qml.qnode(dev)
def circuit():
    qml.Hadamard(wires=0)
    qml.T(wires=0)
    qml.RX(np.pi/3, wires=0)
    qml.RZ(np.pi/6, wires=0)
    return qml.state()

state = circuit()
print("State:", state)
print("Probabilities:", np.abs(state)**2)

PyQuil

from pyquil import Program
from pyquil.gates import H, T, RX, RZ
from pyquil.quilbase import DefGate
import numpy as np

p = Program()
p += H(0)               # Hadamard
p += T(0)               # T gate
p += RX(np.pi/3, 0)     # Rx rotation
p += RZ(np.pi/6, 0)     # Rz rotation

print(p)

Gate Summary Table#

Gate	Matrix	Inverse	Bloch Rotation
\(X\)	\(\begin{pmatrix}0&1\\1&0\end{pmatrix}\)	\(X\)	\(\pi\) about \(\hat{x}\)
\(Y\)	\(\begin{pmatrix}0&-i\\i&0\end{pmatrix}\)	\(Y\)	\(\pi\) about \(\hat{y}\)
\(Z\)	\(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\)	\(Z\)	\(\pi\) about \(\hat{z}\)
\(H\)	\(\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\)	\(H\)	\(\pi\) about \((\hat{x}+\hat{z})/\sqrt{2}\)
\(S\)	\(\begin{pmatrix}1&0\\0&i\end{pmatrix}\)	\(S^\dagger\)	\(\pi/2\) about \(\hat{z}\)
\(T\)	\(\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}\)	\(T^\dagger\)	\(\pi/4\) about \(\hat{z}\)