This post covers the basic units of quantum computation, qubits, and the key quantum gates used to manipulate them, all grounded in developments up to 2019.

Summary

A qubit is the quantum analog of a classical bit but can exist in superposition, typically represented on a Bloch sphere. Quantum gates are unitary operations that change qubits’ states, analogous to logic gates in classical circuits. Together, single-qubit gates (like Pauli, Hadamard, and phase gates) and multi-qubit gates (like CNOT) form a universal set for quantum computation.

Qubits and the Bloch Sphere

A qubit is a two-level quantum system with basis states |0⟩ and |1⟩. Any pure qubit state |ψ⟩ can be written as:

$$ |ψ⟩ = cos\left(\frac{θ}{2}\right) |0⟩ + e^{iφ} sin\left(\frac{θ}{2}\right) |1⟩, $$

where θ and φ are real parameters. This state maps to a point on the Bloch sphere: θ is the polar angle and φ the azimuthal angle [1].

Single‑Qubit Gates

Pauli Gates

The Pauli gates X, Y, and Z correspond to rotations of the Bloch sphere by π about the x-, y-, and z-axes, respectively. Their matrix representations are:

X: $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ flips |0⟩ and |1⟩ [2].

Y: $$ \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$ combines bit- and phase-flip [2].

Z: $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ applies a π phase to |1⟩ [2].

Hadamard Gate

The Hadamard gate (H) creates superposition by mapping:

$$ H|0⟩ = \frac{|0⟩ + |1⟩}{\sqrt{2}},\quad H|1⟩ = \frac{|0⟩ - |1⟩}{\sqrt{2}}. $$

Its matrix is $$H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \ 1 & -1\end{pmatrix}$$ [3].

Phase and T Gates

The phase gate $S$ (also called $P$) applies a $π/2$ phase: $$S = \begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix}$$ The $T$ gate applies a $π/4$ phase: $$T = \begin{pmatrix}1 & 0 \\ 0 & e^{i\pi/4}\end{pmatrix}$$

Both are crucial for achieving universality when combined with other gates [4].

Multi‑Qubit Gates

Controlled-NOT (CNOT)

$CNOT$ acts on two qubits: a control and a target. It flips the target if the control is $|1⟩$. In the computational basis:

$$ CNOT = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix} $$

CNOT generates entanglement and is central to algorithms like Bell state preparation [5].

Universality of Gate Sets

A set of gates is universal if any unitary operation can be approximated using them. Single‑qubit unitaries plus CNOT form a universal set [6]. Alternative universal sets include ${H,T,CNOT}$ and Toffoli plus Hadamard [7].

References

[1] Wikipedia contributors. (2019). Bloch sphere. Wikipedia. https://en.wikipedia.org/wiki/Bloch_sphere

[2] ShareTechnote. (2019). Pauli gates (X, Y, Z). https://www.sharetechnote.com/html/QC/QuantumComputing_Gate_X.html

[3] Quantum Inspire. (2019). Hadamard gate. https://www.quantum-inspire.com/kbase/hadamard/

[4] Wikipedia contributors. (2019). Quantum logic gate. Wikipedia. https://en.wikipedia.org/wiki/Quantum_logic_gate

[5] Wikipedia contributors. (2019). Controlled NOT gate. Wikipedia. https://en.wikipedia.org/wiki/Controlled_NOT_gate

[6] Schmassmann, A. (2007). Universality of two-level gates. ETH Zürich. https://qudev.phys.ethz.ch/static/content/courses/QSIT07/presentations/Schmassmann.pdf

[7] Quantum Computing StackExchange. (2019). What are quantum gates necessary for universal computation? https://quantumcomputing.stackexchange.com/questions/37061/what-are-the-quantum-gates-necessary-for-universal-quantum-computation

[8] Nielsen, M. A., & Chuang, I. L. (2000). Quantum Computation and Quantum Information. Cambridge University Press.

[9] PennyLane. (2019). What is a Hadamard gate? https://pennylane.ai/qml/glossary/what-is-a-hadamard-gate

[10] Quantum Inspire. (2019). CNOT Gate. https://www.quantum-inspire.com/kbase/cnot/