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Topic 2 of 12

Quantum Gates & Circuits

Quantum gates are the quantum analogues of classical logic gates. They are represented by unitary matrices — invertible operations that preserve the normalization of quantum states. Every quantum computation is a sequence of such gates, and a small universal set suffices to approximate any quantum operation to arbitrary precision.

Key Concepts

Unitary Matrix

A matrix

U

satisfying

U^\dagger U = UU^\dagger = I

, where

U^\dagger

is the conjugate transpose. Unitarity ensures that gate operations are reversible and preserve the norm of quantum states — a physical necessity.

Pauli Gates

Three fundamental single-qubit gates:

X = \begin{pmatrix}0&1\\1&0\end{pmatrix}

(bit flip, quantum NOT),

Y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}

, and

Z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}

(phase flip). Together with the identity they form a basis for all

2\times2

Hermitian matrices.

Hadamard Gate

The gate

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

maps

|0\rangle \to |{+}\rangle = \frac{|0\rangle+|1\rangle}{\sqrt{2}}

and

|1\rangle \to |{-}\rangle = \frac{|0\rangle-|1\rangle}{\sqrt{2}}

. It is its own inverse (

H^2 = I

) and creates equal superpositions — the first step of almost every quantum algorithm.

CNOT Gate

A two-qubit gate that flips the target qubit if and only if the control qubit is

|1\rangle

|c\rangle|t\rangle \to |c\rangle|t \oplus c\rangle

. CNOT is the prototypical entangling gate and, combined with single-qubit gates, forms a universal gate set.

Universality

A gate set is universal if any unitary operation on

n

qubits can be approximated to arbitrary accuracy using gates from the set. The set

\{H, T, \text{CNOT}\}

is universal, where

T = \text{diag}(1, e^{i\pi/4})

. The Solovay-Kitaev theorem guarantees efficient approximation.

Quantum Circuit Model

Quantum computation is represented as a sequence of quantum gates applied to qubits in a circuit diagram. Time flows left to right. Unlike classical circuits, all gates are reversible. Measurements (irreversible) appear at the end or mid-circuit.

Key Equations

Pauli-X (NOT) Gate

X = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \quad X|0\rangle = |1\rangle, \quad X|1\rangle = |0\rangle

Bit flip gate: swaps |0⟩ and |1⟩ amplitudes.

Hadamard Gate

H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix}, \quad H|0\rangle = |{+}\rangle, \quad H|1\rangle = |{-}\rangle

Creates equal superposition; H² = I.

CNOT Gate

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

Two-qubit gate: flips target qubit when control is |1⟩.

Phase Gate T

T = \begin{pmatrix}1 & 0 \\ 0 & e^{i\pi/4}\end{pmatrix}

π/8 gate; adds a relative phase of e^{iπ/4} to |1⟩. Together with H and CNOT forms a universal set.

Unitarity Condition

U^\dagger U = I \implies \text{all quantum gates are reversible}

Every quantum gate has an inverse gate, unlike classical irreversible gates like AND or OR.

Worked Example

Applying the Hadamard Gate

Problem

Show that applying $H$ twice to $|0\rangle$ returns $|0\rangle$ , demonstrating $H^2 = I$ .

Solution

Apply H to |0⟩:

H|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix} = |{+}\rangle

Apply H again to |+⟩:

H|{+}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix} = \frac{1}{2}\begin{pmatrix}2\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix} = |0\rangle

Answer H²|0⟩ = |0⟩, confirming H² = I

Practice

Exercises

5 problems

1 of 5

Apply the $X$ gate to $|0\rangle$ . What is $P(|1\rangle)$ after measurement?

Your answer

2 of 5

Apply $H$ to $|0\rangle$ to get $|{+}\rangle$ . What is $P(|0\rangle)$ upon measurement?

Your answer

3 of 5

Apply $H$ then $H$ again to $|0\rangle$ (since $H^2=I$ ). What is $P(|0\rangle)$ ?

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4 of 5

Apply $Z$ to $|1\rangle \to -|1\rangle$ . What is $P(|1\rangle)$ ? (Global phase has no effect.)

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5 of 5

A general 2-qubit unitary can be decomposed into at most how many CNOT gates (Shende et al. 2004 tight bound)?

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Key Takeaways

Quantum gates are unitary matrices; $U^\dagger U = I$ ensures reversibility and norm preservation.
Pauli gates X, Y, Z are the single-qubit building blocks; X is the quantum NOT gate.
The Hadamard $H$ creates equal superpositions and is its own inverse: $H^2 = I$ .
CNOT entangles two qubits — it is the essential two-qubit gate for quantum algorithms.
The set $\{H, T, \text{CNOT}\}$ is universal: any quantum algorithm can be compiled into these three gates.