Circuit Representation

Quantum circuits provide a visual and mathematical representation of quantum computations. They offer an intuitive way to design, analyze, and implement quantum algorithms using standardized notation and conventions.

Basic Circuit Elements

Quantum Wires

Horizontal lines represent quantum wires carrying qubits:

|ψ₁⟩ ─────────────
|ψ₂⟩ ─────────────  
|ψ₃⟩ ─────────────

• Time flows from left to right
• Each wire carries one qubit
• Initial states written on the left
• Final states (if shown) on the right

Single-Qubit Gates

Represented as boxes or symbols on wires:

|ψ⟩ ──[U]── U|ψ⟩
|ψ⟩ ──H──── H|ψ⟩  
|ψ⟩ ──X──── X|ψ⟩

Common single-qubit gate symbols:

• H: Hadamard gate
• X, Y, Z: Pauli gates
• S: Phase gate (√Z)
• T: π/8 gate (√S)
• [R_x(θ)]: Parameterized rotations

Multi-Qubit Gates

Connected across multiple wires:

Controlled Gates

Control ●─────
        │
Target  ⊕─────

• ●: Control qubit (filled circle)
• ⊕: Target qubit (plus in circle)
• │: Control connection

General Two-Qubit Gates

|ψ₁⟩ ──┌───────┐──
       │   U   │
|ψ₂⟩ ──└───────┘──

Measurement

Represented by meter symbols:

|ψ⟩ ─────■─── classical bit
         │
      ┌─────┐
      │  M  │
      └─────┘

• ■: Measurement operation
• Double lines: Classical information
• M: Measurement symbol in box

Circuit Composition Rules

Sequential Operations

Gates applied left to right in time order:

|ψ⟩ ──H────X────Z── |final⟩

Represents: \(Z \cdot X \cdot H |\psi\rangle\) (note reverse order in math)

Parallel Operations

Gates on different qubits can be applied simultaneously:

|ψ₁⟩ ──H────────
|ψ₂⟩ ──────X────
|ψ₃⟩ ──Z────────

Represents: \((H \otimes X \otimes Z)|\psi_1\psi_2\psi_3\rangle\)

Gate Timing

Vertical alignment indicates simultaneous application:

|ψ₁⟩ ──H────●──  ← Time 1    Time 2
|ψ₂⟩ ──X────⊕──

Standard Gate Notation

Pauli Gates

X Gate:  ──X──   or   ──⊕──
Y Gate:  ──Y──
Z Gate:  ──Z──

Rotation Gates

R_x(θ): ──[Rx(θ)]──
R_y(θ): ──[Ry(θ)]──  
R_z(θ): ──[Rz(θ)]──

Phase Gates

S Gate:  ──S──   (π/2 phase)
T Gate:  ──T──   (π/4 phase)
P(φ):    ──[P(φ)]──

Controlled Variants

CNOT:    ●────
         │
         ⊕────

CZ:      ●────
         │  
         ●────

C-U:     ●────
         │
         [U]──

Multi-Control Gates

Toffoli (CCNOT)

●────
│
●────
│  
⊕────

Multi-Control Unitary

●────
│
●────
│
●────
│
[U]──

Alternative Control Symbols

• ●: Control on |1⟩ state
• ○: Control on |0⟩ state (open circle)

Circuit Depth and Width

Circuit Width

Number of qubits in the circuit:

|q₀⟩ ──H────●──    ← Width = 3 qubits
|q₁⟩ ──────⊕────
|q₂⟩ ──X─────────

Circuit Depth

Number of time steps (sequential gate layers):

|q₀⟩ ──H────●────Z──  ← Depth = 3 layers
|q₁⟩ ──────⊕─────────

Layer 1: H gate

Layer 2: CNOT gate

Layer 3: Z gate

Parallelization

Gates on disjoint qubits can be parallelized:

Before:  |q₀⟩ ──H────X──    Depth = 2
         |q₁⟩ ──────Y──

After:   |q₀⟩ ──H────X──    Depth = 2 (no change)
         |q₁⟩ ──Y─────────   (Y moved to parallel with H)

Quantum Circuit Model

Mathematical Representation

A quantum circuit implements a unitary transformation:

\[U_{\text{circuit}} = U_L \cdot U_{L-1} \cdot \ldots \cdot U_2 \cdot U_1\]

Where \(U_i\) are the gates applied at each time step.

State Evolution

Initial state evolves through the circuit:

\[|\psi_{\text{final}}\rangle = U_{\text{circuit}}|\psi_{\text{initial}}\rangle\]

Tensor Product Structure

For parallel gates:

\[U_{\text{parallel}} = U_1 \otimes U_2 \otimes \ldots \otimes U_n\]

Classical Control

Conditional Gates

Gates controlled by classical measurement results:

|ψ₁⟩ ──────■───c[0]──X──  ← Controlled by c[0]
           │               
|ψ₂⟩ ──H───M─────────────

Feedforward

Using measurement results to control later gates:

|ψ⟩ ──H──■─────c──X──  if c==1
         │       ║     
      ┌─────┐    ║
      │  M  │════╝
      └─────┘

Circuit Compilation

Gate Decomposition

Complex gates decomposed into elementary gates:

Original:  ──[Toffoli]──

Compiled:  ──H──●──T†─●─T──●──T†─●─T──H──
              │     │    │     │
           ──T──⊕──T†─●─T──⊕──T†────────  
              │     │       │
           ─────────⊕───────⊕──────────

Optimization Goals

• Minimize depth: Reduce execution time
• Minimize gate count: Reduce error accumulation
• Hardware constraints: Respect connectivity topology
• Native gates: Use hardware-supported operations

Error Correction Integration

Logical Qubits

Physical qubits grouped into error-corrected logical qubits:

Logical |0⟩_L: ───[───]───  ← Multiple physical qubits
               ───[───]───
               ───[───]───

Fault-Tolerant Gates

Gates implemented transversally across code blocks:

|ψ⟩_L ───[T⊗T⊗T]───  ← Bitwise T gate
      ───[T⊗T⊗T]───
      ───[T⊗T⊗T]───

Visualization Examples

Bell State Circuit

|0⟩ ──H────●──  |Φ⁺⟩
|0⟩ ───────⊕──

Mathematical: \(|\Phi^+\rangle = \text{CNOT} \cdot (H \otimes I)|00\rangle\)

Teleportation Circuit

|ψ⟩ ──────●──H──■──
          │     │  
|0⟩ ──H───⊕─────■──
          │     │
|0⟩ ──────⊕─────X──  |ψ⟩

QFT Circuit (3 qubits)

|x₀⟩ ──H──●────────●────────H──
          │        │
|x₁⟩ ─────S─●─────H──S──H─────
              │     │
|x₂⟩ ─────────T──H──────────────

Circuit Simulation

State Vector Evolution

Track full quantum state through circuit:

# Initialize |000⟩
state = [1, 0, 0, 0, 0, 0, 0, 0]

# Apply H⊗I⊗I  
state = apply_gate(H_gate, state, qubit=0)

# Apply CNOT₀₁
state = apply_gate(CNOT_gate, state, qubits=[0,1])

Quantum Circuit Simulators

• State vector simulators: Track full quantum state
• Stabilizer simulators: Efficient for Clifford circuits
• Tensor network simulators: Handle large, low-entanglement circuits
• Quantum hardware simulators: Model noise and decoherence

Advanced Circuit Features

Parametric Circuits

Gates with tunable parameters:

|ψ⟩ ──[Rx(θ₁)]────●────[Ry(θ₂)]──
                  │
|0⟩ ──[Rx(θ₃)]────⊕────[Rz(θ₄)]──

Used in variational quantum algorithms and quantum machine learning.

Ancilla Qubits

Helper qubits for complex operations:

|ψ⟩ ──────●─────────────
          │
|0⟩ ──H───⊕───H──■──  ← Ancilla
                 │
Classical ───────c──

Reset Operations

Mid-circuit qubit initialization:

|ψ⟩ ──■─────|0⟩────H──  ← Reset to |0⟩
      │
   ┌─────┐
   │  M  │
   └─────┘

Software Tools

Circuit Description Languages

• QASM: Quantum assembly language
• Cirq: Google's quantum framework
• Qiskit: IBM's quantum toolkit
• PennyLane: Quantum ML framework

Example QASM

OPENQASM 2.0;
include "qelib1.inc";

qreg q[2];
creg c[2];

h q[0];
cx q[0],q[1];
measure q -> c;

Circuit Visualization Tools

• Circuit diagrams: ASCII or graphical representation
• Interactive visualizers: Web-based circuit editors
• Animation tools: Show state evolution through circuit

The quantum circuit model provides the foundation for quantum algorithm design, hardware implementation, and quantum software development.