QAOA Mixer Ansatz

Overview

The Quantum Approximate Optimization Algorithm (QAOA) uses alternating cost and mixer layers. This circuit explores different mixer designs for various problem constraints.

Property	Value
Category	Ansatz
Difficulty	Advanced
Framework	Cirq
Qubits	4
Depth	O(p × n)
Gates	H, RX, RZZ, ISWAP

Structure

CODE
|ψ(γ,β)⟩ = [Π_p U_M(β_p) U_C(γ_p)] |+⟩^⊗n

Where:

U_C(γ) = exp(-iγC): Cost unitary
U_M(β) = exp(-iβB): Mixer unitary

Mixer Types

Standard X Mixer

CODE
B = Σ X_i
U_M(β) = Π exp(-iβX_i) = Π RX(2β)

Explores all states freely.

XY Mixer

CODE
B = Σ (X_i X_{i+1} + Y_i Y_{i+1})

Preserves Hamming weight - for constrained problems.

Circuit Diagram

CODE
┌───┐ ┌─────────────┐ ┌────────┐
q_0: ┤ H ├─┤ Cost(γ)    ├─┤ RX(2β) ├───
     ├───┤ │             │ ├────────┤
q_1: ┤ H ├─┤  ZZ gates   ├─┤ RX(2β) ├───
     ├───┤ │             │ ├────────┤
q_2: ┤ H ├─┤             ├─┤ RX(2β) ├───
     ├───┤ └─────────────┘ ├────────┤
q_3: ┤ H ├─────────────────┤ RX(2β) ├───
     └───┘                 └────────┘

Running the Circuit

PYTHON
from circuit import run_circuit

# Standard mixer
result = run_circuit(n_qubits=4, p=2, mixer='x')

# XY mixer for constrained problems
result = run_circuit(n_qubits=4, p=2, mixer='xy')

print(f"Best cut: {result['best_cut_found']}")
print(f"Approx ratio: {result['approximation_ratio']:.2%}")

Expected Performance

p	Approx Ratio
1	~0.69
2	~0.79
3	~0.84
∞	1.0

Applications

MaxCut: Graph partitioning
Scheduling: With XY mixer
Routing: TSP, VRP
Portfolio: Asset selection