UCCSD Ansatz

Overview

The Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz is the gold standard for quantum chemistry simulations. It systematically includes electron correlation effects through single and double excitation operators.

Property	Value
Category	Ansatz
Difficulty	Advanced
Framework	Cirq
Qubits	4
Depth	O(n⁴)
Gates	RY, RZ, CNOT

Mathematical Form

CODE
|ψ(θ)⟩ = exp(T - T†)|HF⟩

Where:

CODE
T = T₁ + T₂
T₁ = Σ θ_ia a†_a a_i       (single excitations)
T₂ = Σ θ_ijab a†_a a†_b a_j a_i  (double excitations)

Circuit Structure

Single Excitation (Givens Rotation)

CODE
q_i: ──●──RY(θ)──●──
       │         │
q_a: ──X─────────X──

Double Excitation (Simplified)

CODE
q_0: ──●────────────●──
       │            │
q_1: ──X──●─────●──X──
          │     │
q_2: ─────X─RZ──X─────
          │     │
q_3: ─────●─────●─────

Running the Circuit

PYTHON
from circuit import run_circuit

result = run_circuit(n_qubits=4, n_electrons=2)

print(f"Singles: {result['n_singles']}")
print(f"Doubles: {result['n_doubles']}")
print(f"Parameters: {result['n_params']}")

Parameter Scaling

System	Singles	Doubles	Total
H₂ (4q)	4	1	5
LiH (6q)	8	4	12
H₂O (10q)	20	45	65

Comparison with HEA

Aspect	UCCSD	HEA
Physical motivation	Yes	No
Parameter efficiency	Higher	Lower
Circuit depth	Higher	Lower
Convergence	Better	Variable

Applications

Ground state chemistry: Molecular energies
Bond dissociation: Reaction pathways
Excited states: Via VQD extension