Ansatzes¶

VQE Ansatzes¶

This document describes the parameterized quantum circuits (ansatzes) used in the VQE workflows in this repository.

Implemented ansatz names (as accepted by vqe.engine.build_ansatz):

TwoQubit-RY-CNOT
RY-CZ
Minimal
StronglyEntanglingLayers
UCCSD (UCC-SD)
UCC-D (UCCD)
UCC-S (UCCS)

These are constructed via:

ansatz_fn, params0 = build_ansatz(...)

and used inside QNodes as:

ansatz_fn(params, wires=range(num_wires), ...)

Scope and Role in VQE¶

In VQE, the ansatz defines the trial state

\[ |\psi(\theta)\rangle, \]

and the optimization problem is

\[ \min_{\theta \in \mathbb{R}^d} E(\theta) \]

where

\[ E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle \]

Definitions:

$\theta \in \mathbb{R}^d$: variational parameter vector $d$: number of parameters (ansatz-dependent) $H$: qubit Hamiltonian $|\psi(\theta)\rangle$: quantum state prepared by the ansatz

Common Notation¶

$n$: number of qubits (num_wires)

wires: ordered list of qubit indices $\{0, \dots, n-1\}$ $\theta_i$: individual variational parameters
layers: repeated circuit blocks (if applicable)

All ansatz functions have the interface:

ansatz(params, wires, ...)

Additional arguments (symbols, coordinates, basis) are only used by chemistry ansatzes (UCC family).

Ansatz Registry¶

ANSATZES = {
    "TwoQubit-RY-CNOT": ...,
    "RY-CZ": ...,
    "Minimal": ...,
    "StronglyEntanglingLayers": ...,
    "UCCSD": ...,
    "UCC-SD": ...,
    "UCC-D": ...,
    "UCCD": ...,
    "UCC-S": ...,
    "UCCS": ...,
}

Toy and Hardware-Efficient Ansatzes¶

These ansatzes:

do not depend on molecular structure
ignore symbols, coordinates, basis
use simple gate patterns

1. Minimal Ansatz¶

Definition¶

A 2-qubit circuit embedded in a larger register:

\[ |\psi(\theta)\rangle = \mathrm{CNOT}_{0,1}\, R_Y(\theta)_0\, |0\dots 0\rangle. \]

Circuit (first two wires only)¶

wire 0: ──RY(θ)────●──
                   │
wire 1: ───────────X──

Remaining wires (if any) are untouched.

Parameters¶

$d = 1$

params shape: (1,)

Initialization¶

random normal: $\theta \sim \mathcal{N}(0, \text{scale}^2)$

Interpretation¶

simplest nontrivial entangling circuit
useful for debugging and visualization

2. TwoQubit-RY-CNOT¶

Definition¶

Applies a 2-qubit motif to each adjacent pair:

For each pair $(i, i+1)$:

\[ R_Y(\theta_i)_i \rightarrow \text{CNOT}_{i,i+1} \rightarrow R_Y(-\theta_i)_{i+1} \rightarrow \text{CNOT}_{i,i+1} \]

Circuit (for 3 wires)¶

wire 0: ──RY(θ₀)──●────────────●──
                  │            │
wire 1: ──────────X──RY(-θ₀)───X──RY(θ₁)──●──
                                          │
wire 2: ──────────────────────────────────X──RY(-θ₁)──●──
                                                      │
                                                     X──

Parameters¶

$d = n - 1$

one parameter per adjacent pair

Initialization¶

random normal

Interpretation¶

scalable extension of a 2-qubit motif
introduces structured entanglement along a chain

3. RY-CZ Ansatz¶

Definition¶

Single layer:

Local rotations $$ \prod_{i=0}^{n-1} R_Y(\theta_i)_i $$
CZ chain $$ \prod_{i=0}^{n-2} \text{CZ}_{i,i+1} $$

Circuit (n wires)¶

wire 0: ──RY(θ₀)──●──────────────
                  │
wire 1: ──RY(θ₁)──●──●───────────
                     │
wire 2: ──RY(θ₂)─────●──●────────
                        │
...

Parameters¶

$d = n$

Initialization¶

random normal

Interpretation¶

simple hardware-efficient ansatz
separates local rotations and entangling layer

4. StronglyEntanglingLayers¶

Definition¶

Uses PennyLane template:

qml.templates.StronglyEntanglingLayers(params, wires)

In this repository:

fixed to 1 layer
each wire has 3 rotation parameters

Parameter shape¶

\[ \text{params.shape} = (1, n, 3) \]

Conceptual structure¶

Each layer consists of:

arbitrary single-qubit rotations
dense entangling pattern

Schematic¶

Layer:
  [Rotations on all qubits]
  [Entangling pattern across qubits]

Initialization¶

random normal with scale ~ $\pi$

Interpretation¶

expressive, hardware-efficient ansatz
not chemistry-informed
can be harder to optimize

UCC Chemistry Ansatz Family¶

These ansatzes are chemistry-inspired and depend on:

molecular symbols
geometry (coordinates)
basis set

They use excitation operators derived from quantum chemistry.

General Form¶

All UCC ansatzes follow:

\[ |\psi(\theta)\rangle = e^{T(\theta) - T^\dagger(\theta)} |HF\rangle, \]

where:

$|HF\rangle$: Hartree–Fock reference state
$T(\theta)$: excitation operator

Excitation Construction¶

From:

singles, doubles = qchem.excitations(electrons, spin_orbitals)

singles: $a_a^\dagger a_i$
doubles: $a_a^\dagger a_b^\dagger a_j a_i$

These are mapped to qubit operations via PennyLane templates:

qml.SingleExcitation
qml.DoubleExcitation

Reference State Preparation¶

By default:

qml.BasisState(hf_state, wires=wires)

So:

\[ |\psi_0\rangle = |HF\rangle \]

This is handled inside the ansatz, not externally.

Optional overrides:

reference_state
prepare_reference=False

Parameter Ordering (Important)¶

Parameters are ordered as:

[singles..., doubles...]

first all single-excitation parameters
then all double-excitation parameters

5. UCC-S (Singles Only)¶

Definition¶

\[ T(\theta) = \sum_i \theta_i T_i^{(1)} \]

Parameters¶

$d = \text{number of single excitations}$

Behavior¶

applies only qml.SingleExcitation

6. UCC-D (Doubles Only)¶

Definition¶

\[ T(\theta) = \sum_j \theta_j T_j^{(2)} \]

Parameters¶

$d = \text{number of double excitations}$

Behavior¶

applies only qml.DoubleExcitation

7. UCCSD (Singles + Doubles)¶

Definition¶

\[ T(\theta) = \sum_i \theta_i T_i^{(1)} + \sum_j \theta_j T_j^{(2)} \]

Parameters¶

$d = \text{singles} + \text{doubles}$

Behavior¶

full chemistry ansatz used in most simulations

Initialization (Critical)¶

For all UCC ansatzes:

\[ \theta = 0 \]

i.e.

vals = np.zeros(...)

Interpretation¶

starts exactly at Hartree–Fock state
consistent with quantum chemistry workflows
gradients drive initial updates

Dependence on Molecular System¶

For UCC ansatzes:

parameter count depends on:
- number of electrons
- number of spin-orbitals
therefore varies with:
- molecule
- basis
- geometry

Parameter Initialization Summary¶

Ansatz	Initialization
Minimal	small random
TwoQubit-RY-CNOT	small random
RY-CZ	small random
StronglyEntanglingLayers	random ~ $\pi$
UCC family	all zeros

When to Use Each Ansatz¶

Minimal / Toy ansatzes¶

Use when:

debugging
visualizing energy landscapes
testing optimizers

RY-CZ / TwoQubit-RY-CNOT¶

Use when:

comparing optimizers
studying trainability
hardware-oriented experiments

StronglyEntanglingLayers¶

Use when:

high expressibility is needed
not focusing on chemistry accuracy
exploring general variational behavior

UCC-S / UCC-D / UCCSD¶

Use when:

performing quantum chemistry simulations
comparing with classical methods
studying physically meaningful states

Typical hierarchy:

UCC-S → simplest
UCC-D → captures correlation
UCCSD → most complete (default)

Summary Table¶

Ansatz	Type	Parameters	Uses HF reference	Chemistry-aware
Minimal	Toy	$1$	No	No
TwoQubit-RY-CNOT	Toy / scalable	$n-1$	No	No
RY-CZ	Hardware-efficient	$n$	No	No
StronglyEntanglingLayers	Hardware-efficient	$3n$	No	No
UCC-S	Chemistry	system-dependent	Yes	Yes
UCC-D	Chemistry	system-dependent	Yes	Yes
UCCSD	Chemistry	system-dependent	Yes	Yes

Ansatz	Type	Parameters	Uses HF reference	Chemistry-aware
Minimal	Toy	\(1\)	No	No
TwoQubit-RY-CNOT	Toy / scalable	\(n-1\)	No	No
RY-CZ	Hardware-efficient	\(n\)	No	No
StronglyEntanglingLayers	Hardware-efficient	\(3n\)	No	No
UCC-S	Chemistry	system-dependent	Yes	Yes
UCC-D	Chemistry	system-dependent	Yes	Yes
UCCSD	Chemistry	system-dependent	Yes	Yes

Ansatzes¶

VQE Ansatzes¶

Scope and Role in VQE¶

Common Notation¶

Ansatz Registry¶

Toy and Hardware-Efficient Ansatzes¶

1. Minimal Ansatz¶

Definition¶

Circuit (first two wires only)¶

Parameters¶

Initialization¶

Interpretation¶

2. TwoQubit-RY-CNOT¶

Definition¶

Circuit (for 3 wires)¶

Parameters¶

Initialization¶

Interpretation¶

3. RY-CZ Ansatz¶

Definition¶

Circuit (n wires)¶

Parameters¶

Initialization¶

Interpretation¶

4. StronglyEntanglingLayers¶

Definition¶

Parameter shape¶

Conceptual structure¶

Schematic¶

Initialization¶

Interpretation¶

UCC Chemistry Ansatz Family¶

General Form¶

Excitation Construction¶

Reference State Preparation¶

Parameter Ordering (Important)¶

5. UCC-S (Singles Only)¶

Definition¶

Parameters¶

Behavior¶

6. UCC-D (Doubles Only)¶

Definition¶

Parameters¶

Behavior¶

7. UCCSD (Singles + Doubles)¶

Definition¶

Parameters¶

Behavior¶

Initialization (Critical)¶

Interpretation¶

Dependence on Molecular System¶

Parameter Initialization Summary¶

When to Use Each Ansatz¶

Minimal / Toy ansatzes¶

RY-CZ / TwoQubit-RY-CNOT¶

StronglyEntanglingLayers¶

UCC-S / UCC-D / UCCSD¶

Summary Table¶

See Also¶