VQE, Ansatzes, and ADAPT

VQE uses the variational objective (2) from the theory overview. The sections below cover the core VQE loop, ansatz choices, optimizers, fermion-to-qubit mappings, and adaptive ansatz growth.

The Variational Quantum Eigensolver couples:

  • parameterized quantum circuits

  • classical optimization

Workflow:

  1. prepare ansatz state

  2. measure expectation value

  3. update parameters

  4. iterate to convergence

optimizer → parameters → circuit → expectation → update

Performance depends on:

  • ansatz expressibility

  • optimization landscape

  • Hamiltonian structure

Ansatz Families

Ansätze define the accessible variational manifold.

Tradeoffs:

  • physical structure

  • circuit depth

  • parameter count

  • trainability

UCCSD

Unitary Coupled Cluster Singles and Doubles:

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

with

\[ T = T_1 + T_2 \]

Properties:

  • chemistry motivated

  • interpretable excitation structure

  • strong performance for small molecules

Hardware-efficient ansätze

Example: RY–CZ layered circuits.

Motivations:

  • shallow depth

  • tunable expressibility

  • hardware compatibility

  • useful for benchmarking optimizer behaviour

Minimal ansätze

Low-parameter circuits used for:

  • visualization

  • pedagogical examples

  • landscape analysis

Optimizers

Optimization minimizes:

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

Supported optimizers:

  • Adam

  • Gradient Descent

  • RMSProp

  • Adagrad

  • Momentum / NesterovMomentum

Differences:

  • adaptive learning-rate scaling

  • momentum accumulation

  • noise robustness

Implementation uses a unified optimizer interface shared across ansätze.

Fermion-to-Qubit Mappings

Mapping choice affects circuit structure and optimization behaviour.

Jordan–Wigner

  • direct occupation encoding

  • simple construction

  • longer Pauli strings

Bravyi–Kitaev

  • balanced parity/occupation encoding

  • shorter average Pauli strings

Parity mapping

  • parity-based encoding

  • can expose symmetries

  • may reduce circuit depth

ADAPT-VQE

Adaptive ansatz construction.

Instead of fixing the ansatz size, operators are added iteratively.

Ansatz:

\[ U_k(\theta) = \prod_j e^{\theta_j A_j} \]

Operators selected by gradient magnitude.

Workflow:

  1. optimize current parameters

  2. evaluate gradients for operator pool

  3. append best operator

  4. repeat until convergence

Advantages:

  • compact circuits

  • interpretable operator growth

  • convergence diagnostics