Chemistry and Background

This page covers the shared chemistry layer and the basic spectral problem used by all algorithm families.

All algorithms operate on a unified molecular and Hamiltonian infrastructure to ensure physically consistent comparisons.

Molecule

Typical uses

Basis

Approx. mapped qubits

H₂

ansatz studies, optimizer behaviour, QPE examples

STO-3G

4

LiH

bond-length scans

STO-3G

12

H₂O

bond-angle scans

STO-3G

14

H₃⁺

mapping comparisons, ADAPT-VQE, excited states

STO-3G

6

Shared assumptions:

  • molecular registry → common/molecules.py

  • geometry generation → common/geometry.py

  • Hamiltonian construction → common/hamiltonian.py

  • STO-3G basis used for consistency across examples

Using a shared chemistry layer ensures that differences between VQE, QPE, VarQITE, and VarQRTE reflect algorithmic behaviour rather than inconsistent physical inputs.

Background

Quantum chemistry algorithms in this repository approximate eigenvalues and eigenstates of electronic Hamiltonians for small molecular systems.

Workflow:

  1. choose molecule and geometry

  2. construct electronic Hamiltonian

  3. map fermionic operators to qubit operators

  4. apply quantum algorithm to estimate spectral properties

Centralizing steps 1–3 ensures consistent physical inputs across algorithm families.

Variational Principle

For normalized states:

\[ E_0 \le \langle \psi | H | \psi \rangle \]

Use a parameterized state family:

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

Minimizing the expectation value over that family produces the best approximation to the ground state accessible within the chosen ansatz manifold.

Variational methods rely on:

  • differentiable parameterized circuits

  • classical optimization loops

  • expectation-value estimation

Why qubit mappings are needed

Electronic Hamiltonians are expressed using fermionic creation and annihilation operators. Quantum circuits act on qubits, so the Hamiltonian must be mapped to qubit operators.

Common mappings:

  • Jordan–Wigner

  • Bravyi–Kitaev

  • parity mapping

These mappings preserve physical observables but affect:

  • Pauli-string locality

  • circuit depth

  • measurement grouping

  • optimizer conditioning

Hartree-Fock Reference State

Many workflows begin from a Hartree–Fock (HF) reference determinant.

\[ |HF\rangle = |\phi_1 \phi_2 \cdots \phi_n\rangle \]

In qubit form, this becomes an occupation-number bitstring.

Example:

|111100000000⟩

Roles of the HF reference:

  • initialization for chemistry-inspired ansätze

  • reference state for ADAPT-VQE

  • approximate eigenstate input for QPE

  • starting state for imaginary-time evolution

From classical chemistry to quantum algorithms

Once the qubit Hamiltonian is constructed, algorithm families differ in how they extract spectral information:

method

mechanism

VQE

variational energy minimization

post-VQE methods

linear algebra in reduced manifolds

QPE

phase extraction from unitary evolution

VarQITE

imaginary-time filtering

VarQRTE

projected real-time dynamics

All share identical Hamiltonians, enabling consistent algorithm comparisons.