The Hamiltonian defines what a quantum computer measures — while the ansatz defines how. This page explores how molecular Hamiltonians are built, compressed, and how they transform as you stretch a chemical bond.
A quantum chemistry Hamiltonian encodes a molecule's energy structure as a sum of Pauli operators. Building one is a four-stage pipeline: from atoms, through electron integrals, to qubit operators, to a circuit you can run.
The 2-qubit H₂ Hamiltonian at R=0.735 Å has six Pauli terms. Bars grow left (negative) or right (positive). Hover a term to highlight it. This chart is linked to the bond-stretching slider below.
H₂ in the STO-3G basis has 4 spin-orbitals, giving a 4-qubit Hamiltonian via the Jordan-Wigner transform. Symmetry reduction (parity, spin, particle number) compresses it to just 2 qubits — no information lost.
Drag the slider to stretch the H₂ bond from 0.3 to 3.0 Å. Watch how the Hamiltonian coefficients, energy landscape, and correlation energy all transform in real time. Three views, one control.
H₂ is the simplest molecule. Bigger molecules explode in complexity. Each doubling of qubits roughly quadruples the number of Pauli terms.
See the full glossary for more definitions.
The operator encoding a system’s total energy. In quantum chemistry, H = ∑ gᵢ Pᵢ where Pᵢ are Pauli strings and gᵢ are real coefficients derived from electron integrals.
One of {I, X, Y, Z} acting on a qubit. Multi-qubit Pauli strings like X₀X₁ describe correlated measurements on multiple qubits simultaneously.
Maps fermionic creation/annihilation operators to qubit Pauli operators. Preserves anti-commutation relations at the cost of O(N) Pauli weight (long Z-strings).
An alternative fermion-to-qubit mapping with O(log N) Pauli weight. More efficient than Jordan-Wigner for large systems, but the resulting terms are harder to interpret physically.
Uses molecular symmetries (parity, spin, particle number) to eliminate qubits. Each Z₂ symmetry halves the Hilbert space. H₂ goes from 4 qubits to 2.
FCI — the exact solution within a given basis set. Exponentially expensive classically. For H₂/STO-3G, FCI = -1.1373 Ha at equilibrium.
A classical mean-field approximation that treats electrons independently. Misses correlation energy. The gap between HF and FCI is exactly what VQE must capture.
Stretching a bond until it breaks. At large R, electrons become strongly correlated, HF fails catastrophically, and quantum advantage is most pronounced.
A spatial orbital combined with a spin label (α or β). H₂ in STO-3G has 2 spatial orbitals × 2 spins = 4 spin-orbitals.
⟨ψ|H|ψ⟩ — the average energy measured when running the circuit. VQE minimizes this over parameters θ. Each Pauli term requires separate measurement circuits.
1 kcal/mol (1.6 mHa) — the precision threshold needed for quantum chemistry to predict reaction outcomes. Our best hardware result is 4.1 kcal/mol on Tuna-9.
E_FCI - E_HF: the energy that mean-field theory misses. Grows dramatically during bond dissociation. For H₂ at R=3.0Å, it reaches 174 kcal/mol.
S. McArdle, S. Endo, A. Aspuru-Guzik, S.C. Benjamin, X. Yuan, “Quantum computational chemistry,” Rev. Mod. Phys. 92, 015003 (2020).
arXiv:1808.10402 ↗A. Szabo & N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover (1996).
S. Bravyi, J.M. Gambetta, A. Mezzacapo, K. Temme, “Tapering off qubits to simulate fermionic Hamiltonians,” arXiv:1701.08213 (2017).
arXiv:1701.08213 ↗Sagastizabal 2019 — H₂ VQE with symmetry verification. 6.2 kcal/mol on Tuna-9 (3/4 claims pass)
View replication →Peruzzo 2014 — HeH⁺ VQE bond sweep (first VQE paper). Emulator PASS, IBM 91 kcal/mol (3/5 pass)
View replication →Kandala 2017 — H₂ PES with hardware-efficient ansatz. 10/10 chemical accuracy on emulator (3/3 pass)
View replication →The variational loop, energy landscape, and real hardware results
Circuit architectures from 4 papers, mapped to 3 quantum processors
5 papers, 19 claims tested across emulator, Tuna-9, Garnet, and IBM
50+ experiments: Bell, GHZ, VQE, QAOA, QV, RB results
37 terms across 7 categories with clear definitions
To simulate a molecule on a quantum computer, its Hamiltonian (energy operator) must be decomposed into a sum of Pauli operators — tensor products of I, X, Y, and Z matrices. Each Pauli term can then be measured on quantum hardware, and their weighted sum gives the total energy.
For hydrogen (H₂), the Hamiltonian has 5 Pauli terms whose coefficients change as the bond stretches. At equilibrium (0.735 Å), the molecule sits at the bottom of its potential energy surface. Stretching the bond raises the energy until the atoms dissociate. The full configuration interaction (FCI) energy is the exact answer; Hartree-Fock is the best classical single-determinant approximation.
This interactive shows how the Pauli decomposition coefficients evolve with bond distance, and how VQE results from real quantum hardware compare to the exact energy curve.