Quantum Chemistry on Quantum Computers

Learn how molecular Hamiltonians are mapped to qubit operators and why quantum computers offer an exponential advantage for chemistry.

Quantum Chemistry on Quantum Computers

The most commercially promising near-term application of quantum computing is simulating molecular systems, finding ground state energies of molecules to design drugs, catalysts, and materials.

Why Classical Computers Struggle

The electronic structure Schrödinger equation: H|Ψ⟩ = E|Ψ⟩

For a molecule with N electrons, the many-body wavefunction has exponentially many terms. Exact solution (Full CI) requires diagonalizing a matrix of size C(2M, N) × C(2M, N) where M is the number of basis orbitals:

•H₂ (2 electrons, 4 spin-orbitals): 6 × 6 → tractable
•H₂O (10 electrons, 20 spin-orbitals): 184,756 × 184,756 → barely tractable
•Fe-nitrogenase (active site: 8 Fe, 1 Mo, 9 S, 113 electrons): $\sim 1 0^{120}$ → classically impossible

Quantum computers store this wavefunction naturally in their quantum state.

The Pipeline: Molecule → Qubits

Step 1: Second Quantization Express the Hamiltonian in terms of fermionic creation/annihilation operators (a†, a):

H = pq \sum h_{pq} a_{p}^{†} a_{q} + \frac{1}{2} pq r s \sum g_{pq r s} a_{p}^{†} a_{q}^{†} a_{r} a_{s}

where $h_{pq}$ and $g_{pq r s}$ are one- and two-electron integrals (computed classically by packages like PySCF, PSI4).

Step 2: Qubit Mapping Convert fermionic operators to qubit operators using a mapping (Jordan-Wigner or Bravyi-Kitaev):

Jordan-Wigner:

a_{j}^{†} = (Z_{0} \otimes Z_{1} \otimes \dots \otimes Z_{j - 1}) \otimes (X_{j} - i Y_{j}) /2

Maps each spin-orbital to one qubit. Preserves fermion anti-commutation at the cost of long Pauli strings.

Bravyi-Kitaev: More complex but reduces Pauli string length from O(N) to O(log N).

Step 3: Pauli Decomposition Any qubit Hamiltonian decomposes into a sum of Pauli operators:

H = i \sum c_{i} P_{i}

where $P_{i} \in {I, X, Y, Z}^{\otimes n}$ and $c_{i}$ are real coefficients.

Example for H₂ in minimal basis (STO-3G, 4 qubits after reduction): H ≈ g₀ I + g₁ Z₀ + g₂ Z₁ + g₃ Z₀Z₁ + g₄ Y₀X₁X₂Y₃ + g₅ X₀X₁Y₂Y₃ + ...

This is the opening of the lesson. The full walkthrough, the interactive circuit, and the graded challenge continue inside myqubit.

How this lesson works

A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.

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