Hamiltonian & Ansatz

The H2 electronic Hamiltonian

In second quantization, a molecule's electronic Hamiltonian in a finite orbital basis is

where the one- and two-electron integrals $h_{pq}$ and $h_{pqrs}$ come from the molecular orbitals, and the Jordan-Wigner transform maps each fermionic operator to a string of Pauli operators (each $a_p^\dagger$ becomes a weighted Pauli string acting on qubit $p$ and a $Z$-string over all lower-indexed qubits, so that the fermionic anticommutation relations survive the mapping onto qubits).

For H₂ in a minimal (STO-3G) basis, applying Jordan-Wigner to the 4 spin-orbitals and then a symmetry (parity) reduction — using that the ground state has fixed particle number and $S_z$ — collapses the problem onto 2 qubits. This site uses the reduced 2-qubit Hamiltonian exactly as published, rather than re-deriving the parity-tapering step:

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export const H2_COEFFICIENTS = {
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  g0: -0.4804, g1: 0.3435, g2: -0.4347,
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  g3: 0.5716,  g4: 0.091,  g5: 0.091,
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};
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export function h2PauliSum(): PauliSum {
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  const { g0, g1, g2, g3, g4, g5 } = H2_COEFFICIENTS;
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  return [
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    { coefficient: g0, paulis: ["I", "I"] },
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    { coefficient: g1, paulis: ["I", "Z"] },
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    { coefficient: g2, paulis: ["Z", "I"] },
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    { coefficient: g3, paulis: ["Z", "Z"] },
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    { coefficient: g4, paulis: ["Y", "Y"] },
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    { coefficient: g5, paulis: ["X", "X"] },
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  ];
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}

Nuclear repulsion is added as a classical constant, computed live rather than folded into a lookup table:

with $R_{\text{Bohr}} = R / a_0$ and $a_0 = 0.529177210903$ Å (CODATA Bohr radius). At $R = 0.75$ Å this gives $E_{nn} \approx 0.70557$ Hartree — see nuclearRepulsion() in src/lib/physics/h2Hamiltonian.ts.

Why the ground state lives in a 2-dimensional subspace

Write basis states as $|q_0 q_1\rangle$. $I$, $Z_0$, $Z_1$, and $Z_0 Z_1$ are all diagonal in this basis, so they never mix different computational-basis states. $Y_0 Y_1$ and $X_0 X_1$ each flip both qubits at once, so they only connect $|00\rangle \leftrightarrow |11\rangle$ and $|01\rangle \leftrightarrow |10\rangle$ — the Hamiltonian splits into two independent 2×2 blocks and never mixes them.

Evaluating both diagonals: $\langle 00|\hat H|00\rangle = g_0+g_1+g_2+g_3 = 0$ and $\langle 11|\hat H|11\rangle = g_0-g_1-g_2+g_3 \approx 0.1824$ — both non-negative, so that block cannot contain the ground state. The other block, in the $\{|01\rangle, |10\rangle\}$ basis, is

with closed-form eigenvalues (see eigen2x2Symmetric() in src/lib/physics/linalg.ts, used directly — not a general iterative solver, since a 2×2 symmetric matrix has an exact quadratic-formula solution):

which gives a minimum electronic eigenvalue $\lambda_- \approx -1.8512$ Hartree. Adding nuclear repulsion, $E_{\text{total}} = \lambda_- + E_{nn} \approx -1.1456$ Hartree — matching the well-known H₂/STO-3G full-CI benchmark of $\approx -1.137$ Hartree to within the expected residual from the R = 0.75 Å vs. 0.735 Å bond-length difference between the source data and the textbook value.

The ansatz circuit

Since only the $\{|01\rangle, |10\rangle\}$ subspace matters, a single real parameter is enough to reach the exact ground state — no expressibility is wasted on states the Hamiltonian can never select. The ansatz is Hartree-Fock state preparation followed by a controlled rotation:

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export function h2AnsatzGates(): AnsatzGate[] {
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  return [
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    { kind: "X", qubit: 0 },
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    { kind: "CNOT", control: 1, target: 0 },
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    { kind: "RY", qubit: 1, parameterized: true },
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    { kind: "CNOT", control: 1, target: 0 },
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  ];
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}

Tracing the circuit by hand: $X(q_0)$ takes $|00\rangle$ to $|q_0{=}1,q_1{=}0\rangle$. The first $\text{CNOT}(q_1 \to q_0)$ is a no-op (control qubit is $|0\rangle$). $RY(\theta)$ on $q_1$ gives $\cos\frac{\theta}{2}|q_0{=}1,q_1{=}0\rangle + \sin\frac{\theta}{2}|q_0{=}1,q_1{=}1\rangle$. The second CNOT now fires on the second term (control $q_1=1$), flipping $q_0$ there and landing exactly on the coupled subspace:

Optimization uses the exact parameter-shift rule for this single $RY(\theta)$ generator (no finite-difference approximation):

Running this in the browser (runVqe() in src/lib/physics/vqe.ts) converges to $\theta^\ast \approx -0.2297$, $E \approx -1.145630$ Hartree — matching the exact diagonalization above to machine precision, confirmed against exactGroundStateEnergy() at build-verification time. See it run live on the VQE Suite playground.

Source: src/lib/physics/h2Hamiltonian.ts, h2Ansatz.ts, vqe.ts, linalg.ts