Quantum Chemistry with VQE

Estimate the ground-state energy of a molecule by mapping its electronic Hamiltonian onto qubits and minimising the energy with a variational ansatz. The variational quantum eigensolver (VQE) is the workhorse method for quantum chemistry in the NISQ era.

The problem it solves — electronic structure

Almost every chemical property of a molecule — its stable geometry, reaction barriers, spectra — depends on the ground-state energy of its electrons, the lowest eigenvalue of the electronic Hamiltonian H. Solving this electronic structure problem is the heart of computational chemistry.

The difficulty is that the Hilbert space needed for an exact solution grows exponentially with the number of electrons (orbitals). Exact classical methods (full configuration interaction) quickly become infeasible, so chemists have long relied on approximations. The appeal of a quantum computer is that it natively represents a wavefunction as a quantum state and can, in principle, encode these exponentially large states efficiently.

From molecule to qubit Hamiltonian

The electronic Hamiltonian is first written as a sum of fermionic operators (creation and annihilation operators), because electrons are fermions obeying antisymmetry. Qubits are not fermions, so a mapping is needed to translate fermionic operators into Pauli operators.

The classic mapping is the Jordan–Wigner transformation: it assigns each spin orbital to one qubit (occupied = 1, empty = 0) and properly encodes the fermionic antisymmetry signs through Pauli strings. Other mappings (Bravyi–Kitaev, parity) make different trade-offs between locality and qubit count. After mapping, the Hamiltonian becomes a sum of Pauli strings, each with a real coefficient:

H = Σk ck · Pk, where each P_k is a tensor product of I, X, Y, Z.

The variational principle — why minimizing the energy suffices

The theoretical foundation of VQE is the variational principle: for any normalized trial state |ψ(θ)⟩, its energy expectation value can never fall below the true ground-state energy:

E(θ) = ⟨ψ(θ)| H |ψ(θ)⟩ ≥ E0

This gives a clear strategy: scan a family of parameterized trial states and go as low as you can. The lowest E(θ) you reach is an upper bound on E₀, and the closer it gets, the better your ansatz. So finding the ground state becomes a minimization problem — the quantum circuit prepares |ψ(θ)⟩ and measures E(θ), while a classical optimizer tunes θ.

The circuit ansatz — writing electron correlation into parameters

The ansatz is the parameterized circuit that prepares |ψ(θ)⟩; it determines how large and how faithful a space of states you can search. Two common choices:

• UCCSD (unitary coupled cluster with singles and doubles) — chemically motivated, exciting electron pairs out of a Hartree–Fock reference; accurate but with a deeper circuit.
• Hardware-efficient ansatz — alternating single-qubit rotations and native entangling gates; shallow and hardware-friendly, but lacking chemical priors and more prone to barren plateaus.

A good ansatz balances expressive power (can it cover the true ground state) against trainability / executability (is the circuit shallow enough, are the gradients optimizable).

Measurement cost — an underrated bottleneck

The energy E(θ) is the weighted sum of the expectation value of each Pauli string in the Hamiltonian: E(θ) = Σk ck ⟨Pk⟩. Each ⟨P_k⟩ must be estimated by repeated sampling in the appropriate measurement basis. As the molecule grows, the number of Pauli strings can reach order O(N4) (N = number of orbitals), and estimating every term to sufficient precision demands an enormous number of measurements.

This is one of the real costs of VQE and has spawned many mitigation techniques: grouping commuting Pauli strings for joint measurement, allocating the measurement budget by coefficient, classical shadows, and more. Understanding this is essential to a clear-eyed view of VQE's practical scalability.

What it is good at — and what it isn't

• Built for NISQ. VQE shifts the burden of deep circuits onto an outer classical optimizer, keeping circuits relatively shallow and suited to today's noisy hardware — which is why it is popular.
• Limited by ansatz quality. If the ansatz cannot represent the true ground state, the energy stalls at a too-high upper bound; expressive power and trainability are hard to have at once.
• Huge measurement overhead. The number of Pauli strings grows fast with molecule size, and sampling cost is often the main obstacle to practicality.
• Hard optimization, eroded by noise. Barren plateaus, local minima, plus hardware noise biasing the expectation values, make convergence to chemical accuracy challenging.
• Advantage not yet confirmed at practical scale. For small molecules, classical methods are still fast and accurate; when and for which systems VQE truly wins is an active research question.

The algorithm, step by step

1. A chemistry frontend produces a fermionic Hamiltonian, mapped (e.g. via Jordan–Wigner) to a sum of Pauli strings.
2. Choose a parameterized ansatz |ψ(θ)⟩ (UCCSD or hardware-efficient).
3. Prepare |ψ(θ)⟩ on the quantum circuit and measure each Pauli string to estimate E(θ).
4. Hand E(θ) to the classical optimizer, which updates θ toward lower energy per the variational principle.
5. Repeat until convergence; the lowest E(θ) is the best upper-bound estimate of the ground-state energy.

Seeing it in code

Below, the H₂ Hamiltonian is written directly as a sum of Pauli strings — exactly the post-mapping form above. Each coefficient ck corresponds one-to-one with a Pauli string (such as spin.z(0) * spin.z(1)); here parity tapering has already reduced the qubit count to 2.

import qalgora
from qalgora import spin

# H2 molecule at 0.735 A bond length, mapped to 2 qubits (parity tapering)
hamiltonian = (-1.0524 + 0.3979 * spin.z(0) - 0.3979 * spin.z(1)
               - 0.0112 * spin.z(0) * spin.z(1)
               + 0.1809 * spin.x(0) * spin.x(1))

These coefficients are illustrative values for one specific molecule, basis, and fermion-to-qubit mapping (H₂ at 0.735 Å, minimal basis, parity tapering) — they are not a universal VQE Hamiltonian, and any other molecule, geometry, basis, or mapping yields different terms.

This Hamiltonian represents only the electronic energy; it does not include the nuclear-repulsion term. The total molecular energy requires adding the nuclear-repulsion energy at this bond length.

This is a deliberately minimal one-parameter teaching ansatz for the two-qubit reduced model of H₂: x(q[0]) prepares the Hartree–Fock reference state |10⟩, and the following ry(theta, …) plus controlled gate introduce correlation near that reference with a single parameter θ. It reflects the spirit of UCC-type ansätze, but it is not a full UCCSD circuit; real chemistry uses richer parameterizations.

@qalgora.kernel
def ansatz(theta: float):
    q = qalgora.qvector(2)
    x(q[0])                 # Hartree-Fock reference |10>
    ry(theta, q[1])
    x.ctrl(q[1], q[0])

The final step is the variational loop: inside qalgora.vqe, the ansatz is repeatedly prepared, the Hamiltonian measured to obtain E(θ), and COBYLA pushes θ toward lower energy per the variational principle. The returned energy is the upper-bound estimate of the ground-state energy. Under ideal, noise-free conditions with exact expectation-value estimation, the VQE energy is an upper bound on the ground-state energy; on real hardware and with finite sampling this upper-bound property can be broken by noise and statistical error.

optimizer = qalgora.optimizers.COBYLA()
energy, params = qalgora.vqe(ansatz, hamiltonian, optimizer, parameter_count=1)

print(f"ground-state energy: {energy:.6f} Hartree")
# electronic energy ~ -1.857 Ha; this Hamiltonian excludes nuclear repulsion
# (+0.72 Ha at this bond length), giving ~ -1.137 Ha total

Scaling to larger molecules

• Use the Solvers library's adapt_vqe to grow the ansatz automatically.
• Offload the expectation evaluation to the gpu target as qubit counts grow.
• Batch multiple bond lengths to trace the full dissociation curve.

References

• A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, J. L. O'Brien, "A variational eigenvalue solver on a photonic quantum processor," Nat. Commun. 5, 4213 (2014). doi:10.1038/ncomms5213