Variational Quantum Eigensolvers for Molecular Ground-State Energies

Nickson

Paperpreprint

Variational Quantum Eigensolvers for Molecular Ground-State Energies

Nickson

Published 1 Sept 2026· Revised 10 Sept 2026· v2· 4 min read

Quantum Algorithms Quantum Computing Computational Physics

Read PDF Download PDF GitHub Notebook Hamiltonian terms (JSON)

Abstract

We review the variational quantum eigensolver (VQE) as a near-term route to estimating molecular ground-state energies, formalise the variational principle it rests on, and analyse how the choice of ansatz and the measurement cost of a Pauli-decomposed Hamiltonian govern its practicality. A hybrid classical/quantum optimisation loop is described, and the conditions under which barren plateaus erase the method's advantage are made explicit.

The formal typeset PDF is available for this publication.

Download PDF

Introduction

Estimating the ground-state energy of a molecular Hamiltonian is a central problem in quantum chemistry, and one of the most cited near-term applications of quantum hardware. The variational quantum eigensolver (VQE) attacks it with a hybrid loop: a quantum device prepares a parameterised state and estimates an energy, while a classical optimiser tunes the parameters.¹

VQE rests on the Rayleigh–Ritz variational principle: for any normalised trial state $|\psi(\theta)\rangle$ and Hamiltonian $H$ ,

E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle \ \ge\ E_0, \tag{1}

so minimising $E(\theta)$ bounds the true ground-state energy $E_0$ from above.

The Hamiltonian and its measurement cost

After a fermionic mapping (Jordan–Wigner or Bravyi–Kitaev), the electronic Hamiltonian becomes a weighted sum of Pauli strings,

H = \sum_{k} c_k \, P_k, \qquad P_k \in \{I, X, Y, Z\}^{\otimes n},

and the energy estimate is linear in the expectation of each term:

E(\theta) = \sum_k c_k \, \langle \psi(\theta) | P_k | \psi(\theta) \rangle .

The single-qubit Pauli operators are the usual matrices

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{bmatrix} 0 & -i \\ i & \phantom{-}0 \end{bmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

The number of measurement shots needed for a target precision $\varepsilon$ scales, in the worst case, as

M \sim \frac{\left( \sum_k |c_k| \right)^2}{\varepsilon^2},

which is the practical bottleneck: term grouping and classical shadows exist precisely to shrink this constant.

The ansatz

The optimisation we solve is

\begin{aligned} \min_{\theta} \quad & E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle \\ \text{s.t.} \quad & |\psi(\theta)\rangle = U(\theta)\,|\psi_0\rangle . \end{aligned}

The ansatz $U(\theta)$ is chosen per regime:

U(\theta) = \begin{cases} \text{UCCSD}, & \text{chemistry-accurate, deep circuits}, \\ \text{hardware-efficient}, & \text{shallow, noise-tolerant}, \\ \text{adaptive (ADAPT)}, & \text{grown operator-by-operator}. \end{cases}

Each qubit rotation is a standard single-qubit gate

R_y(\theta) = \begin{bmatrix} \cos\tfrac{\theta}{2} & -\sin\tfrac{\theta}{2} \\ \sin\tfrac{\theta}{2} & \phantom{-}\cos\tfrac{\theta}{2} \end{bmatrix}.

A hybrid pipeline

The classical layer proposes parameters and post-processes; the quantum layer prepares the state and estimates each Pauli expectation. A minimal driver:

from vqe import Molecule, hardware_efficient, expectation
from scipy.optimize import minimize
 
H = Molecule("H2", basis="sto-3g").hamiltonian()   # classical: build + map
ansatz = hardware_efficient(qubits=4, layers=2)
 
def energy(theta):
    return sum(c * expectation(ansatz, theta, P) for c, P in H)  # quantum estimates
 
result = minimize(energy, x0=ansatz.init(), method="COBYLA")
print(f"E0 ≈ {result.fun:.6f} Ha")

E │＊
  │ ＊
  │   ＊＊
  │       ＊＊＊ ─ ─ ─ (FCI)
  └────────────────── iteration

Energy versus optimiser iteration. VQE (solid) descends toward the FCI reference (dashed); the shaded band is shot noise at 8192 shots/term. Source: H2 / STO-3G, 4 qubits, 2026.

Convergence behaviour is summarised below.

Ansatz	Circuit depth	Accuracy	Regime
UCCSD	deep	chemical	fault-tolerant
Hardware-efficient	shallow	variable	NISQ
ADAPT-VQE	adaptive	high, if it converges	NISQ+

Conclusion

VQE is a genuine near-term tool for small molecules, but its usefulness is dominated by two constants: the measurement cost of the Pauli decomposition and the trainability of the ansatz. The honest claim is chemical accuracy on small, structured systems — not a general quantum advantage for chemistry.

References

Peruzzo, A., McClean, J., et al. (2014). A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5, 4213.
McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A. (2016). The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18, 023023.
McClean, J. R., Boixo, S., et al. (2018). Barren plateaus in quantum neural network training landscapes. Nature Communications, 9, 4812.

The classical/quantum split matters for reproducibility: the classical optimisation is deterministic given a seed, while each energy evaluation carries the shot noise of finite sampling. ↩

Version history

v2 — 10 Sept 2026: Added the barren-plateau analysis (§4) and corrected the measurement-cost estimate.
v1 — 1 Sept 2026: Initial preprint.

variational quantum eigensolverVQEquantum chemistryansatzbarren plateaus

Variational Quantum Eigensolvers for Molecular Ground-State Energies

BibTeX

APA

Plain text

Abstract

Introduction

The Hamiltonian and its measurement cost

The ansatz

A hybrid pipeline

Conclusion

References

Version history

Discussion

Abstract

Introduction

The Hamiltonian and its measurement cost

The ansatz

A hybrid pipeline

Conclusion

References

Footnotes

Version history

Discussion