Can a quantum computer find the ground state of a nucleus — or the minimum-risk portfolio — faster than a classical one?
University of Oslo · Spring 2026 · Egil Furnes
Quantum systems are hard to simulate classically. As the number of particles grows, the Hilbert space explodes exponentially — and brute-force diagonalization becomes impossible. Quantum computers don't simulate quantum mechanics; they are quantum mechanics. The variational principle — find the state that minimises the energy — becomes a feedback loop between a parameterised quantum circuit and a classical optimiser. That's the Variational Quantum Eigensolver.
The same logic applies to combinatorial optimisation. A portfolio selection problem — pick k assets to minimise volatility — maps naturally to a QUBO. Encode the constraints into a cost Hamiltonian, evolve the quantum state through alternating layers of cost and mixer unitaries, and let the algorithm find the ground state. That's QAOA.
Both algorithms are near-term. Both are variational. Both are genuinely quantum. And both are studied here from scratch.
The Lipkin model is a minimal universe. The Lipkin–Meshkov–Glick model packs the essential physics of a many-body quantum system — phase transitions, entanglement, symmetry breaking — into a Hamiltonian you can hold in your head. It's the perfect laboratory for benchmarking VQE before scaling up.
Avoided crossings are where quantum mechanics gets weird. Near a phase transition, energy levels repel each other instead of crossing. Entanglement entropy peaks here. Classical methods struggle precisely where quantum effects are strongest. VQE finds the ground state here without needing to diagonalise the full Hamiltonian.
Portfolios are physics problems in disguise. Covariance matrices and correlation functions are structurally identical. The minimum-volatility portfolio is the ground state of a quadratic energy functional. QAOA treats it accordingly — and the connection to quantum phase estimation is not accidental.
Crisis windows test the algorithm honestly. Covariance matrices estimated during the 2008 crash, 2020 pandemic shock, and 2022 rate spike are not well-behaved. If QAOA can find low-volatility portfolios under those conditions, it means something.
Everything is built from scratch. No black-box imports. The VQE loop, the ansatz, the gradient descent, the Hamiltonian construction — all explicit, all inspectable. The goal is understanding, not just results.
How precisely can a variational quantum circuit reproduce the ground state of a nuclear model — and where does it break?
The Lipkin–Meshkov–Glick model is a benchmark for many-body quantum methods. This project implements VQE from scratch and pushes it through progressively harder systems: single-qubit gates, Bell state entanglement, the J=1 Lipkin Hamiltonian on two qubits, and finally the J=2 system on four qubits.
The results are striking. For one qubit, the Ry ansatz is exact — errors below 10⁻¹⁰ across the full parameter range. For two qubits, precision holds until λ ≈ 0.4, where the optimizer undergoes a catastrophic branch-tracking failure at the avoided crossing: errors jump twelve orders of magnitude, from 10⁻¹⁴ to 10⁻¹. The same failure occurs in Qiskit. It's not a bug — it's a systematic consequence of the optimization landscape. The optimizer stays on the continuously connected upper branch and never crosses to the ground state.
For four qubits, the problem shifts: the hardware-efficient ansatz lacks the symmetry of the Lipkin Hamiltonian, the optimizer wanders into unwanted sectors, and barren plateaus cause gradients to vanish near the phase transition. Error plateaus at 10⁻².
What makes this interesting isn't where VQE succeeds — it's exactly where and why it fails.
Full report: claudoverius
Can a quantum circuit find the minimum-volatility portfolio during a financial crisis?
Quantum Approximate Optimization Algorithm applied to cardinality-constrained minimum-volatility portfolio selection. The optimization problem is cast as a QUBO — structurally identical to finding the ground state of an Ising Hamiltonian — and solved via parameterized quantum circuits on Qiskit simulators. Covariance matrices are estimated from historical crisis windows (2008, 2020, 2022) to stress-test performance where classical assumptions break down.
Full report: fys5419-overleaf-2
Python · NumPy · SciPy · Matplotlib · Qiskit · PennyLane
Department of Physics · University of Oslo