Benchmarking quantum algorithms for real quantum chemistry

The problem

Computing the ground-state energy of a molecule is the canonical hard problem in quantum chemistry — and the one quantum computing is most often promised to solve. We wanted to know how much of that promise currently survives contact with real hardware.

So we picked one genuine molecular problem, the ground state of CO₂, and ran four different solvers at it: two quantum-native approaches and two classical references.

My role

I led the project. That started with the parts nobody puts in a portfolio — setting up the team's collaboration framework and issue-tracking process on day one, and keeping the project plan current throughout, tracking progress and risk so that we hit our milestones on schedule across the full project.

On the technical side I implemented VQE and Exact Diagonalization myself and contributed to the first iteration of SQD, together around 4,000 lines of production code.

What we found

The spread was larger than any of us expected, which is why the chart above needs a logarithmic axis to be readable at all.

SQD converged to -185.0648 Ha in 3.24 seconds on a laptop CPU, using a real quantum computer for the sampling. Exact Diagonalization confirmed that number was correct — and took 6.4 hours and 50 GB of RAM to say so. DMRG hit a hard wall: exponential scaling put a full run at an estimated 72 years, which is about as clean an illustration of classical breakdown as you could ask for.

VQE was fully built — ansatz, Hamiltonian, optimizer — and never ran, because the measurement count exceeded IBM Quantum's free monthly limit. We validated the SQD circuit on real IBM Quantum hardware and the results matched simulation.

Conclusion

For NISQ-era quantum chemistry, SQD is the practical winner: fast, memory-efficient and accurate. ED remains the indispensable exact reference for small systems — you need something to check the quantum answer against.

The wider lesson was about how far current hardware still is from practical advantage. Free-tier access limits alone were enough to stop VQE running at all, while SQD's noise resilience made it the only method we could validate end to end on real quantum hardware.

Thanks

A special thanks to Taha Selim for being an outstanding client. He always made time to answer our questions and explain the core theory when we needed it, and pointed us toward using CCSDT to freeze the outer electrons — trading some measurement accuracy for a significant speed boost. His genuine excitement about the project made this one of the most rewarding things I have worked on.