Real-Space Chemistry on Quantum Computers: teaching algorithms to think like chemists

Quantum computers are often sold on their power to change chemistry. If we could model molecules with “chemical accuracy”, predicting energies to within a fraction of an electronvolt, we could speed up drug discovery, design new materials, or unlock cleaner energy. However, the algorithms we currently use must balance accuracy against computation cost.

Quantum chemistry is dominated by uniform grids or plane waves when representing space. This helps in maths as it's neat and tidy but doesn't work in the physical world as molecules are clumpy and electrons gravitate towards bonds. This is known as the wavefunction and when it's near the nucleus of an atom, the wavefunction changes dramatically and has steep so-called "cusps".

However, uniform grids waste qubits when executing computations as they oversample low-density areas, so in order to calculate where the electrons are likely to be, you need a very fine grid, which wastes huge amount of processing power because you are sampling regions where nothing noteworthy is happening.

The findings of a new paper "Real-Space Chemistry on Quantum Computers: A Fault-Tolerant Algorithm with Adaptive Grids and Transcorrelated Extension" promises to solve the challenge of sacrificing accuracy or cost.

The researchers have placed the computational points where the electrons can actually be found, near nuclei and bonds, and then iron out the extremes of the cusps by removing the most challenging mathematical problems (the transcorrelation). This makes the wavelengths less tricky to deal with and maintain accuracy minimising wasteful processing power.

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Following the electron density

Many chemists have little faith in plane-wave-based approaches because they favour the algorithm versus the underlying physics. This paper attempts to close that gap with so-called molecule-adaptive grids. They cluster around bonds and nuclei instead of being evenly distributed, which often includes stretches of empty space.

To use an analogy, when you take a photo of someone's face, you want the subject's features to have a higher resolutions than the blurred background behind them. The adaptive approach means that the algorithm can achieve the same accuracy with mcuh smaller number of points.

Benchmarks on hydrogen and helium molecules show that adaptive grids hit chemical accuracy without the waste of oversampling with uniform grids, where every bit of space is sampled equally regardless of the need to do so.

Why cusps are such a headache

Despite smart grids, one problem remains. When electrons and nuclei get close, the wavefunction develops cusps, which are sharp kinks caused by Coulomb interactions. Mathematically, they need a very fine resolution in order to capture the shape accurately but they are expensive from a processing power perspective.

If chemistry behaved like smooth sine waves, it would be much easier. But Coulomb potentials give you peaks — and peaks break the nice assumptions that keep resource costs low. However, capturing cusps is crucial for strongly correlated systems as they are exactly the kinds of molecules (like hemoglobin or complex enzymes) that pharmaceutical and materials science care about.

Smoothing with a transcorrelated Hamiltonian

The researchers used a transcorrelated Hamiltonian to overcome the cusp problem, which replaces the Coulomb approach to produce smoother interactions. It creates cusp-free eigenfunctions that are far easier to represent or, more simply, it irons out the jagged parts of the wavefunction, which are easier for a computer to handle.

Yet, when you use this trick, the Hamiltonian becomes messier and loses the usual symmetry that makes calculations easy to execute. Furthermore, it picks up extra interactions involving three participles at once. Therefore, the Quantum Phase Estimation, a standard quantum algorithm, can't be used.

To overcome this, the researchers switched to a newer algorithm called the Quantum Eigenvalue Estimation, that can handle more complicated equations. It allows for smoother wavefunctions, which means fewer grid points and fewer operations, so less computing power.

Combined with the adaptive grid, the two tricks work in tandem, fewer points where they aren’t needed, and smoother behaviour where cusps once demanded brute-force resolution.

Why it matters for real chemistry

Conceptually, this produces a new path. It brings the reality of physics into algorithms instead of trying to crowbar chemistry in mathematically convenient boxes. Adaptive grids and transcorrelated Hamiltonians, together, could:

• Accelerate drug discovery as biomolecules hemoglobin are difficult to model. By modelling their cusps more accurately, we get better predictions of the binding energies and reaction pathways.
• Create new materials as the cost of simulating new electron interactions would be lower and would allow for engineers to research more superconductors, catalysts or batteries.
• Lower environmental costs with better stimulations that require less energy-intensive lab experiments. This will also shorten R&D cycles by making "in silico" trials much more realistic.

The adaptive-grid approach allows cusps to be captured more naturally and makes high accuracy simulations available earlier than expected once the hardware catches up.

What’s next?

The next challenges include creating adaptive grids for large, irregular molecules or factoring in the overheads of quantum error correction. The development here is that chemistry doesn't need incremental tweaks, it needs fresh approaches because molecules aren't uniform. Algorithms shouldn't be either, so research needs to embrace physical realism.

If algorithms can closely resemble real chemistry, we bring quantum computers a leap closer towards unlocking the quantum advantage to transform drug design and material science.

Access the full paper here