## Tuesday, August 30, 2022

### Is there evidence for exponential quantum advantage in quantum chemistry?

Figure 1 from the paper. (c) 2022 the authors. Reproduced under the CC-BY licence.

Quantum chemical calculations are widely seen as one of quantum computings killer app's. This paper examines the available evidence for this assertion and doesn't find any.

The potential of quantum computing rests on two assumptions: that the cost of quantum computer calculations on chemical systems scales polynomially with system size, while the corresponding calculations on classical computers scale exponentially.

The former assumption is true for the actual quantum "computation" and the latter assertion is true for the Full CI solution. However, this paper suggests that preparing the state for the quantum "computation" may scale exponentially with system size, and that we don't need Full CI accuracy and that chemically accurate methods such as coupled-cluster based method scale polynomially with system size for a given desired accuracy.

The argument for the potential exponential scaling for system preparation is as follows: If you want the energy of the ground state you have to provide a guess at the ground state wavefunction that resembles the exact wavefunction as much as possible. More precisely, the probability of obtaining the ground state energy scales as $S^{-2}$, where S is the overlap between the trial and exact wavefunction. The authors show that $S$  scales exponentially with system size for a series of Fe-S clusters, which suggests an overall exponential dependence for the quantum computations.

The argument for polynomial scaling of chemically accurate quantum chemistry calculations has two parts: "normal" organic molecules and strongly correlated systems.

The former is pretty straight-forward: no one knowledgeable is really arguing that CCSD(T)-level accuracy is insufficient for ligand-protein binding energies and CCSD(T) scales polynomially with system size. So the simple notion of accelerating drug discovery by computing this with quantum computers does not hold water.

However, CCSD(T) does not work for strongly correlated systems and we don't have any real practical alternative for which we can test the scaling. Instead the authors look at simpler model of strongly correlated systems and demonstrate polynomial scaling with system size.

As the authors are carefull to point out, none of this represents a rigorous proof of anything. But it is far from obvious that quantum chemistry is the killer app for quantum computing that most people seem to think it is.