Short answer: Quantum phase estimation is one of the most useful quantum algorithms for chemistry, because it can in principle reach the precision needed to find ground state energies. What it costs is the open question, so researchers have built software tools that estimate the resources. A 2021 tool called TFermion estimates T-gate cost for any molecule, and a 2023 tool called QREChem gives logical resource estimates between about 10^7 and 10^15 T gates for small molecules. Today's devices still struggle to do chemistry that classical computers cannot.
Why phase estimation for chemistry
A 2021 arXiv paper describes the goal: quantum computers might be particularly useful for preparing ground states of electronic Hamiltonians and finding their energies, which matters for chemical reaction rates, battery properties and biological enzymes. Classical techniques such as density functional theory often rely on approximations, for instance the Kohn-Sham exchange-correlation functional, which may struggle to reach the accuracy some problems need. Chemical reaction rates depend exponentially on energy differences, and the authors say quantum phase estimation in principle allows the precision required.
The resource estimation problem
A 2023 paper in Frontiers in Quantum Science and Technology, also on arXiv, says that even with rapid hardware progress, quantum devices still struggle to perform chemistry calculations that classical computers cannot. A systematic way of estimating the resources needed for specific problems is therefore important. The authors argue that standard complexity arguments give hope that quantum computers will be useful for chemistry but obscure the true impact of many algorithmic overheads, which will determine when quantum computers outperform classical ones.
Two estimation tools
The 2021 paper introduces TFermion, a library for estimating the T-gate cost of phase estimation algorithms for an arbitrary molecule. As examples, it estimates the T-gate cost of a few simple molecules and compares Taylorization algorithms using Gaussian and plane-wave bases.
The 2023 paper presents QREChem, which provides logical resource estimates for ground state energy estimation through a Trotter-based phase estimation approach. It includes heuristic estimates of the number of Trotter steps and ancilla qubits needed, so the estimates for total gate count include overheads specific to chemistry. For a variety of small molecules in various basis sets, it obtains estimates in the range of 10^7 to 10^15 total T gates.
Reading the numbers
The span of 10^7 to 10^15 T gates is eight orders of magnitude, which shows how strongly the answer depends on the molecule and the basis set. The 2023 paper builds the estimates from heuristic counts of Trotter steps and ancilla qubits, so they should be read as planning figures that can move as better algorithms appear. The 2021 comparison between Gaussian and plane-wave bases makes the same point on a smaller scale: the same family of algorithms can have quite different costs depending on how the molecule is represented. Neither paper claims a calculation beyond classical reach has been run.
What this does not tell you
These are resource estimates under stated assumptions, with heuristics for quantities such as Trotter steps. They are counts of logical gates, not predictions of when a machine will run them, and they cover small molecules.
FAQ
Why use phase estimation for chemistry?
It can in principle achieve the precision needed for ground state energies, which classical approximations may miss.
What is a resource estimate?
A count of gates or qubits an algorithm needs, such as T gates.
How large are the estimates?
QREChem gives about 10^7 to 10^15 T gates for small molecules in various bases.
Can devices do this now?
The 2023 paper says devices still struggle with chemistry calculations classical computers cannot do.
Do the tools agree?
They use different algorithms and assumptions, so their numbers are not directly comparable.
All field notes