Feasibility of performing quantum chemistry calculations on quantum computers

TL;DR

The paper argues that achieving chemical accuracy for molecular ground-state energies with quantum computers faces two fundamental hurdles: decoherence and noise severely limit VQE on noisy hardware, while the overlap required for effective QPE on fault-tolerant hardware decays exponentially with system size due to the orthogonality catastrophe. It introduces a concrete decoherence/precision bound for VQE and a practical overlap proxy for QPE, demonstrating that current approaches would require fault-tolerant capabilities well beyond near-term devices, with benzene serving as a demanding benchmark. The work highlights the scaling of hardware-induced energy biases and the exponential shot-cost of error mitigation, concluding that ground-state chemistry may not be the most fruitful target for early quantum advantage and pointing to alternative tasks or hybrid schemes, such as quantum dynamics or configuration-interaction–style workflows guided by classical methods. Together, the criteria provide a framework to assess future hardware and algorithmic developments for quantum chemistry applications. The study emphasizes the need for new approaches or targets beyond straightforward ground-state calculations to realize practical quantum benefits in chemistry.

Abstract

Quantum chemistry is envisioned as an early and disruptive application for quantum computers. Yet, closer scrutiny of the proposed algorithms shows that there are considerable difficulties along the way. Here, we propose two criteria for evaluating two leading quantum approaches for finding the ground state of molecules. The first criterion applies to the variational quantum eigensolver (VQE) algorithm. It sets an upper bound to the level of imprecision/decoherence that can be tolerated in quantum hardware as a function of the targeted precision, the number of gates and the typical energy contribution from states populated by decoherence processes. We find that decoherence is highly detrimental to the accuracy of VQE and performing relevant chemistry calculations would require performances that are expected for fault-tolerant quantum computers, not mere noisy hardware, even with advanced error mitigation techniques. Physically, the sensitivity of VQE to decoherence originates from the fact that, in VQE, the spectrum of the studied molecule has no correlation with the spectrum of the quantum hardware used to perform the computation. The second criterion applies to the quantum phase estimation (QPE) algorithm, which is often presented as the go-to replacement of VQE upon availability of (noiseless) fault-tolerant quantum computers. QPE requires an input state with a large enough overlap with the sought-after ground state. We provide a criterion to estimate quantitatively this overlap based on the energy and the energy variance of said input state. Using input states from a variety of state-of-the-art classical methods, we show that the scaling of this overlap with system size does display the standard orthogonality catastrophe, namely an exponential suppression with system size. This in turns leads to an exponentially reduced QPE success probability.

Figures (5)

• Figure 1: Schematic of the difference between the hardware spectrum and that of the studied molecule. In terms of the target eigenstates, the VQE ansatz is close to the ground state of the molecule; in terms of the hardware eigenstates, it is made of arbitrary, both low- and high-energy states. On the other hand, the hardware noise can populate arbitrarily high excited states of the studied molecule. For instance, relaxation can populate the hardware ground state, which consists of arbitrarily high energy states in terms of the target eigenstates.
• Figure 2: Difference between the energy of the infinite temperature state $E_\infty$ and the energy of the ground state, computed within the Hartree-Fock approximation $E_{HF}$, per electron, for the first atoms in the periodic table, from H to Fe for three common basis sets.
• Figure 3: Different energy scales of a chain of $N$ hydrogen atoms. Left panel: symmetric setting: all atoms are treated with the same basis set. Shown are the Hartree-Fock energy $E_{HF}$ and infinite temperature energy $E_\infty$ per atom, for the "STO-3G" ($2$ qubits per atom), "6-31G" ($4$ qubits per atom) Ditchfield1971, "cc-pVDZ" ($10$ qubits per atom) and "cc-pVTZ" basis ($28$ qubits per atom) Dunning1989. For the "STO-3G" basis we have also plotted the highest excited state energy $E_{max}$ up to $N=10$ (regime where exact diagonalization is possible). In all curves, the energy per atom is offset by its Hartree-Fock value for $N=2$ in the "STO-3G" basis. (Right) Effect of an assymetry on the large energy spectrum. For the rightmost-half of the chain we use the 'sto-3g' basis while for the leftmost-half we use more precise basis sets: "cc-pvdz", "dzp", "cc-pvtz", "tzp". This calculation shows that any asymetry, originating from the presence of different atoms or here simply on the different treatment of the same atom, leads to the formation of a large electric dipole in the energy spectrum.
• Figure 4: (a) Sketch of the energy $E(\tau)$ vs imaginary time. The area $\kappa$ under the energy curve directly provides the overlap $\Omega=e^{-\kappa}$. We approximate $\kappa$ with the more easily accessible orange-shaded area $I_\Omega$. [(b) - (d)]: Energy $E_V$ vs optimization step in VQE simulation, the "error bar" corresponds to the standard deviation $\sigma_V$. [](e) - (g)]: Overlap $\Omega$ and $e^{-I_\Omega}$ vs optimization step.
• Figure 5: Scaling of the Hartree-Fock energy (a) and the energy variance (b) vs number of atoms in a hydrogen chain. (c): Overlap index $I_\Omega$ versus energy error $|E-E_0|$ for the variational ansatz of the Wu2023 data set. The dashed line is a linear fit $I_\Omega \approx 27.8 |E-E_0|$.