VQE Isn't Just for Near-Term Quantum Computing

There's a persistent misconception circulating in quantum computing circles that I keep encountering—both from industry professionals and in academic peer review. The claim goes something like this: "VQE (Variational Quantum Eigensolver) is just a near-term algorithm for noisy intermediate-scale quantum (NISQ) devices, and it won't be relevant once we have fault-tolerant quantum computers." Amazingly, I know researchers whose manuscripts have been desk-rejected by editors under this line of reasoning.

This claim represents a misunderstanding of how fault-tolerant quantum algorithms will actually work in practice.

The Reality: VQE will be a common first subroutine

The fact is, VQE will be an essential component of the long-term quantum algorithms that researchers are developing today. Far from being relegated to the NISQ era, variational methods like VQE—or something extremely similar—will play a crucial role in fault-tolerant quantum computing. In short, it will be used as a first sub-routine, feeding into the standard long-term algorithms.

Consider the fundamental challenge that appears across virtually all quantum algorithms: state preparation. Whether you're running quantum phase estimation (QPE), Krylov subspace methods, or quantum imaginary time evolution (QITE), you need a good starting state.

You don't necessarily need something exactly equal to the ground state, but you need a state with substantial overlap with your target state. In the general case, how do you prepare such a state? The answer is obvious: VQE or something functionally equivalent. Even in cases where you can prepare good overlap, in most cases it still would be worth trying to “pre-optimize” your starting state with VQE.

The Long-Term Algorithm Connection

Let's examine how this plays out across different algorithmic approaches:

Quantum Phase Estimation (QPE): Success requires sufficient overlap between your initial state and the eigenstates of interest. Without adequate overlap, the probability of measuring the correct phase becomes vanishingly small. VQE provides exactly the kind of state preparation needed to increase that success probability.

Krylov Subspace Methods: These algorithms construct a subspace (basically, a smaller eigenvalue problem) that should contain the eigenvalue of interest. The quality of your initial state directly determines whether your Krylov subspace will contain that eigenvalue. Again, VQE-prepared states provide the necessary overlap.

Quantum Imaginary Time Evolution (QITE): Similar principles apply—you need a reasonable starting point to ensure convergence to the desired state.

The Broader Perspective

All the research and development effort currently going into VQE isn't just about making the best of noisy quantum hardware. These advances are building the foundation for state preparation routines that will be essential components of the sophisticated quantum algorithms we'll eventually run on error-corrected quantum computers.

The variational principles, optimization strategies, and ansatz design techniques being developed today will translate directly to tomorrow's fault-tolerant systems. If anything, the removal of noise constraints will make these methods even more powerful and precise.

Conclusion

Rather than viewing VQE as a temporary bridge to better quantum computing, we should recognize it as a fundamental building block of quantum algorithm design. The work happening in variational quantum algorithms today is laying the groundwork for the quantum computing breakthroughs of tomorrow.

The next time someone dismisses VQE as "just a NISQ algorithm," remind them that the best quantum algorithms of the future will likely have VQE components at their core.

Further Reading

McClean, J. R., et al. "The theory of variational hybrid quantum-classical algorithms." New Journal of Physics 18, 023023 (2016). https://doi.org/10.1088/1367-2630/18/2/023023

Tilly, J., et al. "The Variational Quantum Eigensolver: A review of methods and best practices." Physics Reports 986, 1-128 (2022). https://doi.org/10.1016/j.physrep.2022.08.003

Ni, H., Li, H., and Ying, L. "On low-depth algorithms for quantum phase estimation." Quantum 7, 1165 (2023). https://doi.org/10.22331/q-2023-11-06-1165

Motta, M., et al. "Subspace methods for electronic structure simulations on quantum computers." Electronic Structure 6, 013001 (2024). https://doi.org/10.1088/2516-1075/ad3592

Motta, M., et al. "Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution." Nature Physics 16, 205-210 (2020). https://doi.org/10.1038/s41567-019-0704-4

Sawaya, N. P. D., Paesani, F., and Tabor, D. P. "Near-and long-term quantum algorithmic approaches for vibrational spectroscopy." Physical Review A 104, 062419 (2021). https://doi.org/10.1103/PhysRevA.104.062419

Sawaya, N. P. D., et al. "Error sensitivity to environmental noise in quantum circuits for chemical state preparation." Journal of Chemical Theory and Computation 12, 3097-3108 (2016). https://doi.org/10.1021/acs.jctc.6b00220

Gustafson, E. J., et al. "Surrogate optimization of variational quantum circuits." arXiv:2404.02951 (2024). https://arxiv.org/abs/2404.02951

Bauer, B., et al. "Quantum algorithms for quantum chemistry and quantum materials science." Chemical Reviews 120, 12685-12717 (2020). https://doi.org/10.1021/acs.chemrev.9b00829

Cao, Y., et al. "Quantum chemistry in the age of quantum computing." Chemical Reviews 119, 10856-10915 (2019). https://doi.org/10.1021/acs.chemrev.8b00803