Transcription
Finally, let's talk about strategies to solve "quote-unquote" the many-body problem. So, we've talked about the many-body problem, the difficulties in solving it. So, how, in practice, do we go forward for solutions of the many-body problem, both in physical systems and in devices such as quantum computers?
Well, before I talk about numerical techniques, techniques like the extensions of diagonalization and so on, I just want to mention that there's a rich history of analytical techniques that have been used for decades to solve many-body problems in quantum mechanics. And those include things like perturbation theory, where you treat the electron-electron interaction as weak, variational principles, like I talked about in helium, and other methods that may altogether neglect interactions or treat them very perturbatively, and in the regime where they are weak. And this has been very successful in semiconductor physics, solid-state physics, and the theory of metals and so on. This is very important because we have to remember that our current, you know, silicon-based classical computing technology is, in fact, a quantum many-body problem. It's just one where you can neglect many of the electron-electron or spin-spin interactions that I talked about previously.
So, now, as we move forward into 21st-century quantum problems where the electron-electron or the quantum interactions are important, these analytical approaches based on then neglecting those interactions tend to break down. So, in the strongly correlated regime of materials, quantum chemistry, and devices, we're much more reliant on numerical methods. So, I mentioned exact diagonalization, which has an exponential scaling problem. In many cases, that is still your best technology, if you will, for solving the many-body problem. Various numerical sort of approximations can be made even in the regime of diagonalization, such as taking a Hilbert space or a spectra and energy spectra that's exponential in size and finding a way to truncate that down to something perhaps that's polynomial in the number of qubits. These types of truncation compression techniques, where you modify the Hilbert space, can have varying degrees of success based on whether or not, sort of, the problem is amenable to this type of truncation. If you're interested only in low-lying states, diagonalization routines such as Lanczos iterative solvers will give you better scaling than the brute-force O to the M cubed that I talked about previously. Advanced methods such as the density matrix renormalization group involve sophisticated decimation schemes based on the entanglement structure of your wavefunction and basically are known to work very well, for example, in one dimension.
Another main sort of strategy for solving many-body interacting systems is stochastic approaches based on Markov chain Monte Carlo. So, these require a statistical sampling of the configurations that make up the wavefunction, which I talked about previously. So, in these methods, you imagine that some probabilistic technique samples configurations according to a Markov process. The problem with these stochastic methods is that the coefficient of the wavefunction for each one of these configurations has to be interpreted as a probability distribution. Only then can you update configurations stochastically, giving yourself access to estimators for quantum objects, which are basically just made up of, say, for example, the magnetization along that Markov chain. And as you know, the coefficient of the wavefunction in linear superposition or that can be complex by nature. So, there are many cases where stochastic approaches don't work. In many-body problems where these methods do work, you can get N's in the thousands, say for DMRG, and for stochastic quantum Monte Carlo methods, you can get N's in the millions or more. So, when these methods work, we should definitely use them.
There are many cases where we know they don't work, and what we're doing now in physics, in computer science, in quantum chemistry, in academia, industry around the world, is turning to new approaches based on machine learning. The motivation behind using machine learning is essentially because if we take the spin configurations that I talked about in terms of either the Ising model or the transverse field Ising model, and we map these to either binary numbers or say, black and white pixels, we immediately see that this turns into, sort of, an image problem in the sense that we can have a large number of images made up of these pixels that we're interested in processing in some way. So, a big data problem from quantum mechanics can turn into a machine learning problem.
There are various different strategies that were used to employing in machine learning that we can turn to the quantum many-body problem. Supervised learning is when we have data with labels. So, if we can take configurations and label them, say, ferromagnetic, paramagnetic, quantum error correcting, or whatever, then we can essentially do many tasks and classification, etc., with supervised learning techniques. This, of course, requires labels. A priori labels in many cases in the quantum many-body problem will not have labels. We would turn to unsupervised learning. Unsupervised learning might be looking at a quantum wavefunction and trying to find clustering or associative rules that govern the behavior. Unsupervised learning can also be taking measurement data, which is measurement performed on the wavefunction. Say, projective measurement collapses the wavefunction and essentially reconstructing that wavefunction from individual images or snapshots. That's like a reconstruction of, say, a probability distribution in the classical case, or the full wavefunction in the quantum case. These reconstruction techniques based on machine learning are now being studied and compared to traditional techniques based on quantum state and quantum process tomography.
Finally, reinforcement learning, which is known to be a very powerful machine learning technique which doesn't explicitly require datasets, either labeled or unlabeled, is also finding its place in the quantum many-body problem, especially as applied to quantum computing and in particular for quantum error correction. So, in quantum error correction, we we try to find errors in the quantum device and correct them before logical qubits are corrupted. And there are many reinforcement learning techniques that are coming out that will be able to do this sort of quantum error correction very fast on classical processors.
So, as we go into the future, we have a toolbox made up of traditional numerical approaches which have been developed for decades, but we also have a new toolbox based on machine learning that will be crucial in the characterization, preparation, and design of future quantum devices.