## Quantum-enhanced Markov chain Monte Carlo

The not-so-mathematical journal Nature recently published a paper proposing a new Markov chain Monte Carlo method:

• 2023 – Layden, Mazzola, Mishmash, Motta, Wocjan, Kim, and Sheldon – Quantum-enhanced Markov chain Monte Carlo.

Appearing earlier as this preprint, the paper’s publication in such a journal is a rare event indeed. This post notes this, as well as the fact that we can already simulate perfectly1For small instances of the model, we can do this directly. For large instances, we can use coupling from the past proposed by Propp and Wilson. the paper’s test model, the Ising or Potts model.2Wilhelm Lenz asked his PhD student Ernst Ising to study the one-dimensional version of the model. Renfrey Potts studied the generalization and presented it in his PhD. But this is a quantum algorithm, which is exciting and explains how it can end up in that journal.

##### The algorithm

The paper’s proposed algorithm adds a quantum mechanical edge or enhancement to the classic Metropolis-Hastings algorithm.3More accurately, it should be called the Metropolis-Rosenbluth-Rosenbluth-Teller-Teller-Hastings algorithm. As I covered in a recent post, the original algorithm uses a Markov chain defined on some mathematical space. Running it on a traditional or classical computer, at each time step, the algorithm consists of proposing a random jump and then accepting the proposed jump or not. Owing to the magic of Markov chains, in the long run, the algorithm simulates a desired probability distribution.

The new quantum version of the algorithm uses a quantum computer to propose the jump, while still using a classical computer to accept the proposal or not.4In my Metropolis-Hastings post, the classical jumper process, a discrete-time Markov chain, is replaced with a quantum mechanical variant. The quantum jump proposals are driven by a time-independent Hamiltonian, which is a central object in quantum and, in fact, all physics. This leads to a Boltzmann (or Gibbs) probability distribution for the jumping process.

Then, running the quantum part on a quantum computer, the algorithm will hopefully outperform its classical counterpart. The paper nurtures this hope by giving empirical evidence of the algorithm’s convergence speed. The researchers performed the numerical experiments on a 27-qubit quantum processor at IBM using the platform Qiskit.

##### Quantum is so hot right now

In recent years researchers have been focusing on such algorithms that exploit the strangeness and spookiness of quantum mechanics. You will see more and more quantum versions of algorithms that appear in statistics, machine learning, and related fields, as suggested by this survey paper, which also appeared in Nature.

##### Quantum lite

Sometimes quantum mechanics only loosely inspires algorithms and models. In this setting, some of my machine learning work uses determinantal point processes. This kernel-based random model draws direct inspiration from the wave function, a standard object in quantum mechanics. Under suitable simplifying conditions, the model describes the locations of particles known as fermions such as electrons and protons. Still, it’s fascinating that a quantum physics model inspired an interesting random object that has found applications in spatial statistics and machine learning.

## Determinantal thinning of point processes with network learning applications

My colleague and I uploaded a manuscript:

• Błaszczyszyn and Keeler, Determinantal thinning of point processes with network learning applications.

https://arxiv.org/abs/1810.08672

## Details

The paper uses a (relatively) new model framework in machine learning.  This framework is based on a special type of point process called a determinantal point process, which is also called a fermion point process. (This particle model draw inspiration from the form of the wave function in quantum mechanics.) Kulesza and Taskar introduced and developed the framework for using determinantal point processes for machine learning models.

## Code

The MATLAB code for the producing the results in the paper can be found here:

https://github.com/hpaulkeeler/DetPoisson_MATLAB

I also re-wrote (or translated) the MATLAB code into Python:

https://github.com/hpaulkeeler/DetPoisson_Python