Quantum State Preparation of Normal Distributions using Matrix Product States

The Problem

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to predict outcomes of complex scenarios with multiple unknowns. They’re commonly used in the financial industry to evaluate risks, price derivatives, and create long-term strategies, as in the case of retirement planning. The runtime associated with a Monte Carlo simulation can be high, as many samples are required to obtain accurate and reliable results.

Quantum computers are promised to speed up the Monte Carlo process when the problem size becomes large. However, there are still many missing pieces in the existing quantum algorithms, including the best way to prepare a quantum state corresponding to the distribution that we want to sample from.

Here, we focus on preparing the normal distribution as it is a frequently used distribution for Monte Carlo, and market events are mostly modelled using normal distributions.

The Challenge

There exist a few different algorithms to prepare quantum states corresponding to normal distributions. Some of them require a fault-tolerant quantum computer to run, which will not be available for quite a few years. Others use a hybrid, quantum-classical approach or require circuit architectures that are difficult to implement. In both cases it’s unclear how to scale them up when more qubits or higher accuracy are required.

What we learned

We show, both in theory and in experiments, how to prepare such quantum states with great accuracy and a short depth quantum circuit. We also describe ways to scale it up, should that be required by the application.

The Deep Dive

FCAT researchers discovered that polynomial approximations together with matrix product states allow us to prepare quantum states with great accuracy and low depth. The resulting circuit is easy to execute on most gate-based quantum computers. For further details on this project, read the full paper here.