LANL Researchers Tackle the ‘Barren Plateau’ in Quantum Computing

By John Russell

March 24, 2021

Researchers from Los Alamos National Laboratory report a strategy for dealing with the “barren plateau” problem in quantum computing, which would be a significant advance for machine learning on current noisy intermediate scale quantum computers (NISQ). The researchers used a common hybrid approach that leverages classical computers for optimizing model parameters.

Broadly, the iterative calculations when training certain optimization models run into a problem where the resulting gradient used to update weights on each pass become so small – vanishingly so – that the model becomes stuck. LANL scientists, led by Marco Cerezo, have developed a work-around and mathematically proved that it works. Their paper was published in Nature last week and there is also an account of the work posted on the LANL website.

Here’s an excerpt from the LANL article, written by Charles Poling:

“People have been proposing quantum neural networks and benchmarking them by doing small-scale simulations of 10s (or fewer) few qubits,” Cerezo said. “The trouble is, you won’t see the barren plateau with a small number of qubits, but when you try to scale up to more qubits, it appears. Then the algorithm has to be reworked for a larger quantum computer.”

“A barren plateau is a trainability problem that occurs in machine learning optimization algorithms when the problem-solving space turns flat as the algorithm is run. In that situation, the algorithm can’t find the downward slope in what appears to be a featureless landscape and there’s no clear path to the energy minimum. Lacking landscape features, the machine learning can’t train itself to find the solution.”

“If you have a barren plateau, all hope of quantum speedup or quantum advantage is lost,” Cerezo said.”

Machine learning algorithms translate an optimization task—say, finding the shortest route for a traveling salesperson through several cities—into a cost function, a mathematical description of a function that will be minimized. The function reaches its minimum value only if you solve the problem. Most quantum variational algorithms initiate their search randomly and evaluate the cost function globally across every qubit, which often leads to a barren plateau.

“We were able to prove that, if you choose a cost function that looks locally at each individual qubit, then we guarantee that the scaling won’t result in an impossibly steep curve of time versus system size, and therefore can be trained,” said coauthor Lukasz Coles in the LANL account.

“The work solves a key problem of useability for quantum machine learning. We rigorously proved the conditions under which certain architectures of variational quantum algorithms will or will not have barren plateaus as they are scaled up,” said Cerezo. “With our theorems, you can guarantee that the architecture will be scalable to quantum computers with a large number of qubits.”

Their paper (Cost function dependent barren plateaus in shallow parametrized quantum circuits) does nice job explaining their work and its impact.

Here’s an excerpt from their introduction:

“One of the most important technological questions is whether Noisy Intermediate-Scale Quantum (NISQ) computers will have practical applications. NISQ devices are limited both in qubit count and in gate fidelity, hence preventing the use of quantum error correction.

“The leading strategy to make use of these devices is variational quantum algorithms (VQAs). VQAs employ a quantum computer to efficiently evaluate a cost function C, while a classical optimizer trains the parameters θ of a Parametrized Quantum Circuit (PQC) V(θ). The benefits of VQAs are three-fold. First, VQAs allow for task-oriented programming of quantum computers, which is important since designing quantum algorithms is non-intuitive. Second, VQAs make up for small qubit counts by leveraging classical computational power. Third, pushing complexity onto classical computers, while only running short-depth quantum circuits, is an effective strategy for error mitigation on NISQ devices.”

This is from their abstract:

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst polynomially vanishing gradient, so long as the depth of V(θ) is(log )O(logn). Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

Applying principles described by the LANL researchers it may be possible to use VQAs productively to solve practical problems on developing NISQ systems. It’s best to read the paper directly.

Link to Nature paper: https://www.nature.com/articles/s41467-021-21728-w

Link to LANL article: https://www.lanl.gov/discover/news-release-archive/2021/March/0319-barren-plateaus.php

Header image caption: A barren plateau is a trainability problem that occurs in machine learning optimization algorithms when the problem-solving space turns flat as the algorithm is run. Researchers at Los Alamos National Laboratory have developed theorems to prove that any given algorithm will avoid a barren plateau as it scales up to run on a quantum computer. Source: LANL article

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