Finding the fingerprints of noisy distributions in random quantum systems

Imagine shuffling a deck of cards where some of the cards are fading just because of the way they interact with the external environment as you shuffle them.

Your job is to figure out the interplay between the shuffling or “imposed randomness” due to the way you shuffle your cards, and how that relates to the fading as cards are exposed to the external environment.

That’s a rough analogy for what Matthew Duschenes and his PhD advisors were doing in a paper that received an Editors’ Suggestion in the American Physical Society (APS) journal Physical Review A. An Editors’ Suggestion denotes a paper of importance, impact, and intrinsic interest.

Duschenes is currently a postdoctoral researcher at the University of Guelph, and he earned his PhD at Perimeter Institute and the University of Waterloo with the late Ray Laflamme, a Perimeter founding faculty member and founding director of the Institute for Quantum Computing, who passed away in 2025.

The paper bears the names of Laflamme and other advisors — Juan Carrasquilla, a Perimeter visiting fellow who is currently at ETH Zurich in Switzerland, and Roger Melko, a Perimeter associate faculty member who is also at the University of Waterloo.

The research, which was started more than two years ago, is likely one of the last scientific papers to bear Laflamme’s name. “I feel very honoured to have been Raymond's second-last PhD student,” Duschenes says.

We all know about randomness in our everyday world, such as the spin of a roulette wheel or the throw of the dice.

But in the quantum world, randomness is much more intrinsic. There is an inherent probabilistic nature in systems described by quantum mechanics, meaning any properties of these systems are characterized not by specific measurement outcomes, but by probabilities of observing these outcomes.

But beyond that, as Duschenes explains, there can also be an “imposed randomness” in quantum computing systems. “Our systems of study involve imposing that the systems evolve randomly, due to random protocols that are applied.”   

A quantum computation model runs on a huge number of possible circuits, representing sequences of quantum gates, measurements and qubit initializations. Instead of studying just one circuit, researchers need to pick many at random to describe a complex quantum process. To extend the card-shuffling analogy: It’s a bit like learning about card games by studying many well-shuffled decks instead of one hand.

But also, there is “noise” caused by the quantum systems interacting with their environment. One can imagine ensembles of quantum bits – qubits – being like soap bubbles, each holding delicate information in its shape and colour. But any touch from the outside will disturb them, causing some of them to disappear.

This interplay – between the imposed randomness and the noise – is what Duschenes and his colleagues were studying. Understanding it will help researchers determine if a quantum device is doing something truly special that cannot easily be done on a regular computer.

Laflamme was an exceptional scientist and mentor to his students. To honour his legacy, Perimeter has established an early career chair and postdoctoral fellowship in his name, with appointments forthcoming.

This paper is one that Laflamme would have been happy about.

“We were initially interested in understanding ways of trying to simulate these quantum systems using conventional computers — so running computer simulation codes and trying to find more efficient ways to simulate these systems. But that involved different ways of measuring and representing this information within these quantum systems and trying to understand more about what we can measure about these systems and what we can extract from them.” Duschenes said.

 “That led us into more fundamentally understanding these measurement procedures, especially in these randomly behaving quantum systems and how that affects what we can measure.”

They were also trying to understand the effects of noise and errors in quantum systems, he added.

 “We’re trying to understand how these systems change over time, how we can describe that mathematically,” Duschenes said. “By understanding the details of how these sources of randomness are affected by noise, you can get a better sense of whether these quantum computers will be useful for a specific task.”

Duschenes tackled the problem using combinatorics, the branch of math that is all about counting, arranging, and combining things, especially when there are too many possibilities to list one by one.

“Previous people had used different approaches that were more rooted in something called complex analysis, but we used an approach that’s related to combinatorics instead. We think this approach is a little cleaner and is a little more intuitive as you progress through the steps when you’re deriving expressions for these statistics and these quantities,” Duschenes explained.

Carrasquilla said the approach that Duschenes took was “brilliant.” The more general measurement approach provided a more complete picture — much like the difference between looking at an object from one side only and seeing only that surface, versus seeing it in 3D and having a more informative picture or film, he added.

“He tackled the problem of seeing what it would look like from different angles when the state is random.”

The work produced formulas – statistical fingerprints – for the noisy distributions in more realistic settings called mixed states (quantum systems that are partly entangled with an environment that can’t be fully controlled, so some information is lost).

They were able to test those formulas against the simulations of “brickwork” quantum circuits, so called because the quantum operations are laid out like layers of bricks that have noise mixed into each layer, therefore more realistically mimicking the imperfections that plague today’s quantum processors.

“We were able to recreate what these distributions would look like if we were to repeat these experiments many times. We can plot them, like histograms (graphical charts showing the frequency of quantitative, continuous data) and we can vary the different parameters, such as how large the system is, or how long we’re evolving it for, or how many errors and how much noise is in the system. We can then visualize what these different distributions look like,” Duschenes said.

The picture that emerged was interesting. Instead of just getting one “peak” for the noisy distributions, Duschenes and his colleagues found that some kinds of measurements – those built from mismatched, or non-symmetric sets of measurement operators – produced distributions with multiple peaks.

In other words, they got a much richer statistical picture than had previously been observed.

“It leads to nice questions about why there are these multiple peaks and where they might show up in other areas of quantum computing,” Duschenes said.

Carrasquilla said it was surprising that it was even possible to do this using combinatorics. “It was really nice, curiosity-driven research led by Matthew, and it led to these interesting observations.”

The work raised future interesting research questions to be explored. “Instead of simulating these brickwork circuits, you could look at other quantum computation models or other kinds of algorithms and see if they exhibit these new types of distributions we hadn’t seen before,” Duschenes said.

Quantum computing scientists are trying to achieve “quantum advantage,” where a quantum computer solves a real-world, useful problem significantly faster, cheaper, or more accurately than a classical computer. But how do you measure that and figure out where quantum computers can demonstrate an advantage? Much of the current debate rests on assumptions built around the simpler, single-peaked noise cases. The more realistic quantum noise distribution picture that Duschenes and his colleagues have developed might offer better clues.

The work done by Duschenes and his colleagues in this paper is a beautiful way to commemorate the contributions of Laflamme as a mentor and a scientist. One of the last papers to bear his name contains within it the attributes of creativity and ingenuity that Laflamme would have deeply appreciated.

“Raymond always asked incredibly poignant questions and uniquely understood the fundamental mathematical details and experimental implications of quantum information,” Duschenes said.

“Few have ever accomplished breakthroughs in both theory, experiment, and mentorship like he did throughout his career. He was fascinated by whether we may in the future find out that we are fundamentally able or unable to make quantum systems useful and robust to their noisy environment, and what implications that has on the physics that governs our world,” he added.

Laflamme also had an amazing balance between the technical detail and the ability to step back and think about what it meant for quantum computing. He set a high scientific bar for being thorough, along with a boundless curiosity, the paper’s co-authors said.

“He would appreciate that this work contributes to understanding these noise-induced phenomena, and opens up new questions about randomness, noise, and robustness in these more general settings,” Duschenes said.