Mapping the Statistical Fingerprints of Noise in Random Quantum Circuits

A deck of cards is being shuffled, and as it is shuffled, some of the cards are fading from contact with the air around them. That is the picture Matthew Duschenes uses for the problem he spent his doctoral years on: how do you separate the randomness you impose on a system from the damage the environment does to it? The answer, published in Physical Review A with an Editors' Suggestion, is a set of statistical formulas for noisy random quantum circuits.

Duschenes is now a postdoctoral researcher at the University of Guelph. He earned his Ph.D. at Perimeter Institute and the University of Waterloo under Ray Laflamme, a Perimeter founding faculty member and founding director of the Institute for Quantum Computing, who died in 2025. Laflamme's name is on the paper alongside Juan Carrasquilla, a Perimeter visiting fellow now at ETH Zurich, and Roger Melko, a Perimeter associate faculty member who is also at Waterloo. The work began more than two years ago and is likely among the last papers to carry Laflamme's name. Duschenes says he feels honored to have been Laflamme's second-to-last Ph.D. student. Perimeter has created an early career chair and a postdoctoral fellowship in Laflamme's name, with appointments still to come.

Two kinds of randomness are at play. Quantum mechanics is probabilistic by nature, so any property of a system is described by the probabilities of measurement outcomes. On top of that, quantum computing research often imposes randomness deliberately, letting systems evolve under random protocols. A quantum computation runs over an enormous number of possible circuits, sequences of gates, measurements and qubit initializations, so researchers sample many circuits at random rather than studying one. It is like learning about card games from many well-shuffled decks rather than a single hand.

Then there is noise, which comes from qubits interacting with their surroundings. The source compares qubits to soap bubbles holding delicate information, any outside touch making some of them vanish. How imposed randomness and noise interact matters because it helps determine whether a quantum device is doing something that a regular computer cannot easily do.

The project began as an effort to find more efficient ways to simulate quantum systems on conventional computers. That led Duschenes and his colleagues into the details of measurement in randomly behaving quantum systems, and into how noise and errors change what can be measured. Where earlier researchers had used methods rooted in complex analysis, Duschenes turned to combinatorics, the mathematics of counting and arranging when there are too many possibilities to list. He considers it cleaner and more intuitive when deriving the expressions. Carrasquilla called the approach brilliant and said it was surprising that combinatorics could do the job. He likened the broader measurement approach to seeing an object in three dimensions rather than from one side only.

The formulas describe noisy distributions for mixed states, quantum systems partly entangled with an environment that cannot be fully controlled, so some information is lost. The team tested them against simulations of brickwork circuits, named because their operations are laid out like layers of bricks, with noise mixed into each layer to mimic the imperfections of today's quantum processors. By varying system size, evolution time and the amount of noise, they plotted histograms of what repeated experiments would produce.

The result was unexpected. Rather than a single peak, some measurements, those built from mismatched or nonsymmetric sets of measurement operators, produced distributions with multiple peaks. That is a richer statistical picture than had been observed before. Duschenes says it raises questions about why the peaks appear and where else in quantum computing they might show up.

That matters for the debate over quantum advantage, the point where a quantum computer solves a useful problem significantly faster, cheaper or more accurately than a classical one. Much of that debate, the source says, rests on assumptions built around simpler single-peaked noise, and the new picture may offer better clues. Duschenes suggests next testing other quantum computation models and algorithms for these new distributions.

Of Laflamme, Duschenes said he asked incisive questions and was fascinated by whether quantum systems can ultimately be made useful and robust against noise. The co-authors say he set a high bar for thoroughness alongside boundless curiosity. Duschenes believes he would appreciate that the work opens new questions about randomness, noise and robustness.