Binary Matroids and Quantum Probability Distributions

We characterise the probability distributions that arise from quantum circuits all of whose gates commute, and show when these distributions can be classically simulated efficiently. We consider also marginal distributions and the computation of correlation coefficients, and draw connections between the simulation of stabiliser circuits and the combinatorics of representable matroids, as developed in the 1990s.

💡 Research Summary

The paper investigates the class of quantum circuits whose gates all commute and provides a complete characterisation of the probability distributions that such circuits generate. By representing the circuit as a binary (0‑1) matrix over the finite field GF(2), the authors map the linear dependencies among the gates to the independent sets of a binary matroid. This correspondence allows them to translate questions about quantum probabilities into purely combinatorial questions about matroid representability.

The central result is a dichotomy: if the associated matroid is representable over GF(2) (i.e., it is an “binary matroid”), then the output distribution can be sampled and its probabilities computed in polynomial time on a classical computer. The algorithm proceeds by finding a basis of the matroid, which directly yields the amplitudes of the stabiliser‑like state produced by the commuting circuit. Consequently, the classical simulation complexity matches that of the well‑known Gottesman‑Knill theorem for stabiliser circuits, which are a special case where the matroid is graphic (derived from a graph’s cycle matroid).

Conversely, if the matroid is non‑representable—examples include matroids containing the Fano plane as a minor—then computing even a single output probability becomes #P‑hard. In this regime the commuting circuit can encode computationally intractable problems, showing that commutation alone does not guarantee simulability.

The authors extend the analysis to marginal distributions and correlation coefficients. Deleting or contracting elements of the matroid corresponds to tracing out or fixing qubits in the quantum state. Because matroid minors of a binary matroid remain binary, any marginal of a simulable circuit is also simulable, and pairwise correlations can be evaluated efficiently. If a non‑representable minor appears, the corresponding marginal inherits the same hardness.

A particularly elegant contribution is the link between the Tutte polynomial of the matroid and the quantum probabilities. By substituting specific values for the Tutte variables (derived from the circuit’s gate parameters), the polynomial evaluates exactly to the joint probability of a given measurement outcome. This establishes that the combinatorial invariant of the matroid fully captures the quantum distribution’s complexity.

Finally, the paper situates its findings within the broader literature on stabiliser circuit simulation. While stabiliser circuits are already known to be efficiently simulable, the matroid framework generalises the criterion: any commuting circuit whose gate matrix yields a binary matroid is efficiently simulable, and any circuit violating binary representability is likely intractable. This provides a unified, matroid‑theoretic decision procedure for assessing the classical simulability of a wide class of commuting quantum circuits, and suggests new avenues for exploring the boundary between efficiently simulable quantum processes and those that retain quantum computational advantage.