Simulation of quantum computation with magic states via Jordan-Wigner transformations

Negativity in certain quasiprobability representations is a necessary condition for a quantum computational advantage. Here we define a quasiprobability representation exhibiting this property with respect to quantum computations in the magic state model. It is based on generalized Jordan-Wigner transformations, and it has a close connection to the probability representation of universal quantum computation based on the $Λ$ polytopes. For each number of qubits, it defines a polytope contained in the $Λ$ polytope with some shared vertices. It leads to an efficient classical simulation algorithm for magic state quantum circuits for which the input state is positively represented, and it outperforms previous representations in terms of the states that can be positively represented.

💡 Research Summary

The paper addresses a central question in the theory of quantum computation with magic states (QCM): how to delineate the boundary between classically efficiently simulable circuits and those that retain a quantum advantage. It builds on the well‑known observation that negativity in quasiprobability representations (e.g., the discrete Wigner function) is a necessary condition for quantum speed‑up. Existing frameworks fall into two extremes. The CNC (Closed and Non‑Contextual) construction provides an efficiently updatable quasiprobability description but only for a limited set of states, essentially stabilizer states and a few extensions. The Λ‑polytope model, by contrast, offers a fully probabilistic hidden‑variable representation for all quantum states, but the stochastic update rules for Clifford gates and Pauli measurements are generally intractable, rendering the simulation inefficient.

The authors introduce a new quasiprobability representation that sits between these two extremes. The construction is based on a generalized Jordan‑Wigner transformation that maps Pauli operators to Majorana fermion operators. Crucially, the set of Pauli operators that can be expressed as quadratic polynomials in Majorana operators is characterized by the line graph of an underlying “root” graph. Each edge of the root graph becomes a vertex of the line graph, and adjacency in the line graph corresponds to shared endpoints in the root graph. By selecting appropriate root graphs (e.g., the complete graph K₄ and its line graph), the authors generate a family of phase‑space points that go beyond the maximal CNC sets while still being a subset of the vertices of the Λ‑polytope.

For each phase‑space point (Ω, γ) – where Ω is a set of commuting/anticommuting Pauli operators defined via the line‑graph construction and γ is a non‑contextual sign assignment satisfying β‑constraints – they define an operator \