The idea that a system obeying interpolating statistics can be described by a deformed oscillator algebra has been an outstanding issue. This original concept introduced long ago by Greenberg is the motivation for this investigation. We establish that a q-deformed algebra can be used to describe the statistics of particles (anyons) interpolating continuously between Bose and Fermi statistics, i.e., fractional statistics. We show that the generalized intermediate statistics splits into the Boson-like and Fermion-like regimes, each described by a unique oscillator algebra. The B-anyon thermostatistics is described by employing the q-calculus based on the Jackson derivative but the F-anyons are described by ordinary derivatives of thermodynamics. Thermodynamic functions of both B-anyons and F-anyons are determined and examined.
Braiding and fusing anyons are fundamental to topological quantum computation, yet existing circuit protocols often employ ancillas and measurements. We determine the minimal circuit resources for performing such anyon operations in quantum double models of generic finite groups. Our approach braids anyons through a sequential unitary circuit without ancillas. Intuitively, this is obtained by deforming the underlying graph on which the anyons live. More generally, we introduce graph rewiring, a diagrammatic calculus that assigns a geometric interpretation to individual two-qudit gates and tracks their action on stabilizers. For fusion, we prove a general no-go theorem excluding purely unitary protocols for fusing unknown anyon types without introducing ancillas. Remarkably, we show such "input-agnostic" fusion is enabled by simply using a qudit reset. Hence, contrary to folklore, ancilla resources are not needed to braid or fuse non-Abelian anyons (non-Abelions). This work has implications in experimental efforts in error correction: our fusion protocol highlights reset as an underutilized primitive for error correction without the spacetime-overhead of ancillas and measurements.
Abstract Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as <?CDATA
π/2?> π / 2 Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.