Fig. 1: Principle of the dynamical string-net preparation (DSNP) approach and experiment. | Nature Communications

Fig. 1: Principle of the dynamical string-net preparation (DSNP) approach and experiment.

From: Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials

Fig. 1

a Typical string-net configurations of spins in \(\left\vert 1\right\rangle\) state, for a trivial state (left) and for a string-net condensate (right). b Branching rule for the Fib-SNC. A dashed line represents a qubit in the \(\left\vert 0\right\rangle\) state while a solid line represents a qubit in the \(\left\vert 1\right\rangle\). c Five-qubit F-move relating allowed string-net configurations among five qubits. When one or two pairs of four outer legs are identified, this becomes four-qubit or three-qubit F-moves. d Logical qubit encoding using a triplet of τ1 anyons. Logical \(\overline{\left\vert 0\right\rangle }\) and \(\overline{\left\vert 1\right\rangle }\) differ by the fusion outcomes of the first two τ1 anyons. The red lines in the figure represent the space-time trajectory of anyons. e Pairwise braiding among the triplet of τ1 anyons implements a non-Clifford gate on the logical state encoded on the triplet of anyons(see SM Sec. I A). f Schematic outline of DSNP. Yellow dots represent qubits in SNC, and empty dots represent reserve qubits in \(\left\vert 0\right\rangle\) state. (see SM Sec. IV for details). g The Fib-SNC can be visualized with a pair of two-dimensional surfaces representing two time-reversed copies of TQFT. To create two τ1 anyons, we bring in an open string from above. F- and R-moves bring the ends of the open string to join the two copies of TQFT through wormholes, with the ends piercing the wormholes and localizing anyons. h The R-moves (or its complex conjugate) to resolve the over-crossing (see SM Sec. I A for more details). i A deep quantum circuit for braiding two τ1 anyons using hardware-native gates (see Fig. 3). j DSNP for the smallest Fib-SNC. k Quantum circuit corresponding to two \({\mathcal{S}}\) gates followed by the 3-qubit F-move, implementing all the steps of panel (j). l The probability weight of different string-net configurations corresponding to the depicted string nets for the minimal Fib-SNC. The red  × marks the vertices violating the branching rule. Only 5% of the shots violate the branching rule.

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