A computer-implemented method configured to precompile a quantum circuit such that it can be executed by a quantum processor, the method comprising: evaluating a single qubit unitary gate to determine if it is representable exactly as a Fibonacci anyon braid pattern by expressing the single qubit unitary gate as an exact unitary of the form U u v k: = u v ∗ τ ω k v τ − u ∗ ω k where k is an integer, and u, v ∈ Z ( ω), wherein Z ( ω) is the ring of integers extended by the primitive fifth root of unity, i.e., Z ( ω) = { a + bω + cω 2 + dω 3}, a, b, c, d ∈ Z, Z is the ring of integers, and | u 2 | + τ | v 2 | = 1; and if so, synthesizing a sequence of elementary gates to exactly represent the single qubit unitary gate; if the single qubit unitary gate is not exactly representable as a Fibonacci anyon braid pattern, obtaining a synthesized approximation of the gate by a gate that is exactly representable as a Fibonacci anyon braid pattern wherein the approximation is determined based on a requested precision by building an asymptotically optimal approximation of unitary rotations of the form u 0 0 u ∗ and 0 v − v ∗ 0 by Fibonacci anyon braid pattern circuits by approximating each unitary rotation as an exact unitary of the form U[u,v,0] and synthesizing a sequence of elementary gates to exactly represent each exact unitary of the form U[u,v,0]; wherein the synthesizing the sequence of elementary gates comprises iterative reduction of a complexity measure of the exact unitary, wherein the complexity measure is the Gauss complexity of a selected matrix element associated with the unitary gate, by including gates of the form FT k to the sequence until the residual complexity is less than or equal to two, whereupon a gate of the form ω k T j is added to the beginning of the sequence, wherein k and j are non-negative integers, with k less than or equal to nine, and F = τ τ τ − τ, τ = 5 − 1 2, T = 1 0 0 ω, and ω = e iπ /5; defining, by the synthesized sequence of elementary gates, the quantum circuit defined by the Fibonacci anyon braid representations configured to be executed by the quantum processor. evaluating a single qubit unitary gate to determine if it is representable exactly as a Fibonacci anyon braid pattern by expressing the single qubit unitary gate as an exact unitary of the form U u v k: = u v ∗ τ ω k v τ − u ∗ ω k where k is an integer, and u, v ∈ Z ( ω), wherein Z ( ω) is the ring of integers extended by the primitive fifth root of unity, i.e., Z ( ω) = { a + bω + cω 2 + dω 3}, a, b, c, d ∈ Z, Z is the ring of integers, and | u 2 | + τ | v 2 | = 1; and if so, synthesizing a sequence of elementary gates to exactly represent the single qubit unitary gate; if the single qubit unitary gate is not exactly representable as a Fibonacci anyon braid pattern, obtaining a synthesized approximation of the gate by a gate that is exactly representable as a Fibonacci anyon braid pattern wherein the approximation is determined based on a requested precision by building an asymptotically optimal approximation of unitary rotations of the form u 0 0 u ∗ and 0 v − v ∗ 0 by Fibonacci anyon braid pattern circuits by approximating each unitary rotation as an exact unitary of the form U[u,v,0] and synthesizing a sequence of elementary gates to exactly represent each exact unitary of the form U[u,v,0]; wherein the synthesizing the sequence of elementary gates comprises iterative reduction of a complexity measure of the exact unitary, wherein the complexity measure is the Gauss complexity of a selected matrix element associated with the unitary gate, by including gates of the form FT k to the sequence until the residual complexity is less than or equal to two, whereupon a gate of the form ω k T j is added to the beginning of the sequence, wherein k and j are non-negative integers, with k less than or equal to nine, and F = τ τ τ − τ, τ = 5 − 1 2, T = 1 0 0 ω, and ω = e iπ /5; defining, by the synthesized sequence of elementary gates, the quantum circuit defined by the Fibonacci anyon braid representations configured to be executed by the quantum processor.