Topological Quantum Hashing with the Icosahedral Group

We study an efficient algorithm to hash any single-qubit gate into a braid of Fibonacci anyons represented by a product of icosahedral group elements. By representing the group elements by braid segments of different lengths, we introduce a series of pseudogroups. Joining these braid segments in a renormalization group fashion, we obtain a Gaussian unitary ensemble of random-matrix representations of braids. With braids of length $O\mathbf{(}{log}^2(1/\epsilon )\mathbf{)}$, we can approximate all SU(2) matrices to an average error $\epsilon$ with a cost of $O\mathbf{(}\log (1/\epsilon )\mathbf{)}$ in time. The algorithm is applicable to generic quantum compiling.

Figure 1

Probability distribution of distance $d$ [13] to the targeted identity matrix in the set of nontrivial braids that one samples in different algorithms. $P_{\mathrm{BF}}(d)$ of the brute-force search (red solid squares) roughly follows $(4/\pi )d^2\sqrt{1-(d/2{)}^2}$, reflecting the three-sphere nature of the unitary matrix space (three independent parameters apart from an unimportant phase). In the pseudoicosahedral group approach ($n=4$), distributions for $L=8$ ($P_8$, black empty circles) and $L=24$ ($P_{24}$, blue solid trangles) agree very well with the energy-level-spacing distribution of the unitary Wigner-Dyson ensemble of random matrices, $P_L(d)=(32/{\pi }^2)(d^2/d_L^3)e^{-(4/\pi )(d/d_L{)}^2}$. $P_L(d)$ differ only by their corresponding average $d_L$ (not a fitting parameter), which decays exponentially as $L$ increases. Note $P_{24}(d)$ is roughly ten-times sharper and narrower than $P_8(d)$.

Figure 2

Approximation to the $-iX$ gate (an element of the icosahedral group) in terms of braids of the Fibonacci anyons of length $L=24$ in the graphic representation. In this example the error is 0.0031.

Figure 3

The graphic representation of the braid approximating the target gate $iZ$ in the icosahedral group approach with a preprocessor of $l=8$ and $m=3$ and a main processor of $L=24$ and $n=3$. To emphasize the structure, we skip the explicit braid sequence but mark the segments only, among which ${\tilde{g}}_{p1}{\tilde{g}}_{p2}{\tilde{g}}_{p3}\approx iZ$ and ${\tilde{g}}_{q1}{\tilde{g}}_{q2}{\tilde{g}}_{q3}\approx {\tilde{g}}_{q4}^{-1}$ up to a phase. The braid (with a reduced length of 98 due to accidental cancellations where the component braids connect) has an error of 0.000 99 [18].

Figure 4

Probability distribution of $d$ in 10 000 random tests using the icosahedral group approach with a preprocessor of $l=8$ and $m=3$ and a main processor of $L=24$ and $n=3$. The total length of the braids (neglecting accidental cancellations when component braids connect) is 120. The trend agrees with the unitary Wigner-Dyson distribution (solid line) with an average error $7.1\times {10}^{-4}$.