Certain ensembles of metapletic anyons allow for topologically protected encoding and processing of quantum information. Such processing is done by sequences of gates (circuits) drawn from a certain basis of unitary metaplectic gates. A subject unitary operator required for the desired processing can be approximated to any desired precision by a circuit that has to be effectively and efficiently synthesized on a classical computer. Synthesis methods use unitary reflection operators that can be represented either exactly or by ancilla-assisted approximation over the basis of metaplectic gates based on cost-optimizing determinations made by the synthesis algorithm.
Please help me write a proper abstract based on the patent claims. CLAIM: 1-15. (canceled) 16. A computer-implemented synthesis method for quantum computing, comprising: with a processor: selecting a plurality of weakly-integral anyons; and defining an asymptotically optimal quantum circuit approximating a selected unitary operator to a desired precision with a series of gates to be applied to the selected plurality of weakly-integral anyons. 17. The method of claim 16, wherein the weakly-integral anyons are metaplectic anyons. 18. The method of claim 17, wherein gates of the series of gates in the quantum circuit are selected from an extended metaplectic basis. 19. The method of claim 17, wherein gates of the series of gates in the quantum circuit are selected from an augmented metaplectic basis 20. The method of claim 19, wherein the series of gates includes a set of gates corresponding to an axial reflection operator. 21. The method of claim 20, wherein the axial reflection operator is an n-qutrit axial reflection operator, with a recursively-built exact representation in terms of single-qutrit INC and Flip gates and two-qutrit SWAP and SUM gates. 22. The method of claim 20, wherein if the axial reflection operator is an n-qutrit axial reflection operator for n>5, then n-qutrit axial reflection operator is synthesized as an approximate n-qutrit reflection operator using (n−1) ancillary qutrits. 23. The method of claim 20 wherein the selected unitary is a two-level multi-qutrit unitary, and further comprising generating an intermediate representation as a product of two-level reflections and, based on the intermediate representation, defining the quantum circuit as a series of metaplectic gates and axial reflections. 24. The method of claim 20, further comprising representing the selected unitary as a product of two-level unitaries, and defining the asymptotically optimal quantum circuit based on the representation as a product of two-level unitaries. 25. The method of claim 20, wherein the methods of claims 8 and 9 are applied to a known exact decomposition of the selected multi-qutrit unitary into a product of several two-level unitaries and at most one diagonal unitary. 26. The method of claim 20, further comprising determining whether to define the asymptotically optimal quantum circuit based on exact axial reflections or as an ancilla-assisted circuit based on approximated reflections. 27. A circuit synthesis tool, comprising: at least one processor; and a memory storing processor-executable instructions for synthesizing a unitary over a metaplectic basis. 28. The circuit synthesis tool of claim 27, wherein the memory stores processor-executable instructions for representing a qutrit with coefficients that are Eisenstein rationals. 29. The circuit synthesis tool of claim 27, further comprising determining whether to define the asymptotically optimal quantum circuit based on exact axial reflections or as an ancilla-assisted circuit based on approximated reflections. 30. The circuit synthesis tool of claim 29, wherein the memory stores processor-executable instructions for synthesizing an n-qutrit circuit using n-ancillas. 31. The circuit synthesis tool of claim 30, wherein the memory stores processor-executable instructions for synthesizing the unitary using at least one two-qutrit entangler. 32. The circuit synthesis tool of claim 27, wherein the memory stores processor-executable instructions for determining if the unitary is to be synthesized without ancillas, and synthesizing the circuit without ancillas based on the determination. 33. The circuit synthesis tool of claim 32, wherein the memory stores processor-executable instructions for synthesizing the circuit so as to include one or more SUM, SWAP, and axial reflection gates. 34. The circuit synthesis tool of claim 27, wherein the circuit is synthesized with respect to at least one qudit. 35. A computer implemented quantum circuit synthesis method, comprising: defining a universal metaplectic basis; based on a precision and a number of qutrits, determining whether a selected unitary is to be implemented with ancillas or without ancillas; if synthesis without ancillas is selected, synthesizing the unitary based on a series of SUM, SWAP, and axial reflection gates, wherein the axial reflection gates are represented exactly based on a single-qutrit axial reflection gates; and if synthesis with ancillas is selected and the unitary is an n-qutrit unitary, synthesizing the n-qutrit unitary approximately using n ancillas.