Repeat-Until-Success (RUS) circuits are compiled in a Clifford+T basis by selecting a suitable cyclotomic integer approximation of a target rotation so that the rotation is approximated within a predetermined precision. The cyclotomic integer approximation is randomly modified until a modified value can be expanded into a single-qubit unitary matrix by solving one or more norm equations. The matrix is then expanded into a two-qubit unitary matrix of special form, which is then decomposed into an optimal two-qubit Clifford+T circuit. A two-qubit RUS circuit using a primary qubit and an ancillary qubit is then obtained based on the latter decomposition. An alternate embodiment is disclosed that keeps the total T-depth of the derived circuit small using at most 3 additional ancilla qubits. Arbitrary unitary matrices defined over the cyclotomic field of 8th roots of unity are implemented with RUS circuits.
Please help me write a proper abstract based on the patent claims. CLAIM: 1.-15. (canceled) 16. A method of defining a quantum circuit, comprising: establishing a rational cyclotomic approximation of a target single-qubit unitary to a requested precision; expanding the rational cyclotomic approximation into a multi-qubit unitary that implements the target single-qubit unitary upon successful measurement; decomposing the multi-qubit unitary into a Clifford+T circuit; and defining a Repeat-Until-Success (RUS) circuit based on the decomposition. 17. The method of claim 16, further comprising: establishing the rational cyclotomic approximation of the target single-qubit unitary so as to solve a required norm equation; and defining the multiple-qubit unitary based on the rational cyclotomic approximation. 18. The method of claim 16, wherein the multi-qubit unitary is a multi-qubit unitary defined with respect to at least one ancillary qubit and a primary qubit. 19. The method of claim 17, wherein the target unitary is an axial rotation and is approximated by z*/z wherein z is a cyclotomic integer; and modifying z by randomly selecting at least one value of r∈√{square root over (2)}] such that the norm equation is solvable for z replaced by rz. 20. The method of claim 19, wherein a single-qubit matrix defined by z and r is of the form 1 2 L  ( rz y - y * r *  z * ) and the norm equation is |y|2=2L−|rz|2, wherein L is a positive integer. 21. The method of claim 20, wherein at least one value of r is selected so as to boost the one-round probability of success p(r)=|rz|2/2L of the RUS circuit. 22. The method of claim 21, wherein the probability of success is proportional to a square of a magnitude of the product of r and z. 23. The method of claim 20, further comprising defining the multi-qubit unitary based on the single-qubit matrix. 24. The method of claim 23, further comprising expanding the single-qubit matrix corresponding to the target single-qubit unitary into a multi-qubit unitary, defining the multi-qubit unitary as a Clifford+T unitary by identifying the entangling two-qubit gates required for implementation of the RUS circuit. 25. The method of claim 24, further comprising decomposing the single-qubit matrix corresponding to the target single-qubit unitary into a product of axial rotations and implementing the multi-qubit unitary corresponding to each axial rotation as a Clifford+T circuit obtained from a product g3c′g4, wherein g1, g2, g3, g4 are Clifford gates, c is a T-code, and c′ is a decoration of c, and defining the two-qubit unitary as Λ(g5)(Idg1)Lift(c′)(Idg2)Λ(g6) . 26. The method of claim 25, wherein the ancillary qubit or a plurality of qubits associated with the axial rotations are reused for the implementation of at least one additional RUS circuit. 27. The method of claim 24, further comprising decomposing the single-qubit matrix corresponding to the target single-qubit unitary, wherein the target single-qubit unitary performs a rotation around an axis in the x-y plane; implementing the multi-qubit unitary corresponding to the rotation around an axis in the x-y plane with a circuit obtained from a controlled routing operation, wherein the controlled routing operation directs an input state to a unitary operation implementing either U or U\ and prepares one out of two possible EPR states on at least two ancillary qubits. 28. The method of claim 18, wherein the multi-qubit unitary is defined so as to be coupled to the ancillary qubits and the primary qubit, such that measurement of the ancillary qubits can be associated with success or failure of the multi-qubit circuit to produce a primary qubit state corresponding to the target single-qubit unitary. 29. The method of claim 28, further comprising defining a restoring circuit that returns the output state of the primary qubit to an input state of the primary qubit if the ancillary qubits indicate failure of the multi-qubit circuit. 30. A quantum circuit, comprising: a multi-qubit circuit based on a plurality of gates selected from the group of Hadamard gates, CNOT gates, and T-gates selected so as to implement a rotation R based on a ratio z*/z, wherein z is a cyclotomic integer; and at least one ancillary qubit and a primary qubit, coupled to the plurality of gates such that production of a first output state of the ancillary qubits is associated with an output state of the primary qubit that is associated with the rotation R. 31. The quantum circuit of claim 30, wherein the rotation R=R(θ), and z is selected such that |z*/z−eiθ|<ε, wherein ε is a precision. 32. The quantum circuit of claim 31, wherein a second output state of the ancillary qubits is associated with an output state of the primary qubit that is not associated with the rotation R(θ). 33. The quantum circuit of claim 32, wherein a one-qubit circuit corresponding to the rotation R(θ) is a product circuit defined by g3c′g4, wherein g1,g2, g3, g4 are Clifford gates, c is a T-code, and c′ is a decoration of c, and the multi-qubit unitary is Λ(g5)(Idgi)Lift(c′)(Idg2)Λ(g6) . 34. The quantum circuit of claim 33, further comprising a restoring circuit situated to be applied to the primary qubit if the ancillary qubits are measured to be in the second state. 35. A compiler for defining a quantum circuit, comprising: a processor; and at least one computer-readable storage medium having computer-executable instructions for a method comprising: receiving a selected rotation angle and precision; selecting a cyclotomic integer z such that z*/z is associated with the selected rotation angle within the selected precision; performing a random search to find values of r∈[√{square root over (2)}] such that a norm equation (|y|2+|rz|2)/2L=1 is solvable for z=rz and y is a cyclotomic integer; defining a one-qubit unitary as 1 2 L  ( rz y - y * r *  z * ) ; defining a multiple-qubit RUS circuit based on the one qubit unitary and a plurality of Clifford+T gates, the multiple-qubit RUS circuit including at least one ancillary qubit having an output state that is associated with realization of an output state of a primary qubit that is associated with the selected rotation; coupling a restoring circuit so as to return the output state of the primary qubit to an input state if the at least one ancillary qubit is indicates that the output state of the primary qubit does not correspond to the selected rotation; and storing associated circuit definitions.