A Probabilistic Quantum Circuit with Fallback (PQFs) is composed as a series of circuit stages that are selected to implement a target unitary. A final stage is conditioned on unsuccessful results of all the preceding stages as indicated by measurement of one or more ancillary qubits. This final stage executes a fallback circuit that enforces deterministic execution of the target unitary at a relatively high cost (mitigated by very low probability of the fallback). Specific instances of general PQF synthesis method and are disclosed with reference to the specific Clifford+T, Clifford+V and Clifford+π/12 bases. The resulting circuits have expected cost in logb(1/ε(log(log(1/ε)))+const wherein b is specific to each basis. The three specific instances of the synthesis have polynomial compilation time guarantees.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. -15.(canceled) 16. A computer-implemented method of defining a quantum circuit, comprising: establishing a first approximation of a target unitary to a requested precision; expanding the first approximation into a first multi-qubit unitary that implements the target single-qubit unitary in a selected basis upon successful measurement; defining a fallback circuit in the selected basis, wherein the fallback circuit implements the target unitary based upon an unsuccessful measurement; and outputting a circuit definition that includes a definition of the multi-qubit unitary and a definition of the fallback circuit; and based on the circuit definition, constructing the quantum circuit. 17. The computer-implemented method of claim 16, wherein the target unitary is a multi-qubit unitary, and further comprising: establishing a second approximation of the target multi-qubit unitary to a requested precision based on an unsuccessful output of the first multi-qubit unitary; and expanding the second approximation into a second multi-qubit unitary that implements the target single-qubit unitary in the selected basis upon successful measurement, wherein the fallback circuit implements the target unitary based upon an unsuccessful measurement associated with the second multi-qubit unitary. 18. The computer-implemented method of claim 16, further comprising: establishing a series of approximations of the target unitary to a requested precision based on an unsuccessful measurement of a multi-qubit unitary associated with a prior approximation in the series; and expanding the series of approximations into a corresponding series of multi-qubit unitaries that implement the target qubit unitary in the selected basis upon successful measurement, wherein the fallback circuit implements the target unitary based upon an unsuccessful measurement associated with a final multi-qubit unitary in the series. 19. The computer-implemented method of claim 16, wherein the approximation of the target qubit unitary is based on a rational cyclotomic approximation of the target qubit unitary. 20. The computer-implemented method of claim 16, further comprising establishing the rational cyclotomic approximation of the target qubit unitary by solving a norm equation. 21. The computer-implemented method of claim 16, wherein the target unitary is an axial rotation and is approximated by z*/z wherein z is a cyclotomic integer; and 22. The computer-implemented method of claim 16, wherein the multi-qubit unitary is defined with respect to at least one ancillary qubit and at least one primary qubit. 23. The computer-implemented method of claim 17, wherein the target single-qubit unitary is of the form 1 2 L  ( rz y - y * r *  z * ) , wherein z is a cyclotomic rational, r is a probability enhancement factor, and L is the minimal positive integer such that 2L>|r z|2. 24. The computer-implemented method of claim 23, further comprising selecting a value of r ∈[√{square root over (2)}] such that a norm equation is solvable for z replaced by rz. 25. The method of claim 16, wherein the first multi-qubit unitary is coupled to at least one ancillary qubit having a predetermined state. 26. The method of claim 25, wherein the at least one ancillary qubit is used in each of a plurality of probabilistic quantum circuit with fallback (PQF) stages associated with different multi-qubit unitaries and measurements associated with at least one of the PQF stages. 27. A quantum circuit, comprising: a series of multi-qubit sub-circuits based on a plurality of gates selected from a basis set so as to implement an axial rotation R based on respective ratios z*/z for each multi-qubit circuit of the series, wherein z is a cyclotomic integer, the series including at least an initial multi-qubit sub-circuit and a final multi-qubit sub-circuit, the final multi-qubit sub-circuit arranged to implement the rotation R based on an unsuccessful output of at least one sub-circuit of the series; and at least one ancillary qubit and at least one primary qubit coupled to at least the initial multi-qubit sub-circuit such that production of a first output state of the at least one ancillary qubits is associated with an output state of the primary qubit that is associated with the rotation R and a second output state is associated with an output state of the primary qubit that is not the rotation R. 28. The quantum circuit of claim 27, further comprising a fallback sub-circuit coupled to the final multi-qubit sub-circuit that implements the rotation R if an output state of the at least one ancillary qubit produced by the final multi-qubit sub-circuit is associated with an output state of the final multi-qubit sub-circuit that does not correspond to the rotation R. 29. The quantum circuit of claim 27, further comprising a switch that couples an output state of the initial multi-qubit sub-circuit to the final multi-qubit sub-circuit based on the output state of the initial qubit. 30. The quantum circuit of claim 27, further comprising a series of intermediate multi-qubit sub-circuits coupled between the initial and final multi-qubit sub-circuits, each of the intermediate multi-qubit sub-circuits coupled to respective switches that direct output states of the least one primary qubit by each intermediate multi-qubit sub-circuit to a subsequent intermediate multi-qubit circuit or the final multi-qubit sub-circuit based on the output state of the intermediate multi-qubit sub-circuits. 31. The quantum circuit of claim 30, wherein the output states of the intermediate multi-qubit sub-circuits is associated with a result other than the rotation R. 32. The quantum circuit of claim 31, wherein each of the intermediate multi-qubit sub-circuits implements the rotation R if successful based on the output state of the immediately prior multi-qubit sub-circuit. 33. The quantum circuit of claim 32, wherein the output state of the immediately prior multi-qubit sub-circuit corresponds to a state other than an input state at the initial multi-qubit sub-circuit. 34. The quantum circuit of claim 27, wherein the basis is the Clifford+T basis, the Clifford+π/12 basis, or the Clifford+V basis. 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 target rotation, a number of stages k, and a precision; selecting cyclotomic integers z such that z*/z is associated with a stage rotation angle within the selected precision for each stage; performing search to find values of r such that a norm equation or a two squares equation is solvable for z=rz for each stage; defining a one-qubit unitary for each stage based on r,z and the solution of the norm equation; defining a multi-qubit circuit having k stages based on the one qubit unitary and a plurality of gates selected from the Clifford+T basis, the Clifford+π/12 basis, or the Clifford+V basis, the multiple-qubit circuit including at least one ancillary qubit having an output state that is associated with realization of an output state associated with the target rotation; and a fallback circuit coupled to a final stage of the multi-qubit circuit and that implements the target rotation if the at least one ancillary qubit indicates failure of each of the k stages.