Evaluating quantum computing circuits in view of the resource costs of a quantum algorithm

A processor-implemented method for compiling a quantum circuit for execution by a quantum processor, wherein the quantum circuit is for performing an evaluation of a piecewise smooth function, the method comprising:

• receiving an input comprising the piecewise smooth function and a domain;
• partitioning the domain of the piecewise smooth function into a set of sub-intervals by determining whether an input value of the piecewise smooth function falls into any of the set of sub-intervals;
• selecting whether to apply a first method or a second method for each of the set of sub-intervals;
• with respect to a first subset of the set of sub-intervals for which the first method is selected:
• determining a polynomial interpolation for the first subset of the set of sub-intervals corresponding to the input,
• constructing a quantum circuit for performing, in parallel, a polynomial evaluation of a function for each of the first subset of the set of sub-intervals, wherein:
• the quantum circuit includes a label register having a label indicative of at least one of the first subset of the set of sub-intervals,
• the polynomial evaluation is performed in a nested fashion, and
• the polynomial evaluation is performed by applying a gate to load a next set of coefficients for a next sub-interval into a quantum register, while multiplying a previous set of coefficients for a current sub-interval by a variable to produce an output, and then adding the next set of coefficients to the output, and
• outputting the quantum circuit to the quantum processor for execution by the quantum processor, wherein the quantum circuit comprises a label gate for initializing the label register and quantum gates;
• and
• with respect to the second subset of the set of sub-intervals for which the second method is selected, applying Newton's method, wherein the second subset of the set of sub-intervals is different from the first subset of the set of sub-intervals.

A processor-implemented method for compiling a quantum circuit for execution by a quantum processor, wherein the quantum circuit is for performing an evaluation of a piecewise smooth function, the method comprising:

• receiving an input comprising the piecewise smooth function and a domain;
• partitioning the domain of the piecewise smooth function into a set of sub-intervals by determining whether an input value of the piecewise smooth function falls into any of the set of sub-intervals, wherein the partitioning is performed to reduce an approximation error associated with a polynomial evaluation of the piecewise smooth function;
• selecting whether to apply a first method or a second method for each of the set of sub-intervals;
• with respect to a first subset of the set of sub-intervals for which the first method is selected:
• determining a polynomial interpolation for the first subset of the set of sub-intervals corresponding to the input,
• constructing a quantum circuit for performing, in parallel, a polynomial evaluation of a function for each of the first subset of the set of sub-intervals, wherein:
• the quantum circuit includes a label register having a label indicative of at least one of the first subset of the set of sub-intervals,
• the polynomial evaluation is performed in a nested fashion, and
• the polynomial evaluation is performed by applying a gate to load a next set of coefficients for a next sub-interval into a quantum register, while multiplying a previous set of coefficients for a current sub-interval by a variable to produce an output, and then adding the next set of coefficients to the output, and
• outputting the quantum circuit to the quantum processor for execution by the quantum processor, wherein the quantum circuit comprises a label gate for initializing the label register and quantum gates;
• and
• with respect to the second subset of the set of sub-intervals for which the second method is selected, applying Newton's method, wherein the second subset of the set of sub-intervals is different from the first subset of the set of sub-intervals, and wherein the Newton's method comprises:
• (1) determining a number of rounds for the Newton's method, and (2) determining an initial constant for the Newton's method.