A method is provided for solving a computational problem that is reducible to a problem of counting solutions to an associated decision problem. The method includes, using a quantum computer, estimating a number of the solutions to the decision problem by determining if there is at least one solution to the decision problem that lies in a pseudo-random set. The method also includes outputting or using the estimated number of the solutions to the decision problem as a solution to the computational problem. Determining if there is at least one solution to the decision problem that lies in the pseudo-random set could include determining if there is a sequence of solutions to the decision problem that, taken together, lies in the pseudo-random set.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method for solving a computational problem that is reducible to a problem of counting solutions to an associated decision problem, the method comprising: using a quantum computer, estimating a number of the solutions to the decision problem by determining if there is at least one solution to the decision problem that lies in a pseudo-random set; and outputting or using the estimated number of the solutions to the decision problem as a solution to the computational problem. 2. The method of claim 1, wherein determining if there is at least one solution to the decision problem that lies in the pseudo-random set comprises: determining if there is a sequence of solutions to the decision problem that, taken together, lies in the pseudo-random set. 3. The method of claim 2, further comprising generating the pseudo-random set using the quantum computer by: using multiple seeds to generate multiple first component pseudo-random sets; performing a permutation of qubits in the first component pseudo-random sets; using results of the permutation to generate multiple second component pseudo-random sets; and combining the second component pseudo-random sets. 4. The method of claim 3, wherein: the sequence of solutions comprises N solutions, where N is an integer greater than one; each of the first component pseudo-random sets represents a superposition of N pseudo-random values; the permutation comprises a circular shift; and each of the second component pseudo-random sets represents a superposition of N2 sets of pseudo-random values. 5. The method of claim 2, wherein: the sequence of solutions comprises N solutions, where N is an integer greater than one; and the quantum computer has multiple quantum circuits configured to generate the pseudo-random set, the quantum circuits having a size that is linear or approximately linear with N. 6. The method of claim 2, wherein: the sequence of solutions comprises N solutions, where N is an integer greater than one; and an error associated with the estimated number of the solutions to the decision problem decreases at a rate of 1/N. 7. The method of claim 1, wherein determining if there is at least one solution to the decision problem that lies in the pseudo-random set comprises using Grover's quantum search algorithm. 8. The method of claim 1, wherein: the computational problem involves finding a percentile of a function f(x) over inputs x; and the associated decision problem involves deciding whether there is an input x such that f(x)<c for a given value of c. 9. The method of claim 1, wherein: the computational problem involves finding an average, sum, or integral of a function f(x) over inputs x; and the associated decision problem involves deciding whether there is a pair (x, y) for which y<f(x). 10. The method of claim 1, wherein: the computational problem involves finding a conditional expectation p( ) over inputs (x1, . . . , xd), where p(x1, . . . , xd)=f(x1, . . . , xd; averagexd+1p(x1, . . . , xd+1)) for a function f( ) over truncated inputs (x1, . . . , xd; α); and the associated decision problem involves computing averages of p(x1, . . . , xd+1)pi(x1, . . . , xd) and pi(x1, . . . , xd)pj(x1, . . . , xd) over the truncated inputs (x1, . . . ,xd+1), where pi and pj are regression functions. 11. An apparatus comprising: a quantum computer comprising at least one quantum circuit; wherein, to solve a computational problem that is reducible to a problem of counting solutions to an associated decision problem, the quantum computer is configured to use the at least one quantum circuit to estimate a number of the solutions to the decision problem by determining if there is at least one solution to the decision problem that lies in a pseudo-random set; and wherein the estimated number of the solutions to the decision problem represents a solution to the computational problem. 12. The apparatus of claim 11, wherein the quantum computer is configured to use the at least one quantum circuit to determine if there is a sequence of solutions to the decision problem that, taken together, lies in the pseudo-random set. 13. The apparatus of claim 12, wherein the at least one quantum circuit is further configured to generate the pseudo-random set by: using multiple seeds to generate multiple first component pseudo-random sets; performing a permutation of qubits in the first component pseudo-random sets; using results of the permutation to generate multiple second component pseudo-random sets; and combining the second component pseudo-random sets. 14. The apparatus of claim 13, wherein: each of the seeds comprises log2 r qubits and log2 N zero qubits; each of the first component pseudo-random sets comprises approximately (log2 r+log2 N) qubits and represents a superposition of N intermediate values; the permutation comprises a circular shift; and each of the second component pseudo-random sets represents a superposition of N2 intermediate values. 15. The apparatus of claim 12, wherein: the sequence of solutions comprises N solutions, where N is an integer greater than one; and the at least one quantum circuit comprises multiple quantum circuits configured to generate the pseudo-random set, the multiple quantum circuits having a size that is linear or approximately linear with N. 16. The apparatus of claim 12, wherein: the sequence of solutions comprises N solutions, where N is an integer greater than one; and an error associated with the estimated number of the solutions to the decision problem decreases at a rate of 1/N. 17. The apparatus of claim 11, wherein: the computational problem involves finding a percentile of a function f(x) over inputs x; and the associated decision problem involves deciding whether there is an input x such that f(x)<c for a given value of c. 18. The apparatus of claim 11, wherein: the computational problem involves finding an average, sum, or integral of a function f(x) over inputs x; and the associated decision problem involves deciding whether there is a pair (x, y) for which y<f(x). 19. The apparatus of claim 11, wherein: the computational problem involves finding a conditional expectation p( ) over inputs (x1, . . . , xd), where p(x1, . . . , xd)=f(x1, . . . , xd; averagexd+1p(x1, . . . , xd+1)) for a function f( ) over truncated inputs (x1, . . . , xd; α); and the associated decision problem involves computing averages of p(x1, . . . ,xd+1)pi(x1, . . . , xd) and pi(x1, . . . , xd)pj(x1, . . . , xd) over the truncated inputs (x1, . . . , xd+1), where pi and pj are regression functions. 20. A system comprising: a quantum computer comprising at least one quantum circuit; and a classical computer comprising at least one processor configured to execute instructions stored in at least one memory; wherein, to solve a computational problem that is reducible to a problem of counting solutions to an associated decision problem, the quantum computer is configured to use the at least one quantum circuit to estimate a number of the solutions to the decision problem by determining if there is at least one solution to the decision problem that lies in a pseudo-random set; and wherein the quantum computer is also configured to output, to the classical computer, the estimated number of the solutions to the decision problem as a solution to the computational problem.