FIGURE 1. Quantum circuit for Shor's algorithm.

a, Outline of the quantum circuit. Wires represent qubits, and boxes represent operations. Time goes from left to right. (0) Initialize a first register of n = 2log₂ N qubits to |0 ... |0 (for short |0) and a second register of m = log₂ N qubits to |0 ... |0 |1). (1) Apply a Hadamard transform H to the first n qubits, so the first register reaches ^2n-1_{x = 0} |x/(2ⁿ). (2) Multiply the second register by f(x) = a^x mod N (for some random a < N which has no common factors with N), to get |₂ = ^2n-1_{x = 0} |x|1 a^x mod N/(2ⁿ). As the first register is in a superposition of 2ⁿ terms |x, the modular exponentiation is computed for 2ⁿ values of x in parallel. (3) Perform the inverse QFT on the first register¹⁹, giving |₃ = ^2n-1_{y = 0}^2n-1_{x = 0} e^2ixy/2n|y|a^x mod N/2ⁿ, where interference causes only terms |y with y = c2ⁿ/r (for integer c) to have a substantial amplitude, with r the period of f(x). (4) Measure the qubits in the first register. On an ideal single quantum computer, the measurement outcome is c2ⁿ/r for some c with high probability, and r can be quickly deduced from c2ⁿ/r on a classical computer via continued fractions². b, Detailed quantum circuit for the case N = 15 and a = 7. Control qubits are marked by filled circles; represents a NOT operation and 90 and 45 represent rotations over these angles. The gates shown in dotted lines can be removed by optimization, and the gates shown in dashed lines can be replaced by simpler gates (see Methods).