6.16. An Example Quantum Circuit for Shor’s Algorithm

Shor’s algorithm is one of the most important algorithms offered by quantum computers for factoring large numbers. Below, the step-by-step design and optimization of the quantum circuit for Shor’s algorithm are discussed.

6.16.1. Example Circuit Design

6.16.1.1. Initial State and Qubit Preparation

• Number of Qubits:
  • Shor’s algorithm typically uses \(2n + 1\) qubits, where \(n\) is the bit length of the number to be factored.
• Initial State:
  • All qubits are initialized in the \(|0\rangle\) state.
• Transition to Superposition:
  • Hadamard gates are applied to the first \(n\) qubits to create a superposition state.

6.16.1.2. Modular Exponentiation

• Objective:
  • Perform the operation \(a^x \mod N\) on the quantum circuit.
• Circuit Design:
  • Controlled-X gates, Toffoli gates, and other modular arithmetic gates are used.
• Optimization:
  • Minimize the number of gates by reusing common blocks.
  • Reduce execution time with parallel gate applications.

6.16.1.3. Application of Quantum Fourier Transform (QFT)

• Objective:
  • Determine the period of the function \(a^x \mod N\) during the period-finding step.
• Circuit Elements:
  • Hadamard gates and controlled phase gates are used to implement QFT.
• QFT Steps:
  1. Apply Hadamard gates to each qubit.
  2. Establish phase relationships between qubits using controlled phase gates.
  3. Reorder qubits in reverse order using swap gates.

6.16.1.4. Measurement and Result Processing

• Measurement:
  • After QFT, the qubits are measured, and the results are transferred to a classical computer.
• Classical Algorithm:
  • The period is estimated based on the measurement results.
  • Complementary steps and extensive search algorithms are applied to finalize the process.