Shor's Quantum Factoring Algorithm is a groundbreaking quantum algorithm that efficiently factors large composite numbers, which is a problem that is believed to be computationally hard for classical computers. The algorithm utilizes a mathematical technique called period finding to identify the period of a function, which is crucial for the factorization process.
To understand how period finding works in Shor's algorithm, let's first discuss the overall structure of the algorithm. Shor's algorithm consists of two main steps: the quantum Fourier transform (QFT) and the modular exponentiation. The QFT is responsible for creating a superposition of states, while the modular exponentiation is used to extract the period of a function.
The period finding step in Shor's algorithm is performed on a quantum computer with a register of qubits. The input to this step is a function f(x) that maps integers to integers. The function we are interested in for factoring is f(x) = a^x mod N, where a is a randomly chosen integer coprime to N, and N is the composite number we want to factor.
The first step in period finding is to prepare the qubits in a superposition of all possible inputs. This is done by applying a series of Hadamard gates to the input register. The Hadamard gate transforms each qubit from the basis state |0⟩ to the superposition state |+⟩, which is a uniform superposition of |0⟩ and |1⟩. Applying Hadamard gates to all qubits in the input register creates a superposition of all possible inputs.
Next, the function f(x) is evaluated on the superposition of inputs using modular exponentiation. Modular exponentiation is a process that computes a^x mod N for each input state. This is done by applying a series of controlled modular multiplication gates to the output register, where the control qubits are the qubits in the input register. These controlled gates perform the modular exponentiation operation on the output register based on the value of the input register.
After the modular exponentiation step, the output register contains the values of f(x) for all possible inputs in superposition. However, we are interested in finding the period of the function f(x). To extract the period, we perform the quantum Fourier transform (QFT) on the output register.
The QFT is a quantum analogue of the classical discrete Fourier transform, which is used to decompose a function into its frequency components. In the context of period finding, the QFT transforms the superposition of function values in the output register into a superposition of the frequencies that make up the function.
The QFT is implemented by applying a series of controlled phase gates and Hadamard gates to the output register. The controlled phase gates introduce phase shifts to the states in the output register based on their frequency. The Hadamard gates further transform the states into a superposition of all possible frequencies.
After applying the QFT, the output register contains information about the frequencies present in the function f(x). The final step is to measure the output register and obtain a frequency value. This frequency value corresponds to the period of the function f(x).
To summarize, period finding in Shor's Quantum Factoring Algorithm involves preparing the input register in a superposition of all possible inputs, evaluating the function f(x) using modular exponentiation, applying the QFT to the output register to extract the period, and finally measuring the output register to obtain the period value.
The period finding step is crucial in Shor's algorithm because it allows us to find the period of the function f(x), which is directly related to the factors of the composite number N. By finding the period, we can extract the factors and efficiently factorize large composite numbers.
Period finding in Shor's Quantum Factoring Algorithm is a key component that enables the efficient factorization of large composite numbers. Through the use of the quantum Fourier transform and modular exponentiation, the algorithm is able to identify the period of a function, which is essential for the factorization process.
Other recent questions and answers regarding EITC/QI/QIF Quantum Information Fundamentals:
- Are amplitudes of quantum states always real numbers?
- How the quantum negation gate (quantum NOT or Pauli-X gate) operates?
- Why is the Hadamard gate self-reversible?
- If measure the 1st qubit of the Bell state in a certain basis and then measure the 2nd qubit in a basis rotated by a certain angle theta, the probability that you will obtain projection to the corresponding vector is equal to the square of sine of theta?
- How many bits of classical information would be required to describe the state of an arbitrary qubit superposition?
- How many dimensions has a space of 3 qubits?
- Will the measurement of a qubit destroy its quantum superposition?
- Can quantum gates have more inputs than outputs similarily as classical gates?
- Does the universal family of quantum gates include the CNOT gate and the Hadamard gate?
- What is a double-slit experiment?
View more questions and answers in EITC/QI/QIF Quantum Information Fundamentals