mod03lec16 - Quantum Algorithms: Bernstein Vazirani Algorithm

Summary

This lecture segment delves into the Bernstein-Vazirani algorithm, a quantum computing algorithm that demonstrates the superiority of quantum computers over classical ones in solving certain problems. The algorithm is designed to find a hidden bit string 'a' embedded in a function 'f', which is a dot product of 'a' and the input 'x'. While classical computers require exponential time to solve this with certainty, the quantum algorithm can do it with high probability in polynomial time. The lecture explains the classical and quantum solutions, emphasizing the need for 'n' queries in the classical approach and the efficient quantum method using Hadamard gates and an oracle. The implementation of the algorithm in Qiskit is also discussed, showcasing the practical application of quantum computing principles.

Takeaways

• 😀 The Bernstein-Vazirani algorithm is a quantum algorithm that demonstrates quantum computers can solve certain problems with high probability in polynomial time, whereas classical computers cannot solve them with a probability greater than 1/2 in the same time.
• 🧠 The algorithm is a follow-up to Deutsch's algorithm, addressing a problem where quantum computers can solve with certainty in polynomial time, while classical computers require exponential time or settle for probabilistic solutions.
• 🔍 The problem statement involves an oracle function f that takes an n-bit input and outputs a single bit, either 0 or 1, with the guarantee that there exists a hidden bit string a such that f(x) = a · x mod 2.
• 🎯 The goal of the Bernstein-Vazirani algorithm is to determine the hidden bit string a embedded in the function f using quantum computation.
• 📊 Classical solutions require querying the oracle n times, one for each bit of a, which is both necessary and sufficient for classical algorithms to determine a.
• 📚 The quantum solution involves applying Hadamard gates to all qubits, applying the oracle, and then applying Hadamard gates again before measurement, which allows the algorithm to determine a in a single query.
• 🛠️ The implementation of the oracle in the algorithm requires an understanding of how to perform a quantum XOR operation, which is essential for the algorithm to function correctly.
• 📈 The Bernstein-Vazirani algorithm highlights the potential of quantum computing to solve problems more efficiently than classical computing, showcasing an oracle separation between BQP and BPP complexity classes.
• 🔑 The algorithm's success relies on the ability to apply and invert the Hadamard transform, which is crucial for the quantum state to collapse to the desired solution upon measurement.
• 💡 The lecture concludes with a discussion of the algorithm's implementation in Qiskit, providing a practical example of how theoretical quantum algorithms can be realized in existing quantum computing frameworks.

Q & A

What is the Bernstein-Vazirani algorithm?

The Bernstein-Vazirani algorithm is a quantum algorithm that demonstrates a problem which quantum computers can solve with high probability in polynomial time, while classical computers cannot solve with a probability greater than 1/2 using the same time.

How does the Bernstein-Vazirani algorithm differ from Deutsch's algorithm?

While Deutsch's algorithm deals with determining if a function is constant or balanced, the Bernstein-Vazirani algorithm focuses on finding a hidden bit string 'a' embedded in a function f(x) = a·x mod 2, where 'a' is a binary vector and 'x' is the input.

What is the classical solution to the problem addressed by the Bernstein-Vazirani algorithm?

The classical solution involves querying the oracle with a sequence of inputs designed to isolate each bit of the hidden string 'a'. This requires n queries to the oracle to determine the n-bit string 'a'.

Why is the classical solution to the Bernstein-Vazirani problem limited to n queries?

The classical solution is limited to n queries because it is an underdetermined system of equations with n variables and n-1 equations, making it impossible to find a unique solution for 'a' with fewer than n queries.

How does the quantum solution to the Bernstein-Vazirani problem work?

The quantum solution uses a quantum circuit that applies Hadamard gates, the oracle, and another round of Hadamard gates to transform the initial state into the hidden bit string 'a'. This is achieved with fewer than n queries to the oracle.

What is the role of the oracle in the Bernstein-Vazirani algorithm?

The oracle in the Bernstein-Vazirani algorithm is a black box that takes an input x and outputs x XOR f(x), where f(x) is the function that encodes the hidden bit string 'a' as f(x) = a·x mod 2.

What is the significance of the Hadamard gate in the Bernstein-Vazirani algorithm?

The Hadamard gate is used to create a superposition of all possible states, which allows the quantum algorithm to query the oracle in a superposed state, enabling the extraction of information about the hidden bit string 'a' with fewer queries than the classical method.

How does the Bernstein-Vazirani algorithm demonstrate the advantage of quantum computing over classical computing?

The Bernstein-Vazirani algorithm shows that quantum computers can solve certain problems more efficiently than classical computers by leveraging quantum superposition and entanglement, which allows for the extraction of information with fewer queries to the oracle.

What is the output of the Bernstein-Vazirani algorithm?

The output of the Bernstein-Vazirani algorithm is the hidden n-bit string 'a' that is embedded in the function f(x). This is achieved by measuring the quantum state after applying the quantum circuit.

How is the implementation of the oracle in the Bernstein-Vazirani algorithm described in the script?

The implementation of the oracle in the Bernstein-Vazirani algorithm is described by showing how to apply a controlled-NOT gate for each bit of the input 'x' where the corresponding bit in the hidden string 'a' is 1. This effectively computes the dot product a·x mod 2.