Deutsch Function

What is the Deutsch function?

 

The Deutsch function is a fundamental concept in quantum computing that serves as a building block for more complex quantum algorithms. It is a two-qubit function that takes two input qubits (x and y) and returns a single output qubit (z). The function is defined as follows:

Deutsch(x, y) = x XOR y, where XOR is the exclusive OR operation.

The Deutsch function can be implemented using a quantum circuit consisting of a Hadamard gate, a controlled-NOT gate, and another Hadamard gate. Here’s a visual representation of the Deutsch circuit:

The circuit works as follows:

  1. Hadamard gates: The input qubits x and y are initialized to the state |0>. Hadamard gates are applied to both qubits to put them into a superposition of |0> and |1>.
  2. Controlled NOT gate: The controlled-NOT gate is applied with qubit x as the control qubit and qubit y as the target qubit. This gate flips the target qubit (y) if the control qubit (x) is |1>.
  3. Hadamard gates: Hadamard gates are applied again to both qubits.

After the circuit completes, the output qubit z will be in the state |1> if the Deutsch function is evaluated to 1, and |0> if it is evaluated to 0.

The Deutsch function is a simple example of a quantum algorithm that can be used to demonstrate the power of quantum computing. It shows how quantum superposition and entanglement can be used to perform calculations that would be difficult or impossible on a classical computer.

 

How is the Quantum Circuit for the Deutsch function represented?

The quantum circuit to solve this problem involves the following steps:

1. Initialization: Prepare two qubits in the state |0⟩|1⟩.

2. Hadamard Transformation: Apply a Hadamard gate to each qubit, creating a superposition of states:

 

|0⟩|1⟩ --> (|0⟩ + |1⟩)/√2 * (|0⟩ - |1⟩)/√2

3. Oracle: Apply the oracle, which implements the function f(x). The oracle can be represented as a unitary matrix U_f:

U_f |x⟩|y⟩ = |x⟩|y ⊕ f(x)⟩

This step introduces a phase shift to the second qubit, depending on the value of f(x).

  • Hadamard Transformation:

Apply a Hadamard gate to the first qubit: (H ⊗ I)U_f(H ⊗ I)|0⟩|1⟩

  • Measurement: Measure the first qubit.

Mathematical Analysis

The key to understanding the algorithm lies in the phase shifts introduced by the oracle. If the function is constant, the phase shifts cancel out, and the measurement always yields 0. If the function is balanced, the phase shifts interfere constructively or destructively, leading to a measurement of 1.

For a Constant Function:

  • The oracle applies the same phase shift to both |0⟩ and |1⟩ states of the second qubit.
  • After the final Hadamard gate, the first qubit will be in the |0⟩ state.

For a Balanced Function:

  • The oracle applies opposite phase shifts to the |0⟩ and |1⟩ states of the second qubit.
  • After the final Hadamard gate, the first qubit will be in the |1⟩ state.

Quantum Advantage

The quantum algorithm solves the Deutsch problem with a single query to the oracle, while a classical algorithm would require two queries in the worst case. This demonstrates a quantum advantage, even for a simple problem.

This quantum speedup is a result of quantum interference, which allows the quantum algorithm to explore multiple possibilities simultaneously.