HPC-Tools-for-Quantum-Computing/book/src/algorithms/deutsch-jozsa.md

Deutsch-Jozsa Algorithm

This algorithm was the first to show that quantum computers could have a speedup over classical computers.

Consider a function \(f(x)\) which takes an input of an \(n\)-bit string \(x\) and returns 0 or 1.

Suppose that \(f(x)\) is either a constant function that has the same value \(c \in {0, 1}, \forall x\), or a balanced function, where the value is 0 for half of the inputs, and 1 for the other half.

The Deutsch-Jozsa problem is to find whether \(f\) is constant or balanced, in as few function evaluations as possible.

Using classical computing, in the worst case, this requires \(2^{n-1}+1\) function evaluations. Using quantum computing, this can be done with just one function evaluation.

The function \(f\), to be used in a quantum computer, must be specified by an oracle circuit \(U{f}\) such that \(U{f} \lvert x \rangle = (-1)^{f(x)}\lvert x \rangle\).

The Algorithm

1. Initialize \(n\) qubits in the state \(\lvert 0, \dots, 0\rangle\).
2. Apply the Hadamard gate \(H\) to each qubit.
3. Apply the oracle circuit \(U_f\).
4. Apply the Hadamard gate \(H\) to each qubit.
5. Measure each qubit. Let \(y = (y{1}, \dots, y{n})\) be the list of measurement outcomes.

From this procedure, we find that \(f\) is constant if \(y\) is the all zero string.

extern crate qcgpu;

use qcgpu::State;
use qcgpu::gates::h;

fn main() {
    // 3 qubits, f(x) = x_0 NOT x_1 x_2
    // Balanced
    let mut balanced_state = State::new(3, 1);

    balanced_state.apply_all(h());

    // Oracle U_f
    balanced_state.h(2);
    balanced_state.z(0);
    balanced_state.cx(1, 2);
    balanced_state.h(2);

    balanced_state.apply_all(h());

    println!(
        "{}",
        if balanced_state.measure() == 0 {
            "constant"
        } else {
            "balanced"
        }
    );

    // 3 qubits, f(x) = 0
    // Constant
    let mut constant_state = State::new(3, 1);

    constant_state.apply_all(h());

    // Oracle is equivilant to the identity gate, thus has no effect on the
    // state.

    constant_state.apply_all(h());

    println!(
        "{}",
        if constant_state.measure() == 0 {
            "constant"
        } else {
            "balanced"
        }
    );
}