Quantum Random Walk

Quantum Walks

The quantum walk is a quantum analog of the classical random walk. Unlike a classical walker that randomly chooses left or right at each step, a quantum walker exists in a superposition of positions and evolves unitarily through the lattice.

Discrete-Time Quantum Walk

The walker has an internal "coin" degree of freedom (spin up |↑⟩ or down |↓⟩) and a position on the lattice. At each time step:

• Coin flip: Apply a unitary coin operator C to the internal state
• Conditional shift: Move left if spin is down, right if spin is up

Key Observations

• Ballistic spreading: The quantum walk spreads quadratically faster than the classical random walk (√t vs √t in position variance).
• Interference patterns: The probability distribution shows characteristic peaks and valleys due to quantum interference.
• Coin dependence: Different coin operators (Hadamard, Grover, DFT) produce different interference patterns and spreading rates.
• Initial state sensitivity: The final distribution depends strongly on both the initial position and the initial coin state.

Applications

Quantum walks provide a framework for quantum algorithms (including search algorithms), quantum simulation of physical systems, and understanding quantum transport phenomena. They have been implemented experimentally in optical systems, trapped ions, and other quantum platforms.