Exploring Quantum Random Walks Through Simulation

Quantum Random Walks (QRWs) are fascinating phenomena at the intersection of quantum computing and physics. Unlike classical random walks, which rely on straightforward probabilistic rules, QRWs harness quantum principles such as superposition and interference. In this blog post, we’ll delve into the mechanics of QRWs, explore how interference shapes their behavior, and discuss a simulation that models these concepts—even if it doesn’t fully capture interference in its current form.

What is a Quantum Random Walk?

A Quantum Random Walk extends the classical concept of a random walk by incorporating quantum mechanics. In a classical random walk, a particle moves to a neighboring position based on predefined probabilities. In contrast, a QRW allows the particle to exist in a superposition of positions, and its movement is governed by quantum amplitudes and interference effects.

Key Quantum Principles in QRWs

1. Superposition

• Definition: A fundamental principle of quantum mechanics where a quantum system exists in multiple states simultaneously.
• In QRWs: The walker is in a superposition of positions, meaning it has a probability amplitude for being at each possible location.

2. Interference

• Definition: When quantum amplitudes combine, they can interfere constructively or destructively, affecting the probability of finding the particle in certain positions.
• In QRWs: Interference patterns emerge as the walker’s amplitudes from different paths combine, leading to non-classical probability distributions.

Mathematics of Quantum Random Walks

State Representation

The state of the quantum walker is represented as a superposition of position states:

Equation: ψ = Σₓ A(x) * |x⟩

• ψ: The quantum state of the system.
• A(x): The complex amplitude of the walker being at position x.
• |x⟩: The basis state corresponding to position x.

Probability Calculation

The probability of finding the walker at position x is given by the square of the magnitude of its amplitude:

Equation: P(x) = |A(x)|² = [Re(A(x))]² + [Im(A(x))]²

• Re(A(x)) and Im(A(x)) are the real and imaginary parts of the amplitude A(x).

Quantum Evolution with Interference

The amplitude at each position evolves based on contributions from neighboring positions, incorporating interference through phase shifts:

Equation: A(x, t+1) = Σᵧ Uₓᵧ * A(y, t)

• A(x, t+1): Amplitude at position x at time t+1.
• Uₓᵧ: Unitary operator representing the transition from position y to x, including phase factors.
• The summation over y accounts for all possible paths leading to x, allowing interference to occur.

Understanding Interference in QRWs

Interference arises when amplitudes from different paths to the same position combine. The phase differences between these amplitudes determine whether they interfere constructively or destructively.

Constructive Interference

• Occurs when amplitudes have phases that align.
• The amplitudes add together, increasing the probability at that position.

Destructive Interference

• Occurs when amplitudes have opposing phases.
• The amplitudes partially or completely cancel out, decreasing the probability at that position.

Mathematical Representation:

For two paths leading to position x:

Equation: A_total(x) = A₁(x) + A₂(x)

• A₁(x) = |A₁(x)| * exp(i * φ₁)
• A₂(x) = |A₂(x)| * exp(i * φ₂)
• The total amplitude depends on the phase difference Δφ = φ₂ - φ₁.

The resulting probability is:

Equation: P(x) = |A_total(x)|²

An Example of Interference

Consider two paths leading to position x with amplitudes:

• A₁(x) = (1/√2) * exp(i * 0)

• A₂(x) = (1/√2) * exp(i * π)  (phase difference of π)

Total amplitude:

• A_total(x) = (1/√2) + (-1/√2) = 0

Resulting probability:

• P(x) = |0|² = 0

This destructive interference results in zero probability of finding the walker at position x.

The Simulation

Overview

The simulation models a walker moving on a 2D grid. While it introduces concepts inspired by QRWs, such as random amplitudes and probabilistic transitions, it does not fully implement quantum interference.

Key Features

• Random Amplitudes: The walker assigns random amplitudes to neighbouring positions to determine movement probabilities.
• Probability Calculation: Probabilities are normalised but derived from random amplitudes rather than complex amplitude sums.
• Visualisation: The grid displays probabilities using a colour scale, with higher probabilities shown in darker reds.

Limitations of the Current Model

Despite incorporating quantum-inspired elements, the simulation lacks key aspects of true QRWs:

1. No Interference Modeling

• Amplitudes from different paths do not combine with phase information.
• The absence of phase factors means interference effects are not captured.

2. Real vs. Complex Amplitudes

• The simulation uses real-valued amplitudes, ignoring the imaginary components crucial for interference.

3. Unitary Evolution

• Without unitary operators governing transitions, the model does not preserve quantum coherence or exhibit true quantum evolution.

Implications of Not Modelling Interference

• Classical Behaviour: The walker behaves more like a classical random walker with probabilistic movement rather than exhibiting quantum characteristics.
• Probability Distribution: The resulting probability distribution lacks the non-Gaussian, oscillatory patterns typical of QRWs influenced by interference.
• Spreading Rate: The walker does not demonstrate the ballistic spreading (variance proportional to n²) associated with QRWs.

Key Differences: Classical Random Walk vs. This Simulation

Conclusion

While the current simulation provides an interesting exploration of random walks with quantum-inspired elements, it falls short of modelling a true Quantum Random Walk due to the absence of interference and complex amplitudes. Understanding and implementing interference is crucial for capturing the unique behaviours of QRWs, such as non-classical probability distributions and ballistic spreading.