### Abstract:

We examine a discrete-time quantum walk with two-step memory for a particle on a one-
dimensional infinite space. The walk is defined with a four-state memory space analogous to the
two-state coin space commonly used in discrete time quantum walks, and a method is presented
for calculating the time evolution by using the Fourier transform. An integral expression for
the probability is calculated, and this is used to produce numerical solutions for the probability
distribution as a function of the time step and position. The results show two peaks in the
probability distribution. One peak propagates ballistically with time, which is a common feature
of quantum walks. The other peak is stationary with time and located at the initial site of the
particle. This feature is not common in quantum walks and suggests that tracing the immediate
history of the particle using two-step memory may represent the beginning of a transition to a
classical system.

### Description:

Contents:
List of Figures -- page 2; 1) Introduction -- page 3; 1.1) Quantum Walks -- page 4; 1.2) Quantum Walks With Memory -- page 6; 2) Methods -- page 8; 2.1) Quantum Walk on an Infinite Position Space -- page 8; 2.2) Types of Tails -- page 8; 2.3) Evolution Operator -- page 9; 2.4) Fourier Transform -- page 12; 2.5) Diagonalizing the Operator M -- page 14; 2.6) Inverse Fourier Transform -- page 16; 3) Results -- page 17; 3.1) Tail Probability Amplitudes -- page 17; 3.2) Probability Distribution -- page 20; 4) Discussion -- page 21; 4.1) Tail Probability Amplitudes -- page 21; 4.2) Probability Distribution -- page 22; 5) Conclusion -- page 23; 6) Acknowledgements -- page 25; 7) References