Quantum motion / Going both ways at once

Quantum Walk & Phase Traps

How does a quantum walk differ from an ordinary random walk? A quantum walker steps both ways at once. Paths that meet again add their amplitudes before squaring, so they reinforce or cancel. The result is two wavefronts racing outward, and a phase defect can steer them. Measure the walker at every step and it turns back into an ordinary random walk.

Turn the quantum coin, shift the initial phase and put a phase defect at the origin. Compare the coherent spread with a reference walk that starts dephased and is measured at every step, then build up the final distribution from independent position readouts.

The model is an ideal discrete coined walk on a line, with up to 64 steps. Hardware loss and atom imaging are not modeled. The classical reference measures the coin and the position at the start and after every step; only the balanced coin turns it into the ordinary unbiased random walk.

QUANTUMCLASSICAL

STEP0

TURN PHASE

VIEW RANGE

-64

x/a

64

Probability landscape · height is the square root of probability · both rows share one scale

Preparing the sequence…

Many independent readouts of one preparation

Where does it land?

Collecting pauses the sequence. Each outcome comes from a fresh identical preparation.

Recorded outcomes

0

Second moment · quantum

0

Second moment · classical

0

Second moment · records

\text{—}

Probability, linear scale

$x/a$

Teal: quantum model. Amber: classical reference. White-centered dots: recorded frequency. Purple dashes: pinned quantum model.

Spread after every step

$\langle(x/a)^2\rangle\quad\text{vs.}\quad N$

The second moment is taken about zero, not about the mean, so it includes drift as well as width. The teal and amber curves are computed independently.

The whole sequence at a glance

Horizontal: position or momentum. Down: steps. Click to revisit a step. This is probability, not a trajectory.

Model, units and approximations

One ideal walker begins at the origin in an equal-weight coin superposition. The two internal states shift left and right after a real coin rotation. A phase defect acts only at the origin. The classical comparator dephases position and coin initially and after every step; it retains coin memory when unbalanced.

|\psi_0\rangle=|0\rangle\otimes\frac{|L\rangle+e^{i\phi}|R\rangle}{\sqrt2}

The 3D view groups adjacent bins when needed. Both models share the same square-root height gain and a fixed horizontal range per preparation. The distribution below shows probabilities per native bin. Measurement outcomes sample the quantum model only; no phase reconstruction is claimed.

Primary research：Karski et al. (2009)

Start with the essentials

Focus question: How does a quantum walk differ from an ordinary random walk?
One-sentence intuition: Add the amplitudes first, then square. Paths that meet again can reinforce or cancel each other, and measuring at every step wipes out that interference.

Core mathematical model

One complete step

U=D_\chi S(C_\alpha\otimes I),\quad C_\alpha=\begin{pmatrix}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{pmatrix}

Coin state zero moves the walker one site left, and coin state one moves it right. After the shift, the local phase acts on both coin components at the origin.

A limit you can check exactly

\alpha=\frac\pi4,\ \phi=\frac\pi2,\ \chi=0:\quad\langle x^2\rangle\sim\left(1-\frac1{\sqrt2}\right)N^2

The balanced, symmetric quantum walk spreads ballistically, while the walk measured at every step has a second moment equal to the number of steps. A finite run need not match the long-time coefficient exactly.

What is sampled

\widehat{M_2}=\frac1n\sum_{j=1}^{n}q_j^2

Each final readout comes from a fresh preparation. The reported uncertainty is one standard error of the sample. These records cannot reconstruct the quantum phase.

Common difficulties

A probability landscape

Typical misconceptionEach peak is an individual particle, or the trajectory one particle follows.

Better mental modelThe model is an ideal discrete coined walk on a line, with up to 64 steps. Hardware loss and atom imaging are not modeled. The classical reference measures the coin and the position at the start and after every step; only the balanced coin turns it into the ordinary unbiased random walk.

Run the experiment

01
Watch quantum and classical part ways
Play the sequence. Compare the teal quantum ridge with the amber classical ridge.
What to observe: Both rows use the same horizontal scale and the same square-root height scale. For a quantitative comparison, use the linear plots below.
02
Change only the phase
Keep the coin balanced. Turn the initial phase, then add a phase at the origin.
What to observe: Neither phase affects the measured reference, but both change the coherent walk.
03
Record a distribution
Collect 500 outcomes and pin the preparation. Change the experiment, restore the pinned sample and export its raw CSV.
What to observe: The histogram and the second moment come only from recorded outcomes; the classical ensemble is a separate reference.