Figure 2: The random walk used in the 2p procedure.

The quantum probability amplification construction can be represented via a random walk on a semi-infinite ladder with transitions given by Bell-measurement outcomes on two qubits. The bell procedure gives a result of heads with probability g(p). Initially one flip is used to determine the starting point. After the first flip a result of heads causes a move upwards or downwards with equal probability; a result of tails causes a move to the other side of the ladder. If the bottom left corner is ever reached, we output heads, if the bottom right is reached we output tails. The probability of outputting heads overall is .