**Implementing quantum walks**

The random walk (RW), which is ubiquitous in physics, chemistry, mathematics, and computer science, underpins Brownian motion and diffusion processes, is used in satisfiability proofs, and is intimately connected with the Wiener measure. Quantization of the RW has led to new quantum algorithms and fascinating physics such as decoherence-induced diffusion reduction. Our goal is to see the quantum walk (QW) realized in the laboratory. However, compromises have to be made to the ideal QW in order to realize the QW experimentally, such as side-stepping the requirement of direct coin flipping in cavity quantum electrodynamics (QED) and finding an alternative to measuring the position distribution for a quantum walk in an ion trap. Here we discuss how QW can be implemented by making compromises to the ideal QW but nonetheless demonstrating a true QW in the laboratory.
First, we propose a variant of the QW in cavity QED [1-2], designed to eliminate the onerous requirement of directly flipping the coin. Instead, we propose driving the cavity in such a way that cavity field displacements are minimized and the coin is effectively flipped via this indirect process. An effect of this indirect flipping is that the walker's location in phase space is no longer confined to a single circle in the planar phase space, but we show that the phase distribution nonetheless shows quadratic enhancement of phase diffusion for the quantum versus classical walk despite this small complication. Thus our scheme leads to coined QW behaviour in cavity QED without the need to flip the coin directly. We also show how interpolation from a quantum to a random walk is implemented by controllable decoherence using a two-resonator system. Direct control over the coin qubit is difficult to achieve in either cavity or circuit QED, but we show that a Hadamard coin flip can be effected via direct driving of the cavity, with the result that the walker jumps between circles in phase space, but still exhibits quantum walk behavior over 15 steps.
Second, we have developed a scheme for realizing the first single-walker QW in the laboratory, with the ion's electronic degree of freedom serving as the two-state coin and the motion as the walker's degree of freedom [3]. In contrast to current approaches to developing QW implementations, which would realize QWs on circles in phase space as we mentioned as the former proposal, our approach yields a RW-QW transition in position space. In other words, the walker is truly spreading out over unbounded position space rather than being folded back on itself. Although the walk is over position, we show that the experimentally accessible phonon number is equally revealing of the RW-QW transition. Here we have shown that the phonon number measurement is accessible for up to dozens of phonons by driving the ions at the carrier frequency, then Fourier transforming the ground state population to reveal the Rabi frequencies, hence the phonon number distribution. This approach is similar to the approach of driving at the blue sideband but is much more effective in revealing phonon distribution over a wide range of phonon number. In addition to these new approaches, we introduce an experimentally controllable phase randomization. The RW-QW transition is a key part of any experiment that plans to demonstrate QW behavior, yet the ion trap dynamics are almost perfectly coherent. In conclusion our theory establishes a pathway to realize and study the QW in the laboratory, and our techniques for counting phonons, which can act as a ¡°bus¡± in ion trap quantum computing, should be useful for general quantum information processing protocols.