Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor

We present an experimental implementation of the coined discrete-time quantum walk on a square using a three-qubit liquid-state nuclear-magnetic-resonance (NMR) quantum-information processor (QIP). Contrary to its classical counterpart, we observe complete interference after certain steps and a periodicity in the evolution. Complete state tomography has been performed for each of the eight steps, making a full period. The results have extremely high fidelity with the expected states and show clearly the effects of quantum interference in the walk. We also show and discuss the importance of choosing a molecule with a natural Hamiltonian well suited to a NMR QIP by implementing the same algorithm on a second molecule. Finally, we show experimentally that decoherence after each step makes the statistics of the quantum walk tend to that of the classical random walk.

Figure 1

Comparison of the dynamics of the classical (left) and quantum (right) random walk on a square for three steps. $H$ or $T$ represents the state of the coin and the number the position of the walker at one of the four nodes of the square. $p$ is the probability of each classical state and $A$ is the probability amplitude for the quantum state. Part (a) for each step is the coin flip and part (b) the movement around the square. In both cases the walker starts at node 0 with the coin in the heads state. After one step he has a 50% probability of being at either node 1 or 3. Then, in the second step he goes to either 0 or 2 with a 50% probability. In the third step, however, the two types of walk diverge. The classical walk continues to oscillate and the probability remains spread out. In the quantum walk, on the other hand, the probability amplitudes interfere and cancel out, leaving all the probability in one corner after three steps.

Figure 2

Logical labeling of the nodes on which we implemented the DTQW. With this labeling, flipping the first qubit corresponds to a horizontal move and flipping the second qubit a vertical move.

Figure 3

(Color online) Molecular structure of trans-crotonic acid and its Hamiltonian parameters. The chemical shifts are given as the diagonal elements and the coupling strength (Hz) by the off-diagonal elements. Note that since the darkly shaded unlabeled nuclei are oxygen whose natural abundance of ${}^{16}\mathit{O}$ with 0 spin is close to 100%, the two oxygen nuclei do not couple with the rest of the molecule and can be ignored. Lightly shaded unlabeled nuclei are hydrogen which were decoupled during the experiment.

Figure 4

NMR pulse sequence representing one step of DTQW. The notation $R_i^{\theta }$ means a rotation of an angle $\theta$ around the axis $i$. Refocusing pulses are not shown. Since each nucleus is tracked in its own rotating frame, rotations about the $z$ axis are implemented instantaneously through a change of reference frame.

Figure 5

Examples of the real part of reconstructed density matrices after (a) steps 0, 4, and 8 (left to right) of the quantum walk implemented on crotonic acid; (b) step 3 with no, partial, and full decoherence applied after each step; and (c) steps 0, 4, and 8 of the quantum walk implemented on TCE. The effects of the quantum interference returning all the probability to one corner is clearly evident in steps 4 and 8; however, the fidelity in the TCE case is much worse. The density matrices from the decoherence experiments show how destruction of the off-diagonal coherences prevents all the probability returning to one corner after three steps.

Figure 6

Diagram of ${}^{13}\mathrm{C}$ labeled TCE. The chemical shifts and couplings are given in the table. Note that since the chlorine nuclei (unlabeled) have a spin of $\frac{3}{2}$, they have an electric-quadrupole moment which causes them to decohere quickly and they have a very small coupling to the rest of the molecule which we can ignore in the natural Hamiltonian of the molecule.

Figure 7

Circuit used to implement one step of the DTQW on the TCE. The $z$ rotation on $C_2$ occurs naturally during the coupling with $C_1$. Note also that the refocusing pulses are not shown in this pulse sequence.