Quantum state engineering using one-dimensional discrete-time quantum walks

Quantum state preparation in high-dimensional systems is an essential requirement for many quantum-technology applications. The engineering of an arbitrary quantum state is, however, typically strongly dependent on the experimental platform chosen for implementation, and a general framework is still missing. Here we show that coined quantum walks on a line, which represent a framework general enough to encompass a variety of different platforms, can be used for quantum state engineering of arbitrary superpositions of the walker's sites. We achieve this goal by identifying a set of conditions that fully characterize the reachable states in the space comprising walker and coin and providing a method to efficiently compute the corresponding set of coin parameters. We assess the feasibility of our proposal by identifying a linear optics experiment based on photonic orbital angular momentum technology.

Figure 1

Schematic representation of how the amplitudes are distributed among the sites, at the various steps of the evolution. The red (blue) arrows represent the movement of the walker with coin state $|\uparrow \rangle$ ($|\downarrow \rangle$) after the coin flip. The coin operators ${\mathcal{C}}_i$ determine the weights with which the amplitude at every site is distributed between the red and blue arrows, and thus the two sites of the next layer with which it is connected.

Figure 2

Distribution of projection probabilities computed (1) solving Eq. (15) for the parameters $d_i$, (2) computing the projection probabilities associated to each full state given by one such solution set, and (3) picking the solution set for the $\left\{d_i\right\}$ associated to lowest and highest projection probabilities. The shown data is for the cases of two, three, four, and five steps. For two, three, and four steps the data show the distribution of probabilities from a sample set of $20000$ target states, drawn according to the uniform Haar measure over the set of target states. For five steps only 6000 states were used (this case being much more computationally expensive). On the $y$ axis is shown the fraction of target states in a given bin.

Figure 3

Behavior of projection probability of the solutions found solving Eq. (15) for the target state $|\varphi \rangle \simeq (0.053,-0.078+0.603i,-0.524+0.189i,-0.302+0.363i,0.182+0.099i,0.042-0.224i)$. In each of the plots, the final fidelity is plotted against many coin parameters, each time fixing the value of all of them except for one, whose value is changed by an absolute value $\epsilon$ ($x$ axis). The six figures correspond to the six solutions for this target state, having respectively the projection probabilities 0.0014 (top left), 0.0020 (top right), 0.0036 (middle left), 0.0038 (middle right), 0.14 (bottom left), and 0.398 (bottom right). As clearly illustrated in this case, solutions with low projection probabilities tend to present a higher degree of instability with respect to small changes of the coin parameters.

Figure 4

Fidelity varying the various coin parameters for three (left) and five (right) steps, when the target is the completely balanced superposition over four and six modes, respectively. The variation of the coin parameters is here shown in percentage: the edges of the plots correspond to a variation of 10% of a single parameter with the others kept fixed at their optimal value.

Figure 5

Distribution of projection probabilities for randomly sampled states, computed with the numerical maximization described in Sec. 4b. Both plot shows the probabilities associated to a set of 3000 target states sampled from the uniform Haar distribution, for 15 and 20 steps. The light orange (upper) histograms represent the total number of target states found to correspond to a given range of probability. Starting from these data sets, we progressively removed the states that were found to reproduce the target states with fidelity less than $1\text{–}{10}^{-t}$, for various values of the threshold $t$. Light orange, gray, green, dark orange, and purple (from top to bottom) histograms correspond to thresholds of respectively $t=0,2,5,10,12$ (higher thresholds correspond to only a handful of states and are therefore omitted). This data further suggests a connection between the instability of the found solutions with respect to perturbations of the coin parameters, and the projection probability, as already hinted in Fig. 3: it is harder to find numerically with very good fidelity solutions corresponding to low projection probabilities because of their more unstable nature. It is also important to note that the solutions shown here are generally not the optimal ones, as the optimization algorithm only seeks to optimize the final fidelity, regardless of the corresponding projection probability.

Figure 6

Scheme for the proposed implementation of the quantum walk state engineering protocol using orbital angular momentum (OAM) and spin angular momentum of a photon. The input state is prepared in a superposition of $n$ modes with a spatial light modulator (SLM). At each step, the coin operation is then realized in polarization with a sequence of a quarter wave plate (QWP), a half wave plate (HWP), and a second QWP, arranged to implement an arbitrary transformation. The shift operation in OAM space is implemented with a $q$-plate (QP), which shifts the OAM of $\pm 2q$ conditionally to the polarization state of the photon. For $q=1/2$, the shift is equal to $\pm 1$, with the corresponding evolution schematically shown in the lower box. Finally, the coin is projected in the $|+\rangle$ state by means of a HWP and a polarizing beam splitter (PBS). A second SLM followed by a single-mode fiber performs the measurement of the output state.

Figure 7

Probability given by Eq. (B3) plotted against $u_1$ and $u_2$, for the special case of $u_1,u_2\in \mathbb{R}$.

Figure 8

Probability given by Eq. (B3) plotted against $u_2$, for various choices of $u_1$. Each line corresponds to a slice taken from Fig. 7.

Figure 9

Fidelity vs a relative change of the value of a coin parameter, for different values of $d_I$. Starting from Eq. (B4), we use as target state the normalized vector $\mathit{u}=N(0.5i,0.2,0.5i)$, and for various values of $d_I$ we compute the coin parameters generating the full state shown in Eq. (B4). We then take a single coin parameter, the parameter $\theta$ in the first step, and substitute it with $\epsilon \theta$, plotting the resulting fidelity between target and generated states as a function of $\epsilon$. Larger values of $d_I$ clearly correspond to higher instability.