Quantum information processing by weaving quantum Talbot carpets

Single-photon interference due to passage through a periodic grating is considered in a novel proposal for processing $D$-dimensional quantum systems (quDits) encoded in the spatial degrees of freedom of light. We show that free-space propagation naturally implements basic single-quDit gates by means of the Talbot effect: an intricate time-space carpet of light in the near-field diffraction regime. By adding a diagonal phase gate, we show that a complete set of single-quDit gates can be implemented. We then introduce a spatially dependent beam splitter that allows for projective measurements in the computational basis and can be used for the implementation of controlled operations between two quDits. Universal quantum information processing can then be implemented with linear optics and ancilla photons via postselection and feed-forward following the original proposal of Knill-Laflamme and Milburn. Although we consider photons, our scheme should be directly applicable to a number of other physical systems. Interpretation of the Talbot effect as a quantum logic operation provides a beautiful and interesting way to visualize quantum computation through wave propagation and interference.

Figure 1

(a) The intensity in the transverse direction of the periodic wave function which represents the encoding of a state $|d\rangle$ and an illustrations of the four different wave functions encoding the computational basis in dimension four, $\{0_4\rangle ,|1_4\rangle ,\{2_4\rangle ,\{3_4\rangle \}$. (b) The diffraction pattern in the near field of a periodic grating in the $z\text{−}x$ plane, the Talbot carpet; white regions indicate maximum intensity, and black regions correspond to zero intensity. (c) A cell in the Talbot carpet of sides $\ell /2$ along $x$ and $z_T$ in $z$, where one can identify three states of Talbot qubits along the propagation in $z$.

Figure 2

The plot shows the mean orthogonality $2{\sum }_{d>d^{\prime }}|\langle d_D\left|{d^{\prime }}_D\right.〉{|}^2/D(D-1)$ for the quDit computational states as function of $a/\ell$. For higher dimensions one needs the interslit distance $\ell$ to be much greater than the slit size $a$ in order to guarantee the orthogonality among the states. For fixed $a$ and $\ell$ the largest number of orthogonal states that one can encode is given by $D=\ell /a$, since in this way the space between two slits is evenly divided among nonoverlapping states.

Figure 3

Example of the phase function $\overrightarrow{{\varphi }_d}=({\varphi }_0,{\varphi }_1,{\varphi }_3)$ for $D=3$.

Figure 4

The general scheme for processing quantum information with a Talbot quDit. Staring with a $|0_D\rangle$ at the grating, free-space propagation by distance $q/r$ implements ${\mathsf{U}}_{q/p}$, until a diagonal gate is applied at $q/r$. Repeated applications of this sequence generate arbitrary unitary transformations in $\text{SU}\left(D\right)$ for $D$ prime.

Figure 5

In panel (a) we show the circuit consisting Hadamard gates and rotations in the Bloch sphere along the $z$ axis needed to prepare arbitrary qubit states. In this case we show the preparation of a state defined by the polar and azimuthal angles $(\theta ,\varphi )=(\pi /2.7,\pi /2)$ in the Bloch sphere. The color changes are to illustrate that a phase gate has been applied at the regions marked with vertical arrows. This circuit requires $2z_T$ of propagation in order to complete the state preparation. For comparison we show the Talbot carpet due to free evolution throughout the same distance $2z_T$, which thus implements the identity operator.

Figure 6

Example of a spatially dependent beam splitter (SDBS) for spatial quDits. Shown is an example for $D=3$. The colored regions represent parts of the beam splitter with different transmission coefficients $t_d$ and reflection coefficients $r_d$.

Figure 7

The spatially dependent beam splitter can be used to make measurements in the computational basis. Shown is the case for $D=3$, which requires two devices. The first has coefficients $r_0=t_1=t_2=1$, which filters out the $d=0$ state from the other two. The second has coefficients $t_0=r_1=t_2=1$ to separate the $d=1$ and $d=2$ states. The detectors are area-integrating “bucket” detectors, which detect photons in the entire spatial region.

Figure 8

Several SDBS devices can be used to implement a controlled gate between Talbot quDits (see text). SDBS and SDBS2 are spatially dependent beam splitters with different transmission and reflection coefficients.

Figure 9

Fidelity of the initial state $|0_D\rangle$ with respect to itself after evolution by a distance $m2z_T$ with $m$ an integer, considering the deviations from the paraxial approximation and a finite number of illuminated slits. The results where calculated numerically considering the propagation of a Gaussian beam of width $\sigma$ at the grating and effectively illuminating $l$ slits. While this fidelity decays rapidly for small values of $\sigma$ it is well preserved for values of the order of ${10}^2$ illuminated slits, as is the case for reported experiments, such as that in Ref. [21].