1. Introduction

The capacity to transmit and process classical and quantum information has experienced tremendous growth in the latest years [1]. However the need to continue this trend poses challenges in areas such as computing, nanotechnology, telecommunications, and information processing [2]. One promising direction to handling increasingly huge sets of data is to build information-processing devices based on optical logic gates.

These gates make use of light beams with information encoded in their field amplitude and polarization. At the quantum level they use single photons with the information embedded in their quantum state. A reversible logical gate that has received great attention is the Fredkin gate, or controlled-SWAP (c-SWAP) gate, introduced by Edward Fredkin in the context of computational models to perform any logical or arithmetic operation in the domain of reversible logic-based operations. This gate has three input bits and three output bits and swaps or not the last two bits depending on the value of the first bit that acts as control bit [3]. A generalized version of the Fredkin gate allows direct estimations of linear and nonlinear functionals of a quantum state [4].

There have been experimental and theoretical proposals to implement a Fredkin gate with optical systems [5,6]. Additional theoretical work has considered a quantum version using single atoms and single photons [7–10]. Current experimental work includes nuclear magnetic resonance (NMR) [11], superconducting quantum circuits [12], DNA enzymes [13]and weak coherent pulses [14,15].

Recently there have been the first demonstrations of a quantum Fredkin gate using linear optics with quantum-entangled photons [16–18]. Generally speaking these implementations are probabilistic and experimentally cumbersome, requiring the use of multiple interferometers. The low efficiency of photonic quantum gates makes the success rate of these gates extremely low. For instance, in [16] the experimental setup makes use of three interferometers and the successful operation of the Fredkin gate requires the measurement of fourfold coincidences across four single-photon detectors. They measure a fourfold coincidences rate of $2.2$ per minute. In [17] they use several interferometers that should be perfectly stabilized yielding a low count rate of 10$^{-4}$ Hz. In [18] the experimental setup is composed of several Mach–Zehnder interferometers with seven independent phases that make the whole system prone to imperfections.

One interesting and promising option that attracts a lot of attention is the implementation of classical logical gates whose design is inspired by counterpart quantum gates [19–29]. They are generally not probabilistic, much easier to implement and make use of intense beams. The transformation of protocols and technologies designed with quantum tools into classical protocols and technologies is based on the fact that certain features of quantum physics are also shared by waves in the classical world. This is the case of interference or entanglement between degrees of freedom of a single particle. However, there will be aspects of these quantum-inspired gates, such as non-locality, that will make fundamentally different classical analogs from its counterpart quantum algorithms. How far one can go in this analogy is a matter of discussion and controversy in the scientific community [26,30].

Here we demonstrate in a proof-of-concept experiment a quantum-inspired protocol for comparing strings of data and waveforms without the need to unveil the information contained in the signals. For implementing this gate, we translate key ideas and elements of the quantum fingerprinting protocol [31] to the classical domain. One key element is a Fredkin gate that mimics features of the counterpart quantum-optical Fredkin gate originally proposed by Milburn [32]. The two channels or carriers of information are two strings of orthogonal spatial modes with orbital angular momentum (OAM). Information is encoded in various formats on these two channels. OAM beams allow implementing a crucial element of this system: a polarization-controlled SWAP operation. The c-SWAP operation exchanges or not data carried by different channels depending on the state of polarization of the beam, that acts as control bit. Our system can compare two strings of data and evaluate the degree of similarity of the information encoded in the complex amplitudes. We will show below several examples of this that will validate the capability of our system for estimating the fidelity of streams of data without evaluating the data itself.

Spatial modes of light play a central role in the development of new information technologies, information processing, and secure communications. OAM modes are particularly interesting for quantum and classical communications due to its capacity for carrying large amounts of information [33]. Interferometric methods can be used for measuring and sorting modes with OAM in the classical and quantum domains [34]. When considering polarization-sensitive interferometers with Dove prims capable of performing NOT operations on the sign of OAM modes, one can generate and manipulate OAM beams with high efficiency and robustness [35]. In the last few years a lot of attention has been directed towards using such beams for information transfer in the context of free-space communications [36,37].

2. Fredkin gate in the quantum and classical domains

The circuit representation of the original quantum Fredkin gate is shown in Fig. 1(a). It is a 3-qubit gate that performs a c-SWAP operation conditioned by the state of the control qubit $|C\rangle _3$. At the input we have qubit $|\alpha \rangle _1$ in channel 1 and qubit $|\beta \rangle _2$ in channel 2. If the control bit is $|0\rangle _3$, the qubits in each channel remain the same: $|\alpha ^{\prime } \rangle _1=|\alpha \rangle _1$ and $|\beta ^{\prime } \rangle _2=|\beta \rangle _2$. If the control qubit is $|1\rangle _3$, the qubits are swapped between channels: $|\alpha ^{\prime } \rangle _1=|\beta \rangle _1$ and $|\beta ^{\prime } \rangle _2=|\alpha \rangle _2$.

 

Fig. 1. The Fredkin gate. (a) Quantum Fredkin gate. The quantum state of input channels $1$ and $2$ is either swapped or not depending on the value of the control bit $|C\rangle _3$. (b) Quantum-inspired controlled-swap gate built with spatial modes carrying orbital angular momentum. Indexes $m$ and $-m$ swap their sign or not depending on the polarization of the beam. The state of polarization determines if the gate performs the identity transformation ($\{ A_m^{\prime } \}=\{ A_m \}$ and $\{ B_m^{\prime } \}=\{ B_m \}$) or the swap transformation ($\{ A_m^{\prime } \}=\{ B_m \}$ and $\{ B_m^{\prime } \}=\{ A_m \}$).

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In our quantum-inspired Fredkin gate, the two channels correspond to a set of Laguerre-Gauss spatial modes $\mathrm {LG}^{0}_{m}(\textbf {r}_{\perp })$ with either positive or negative index $m$. Modes with positive index $m$ correspond to channel 1 and modes with negative $m$ correspond to channel 2. This index indicates a varying phase of the field of the form $\sim \exp (i m \varphi )$, where $\varphi$ is the azimuthal angle in cylindrical coordinates. $m$ also designates an OAM content of $m\hbar$ per photon of the mode. $p=0$ is the radial index of the modes and $\textbf {r}_{\perp }=(x,y)$ is the transverse coordinate.

Information in channel 1 is encoded into $N$ complex amplitudes $A_{m}$ and information in channel 2 is similarly encoded into $N$ complex amplitudes $B_{m}$. Each channel can contain one or several modes, and each amplitude ($A_m$ or $B_m$) can take an array of different values. For the sake of clarity let us consider some examples. Bits can be implemented using channels containing a single mode ($m=1$) with each amplitude $A_1$ and $B_1$ taking one of two values. These two values can be two different phases: $0$ and $\pi$. If each amplitude can take one of three values we would be implementing trits. Four possible values would yield quarts. Another option to implement quarts is to consider channels composed of two modes ($m=1$ and $m=2$) with amplitudes $A_1$, $A_2$, $B_1$ and $B_2$ taking one of two values. In general, the amplitude of the electric field writes

Figure 1(b) shows a schematic representation of the c-SWAP gate between modes with positive and negative index $m$ conditioned by the state of polarization. For this, we use a Mach-Zehnder interferometer where each arm of the interferometer bears a different orthogonal polarization. In each arm, the beam experiences a different number of reflections. A single reflection in a mirror changes the OAM of the LG modes $m \Longleftrightarrow -m$. In the arm of the interferometer with vertical polarization the beam experiences an odd number of reflections that implements the SWAP operation, the electric field amplitude changes as

From an experimental point of view, due to the symmetry of channels $1$ and $2$ with respect to the sign of the index $m$, we can effectively perform the polarization-dependent SWAP operation of amplitudes $\left \{A_m,B_m \right \}$ by performing a polarization-dependent CNOT operation on the modes that compose the channel. Information is contained on the amplitudes, while the sign of index $m$ designates the channel. In an alternative scenario where channels $1$ and $2$ would correspond to single spatial modes with OAM indexes $m_1$ and $m_2$, one can exchange information between channels using a spatial light modulator encoded with $-m_1-m_2$ as demonstrated in [10].

3. Quantum-inspired optical device for data and waveform comparison

The system we demonstrate consists of: 1) A Hadamard operation in polarization, the degree of freedom that plays the role of the control bit in our scheme, 2) a Fredkin gate as discussed above, and 3) another Hadamard operation in the polarization degree of freedom. The relevant measurement for data comparison between information encoded in channels $1$ and $2$ is the output power in the horizontal ($P_x$) and vertical ($P_y$) polarizations.

We define the overlap $\gamma$ as

In order to unveil the meaning of the parameter $\gamma$, let us assume that $\{A_m\}$ and $\{B_m \}$ are real and that $p_m \equiv |A_m|^2$ and $q_m \equiv |B_m|^2$ ($m=1,2,\ldots ,N$) correspond to two probability distributions. We obtain that $\gamma =-\sum _{m=1}^{N}\sqrt {p_m q_m}$. This shows that the overlap measure introduced for a series of complex numbers is related to the fidelity or Bhattacharyya coefficient [38], a measure of how different are two probability distributions.

We consider now that signals $A_m$ and $B_m$ can vary in time. One can think of the discretization of signals of interest at times $t_1=0, t_2=\Delta t, t_3=2\Delta t, \ldots , t_{z}=(Z-1) \Delta t$. For the case $N=1$ (single-mode) we consider two functions $\alpha _1(t_i)$ and $\beta _1(t_i)$ ($i=1,\ldots ,Z$) that correspond to two probability distributions. We encode their values into the phases of $A_1$ and $B_1$, i.e., $A_1(t_i)=\exp [i\alpha _1(t_i)]$ and $B_1(t_i)=\exp [i\beta _1(t_i)]$. We obtain $\gamma (t_i)=-\cos [\alpha _1(t_i)-\beta _1(t_i)]$. The Kolmogorov distance $K(\alpha ,\beta )=\sum _{i=1}^{Z} \left | \alpha _1(t_i)-\beta _1(t_i)\right |$ between the two probability distributions is

4. Experimental setup

Experimental implementation of the protocols for waveform and data comparison that includes the quantum-inspired Fredkin gate, is shown in Fig. 2. We use a Gaussian beam from a Helium-Neon laser ($\lambda =633$ nm) with a beam waist of $\sim$ 1.4 mm. The beam shows vertical polarization with the help of a linear polarizer (LP). It is collimated by two lenses, L1 and L2, with focal lengths of 10 cm and separated by 20 cm.

 

Fig. 2. Experimental setup. Beams with OAM are generated with the help of a spatial light modulator (SLM). The Hadamard gates are implemented with half-wave plates oriented at $22.5^{\circ }$ with respect to the horizontal polarization. The c-SWAP gate is a Mach-Zehnder interferometer where each arm bears a different polarization. PBS$_i$= polarizing beam splitter; L$_i$: lenses; HWP$_i$: Half-wave plates; M$_i$: mirrors; D$_i$: photodetectors; $\Delta \varphi$: adjustable phase.

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We generate superpositions of LG modes with positive and negative OAM indexes ($\pm m$) with the help of a Spatial Light Modulator (SLM, Hamamatsu X10768-01, 792 $\times$ 600 pixels with a pixel pitch of 20 $\mu$m). The spatially-dependent phase of the incoming beam is tailored with appropriate computed-engineered phase patterns displayed on the SLM. A half-wave plate (not shown in the figure) changes the polarization orientation of the beam to horizontal as required by the SLM.

The c-SWAP gate is a Mach-Zehnder (MZ) interferometer, where light in each arm of the interferometer shows a different polarization. Prior to entering the MZ interferometer, the first Hadamard operation transforms the polarization of the incoming beam into diagonal with the help of a half-wave plate (HWP$_1$). A polarizing beam splitter (PBS$_1$) splits the input beam into the reflected and transmitted beams that have orthogonal polarizations and experience a dissimilar number of reflections given by the number of mirrors present. The OAM of the beams is reversed for an uneven number of reflections and remains the same for an even number of reflections. The phase difference between the two arms is controlled by the displacement of mirrors M$_3$ and M$_4$.

To verify that the polarization-controlled SWAP gate functions correctly, we measure the transverse intensity of the beams with a CCD camera (1200 $\times$ 1600 pixels of $4.4 \times 4.4$ $\mu$m$^2$) before PBS$_1$ (input beam) and after PBS$_2$ (output beam). We image the beams with a telescope with two lenses of focal length 12.5 cm ($L_3$ and $L_4$) and separated by $25$ cm. The CCD is taken away after recording the spatial shape of the beams. We measure light with horizontal and vertical polarizations.

We use LG modes with index $m=\pm 1$, where the amplitude of the input beam is $\mathrm {LG}_{1}^{0}(\textbf {r}_{\perp }) + i \mathrm {LG}_{-1}^{0}(\textbf {r}_{\perp })$, and the polarization is diagonal. Figure 3 shows the theoretical prediction and the experimental results. The intensity of the input beam is $\sim \rho ^2 \exp (-2\rho ^2/w_0^2) \cos ^2(\varphi -\pi /4)$, where $\rho$ and $\varphi$ are the radial and azimuthal coordinates, respectively, in cylindrical coordinates and $w_0$ is the beam waist. Figures 3(a) and 3(d) show the spatial shape of the input beam, the same for both polarizations. There is a line of zero intensity along $\varphi =3\pi /4$ and $\varphi =-\pi /4$.

 

Fig. 3. Demonstration of the controlled-swap gate. (a) and (d) correspond to the spatial shape of the input beam. (b) and (e) show the output beam with vertical polarization where the effect of swap operation can be observed by the change of orientation of the beam with respect to the input beam. (c) and (f) show the shape of the output beam with horizontal polarization, the same as the one of the input beam. (a), (b) and (c) are theory, (d), (e) and (f) are experimental results.

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Figures 3(b) and 3(c) (theory) and Figs. 3(e) and 3(f) (experiment) show the spatial shape of the output beams. The spatial shape of the beam with horizontal polarization remains unchanged showing the same orientation as the input beam. However, the intensity of the output beam with vertical polarization is $\sim \rho ^2 \exp (-2\rho ^2/w_0^2) \cos ^2(\varphi +\pi /4)]$. It shows zero intensity along the line $\varphi =-3\pi /4$ and $\varphi =\pi /4$, a signature of the effect of the SWAP operation $m \Longleftrightarrow -m$.

The half-wave plate HWP$_2$ performs the second Hadamard operation before detection. Finally, polarizing beam splitter PBS$_3$ separates the horizontal and vertical components of the output beam whose powers ($P_{x}$ and $P_{y}$, respectively) are measured with photodiodes D$_1$ and D$_2$.

5. Examples of waveform and data comparison

Our system allows comparing waveforms and streams of data that vary in time without measuring its content. In a series of experiments, we will consider the case in which the variables $\{A_m(t_i)\}$ and $\{B_m(t_i)\}$ ($i=1,\ldots ,Z$) can take only one of two values: $A_m(t_i),~B_m(t_i)=\pm 1$. This corresponds to encoding logical bits of information "0" and "1" as phases $0$ and $\pi$. The use of this information encoding generates bits in the single mode case ($N=1$) and it generates quarts in the two-modes case ($N=2$). In general there will be $M$ bits (or quarts) whose value will be different, and $Z-M$ bits (quarts) with the same value. We define the mean overlap as

A first example is shown in Fig. 4, where we measure the mean overlap $\bar {\gamma }$ between two equal square pulses but delayed between them (for further details see Appendix 6). When the two pulses coincide (zero pulse separation) one obtains $\bar {\gamma }= -1$ as expected.

 

Fig. 4. Mean overlap $\bar {\gamma }$ between two similar square pulses but delayed one with respect to the other. The value of $\bar {\gamma }$ is shown as a function of the pulse separation (in number of times slots $\Delta t$). $\bar {\gamma }=-1$ corresponds to the case when the pulses are not delayed. Dots: experimental data. Solid line: theoretical prediction. See Appendix 6 for further details. Error bars represent the standard deviation of the value $\bar {\gamma }$.

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A second example of waveform comparison is shown in Fig. 5. A signal $A_1$ with constant phase is compared with another signal with a chirp $B(t_k)=\exp (i \alpha t_k^2)$ (see Appendix 6 for details). $\bar {\gamma }= -1$ corresponds to the case where both waveforms are equal. Increasing the value of the chirp $\alpha$ makes both signals more and more different.

 

Fig. 5. Comparison of two signals with different chirp. Signal $A_1$ is constant and signal $B_1$ shows a temporal chirp. Dots: experimental data. Solid line: theoretical prediction. See Appendix 6 for further details. Error bars represent the standard deviation.

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We can also compare strings of data. Figure 6 shows the experimental result of comparing two strings of random bits at times $t_k$, $A_1(t_k)$ and $B_1(t_k)$, which can take only values of $\pm 1$. $M/Z$ is the fraction of pairs of bits that are different. If the two bits are equal, one obtains $\gamma (t_k)=-1$, while if they are different $\gamma (t_k)=1$. The inset of Fig. 6 shows measurements corresponding to the two cases. If the two series of bits are equal ($M=0$), we have $\bar {\gamma }=-1$. If all bits are different ($M=Z$) we have $\bar {\gamma }=1$. In between, the value of $\bar {\gamma }$ determines the fraction of bits that are different without the need to evaluate the value of each bit. To correct for the deleterious effect of detection noise in the experiment, we made use of a threshold value to decide when two bits are equal or not: two bits are different if the value measured of $\gamma (t_k)$ was over 0.7, and they are equal if the value measured was below $-0.7$.

 

Fig. 6. Mean overlap $\bar {\gamma }$ as a function of the fraction of pairs of bits that are different. The solid line corresponds to the expression $\bar {\gamma }=2\,M/Z-1$ (see Appedix 6) where $M$ is the number of pairs $[A_1(t_i),B_1(t_i)]$ where each bit have a different value ($A_1(t_i) \times B(t_i)=-1$) and $Z$ is the total number of pairs of bits. The inset was obtained using 400 different random bits. The figure made use of a subset of 100 random bits from the 400 bits considered in the inset.

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Figure 7 compares two sets of quarts encoded in the amplitudes of two modes, i.e., [$A_1(t_k)=\pm 1, A_2(t_k)=\pm 1$] and [$B_1(t_k)=\pm 1, B_2(t_k)=\pm 1$]. Differences between quarts can originate from the two bits of the quarts being different (two-bit errors, $\gamma (t_k)=1$) or just one bit of the quarts being different (one-bit errors, $\gamma (t_k)=0$). If the two quarts are equal $\gamma (t_k)=-1$. The inset of Fig. 7 shows experimental results for all of these possibilities. As shown in Appendix 6, for a given fraction of different quarts ($M/Z$), the value of $\bar {\gamma }$ ranges between two well-defined values, $M/Z-1$ and $2\, M/Z-1$, corresponding to one- and two-bits errors, respectively. Again, in order to correct for the deleterious effect of detection noise in the experiment, we made use of a threshold value to decide when the bits are equal or not.

 

Fig. 7. Mean overlap $\bar {\gamma }$ as a function of the fraction of pairs of quarts that are different. The inset shows results for the three possible cases: both quarts are equal ($\bar {\gamma }=-1$), both bits of the quarts are different ($\bar {\gamma }=1$), or just one of the bits of the pair of quarts is different ($\bar {\gamma }=0$). The top solid line $\bar {\gamma }=2\, M/Z-1$ corresponds to the case where all of the errors in a string of quarts are two-bits errors. The lower solid line $\bar {\gamma }=M/Z-1$ corresponds to the case where all errors are one-bit errors. The colored region shows the region of possible events. The inset was obtained using 180 quarts, and for the figure, we considered randomly 60 quarts of the previous 180 quarts.

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6. Conclusions

We have demonstrated a protocol for data and waveform comparison that makes use of a Fredkin gate. The functioning of the protocol is a translation of certain features of a counterpart quantum protocol (quantum fingerprinting). The gate uses light beams carrying orbital angular momentum. Intrinsic characteristics of the spatial shape of these modes allow implementing a c-SWAP operation easily, a gate that is generally difficult to implement and that, on many occasions, can only work with a certain probability of success. Our results provide a method to estimate how close are two signals by calculating the overlap between them with simple power measurements. Notice that we can do this in spite that we do not measure the information contained in the signals. The proposed system is another example of the advantages of using light beams with a spatial shape (i.e., structured light).