1 Introduction

The Quantum computers are expected to perform in such a way that they outsmart classical computers in some tasks. Research groups around the world are working hard for the implementation of such machines. As classical computing needs bits 0 and 1 for the storage of information while Quantum computing transfer information in qubits. The Photonics has a history of providing a platform for the Quantum mechanics approach and now it is emerging into a technology for future Quantum computations. One of the challenges of traditional optical approaches for the implementation of Quantum computation is bulky optical setups. These setups would further grow in size for complex logical Quantum operations and surely difficult to stabilize. The solution to this bulky setup is integrated optics where large numbers of Quantum circuits can be put on a single chip. Quantum information has the property that it can exist in a superposition of 0 and 1 along with this it can also be represented in complex superposition of 0 and 1. Qubits have the property of entanglement and by entangling with one another it can be represented in 2N number of states simultaneously, making quantum computing devices much faster than a classical computing devices. Researchers (Crespi et al. 2011) have demonstrated the silica waveguide on silicon chip to realize two-qubit logic gates but polarization encoded self-guided qubit integrated chip is still lacking. More specifically two-qubit logic gate suitable for Quantum computing need strong interaction between photons which ultimately need high nonlinearity which is not available to photons. Qubits should be manipulated, and at the same time protected from the heat and EMI radiations, which can make severe impact on their superposition state called as de-coherence. To avoid this problem Knill et al. (2001) have demonstrated efficient Quantum operation using linear optics. The two important principles of the KLM proposal are (1) that detection of photon generates nonlinearities in the operation of two photon gate (2) measurement of ancilla photon gives the result of gate operation. The researchers have proved with large number of experimental and theoretical work with success therefore the probability of linear Quantum operation can be increased up to unity with the introduction of large number of ancilla photons.

2 Literature survey

The researchers around the world have been working for the implementation of Quantum chips. The Quantum computing came into spotlight in 1994 when Shor (1994) declared his Quantum algorithm for factoring large numbers with a better efficiency than any classical approach preceding it. This factoring problem is used widely for the encryption of the information message sent over the channel. The CNOT gate is very important in Quantum mechanics same as NAND gate is universal gate for classical computer. In 2001 Pittman et al. (2001). demonstrated probabilistic logic gates using polarization beam splitter, the research was based on Knill et al. (2001) demonstration of Quantum operation and CNOT gate realized by Koashi et al. (2001) using probabilistic manipulation of entangled photons. Before this in the year 1999 Gottesman and Chuang (1999) have shown that concept of Quantum teleportation and realized CNOT gate. In the latter half of 90s lots of researchers (Braunstein et al. 1992; Michler 1996; Bouwmeester et al. 1997; Vaidman and Yoran 1999; Lutkenhaus et al. 1999) have proved that for Quantum teleportation the measurement of Bell states are essential which is nonlinear in nature but using probabilistic method linear calculation of Bell states can be done. The challenge associated with the Quantum computation till 2004 is complexity of linear optical Quantum computing (Muthukrishnan et al. 1999; Pittman et al. 2004; Raussendorf and Briegel 2001) and people were working to reduce the elaborative steps of Quantum teleportation for computation. With the start of new century in 2003 O’Brien et al. (2003) have demonstrated a logic gate using Quantum computing.

Later in 2004 also Gasparoni and Walther (2004) have demonstrated first experimental demonstration of a Quantum controlled-NOT gate for different photons, which was classically feed forwardable. The technique used in the experiment was proved as an important aspect for further experiments as they have used linear optical computing entangled ancillary pair of photons in the experiment. In the very same year 2004 Nielsen et al. (2005) have presented the Quantum computation using cluster state which was somehow based on the concepts proposed by Raussendorf and Briegel (2001). Nielsen’s method works for any non-trivial linear optical gate which succeeds with finite probability, but when it fails, it affects a measurement in the computation. Kok et al. (2006) developed a mathematical technique based on Quantum field theory for the study of entanglement in arbitrary coordinate transformations. Just before this in 2005 Zhao et al. (2005) and Langford et al. (2005) demonstrated a simple optical entangled logic gate using partial polarizing beam splitters. Langford demonstrated controlled not gate can be operated with continues signal and pulsed signal with the use of Quantum process tomography. They have also demonstrated bell state analyzer using this gate which is not non-deterministic and fully resolving. This design proved to be very promising for the people working in area of Quantum optics.

There are numerous architecture available in Quantum computing for different systems like spin based Quantum computing, optical Quantum computing, ion trapping based, magnetic resonance based and superconducting charge and flux qubit, keeping that in mind Spiller et al. (2005) in 2006 gave an detailed view on Quantum information processing and its processing. Kok et al. (2007) also reviewed the work done on linear Quantum computing. In Quantum computation one of the most accepted approaches by most of the researchers is polarization entangled photon as Quantum bits but till 2011 not much work had been proposed for the demonstration of Quantum integrated circuit using the same approach.

As optical computation showed promising results with Quantum approach so with the development in the field of integrated optics in Crespi et al. (2011) have proposed integrated photonic Quantum logic gate for polarization encoded Quantum bits. The results obtained had been based on femtosecond laser waveguide writing. Along with this they have also showed the transformation of states from entangled to separate one and vice versa. In the demonstration they had performed optimization of CNOT gate using Quantum tomography. Cai et al. (2013) presented an algorithm for the solution of 2 variable equation using Quantum logic gates. They had utilized four Quantum and four logic gates for the implementation of subroutines. In 2014 one of the very similar approaches has been presented by Barz et al. (2014) with less number of controlled logic gates. Both of these circuits presented a versatile platform for integrated Quantum optics. In 2015 researchers (Carolan et al. 2015) have demonstrated a reprogrammable photonic quantum circuit which can take input up to six photons and having 12 photon detector circuit for logic operations. It is one of the path showing research for numerous application of quantum computing. In the last few year researchers (Meany et al. 2015; Christof 2016; Zeuner et al. 2018) have reviewed various methods for the development of Quantum circuit. The researchers (Okamoto et al. 2005; Patel et al. 2016; Huang et al. 2017) have also demonstrated various quantum logic gates using integrated optics, they had also gave their detailed perspective on the use of efficient material and its importance in fabrication of Quantum chips. Moving ahead in this direction the researchers (Wang et al. 2017) have demonstrated the interfacing between quantum systems through classical channels which shown satisfactory improvement in the existing models. Along with this Schafer et al. (2018) have demonstrated fast and robust two-qubit gates for trapped-ion qubits and all of these approaches showed very promising results for the development of optical integrated circuits. The Quantum computing are showing new perspective of applications like data analysis (Huang et al. 2018) through high speed processors, overcoming the limitation of information security through blind quantum computing (Huang et al. 2017) with every passing years, The research have also demonstrated the nine qubit system for the error correcting application (Liu et al. 2019). In the very recent articles researchers from University of California (Yang 2018) have developed a theoretical approach for random circuit sampling. The RCS development would help to create 72 qubit computer chips which will open doors of enormous application. In this review we have focused our discussion on implementation of Quantum logic gate using polarization encoded qubits. In Sect. 3 we have discussed Quantum optics concepts, Sect. 4 we have discussed the various methods available to design cnot gates, in Sect. 5 we have discussed the concept of optical interconnect for Quantum chips followed by conclusion, acknowledgement and references.

3 Quantum optics concepts

3.1 Quantum bits

Most of information these days processed in form of bits either 0 or 1. In Quantum mechanics two states of signal are represented as |0〉 and |1〉. These bits can be realized in many forms depending the state of data either it is processed, stored or communicated. A normal bit can have only two states but a qubit has more liberty. It lies in more than one dimensional Hilbert form with the form

$${{\upvarphi }} = {\text{cos}}\uptheta \left| {0\rangle + {\text{exp}}\left( {{\text{i}}\upvarphi} \right){\text{sin}}\uptheta} \right|1\rangle$$
(1)

The normal bits can have position on polesv only but qubits could have any position on sphere as shown in Fig. 1. The state shown in Eq. (1) is called superposition, which can have two amplitudes at the same instant. A qubit resembles spin of electron (-1/2) encoded with the polarization of photon showing the path of photon transfer. It is also possible to estimate the approximate qubit, it is case when two states of the system are so different from all other states occurring in the system, and the system can then be restricted to these two states. A set of n different qubit can have 2n amplitudes in the set. If we somehow be able to process all those amplitudes together this technique will open the doors of enormous possibilities of parallel computing. The researchers (Rocchetto et al. 2019) have already demonstrated the computational learning of quantum states up to 6 qubits experimentally.

Fig. 1
figure 1

Reproduced from Kok et al. (2006)

Quantum bit position.

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3.2 Quantum measurement

The state shown in Eq. 1 may give random values 0 having possibility cos2θ and 1 with possibility sin2θ. Generally Quantum measurements are irreversible it means without disturbance in the system information cannot be gain from the external source. Measurement is the only method to obtain output value from the Quantum system. The measurement can be non-destructive in which qubit is in known state. For n qubits with 2n possibilities the measurement will give only single bit of output. The possible output of any physical quantity O can be represented in the following manner.

$$\langle {\text{O}}\rangle = \langle {{\upvarphi }}\left| {\text{O}} \right|{{\upvarphi }}\rangle \rangle$$
(2)

If O has some Eigen state then this equation will denote eigenvalues of the Eigen state. But if O is in superposition state then the output value will be inaccurate.

3.3 Entanglement

The two Quantum bits (A, B) can coincide in Quantum state represented by

$$|{{\upvarphi }}\rangle_{\text{AB}} = \left( {1/\sqrt 2 } \right) \left( {|0\rangle_{\text{A}} |1\rangle_{\text{B}} - |1\rangle_{\text{A}} |0\rangle_{\text{B}} } \right)$$
(3)

This state is called entangled state because both of the qubit are dependent on each other and not in state of its own. This state also cannot be represented as product for any transformation. This state of Eq. (3) have equal magnitude of superposition coefficient and it is one of the four Bell states. The particular state of above equation is called singlet as its form is invariant for any transformation applied to Quantum bits. Entanglement includes some important characteristics in processing of Quantum information. One of the important aspects is that most pure states of Quantum bits includes entanglement among them. Entanglement also gives the advantage in Quantum information processing of having 2n dimensional spaces for information processing. Another advantage is that if A and B are spatially separable states, the correlation between these two states cannot be explained by conventional methods, entanglement fill the gap and help to find separated measurement outputs. All these advantages help in organizing distributed processes which cannot be done with traditional methods.

4 Linear quantum computing

4.1 Quantum optics

The primary bricks of Quantum optics are obviously beam splitters they can be quaterwave and half wave plates and phase shifting devices etc. The phase shifter is one of the most important element that change the phase of propagating mode. If we consider it physically it is a slab of dielectric material whose refractive index is different from space. Beam splitter is another important component it made up of semi reflective mirror, when light signal incident on this mirror partly it gets reflected while rest is transmitted. If we assume two incoming mode in beam splitter as ain and bin and output mode as aout and bout, then beam splitter output can be shown as

$$a_{out}^{\dag } = cos\theta a_{in}^{\dag } + {\text{ie}}^{{ - {\text{i}}\upphi}} sin\theta b_{in}^{\dag }$$
(4)
$$b_{out}^{\dag } = cos\theta b_{in}^{\dag } + {\text{ie}}^{{{\text{i}}\upphi}} sin\theta a_{in}^{\dag }$$
(5)

The reflection coefficient and the transmission coefficient of the beam splitter are given as R = sin2θ and T = 1 − R. The beam splitter can be represented by above equations with proper choice of phases. The phase shift in beam splitter is represented by i.e. (±) iϕ and θ/ϕ are the angles of rotation. The same description is applicable for polarization rotation done by plates either half wave or quarter-wave. Instead of having two different modes ain and bin having different polarization, we can be represent them as ax and by for coordinates x and y respectively with same equations as above.

One more important element of linear Quantum optics is polarizing beam splitter. If PBS cuts the incoming modes ain and bin into horizontal and vertical polarization as shown Fig. 2a, the transformation gives the following output

$$\begin{aligned} & {\text{a}}_{\text{in}} {\text{H}} \to {\text{a}}_{\text{out}} {\text{H}}\quad {\text{and}}\quad {\text{a}}_{\text{in}} {\text{V}} \to {\text{b}}_{\text{out}} {\text{V}} \\ & {\text{b}}_{\text{in}} {\text{H}} \to {\text{b}}_{\text{out}} {\text{H}}\quad {\text{and}}\quad {\text{b}}_{\text{in}} {\text{V}} \to {\text{b}}_{\text{out}} {\text{V}} \\ \end{aligned}$$
Fig. 2
figure 2

Beam splitter a polarization splitter b directional splitter

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With the use of quarter wave plates and half wave plates we can also make beam splitters for directional polarization, in that case we can left and right instead of horizontal and vertical, it is represented in Fig. 2b.

4.2 Qubit encoding

Two most popular method of qubit encoding are polarisation encoding and spatial encoding. In polarisation encoding the state of qubit is represented by its position orientation i.e. horizontal and vertical. The qubit state can be represented as

$$\left| {0\rangle = } \right|{\text{H}}\rangle$$
(6)
$$\left| {1\rangle = } \right|{\text{V}}\rangle$$
(7)

This polarisation encoded qubit state can be converted into into spatial encoding with the linear Quantum components like quaterwave plate and beam splitter. The polarisation encoded method is also known as single rail as both logical states occupy the same spatial encoding. As qubit encoding is very straight forward similarly qubit rotation can be achieved easily with wave plates and polarisation beam splitters. To perform Quantum operations we need two qubit gate which is CNOT gate, CNOT gate toggle the output bit when the target bit is high, it can be represented in the form of interferometer where providing high bit will change the phase in arm of interferometer. The change in phase would result in change of output. The representation of states of gate in two qubit space on computational basis (|00〉, |01〉, |10〉, |11〉) can be shown as

$${\text{U}}_{{{\text{CNOT}}\left( {\varphi } \right)}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} } \right]$$

This gate is very similar to classical XOR gate, which is one of the challenging gates to implement using linear Quantum computing because we need to have nonlinear phase shift to control the toggle of target bit. The linear Quantum computing only requires single qubit operation to construct a circuit but the trouble is gate is non-deterministic and it only uses some outputs out of total available outputs of the circuit. Polarization encoded qubit have less experimental error as they are easy to manipulate. The polarization encoded qubit helped us to design a CNOT gate with two PBS, two polarization sensitive detector and two ancilla photons and these two ancilla photons are in entangled state. In such Quantum states the value of these two ancilla photons or polarization are always unknown but we can predict the value using measurement methods.

4.3 Two qubit gate

The operation of two qubit gate was demonstrated by many researchers in both destructive (in which logic operation can only be verified by the estimation of target and control qubit) and non-destructive methods. The laser written and wave guided based two qubit gates were also demonstrated. In the operation two qubit CNOT gate |0〉 and |1〉 represents horizontal and vertical polarisations respectively. In order to achieve universal Quantum computation single transformation using two qubit gates is sufficient. The most often implemented two qubit gate is CNOT gate that toggles the target qubit (as shown in Fig. 3) according to the value of control qubit. The unitary matrix of CNOT gate is shown already in previous section. The two qubit CNOT gate needs interaction between photons which carries the information. The interaction requires nonlinearity which is achieved using polarisation beam splitters and wave plates. The polarisation beam splitter produce entanglement of photons to produce polarisation encoded CNOT gate for polarisation encoded qubits. The partial polarisation beam splitter gives different beam splitting operations on horizontal and vertical polarisation modes. Such component is challenging to integrate in optics.

Fig. 3
figure 3

Output of CNOT gate

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4.4 Heralded CNOT gate

The demonstration of CNOT gate (Zeuner et al. 2018; Okamoto et al. 2005) generates nonlinearity using destructive measurement of photons. These types of gates are called unheralded since there operation can only be verified by the qubit measurement. But this type of gate is unsuitable for tandem application where for the implementation large circuit one gate output used as an input for next stage. In this type of gate an additional ancillary photon required to herald the effective implementation of the circuit. In one the recent demonstration of heralded CNOT gate researchers have used laser written waveguide to process the polarisation encoded qubit. The horizontal and vertical polarization components are considered as logic 0 and logic 1 respectively. The Bell states of ancillary entangled photons can be defined as

$$|{{\upvarphi }}^{ \pm } \rangle = \left( {\left| {{\text{H}}, {\text{V}}\rangle \pm } \right|{\text{V}}, {\text{H}}\rangle } \right)/\sqrt 2 )$$
(8)
$$|\phi^{ \pm } \rangle = \left( {\left| {{\text{H}}, {\text{H}}\rangle \pm } \right|{\text{V}}, {\text{V}}\rangle } \right)/\sqrt 2 )$$
(9)

In the operation of CNOT gate four photons are incidented as input to the gate with the polarization |cin,tin〉 ⊗ |a 1in ,a 2in 〉, where |cin,tin〉 is the state of control and target qubit, and |a 1in ,a 2in 〉, is the state of ancillary photons. For the detection of photons four detectors are placed and for the successful operation of CNOT gate one photon is detected by two of the four detector and second photon is detected by rest two detectors which happens with probability ¼, such twofold detection heralded the successful operation of CNOT gate with the presence of output control and target bits. It is not essential to measure control bit and target as they can be used for the implementation of cascade circuits.

4.5 Fredkin and Toffoli gate

Meanwhile, with the demonstration of CNOT gate the researchers have also demonstrated the quantum reversible logic gate Fredkin gate (Patel et al. 2016) and Toffoli gate Huang et al. (2017). The Fredkin gate is demonstrated with 3 qubit inputs and the state of the target qubits are swapped as per the state of control qubit. The path mode entanglement of the photons using SPDC (spontaneous parametric down conversion) is used to obtain the operation of the logic gate. The truth table of quantum Fredkin gate is shown below (Table 1). The operation of another reversible gate Toffoli gate is explained using linear quantum optics. In this demonstration the three qubit Toffoli gate is operated using three different degree of freedom (spin angular momentum, spatial, orbital angular momentum) of a single photon. The demonstrated Toffoli gate performs the bit flip operation on the orbital angular momentum qubit when the other two qubits are in the state |1〉 and |1〉, otherwise the system remains at the same state. The truth table of the quantum Toffoli gate is shown below as Table 2.

Table 1 Truth table of quantum Fredkin gate
Full size table
Table 2 Truth table of quantum Toffoli gate
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5 Quantum optical integrated circuits

As we already mentioned one of the major benefits of optical approach to implement computing is the ability to integrated logic devices using optical fibres. In order to connect these devices we have to find way to interconnect those devices that form relatively simple circuit. Large scale Quantum computers requires series of entangling gates to process large information. The large integrated circuit requires lots of component binding together sticking on a single chip and connecting all those components for all optical processing needs optical interconnection. So we have to look for optical interconnection instead of electrical interconnects for the fast processing of Quantum chips.

5.1 Optical interconnect

The role of Quantum interconnect was initially represented by Kimble. A chip based Quantum computing solution will provide secure Quantum key distribution and Quantum communication. In optical interconnects, there should be the coherent transmission of the qubit state between subsystem. One of the basic need for optical interconnect is to keep the entanglement of the bits throughout the manipulation, conversion and transmission of information in the system. With the development of integrated photonics Quantum computing enabled its progress in the field of communication, sensing and cloud computing for all these applications we have multichip systems. Although currently it is very difficult to integrate such chip due to fragile nature of Quantum states. The main requirement of interconnected systems are processing of entanglement in distributed form among various nodes.

  1. (a)

    Quantum Interconnect through interconversion of path polarization

Path-encoding of a photon across two waveguide is used for robust on chip encoding in Quantum information processing. There are some other more suitable methods which can be used like polarization, special mode, time-bin encoding (Xiong et al. 2015) in fiber and free space for Quantum information transmission and distribution. But still no one get a complete Quantum optical connect system capable of coherently distributing and maintaining entanglement between qubit in two or more Quantum circuits (Silverstone et al. 2014; Nickerson et al. 2014; Harris et al. 2014). In one of the experiment Wang et al. (2016) demonstrated high-fidelity quanta photonic interconnect. In that entangled photons in the range of optical communication were generated, processed and then distributed among two silicon photonic chips connected by an optical fiber. They fabricated these chips using cutting edge technology to integrate all of the requirements of the Quantum interconnects. They had utilized the simpler setup in which they generated path entangled photons. In that setup they included two waveguides which have capabilities to generate spatial entangled Bell states with maximum entanglement. These states were then distributed among chips with the transmission of qubits. To avoid decoherence among the bits grating coupler was used to interconvert path encoding and polarization encoding.

  1. (b)

    Quantum Dots based on-chip PTP (point to point) interconnect medium for high performance computing systems

Quantum dot is semiconductor crystal which confines electrons well in a nanostructure dimension well. Quantum dots can be used for Quantum computing. It is not going to take time that energy efficient high performance Quantum computing system in perspective, scalable Quantum dots based on chip interconnects will be useful. An optical chip interconnect medium based on Quantum dot strip (Ben Ahmed and Ben Abdallah 2016) would be built, which can be extended to provide the interconnect medium of required length. Both light intensity and quality of photons emitted would eventually determine the need of gold nanoparticles to be inserted as repeaters, or any other approach to increase the intensity of light being transmitted with Quantum dots itself as part of constructing the on chip interconnect medium.

6 Conclusion

In the last few decades the progress in Quantum computing took some giant steps. There are lots of techniques available for the demonstration of logic gates and fabrication of high speed processor. In the last few year with the implementation of non-destructive methods of optical computing the scope for integrated circuits reached a new height. The applications of Quantum information processing are already increasing. To progress from here Quantum information technology needs to develop real time application using Quantum simulators. We have briefly discussed the progress associated with Quantum computing with basic concepts. The next few years would be very interesting from progress point of view and we may achieve new milestones in Quantum technology. Right now, Quantum computer is a multi-dollar scientist project that we generally find in R&D departments at large IT companies like Intel, IBM and Google or in physics laboratory of large university, like MIT. The quantum computers are better than classical computers in solving complex problems within no time. But to design those quantum computers we also need complex algorithms having support of machine learning, artificial intelligence, big data and cloud computing. The formulation of such huge tasks needs lots of financial support and time which make it a tougher task to do. Even if we design such complex algorithms the debugging is still a huge issue, even the programmer don’t able to observe the path their data takes from input to output. Along with this data security is still a issue quantum cryptography is still a new born baby. Some companies are still working on quantum resistant cryptography to solve the security issues, China has already launched quantum technology satellite which they claimed is unhackable. First generation of Quantum computer looks too harshly but one should not forget that classical computer has also passed the same path. We therefore believe that technology reviewed in this article will have serious application in future.