Quantum computation based on photonic systems with two degrees of freedom assisted by the weak cross-Kerr nonlinearity

Most of previous quantum computations only take use of one degree of freedom (DoF) of photons. An experimental system may possess various DoFs simultaneously. In this paper, with the weak cross-Kerr nonlinearity, we investigate the parallel quantum computation dependent on photonic systems with two DoFs. We construct nearly deterministic controlled-not (CNOT) gates operating on the polarization spatial DoFs of the two-photon or one-photon system. These CNOT gates show that two photonic DoFs can be encoded as independent qubits without auxiliary DoF in theory. Only the coherent states are required. Thus one half of quantum simulation resources may be saved in quantum applications if more complicated circuits are involved. Hence, one may trade off the implementation complexity and simulation resources by using different photonic systems. These CNOT gates are also used to complete various applications including the quantum teleportation and quantum superdense coding.


Results
To show the encoding independence of the polarization and spatial DoFs of a photon for any quantum tasks, it is necessary to show that all n-qubit quantum operations may be realized on these DoFs. From the universality of the CNOT gate and single-qubit operations in the quantum logic 2,3,8 , it only needs to consider the CNOT gate on all the combinations of two DoFs of photonic systems. From different roles of two DoFs, six CNOT gates should be implemented, i.e., four CNOT gates on the two-photon system (each DoF of one photon is used) and two CNOT gates on the one-photon system. None of these gates require switching these DoFs during the simulations.
Before expounding our schemes of the CNOT gate, we first introduce the weak cross-Kerr nonlinearity [21][22][23][49][50][51][52] . Given a signal field |n a 〉 and a probe beam |α〉, after photons passing through the cross-Kerr medium, the joint state of the combined system will be ) / 2 on two spatial modes a and b of an input photon. An auxiliary probe beam is in the coherent state α 2 49,51 . Another BS denotes a 50:50 polarizing beam splitter to implement the transformation η η η η η η → − + ( )/ 2 ( )/ 2  53 of the coherent photons. PNND denotes the photon number non-resolving detector. PA denotes a quantum parity gate for one photon with four spatial modes.
Scientific RepoRts | 6:29939 | DOI: 10.1038/srep29939 i t a a a a a a in s s p p a where θ = χt and t is the interaction time. Previous works indicated that a cross-Kerr medium and a coherent state can be used to implement the CNOT gate 22,23 and single-photon logic gates with minimal sources 51 and Toffoli gate 52 , and complete entanglement purification and concentration [45][46][47][48] ; generating high-quality entanglement 49,50 and qubits 60-62 . CNOT gate on the polarization DoFs of two photons. Suppose that two photons are initially prepared in the state A j j j j j j j for the simplicity of schematic representation, where {a j , b j } is the basis of the spatial DoF (the paraxial spatial modes (Laguerre-Gauss) carrying −ħ and ħ orbital angular momentum) of photon A j . The same results can be followed for general forms of a two-photon system. Our consideration in this subsection is to realize the CNOT gate on the polarization DoFs of two photons. Schematic circuit is shown in Fig. 1 using the double cross-phase modulation technique 49,[51][52][53] to avoid an impractical interacted-phase shift −θ 21,22 . The controlling photon A 1 from each mode passes through a PS, interacts with the coherent photons with a phase θ, and another PS. And then, the photon A 2 passes through a H and a BS, and interacts with the second coherent pulse with a phase θ. Now, two coherent pulses pass through a phase shifter −θ, BS and QND (a quantum nondemolition module) 53 . In detail, the joint system of two input photons (A 1 and A 2 ) and the auxiliary coherent pulse evolves from the initial state as follows: Scientific RepoRts | 6:29939 | DOI: 10.1038/srep29939 Now, by using a parity analyzer (PA) for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 , see the Method), the state in the Eq. (4) will be 1  2  2  1  2  2   1 1  1 1  2 2  2 2   1  1 Otherwise, n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes ′ a 2 and ′ b 2 , see Method), the state in the Eq. (4) will be |Ψ f 〉 in the Eq. (5) after a Pauli flip Z = |H〉 〈H| − |V〉 〈V| on the photon A 1 . Here, the unmeasured beams in the state α 2 may be reused. If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (3) collapses into In the follow, using a PA for the photon A 2 (similar projection has been performed for two modes 53 ), if the photon A 2 passes through the modes a 2 and b 2 (n ≠ 0 for the new Homodyne detection), the state in the Eq. (6) will be |Ψ f 〉 in the Eq. (5). If the photon A 2 passes through the modes ′ a 2 and ′ b 2 (n ≠ 0 for the new Homodyne detection), the state in the Eq. (6) will be |Ψ f 〉 in the Eq. (5) after a Pauli flip σ z on the photon A 1 . The projection |n〉 〈n| may be approximated by a transition edge sensor-a superconducting microbolometer 49,51 . Thus a CNOT gate has been nearly deterministically implemented on the polarization DoFs of two photons. Here, the unmeasured beams in the state α θ 2 cos may be reused.
CNOT gate on the spatial DoFs of two photons. Our consideration in this subsection is to realize a CNOT gate on the spatial DoFs of two photons. The schematic circuit is shown in Fig. 2 2 , |η〉 and |ζ〉 are defined in the Eq. (3). Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output 49,51 . If the measurement outcome is n = 0, the photonic state in the Eq. (7) collapses into after the photon A 2 passing through a BS. Now, using a PA for the photon A 2 , the state in the Eq. (8) will be if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ). When n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes a′ 2 and b′ 2 ), the state in the Eq. (8) may be changed into |Ψ f 〉 in the Eq. (9) using a phase gate −I on the photon A 1 from the spatial mode b 1 .
If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (7) collapses into  Scientific RepoRts | 6:29939 | DOI: 10.1038/srep29939 here, each switch operation S is a NOT = H · Z · H gate of two spatial modes, which may be realized with two BSs and a waveplate −I on the second mode. Now, by using PA for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ), the state in the Eq. (10) will be |Ψ f 〉 in the Eq. (9). If n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes ′ a 2 and ′ b 2 ), the state in the Eq. (10) will be |Ψ f 〉 in the Eq. (9) using a phase gate −I on the photon A 1 from the spatial mode a 1 . Thus a CNOT gate has been nearly deterministically implemented on the spatial DoFs of two photons.
CNOT gate on the polarization-spatial DoFs of a two-photon system. Our consideration in this subsection is to realize a CNOT gate on the polarization DoF of one photon and the spatial DoF of the other. Schematic circuit is shown in Fig. 3. From the Figs 1 and 2, the photons A 1 and A 2 and the coherent photon will evolve as follows:  Scientific RepoRts | 6:29939 | DOI: 10.1038/srep29939 2 , |η〉 and |ζ〉 are defined in the Eq. (3). Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output 49,51 . If the measurement outcome is n = 0, the photonic state in the Eq. (11) collapses into Now, from a PA for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ), the state in the Eq. (12) will be by switching the modes a 2 and b 2 . Otherwise, n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes ′ a 2 and ′ b 2 ), and the state in the Eq. (12) will be |Ψ f 〉 in the Eq. (13) after performing a Pauli phase flip Z on the photon A 1 and switching the modes ′ a 2 and ′ b 2 . If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (11) collapses into Now, from a PA for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ), the state in the Eq. (14) will be |Ψ f 〉 in the Eq. (13). If n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes ′ a 2 and ′ b 2 ), the state in the Eq. (14) will be |Ψ f 〉 in the Eq. (13) after performing a Pauli phase flip Z on the photon A 1 . Thus a CNOT gate is nearly deterministically implemented on the polarization DoF of one photon and the spatial DoF of the other photon.
CNOT gate on the hybrid spatial-polarization DoF of a two-photon system. Our consideration in this subsection is to realize a CNOT gate on the spatial DoF of one photon and the polarization DoF of the other. Schematic circuit is shown in Fig. 4. From the Figs 1 and 2, two photons A 1 and A 2 and the coherent pulse will evolve as follows: 2 , |η〉 and |ζ〉 are defined in the Eq. (3). Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output 53 . If the measurement outcome is n = 0, the photonic state in the Eq. (15) collapses into Now, from a PA for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ), the state in the Eq. (16) will be If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (15) collapses into Now, from a PA for the photon A 2 , if n = 0 for the new Homodyne detection (the photon A 2 passes through the modes a 2 and b 2 ), the state in the Eq. (18) will be |Ψ f 〉 in the Eq. (17). Otherwise, n ≠ 0 for the new Homodyne detection (the photon A 2 passes through the modes ′ a 2 and ′ b 2 ), and the state in the Eq. (18) will be |Ψ f 〉 in the Eq. (17) after −I being performed on the photon A 1 from the mode a 1 . Thus a CNOT gate is nearly deterministically implemented on the hybrid system consisted of the spatial DoF of one photon and the polarization DoF of the other photon. )( )  Fig. 1.
Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output 53 . If the measurement outcome is n = 0, the photonic state in the Eq. (19) collapses into   A 1 passes through the modes a 1 and b 1 ). If n ≠ 0 for the new Homodyne detection (the photon A 1 passes through the modes ′ a 1 and ′ b 1 ), the state in the Eq. (22) will be |Ψ f 〉 in the Eq. (21) after Z being performed on the photon A 1 .
Quantum teleportation assisted by the weak cross-Kerr nonlinearity. Suppose that Alice wants to teleport an arbitrary n-photon system in the state   where the photons A 1 , ..., A n belong to Alice while the photons B 1 , ..., B n own to Bob. For special case of n = 1, Wang et al. 32 have experimentally teleported a photon with the spin angular momentum and orbital angular momentum DoFs while Graham et al. 33 teleported a specific photon of two DoFs with only phase information. Sheng et al. 44 have proposed a theoretical teleportation using the Bell analysis assisted by the cross-Ker nonlinearity. Luo et al. 64 have proposed a general teleportation of hybrid two-qubit systems assisted by the QED-cavity   nonlinearity. In this subsection, by using present CNOT gates, we can complete the teleportation task with arbitrary n ≥ 1. Schematic circuit is shown in Fig. 6. These photons evolve as follows where CNOT 1 = |0〉 〈0| ⊗ I 2 + |1〉 〈1| ⊗ X p denotes a CNOT gate on the polarization DoFs of two photons and CNOT 2 = |d 0 〉 〈d 0 | ⊗ I 2 + |d 1 〉 〈d 1 | ⊗ X s denotes a CNOT gate on the spatial DoFs of two photons. Now, by measuring each photon j = 1, ..., n under the basis {|Ha j,0 〉, |Ha j,1 〉, |Va j,0 〉, |Va j,1 〉} (using two PSs and four single photon detectors), and measuring each photon A j ,j = 1, ..., n under the basis  Table 1 of the Supplementary information with corresponding recovery operations.
Quantum superdense coding assisted by the weak cross-Kerr nonlinearity. Similarly, with the hyperentanglement, Alice and Bob may complete a general quantum superdense coding 65,66 , as shown in Fig. 7. Here, two photons A and B are prepared in in the Eq. (24) by Alice. One photon B will be sent to Bob along Alice's quantum channel. Now, Bob will perform a single photon operation on the received photon B according his coding of four bits i 1 i 2 i 3 i 4 in the Fig. 7 and send back to Alice. The corresponding quantum measurements of Alice are shown in the Fig. 7. In detail, Alice performs two CNOT gates CNOT 1 and CNOT 2 on the photons A and B, let the output pulse of the photon B pass a H and a BS, and the photons A and B from each mode pass through a PS and be finally detected by single photon detectors. The resulting quantum hyperentanglements of Alice are shown in Table 2 of the Supplementary information. Different from previous quantum superdense coding which has realized two bits per photon transmission 60,61 , four bits can be communicated by sending a single photon.

Quantum computation assisted by the weak cross-Kerr nonlinearity. Previous schemes have
shown that the controlled logic gates may be performed on the polarization state using the spatial DoF as auxiliary quantum resources [12][13][14]22,23 . Although it is easy to switch different DoFs of one photon if only one DoF is used to encode information in quantum application, their conversions may cause confusions when two DoFs or more DoFs are independently used for encoding different information in one quantum task. With the present CNOT gates assisted by the weak cross-Kerr nonlinearity, the polarization and spatial DoFs of photonic states can be used as independent qubits without auxiliary DoFs. It means that two DoFs of each photon may be used as encoding qubits or register qubits simultaneously. In this case, the simulation resources may be saved one half. This may be very important for large-scale simulations such as the Shor algorithm. To show the implementation complexity of our CNOT gates, the comparisons of these CNOT gates with previous photonic implementations are shown in Table 1. All the linear optical elements of wave plates [H, Z, -I] and beam splitters [BS and PS] may be ignored because of their simplicities. It means that the complexity mainly depends of the cross-phase modulation, the interferences, and ancillary photons. From this table, the most of photonic CNOT gates with two DoFs [except CNOT s,p,1 using one wave plate] should involve more interactions with the weak-Ker nonlinearity than other schemes with single DoF 22,23,[51][52][53] . The main difference is derived from an additional DoF in comparison with previous single DoF. In experiment, the added complexity may be reasonable because the perfect single photon is difficult and expensive with the modern physic technique. Using the photon number non-resolving detector for PND, ancillary single photons are avoided for the QND 53 . If this efficient way 53 is used for our QNDs ancillary photons are not required in our CNOT gates, which are different from the qubus mediated CNOT gate in ref. 22. Moreover, the DXPM method [50][51][52][53] are explored in our schemes to avoid the impractical XPM with a shift −θ [21][22][23] . Compared with the scheme in ref. 21, our schemes donot require displacement operations on the qubus beams, which may be hard to implement for large displacement amplitudes. Besides, coherent resources are necessary in all schemes and may be recycled. The complexity of the circuit in ref. 52 is same as these for general two-qubit gates. Generally, one may trade off the implementation complexity and simulation resources by choosing proper photon systems with one DoF and two DoFs.

Discussions and Conclusions
The present CNOT gates on photons with two DoFs may be nearly deterministically performed. These CNOT gates are different from CNOT gates on photonic systems with only one DoF [11][12][13][14][15][22][23][24][25][26][27][28][29] , where the latter is always applied in quantum applications using the polarization DoF while other DoFs such as the momentum and time-bin are not considered or only considered as auxiliary systems 11,15,22,23 . Our CNOT gates show that quantum tasks may be simulated using photonic systems with two DoFs assisted by the weak cross-Kerr nonlinearity. During simulations, each DoF of one photon can be encoded as an independent qubit for storing or transferring quantum information. The key elements are the present CNOT gates which provide us useful primitives to manipulate photons with two DoFs.
Up to now, a well cross-Kerr nonlinearity in the optical single-photon regime is still difficult with current technology even lots of related results have been obtained 67 . In fact, Kok et al. 68 showed that the Kerr phase shift is only τ ≈ 10 −18 to operate in the optical single-photon regime. It may be improved to τ ≈ 10 −5 using electromagnetically induced transparent materials. Recently, Gea-Banacloche 69 shows that it is impossible to obtain large phase shifts via the giant Kerr effect with single-photon wave packets, as pointed out in refs 70,71. Note that −θ is indeed a large phase shift π/2 − θ. The weak cross-Kerr nonlinearity will make the phase shift ±θ of the coherent state become extremely small 72 . To address this problem, we take use of the double cross-phase modulation method 49,[51][52][53] to avoid the impractical −θ. Combining with a photon-number-resolving (PNR) detector, a homodyne detector may be used to discriminate two coherent states 53,73 . The post-selection strategy is useful in order to lower the error probability. PNR has been realized at infrared wavelengths, operating at room temperature and with a large dynamic range 74 , or at an operating wavelength of about 850 nm 75 . New measurement scheme has been realized based on a displacement operation followed by a PNR 76 . PNR has also been discussed with integrated optical circuit in the telecom band at 1550 nm based on UV-written silica-on-silicon waveguides and modified transition-edge sensors 77 . Of course, the PNR capability may be also shown from InGaAs single photon avalanche detectors, arrays of silicon photomultipliers, transition edge sensors and InGaAs with self-differencing circuits. Recently, superconducting nanowire as another candidate may provide free-running single-photon sensitivity from visible to mid-infrared frequencies, low dark counts, excellent timing resolution and short dead time, at an easily accessible temperature. Myoren et al. demonstrate the superconducting nanowire single-photon detectors with series-parallel meander-type configurations to have photon-number-resolving capabilities 78 . Some methods and device configurations are also proposed to obtain PNR capability using superconducting nanowire detectors 79 . By exploiting a superconducting qubit Lecocq et al. measure the photon/phonon-number distributions during these optomechanical interactions which may provide an essential non-linear resource 80 . Moreover, Weng et al. take use of quantum dot coupled resonant tunneling diodes to demonstrate a PNR 81 . Proposed electron-injecting operation may turn photon-switches to OFF state and make the detector ready for multiple-photons detection. Their results showed that the new PNR is better than a homodyne receiver. Hence, the present CNOT gates may be feasible if we choose a suitable Kerr nonlinear media and some good quantum measurement strategies on coherent beams.
In conclusion, we have proposed the parallel quantum computation based on two DoFs of photon systems, without auxiliary spatial or polarization DoFs. We have constructed five nearly deterministic CNOT gates (except one trivial CNOT gate) operating on the spatial and polarization DoFs of the two-photon system or one-photon system. With these CNOT gates, two DoFs of each photon may be independently encoded as different qubits in each task. We also discussed their applications of the quantum teleportation, quantum supertense coding and quantum computation. We concluded that one can teleport arbitrary n-photon in two DoFs when the hyperentanglement channels are set up and present CNOT gates are permitted perfectly. Moreover, we have obtained new quantum supertense coding in which a hyperentanglement is used to transfer four bits per photon transmission. For different quantum computation tasks, one may perform their simulations using photonic systems with two DoFs. In this case, quantum simulation resources are reduced to one half. All these results may be useful in various quantum applications.

Methods
The weak cross-Kerr nonlinearity. The cross-Kerr nonlinearity 21-23 has a Hamiltonian in the form H a a a a s s p p  . Here, † † a a ( ) s p and a s (a p ) represent the creation and annihilation operations, respectively, and the subscript s(p) denotes the signal (probe) mode. χ is the coupling strength of the nonlinearity decided by the cross-Kerr medium. Given a signal field |n a 〉 and a probe beam |α〉, after photons passing through the cross-Kerr medium, the joint state of the combined system will be where θ = χt and t is the interaction time. Thus, by measuring the phase of the probe beam, the photon numbers may be distinguished in the signal mode, that is, the state |Ψ〉 will project into a number state.
The parity gate. To distinguish different outputs of one photon with four modes, a parity gate (PA) is used using an ancillary coherent state α 2 , see the Fig. 1. The detailed evolution is defined as follows for the any initial system are different states of the photon A 2 with four spatial modes a 2 , b 2 , ′ a 2 and ′ b 2 , while {|φ 1 〉, |φ 2 〉, |φ 3 〉, |φ 4 〉} are corresponding states of the other system except the photon A 2 . In detail, the photon A 2 from the modes ′ a 2 and ′ b 2 is firstly interacted with the coherent system in order. One can get a joint system