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

Abstract

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.

From the quantum circuit model¹, quantum controlled gates²,³ play key roles for various quantum applications⁴,⁵,⁶,⁷. It has shown that two-qubit gates, especially the CNOT gate and single-qubit gates are universal for synthesizing quantum tasks based on multiple qubits²,³,⁸. The pioneer model⁹ takes use of single photon sources, linear optical elements including feed forward, and single photon detectors to realize the CNOT gate with the maximum probability of 3/4¹⁰. With this standard model, various schemes are proposed to implement the CNOT gate¹¹,¹²,¹³,¹⁴,¹⁵ and controlled-phase gate¹⁶,¹⁷. Although their upper bounds of the success probability are not thought to be tight¹⁸, however, it has shown that near deterministic gates are impossible using only linear optical elements. Moreover, the multiple-qubit based quantum tasks may be inefficient when lots of probabilistic gates are involved. For an example, the qubit flip coding with three qubits may be constructed using ten CNOT gates (four CNOT gates and one Toffoli gate¹⁹) and some single-qubit gates²⁰, its success probability is only (3/4)¹⁰ = 5.6 × 10⁻² with the maximum probability of a CNOT¹⁰. Hence, more efficient or deterministic gates should be proposed by relaxing constraints in the standard model⁹. Fortunately, with the weak cross-Kerr nonlinearity, a nearly deterministic CNOT gate²¹,²² and multiple-qubit logic gates such as Fredkin gate, Toffoli gate, arbitrary controlled-U gate²³ have been proposed on the polarization DoF. These controlled gates are also implemented using different physical systems such as the ion trap²⁴,²⁵, atom²⁶,²⁷, and nuclear magnetic resonance²⁸,²⁹.

Previous implementations of controlled gates have focused on the systems with only one DoF⁹,¹¹,¹²,¹³,¹⁴,¹⁵,²²,²³. Controlled logic gates are always realized on the polarization DoF using auxiliary spatial DoFs¹²,¹³,¹⁴,²²,²³ or auxiliary polarized photons¹¹,¹⁵. If two DoFs are independently used for encoding different information, their conversions may cause confusions in large-scale quantum applications such as the Shor’s algorithm. Moreover, an experimental system may possess various independent DoFs simultaneously. Different DoFs of physical system may be useful in various quantum applications³⁰. Recent experiment shows that quantum information may be transferred from the polarization DoF of one photon to the orbital angular momentum of the other photon³¹. By using a hyper-entangled photon pair (the simultaneous entanglement in more than one DoF), 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. Here, the hyperentanglement³⁴,³⁵,³⁶ such as polarization momentum, polarization-time-bin, and polarization- and spatial modes-energy-time can be used to assist the Bell-state discrimination³⁷,³⁸,³⁹,⁴⁰,⁴¹,⁴²,⁴³,⁴⁴,⁴⁵,⁴⁶,⁴⁷.

Motivated by the recent experiments³⁰,³¹,³²,³³ and usefulness of different photonic DoFs³⁴,³⁵,³⁶,³⁷,³⁸,³⁹,⁴⁰,⁴¹,⁴²,⁴³,⁴⁴,⁴⁵,⁴⁶, in this paper, we consider the controlled gates on photonic system with two DoFs assisted by the weak cross-Kerr nonlinearity²²,²³,⁴⁶,⁴⁷,⁴⁸,⁴⁹,⁵⁰,⁵¹,⁵²,⁵³. Different from previous schemes on photonic systems with the polarization DoF¹¹,¹²,¹³,¹⁴,¹⁵,²²,²³, where another DoF is used to assist quantum logic gates, we investigate the photonic quantum computation using two DoFs as simultaneous encoding qubits. To show the independence of two photonic DoFs in each quantum task, from the quantum circuit model the CNOT gate will be implemented on all the combinations of the polarization and spatial DoFs of the two-photon or one-photon system. This is beyond previous CNOT gates on the two-photon system with one DoF¹¹,¹²,¹³,¹⁴,¹⁵,²²,²³. By exploiting the weak cross-Kerr nonlinearity⁴⁹,⁵⁰,⁵¹,⁵²,⁵³, all of controlled gates are nearly deterministic without auxiliary DoFs¹¹,¹⁵. In contrast to the hybrid CNOT gates on the photon and stationary electron spins in quantum dots⁵⁴,⁵⁵, our CNOT gates are realized on photonic systems. Our results are also different from previous controlled gates⁵⁴,⁵⁵,⁵⁶,⁵⁷,⁵⁸,⁵⁹, where a CNOT gate is only considered in the same DoF of two photons assisted by a double-sided quantum dot-cavity system⁵⁴,⁵⁵ or one-sided quantum dot-cavity system⁵⁶,⁵⁷,⁵⁸,⁵⁹. Our theoretical results show that two DoFs of a photon system can be independently and simultaneously encoded in each quantum task. With these constructions, one half of quantum resources may be saved for quantum simulations, which are very important in large-scale quantum applications such as the quantum Shor algorithm and network-based quantum communications. To show its applications, we also present faithful teleportation of arbitrary n-photon 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 logic²,³,⁸, 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 nonlinearity²¹,²²,²³,⁴⁹,⁵⁰,⁵¹,⁵². 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

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 gate²²,²³ and single-photon logic gates with minimal sources⁵¹ and Toffoli gate⁵², and complete entanglement purification and concentration⁴⁵,⁴⁶,⁴⁷,⁴⁸; generating high-quality entanglement⁴⁹,⁵⁰ and qubits⁶⁰,⁶¹,⁶².

CNOT gate on the polarization DoFs of two photons

Suppose that two photons are initially prepared in the state

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 technique⁴⁹,⁵¹,⁵²,⁵³ to avoid an impractical interacted-phase shift −θ²¹,²². The controlling photon A₁ from each mode passes through a PS, interacts with the coherent photons with a phase θ, and another PS. And then, the photon A₂ 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 (A₁ and A₂) and the auxiliary coherent pulse evolves from the initial state as follows:

Figure 1. Schematic CNOT gate on the polarization DoFs of two photons.

PS denotes a beam splitter to transmit |H〉 and reflect |V〉 of an input photon. BS denotes a 50:50 polarizing beam splitter to realize and on two spatial modes a and b of an input photon. An auxiliary probe beam is in the coherent state ⁴⁹,⁵¹. Another BS denotes a 50:50 polarizing beam splitter to implement the transformation for the auxiliary coherent photons. H-plate denotes a half-wave plate to perform the Hadamard operation and . QND denotes quantum nondemolition module⁵³ 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.

where , and .

Due to the quantum noise effect, the Homodyne detection cannot work well as its expected⁶³. Afterwards, the quantum nondemolition module is used to discriminate two coherent states. In detail, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output⁵³. If the measurement outcome is n = 0, the photonic state in the Eq. (3) collapses into

Now, by using a parity analyzer (PA) for the photon A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂, see the Method), the state in the Eq. (4) will be

Otherwise, n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes and , 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 A₁. Here, the unmeasured beams in the state 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 A₂ (similar projection has been performed for two modes⁵³), if the photon A₂ passes through the modes a₂ and b₂ (n ≠ 0 for the new Homodyne detection), the state in the Eq. (6) will be |Ψ_f〉 in the Eq. (5). If the photon A₂ passes through the modes and (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 A₁. The projection |n〉 〈n| may be approximated by a transition edge sensor-a superconducting microbolometer⁴⁹,⁵¹. Thus a CNOT gate has been nearly deterministically implemented on the polarization DoFs of two photons. Here, the unmeasured beams in the state 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. The joint system of two photons A₁ and A₂ and the coherent photon evolve from the initial state |Ψ₀〉 as follows:

Figure 2. Schematic CNOT gate on the spatial DoFs of two photons.

PS, BS, QND and PA are defined in the Fig. 1. −I denotes a wave plate to perform the phase operation −|H〉 〈H| − |V〉 〈V|. S denotes the switching operation (NOT gate) of two spatial modes, which may be realized with two BSs and a waveplate −I on the second mode. An auxiliary probe beam is in the coherent state .

where , |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 output⁴⁹,⁵¹. If the measurement outcome is n = 0, the photonic state in the Eq. (7) collapses into

after the photon A₂ passing through a BS. Now, using a PA for the photon A₂, the state in the Eq. (8) will be

if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂). When n ≠ 0 for the new Homodyne detection (the photon A₂ 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 A₁ from the spatial mode b₁.

If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (7) collapses into

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 A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂), the state in the Eq. (10) will be |Ψ_f〉 in the Eq. (9). If n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes and ), the state in the Eq. (10) will be |Ψ_f〉 in the Eq. (9) using a phase gate −I on the photon A₁ from the spatial mode a₁. 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 A₁ and A₂ and the coherent photon will evolve as follows:

Figure 3. Schematic CNOT gate on a hybrid system consisting of the polarization DoF of one photon and the spatial DoF of the other photon.

PS, BS, QND and PA are defined in the Fig. 1. −I and S are defined in the Fig. 2. An auxiliary probe beam is in the coherent state .

where , , |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 output⁴⁹,⁵¹. If the measurement outcome is n = 0, the photonic state in the Eq. (11) collapses into

Now, from a PA for the photon A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂), the state in the Eq. (12) will be

by switching the modes a₂ and b₂. Otherwise, n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes and ), 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 A₁ and switching the modes and .

If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (11) collapses into

Now, from a PA for the photon A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂), the state in the Eq. (14) will be |Ψ_f〉 in the Eq. (13). If n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes and ), the state in the Eq. (14) will be |Ψ_f〉 in the Eq. (13) after performing a Pauli phase flip Z on the photon A₁. 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 A₁ and A₂ and the coherent pulse will evolve as follows:

Figure 4. Schematic CNOT gate on the hybrid system consisting of the spatial DoF of one photon and the polarization DoF of the other photon.

PS, BS, H, QND and PA are defined in the Fig. 1. An auxiliary probe beam is in the coherent state .

where , |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 output⁵³. If the measurement outcome is n = 0, the photonic state in the Eq. (15) collapses into

Now, from a PA for the photon A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂), the state in the Eq. (16) will be

Otherwise, n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes a′₂ and ), and the state in the Eq. (16) will be |Ψ_f〉 in the Eq. (17) after −I being performed on the photon A₁ from the mode b₁.

If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (15) collapses into

Now, from a PA for the photon A₂, if n = 0 for the new Homodyne detection (the photon A₂ passes through the modes a₂ and b₂), the state in the Eq. (18) will be |Ψ_f〉 in the Eq. (17). Otherwise, n ≠ 0 for the new Homodyne detection (the photon A₂ passes through the modes and ), and the state in the Eq. (18) will be |Ψ_f〉 in the Eq. (17) after −I being performed on the photon A₁ from the mode a₁. 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.

CNOT gate on one photon with two DoFs

Our considerations in this subsection is to realize a CNOT gate on one photon A₁ with two DoFs. It is trivial to realize a CNOT gate when the spatial DoF of one photon is the controlling qubit. For the polarization DoF as the controlling qubit, its schematic circuit is shown in Fig. 5. In detail, the photon A₁ and the coherent photon will evolve as follows:

Figure 5. Schematic CNOT gate on one photon with two DoFs.

PS, BS, QND and PA are defined in the Fig. 1. −I and S are defined in the Fig. 2. An auxiliary probe beam is in the coherent state .

where , |ζ〉 are defined in the Eq. (3), and denotes the BS for coherent photons defined in the Fig. 1.

Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output⁵³. If the measurement outcome is n = 0, the photonic state in the Eq. (19) collapses into

After a PA for the photon A₁, if n = 0 for the new Homodyne detection (the photon A₁ passes through the modes a₁ and b₁), the state in the Eq. (20) will be

Otherwise, n ≠ 0 for the new Homodyne detection (the photon A₁ passes through the modes and ), and the state A₁ in the Eq. (20) will be |Ψ_f〉 in the Eq. (21) after Z being performed on the photon A₁.

If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (19) collapses into

After a PA for the photon A₁, the state in the Eq. (22) will be |Ψ_f〉 in the Eq. (21) if n = 0 for the new Homodyne detection (the photon A₁ passes through the modes a₁ and b₁). If n ≠ 0 for the new Homodyne detection (the photon A₁ passes through the modes and ), the state in the Eq. (22) will be |Ψ_f〉 in the Eq. (21) after Z being performed on the photon A₁.

Quantum teleportation assisted by the weak cross-Kerr nonlinearity

Suppose that Alice wants to teleport an arbitrary n-photon system in the state

to Bob, where , a_k,0 and a_k,1 denote the spatial modes of the input photon k. The quantum channel is constructed by hyperentanglements³²

where the photons A₁, ..., A_n belong to Alice while the photons B₁, ..., 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

Figure 6. Schematic teleportation of arbitrary n-photon system with two DoFs.

The quantum channel is constructed by hyperentanglements , j = 1, ..., n. X^p = |0〉 〈1| + |1〉 〈0| and X^s = |r₁〉 〈r₂| + |r₂〉 〈r₁| denote Pauli flips on the polarization DoF and spatial DoF {r₁, r₂} of one photon, respectively. CNOT₁ = |0〉 〈0| ⊗ I₂ + |1〉 〈1| ⊗ X^p and CNOT₂ = |d₀〉 〈d₀| ⊗ I₂ + |d₁〉 〈d₁| ⊗ X^s are preformed on the photon A_j and the input photon j. PS, BS and H-plate are defined in the Fig. 1. D_H and D_V are single photon detectors. The feed-forward operations of Bob are only single photon operations.

where CNOT₁ = |0〉 〈0| ⊗ I₂ + |1〉 〈1| ⊗ X^p denotes a CNOT gate on the polarization DoFs of two photons and CNOT₂ = |d₀〉 〈d₀| ⊗ I₂ + |d₁〉 〈d₁| ⊗ 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 (using two Hs, a BS, two PSs and four single photon detectors), each collapsed state of the photons B₁, ..., B_n may be faithfully transferred into in the Eq. (23) by using local quantum single operations of Bob. Taking n = 2 as an example, all the collapsed states are shown in Table 1 of the Supplementary information with corresponding recovery operations.

Table 1. The comparisons of our CNOT gates with previous photonic implementations.

Type	ni	np	nQND	|ϕa〉	L	Ps
CNOTp,p in ref. 22	4	2	2		XPM	≈1
CNOTp,p in ref. 23	2	0	1	|α〉	XPM	1/2
CNOTp,s in ref. 51	4	0	1		DXPM	≈1
CNOTp,p in ref. 52	12	1	9		DXPM	≈1
CNOTp,p in ref. 53	5	0	2		DXPM	≈1
CNOTp,p in Fig. 1	8	0	2		DXPM	≈1
CNOTs,s in Fig. 2	8	0	2		DXPM	≈1
CNOTs,p in Fig. 3	10	0	2		DXPM	≈1
CNOTp,s in Fig. 4	8	0	2		DXPM	≈1
CNOTp,s,1 in Fig. 5	10	0	2		DXPM	≈1

P_s denotes the success probability. CNOT_p,p denotes the CNOT gate on the polarization DoF of two photons. CNOT_p,s,1 denotes the polarization DoF and spatial mode of one photon. CNOT_p,s denotes the polarization DoF of one photon and the spatial mode of the other photon. CNOT_s,p denotes the spatial DoF of one photon and the polarization DoF of the other photon. CNOT_s,s denotes the spatial DoFs of two photons. n_i denotes the number of the interaction between the input photon and a coherent state with the cross-Kerr nonlinearities. n_p denotes the number of ancillary photons. n_QND denotes the number of the QND. |ϕ_a〉 denotes the auxiliary coherent state. L ∈ {DXPM, XPM} denotes the double cross-phase modulation technique or cross-phase modulation technique. CNOT_p,p is easily followed by combining with the C-path gate and Merging gate⁵³.

Quantum superdense coding assisted by the weak cross-Kerr nonlinearity

Similarly, with the hyperentanglement, Alice and Bob may complete a general quantum superdense coding⁶⁵,⁶⁶, 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 i₁i₂i₃i₄ 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 CNOT₁ and CNOT₂ 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 transmission⁶⁰,⁶¹, four bits can be communicated by sending a single photon.

Figure 7. Schematic quantum superdense coding using the hyperentanglement in the Eq. (24).

PS, BS and H are defined in the Fig. 1. X^p, X^s, CNOT₁ and CNOT₂ are defined in the Fig. 6. Z^p = |H〉 〈H| − |V〉 〈V| and Z^s = |r₁〉 〈r₁| − |r₂〉 〈r₂| denote the Pauli flip and Pauli phase flip on the spatial DoF {r₁, r₂} of one photon. , , , and . are single photon detectors. The feed-forward operations of Bob are only single photon operations.

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 resources¹²,¹³,¹⁴,²²,²³. 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 DoF²²,²³,⁵¹,⁵²,⁵³. 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 QND⁵³. If this efficient way⁵³ 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 method⁵⁰,⁵¹,⁵²,⁵³ are explored in our schemes to avoid the impractical XPM with a shift −θ²¹,²²,²³. 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 DoF¹¹,¹²,¹³,¹⁴,¹⁵,²²,²³,²⁴,²⁵,²⁶,²⁷,²⁸,²⁹, 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 systems¹¹,¹⁵,²²,²³. 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 obtained⁶⁷. 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-Banacloche⁶⁹ 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 small⁷². To address this problem, we take use of the double cross-phase modulation method⁴⁹,⁵¹,⁵²,⁵³ to avoid the impractical −θ. Combining with a photon-number-resolving (PNR) detector, a homodyne detector may be used to discriminate two coherent states⁵³,⁷³. 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 range⁷⁴, or at an operating wavelength of about 850 nm⁷⁵. New measurement scheme has been realized based on a displacement operation followed by a PNR⁷⁶. 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 sensors⁷⁷. 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 capabilities⁷⁸. Some methods and device configurations are also proposed to obtain PNR capability using superconducting nanowire detectors⁷⁹. 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 resource⁸⁰. Moreover, Weng et al. take use of quantum dot coupled resonant tunneling diodes to demonstrate a PNR⁸¹. 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 nonlinearity²¹,²²,²³ has a Hamiltonian in the form . Here, 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 , see the Fig. 1. The detailed evolution is defined as follows for the any initial system

where are different states of the photon A₂ with four spatial modes a₂, b₂, and , while {|ϕ₁〉, |ϕ₂〉, |ϕ₃〉, |ϕ₄〉} are corresponding states of the other system except the photon A₂. In detail, the photon A₂ from the modes and is firstly interacted with the coherent system in order. One can get a joint system

where and . Afterwards, the projection |n〉 〈n| is performed on the first qubus beam to get the proper output⁴⁹,⁵¹. If the measurement outcome is n = 0, the photonic state in the Eq. (27) collapses into

If the measurement outcome satisfies n ≠ 0, the photonic state in the Eq. (27) collapses into

Additional Information

How to cite this article: Luo, M.-X. et al. Quantum computation based on photonic systems with two degrees of freedom assisted by the weak cross-Kerr nonlinearity. Sci. Rep. 6, 29939; doi: 10.1038/srep29939 (2016).