Quantum Computation Based on Photons with Three Degrees of Freedom

Quantum systems are important resources for quantum computer. Different from previous encoding forms using quantum systems with one degree of freedom (DoF) or two DoFs, we investigate the possibility of photon systems encoding with three DoFs consisting of the polarization DoF and two spatial DoFs. By exploring the optical circular birefringence induced by an NV center in a diamond embedded in the photonic crystal cavity, we propose several hybrid controlled-NOT (hybrid CNOT) gates operating on the two-photon or one-photon system. These hybrid CNOT gates show that three DoFs may be encoded as independent qubits without auxiliary DoFs. Our result provides a useful way to reduce quantum simulation resources by exploring complex quantum systems for quantum applications requiring large qubit systems.


Results
To show the encoding independence of the polarization DoF and two spatial DoFs of each photon, it is necessary to prove that all quantum transformations in SU(2 n ) may be implemented on these DoFs. Based on the theory of the universal logic gates [12][13][14] , it is sufficient to consider the CNOT gate on any two DoFs of the photonic system. It means that fifteen CNOT gates should be performed on photonic systems with three DoFs, where nine CNOT gates are on the two-photon system (all combinations of three DoFs) and six CNOT gates are on the one photon system. By exploring optical selection rules of the NV center in the crystal cavity, these CNOT gates may be realized without altering DoFs and auxiliary DoFs during implementations. In this case, each photonic DoF can be encoded as an independent qubit in quantum applications.
Photon with three DoFs. Circularly polarized photon in the state α 1 |L〉 + α 2 |R〉 (left circularly polarized state |L〉 and right circularly polarized state |R〉 ) is created at a degenerate wavelength λ = 2λ p by each BBO crystal along two correlated directions belonging to the lateral surfaces of two SPDC cones, with full aperture angles θ I and θ E , respectively 64,65 , as shown in Fig. 1. The output state is dependent of these angles. I refers to the internal cone whereas E denotes the external cone, corresponding to the first and the second crystal, respectively. The dichotomy existing between the I cone and E cone is thus identified as an independent DoF, i.e., the corresponding mode emission as l(r) by referring to the left (right) side of each cone 64,65 . If the pump coherence length exceeds more than one order of magnitude the total crystal length, the coherence and indistinguishability between two crystal emissions may be guaranteed 64,65 . Two conical emissions are then transformed into two cylindrical ones by a positive lens with focal length f, located at a distance f from the intermediate point of the second crystal device. By selecting four pairs of correlated modes with an eight-hole screen, |l, I〉 and |r, I〉 for the first crystal and |l, E〉 and |r, E〉 for the second crystal emission, a general photonic state is prepared as the product of one polarization state and two longitudinal momentum states (or, equivalently, a ququart state) and is expressed as a 3-qubit state: where |α 1 | 2 + |α 2 | 2 = 1 and |β 1 | 2 + |β 2 | 2 + |β 3 | 2 + |β 4 | 2 = 1. Here, β j are dependent of aperture angles θ I and θ E , and focal length f, which are not goals in this paper 64,65 .  A diamond NV center coupled to an MTR with a WGM. Schematic NV center in a diamond embedded in a photonic crystal cavity is shown in Fig. 2. The negatively charged NV center is consisted of a substitutional nitrogen atom and an adjacent vacancy with six electrons. The Λ -type three-level system is realized using specific excited state |A 2 〉 = (|E − 〉 |m + 〉 + |E + 〉 |m − 〉 ) as an ancillary state 66,34 . Here, |E ± 〉 are orbital states with angular momentum projection along the NV axis. The ground state is an electronic spin triplet with a splitting of 2.88 GHz between the magnetic sublevels |0〉 (m s = 0) and |m ± 〉 (m s = ± 1) 34 . |A 2 〉 may decay into two ground states |m − 〉 and |m + 〉 by exciting the NV center with a polarized 2-ns p-pulse that is shorter than the emission timescale, and the reflection may be separated from fluorescence photons using detection timing 34 . The normal boundary condition κ = +ˆb b a out i n is used to derive the optical selection rule with the input field b in , output field b out and cavity field operator â. If spins stay in the ground states most of the time 67 , the optical reflection coefficient may be approximately defined in the follow (see Methods) where δω c and δω e are frequency detunings satisfying δω c = ω c − ω and δω e = ω e − ω. ω c , ω and ω e are the frequencies of the cavity mode, input photon pulse, and NV center, respectively. g is the coupling strength between the cavity and the NV center. κ, κ s and γ are the damping rate of the cavity, cavity side leakage mode, and spontaneous decay rate of the NV center, respectively. If define the cooperativity C = 2g 2 /(γκ), the photonic reflection probability 68 is determined by the cooperativity C and the cavity tuning as follow Thus the input pulse in the polarized state |L〉 gains a phase shift θ after reflecting from the hot cavity (g > 0) with the NV center |m − 〉 , or a phase shift θ 0 after reflecting from the cold cavity (g = 0) with the NV center |m + 〉 . The input pulse in the state |R〉 gains a phase shift θ 0 after reflecting from the cold cavity with the NV center |m − 〉 , or a phase shift θ after reflecting from the hot cavity with the NV center |m + 〉 . By choosing a proper frequency detuning δω e = 0 66 and the cooperativity C ≫ 1, the reflection coefficients may satisfy |r(ω)| ≈ 1 and |r 0 (ω)| ≈ 1 when the cavity side leakage κ s is negligible. By adjusting the frequencies ω and ω c such that δω c /κ → 0 and C ≫ 1, the phase shifts may be realized as θ = 0 and θ 0 = π. Hence, the following optical transition may be obtained as From this optical transition, an NV center requires a polarization-degenerate cavity mode, which is also suitable in H1 photonic crystals 69,70 and fiber-based cavities 71 . CNOT gate on the same DoF of the two-photon system. Schematic CNOT gate on the same DoF of the two-photon system is shown in Fig. 3. NV centers e i trapped in the photonic crystal NV i are initially prepared in the superposition states Figure 3(a) is used to complete the CNOT gate on the polarization DoFs of two photons, i.e., pp 1 2 In detail, the photon A 1 from each spatial mode (l 1 I 1 , l 1 E 1 , r 1 I 1 or r 1 E 1 ) evolves as CPBS → NV 1 → CPBS to complete the following controlled phase gate pa 1 1 on the polarization DoF and the NV center e 1 (see Appendix A of Supplementary Information for details). And then, after one Hadamard operation H a on the NV center e 1 in the NV 1 , the photon A 2 from each spatial mode evolves as H p → CPBS → NV 1 → CPBS → H p to complete the following hybrid CNOT gate . Thus the CNOT gate C pp (A 1 , A 2 ) has been realized on the photons A 1 and A 2 . Figure 3(b) is a schematic circuit to complete the CNOT gate on the spatial DoF {l, r}s of two photons, i.e., Here, the photon A 1 from each spatial mode r 1 I 1 or r 1 E 1 evolves as CPBS → NV 2 → (X → NV 2 → X) → CPBS to complete the following controlled phase gate on the spatial DoF {l, r} and the NV center e 2 in the state |+〉 (see Appendix B of Supplementary Information for details). Now, after a Hadamard gate H a performed on the NV center e 2 in the NV 2 , the followed circuit CB S → CPBS → NV 2 → (X → NV 2 → X) → CPBS → CBS for each mode pair (l 2 I 2 , r 2 I 2 ) or (l 2 E 2 , r 2 E 2 ) is used to complete the hybrid CNOT gate on the NV center e 2 and the spatial DoF {l, r} of the photon A 2 (see Supplementary Information for details), i.e., may be realized by disentangling the NV center e 2 using the measurement under the basis {|± 〉 }, where Z p is performed on the photon A 1 from each spatial mode r 1 I 1 and r 1 E 1 for the measurement outcome |−〉 e 2 . A similar CNOT gate holds for the spatial DoF {I, E}s of two photons using an NV center e 3 trapped in the optical cavity NV 3 (see Appendix C of Supplementary Information for details).
Hybrid CNOT gate on the different DoFs of the two-photon system. Figure 4(a) is a schematic circuit to implement the CNOT gate on the polarization DoF of the photon A 1 and the spatial DoF {l, r} of the photon A 2 , i.e., In fact, similar to the Fig. 3(a), the first controlled phase flip CZ pa (A 1 , e 1 ) in the equation (7) is used to change the photon A 1 and the NV center e 1 from φ | 〉 |+〉 . And then, after one Hadamard operation H a performed on the NV center e 1 in the NV 1 , the followed circuit CBS → CPBS → NV 1 → (X → NV 1 → X) → CPBS → CBS for each spatial mode pair (l 2 I 2 , r 2 I 2 ) or (l 2 E 2 , r 2 E 2 ) is used to complete the CNOT gate C as (e 2 , A 1 ) in the equation (11) on the NV center e 1 and the spatial mode {l, r} of the photon A 2 (similar to the Fig. 3(b)). After disentangling the NV center e 1 using the measurement under the basis {|± 〉 }, the hybrid CNOT gate C A A ( , ) is realized on the photons A 1 and A 2 , where Z p is performed on the photon A 1 from each spatial mode r 1 I 1 and r 1 E 1 for the measurement outcome − e 1 , see Appendix D of Supplementary Information for details. Similarly, after the controlled-phase flip CZ pa (A 1 , ′ e 1 ) on the photon A 1 and the NV center e 1′ in the state |+ 〉 , a schematic circuit is applied to the photon A 2 from two spatial mode pairs (l 2 I 2 , l 2 E 2 ) and (r 2 I 2 , r 2 E 2 ) to complete the CNOT gate on the NV center e 1′ and the spatial DoF {I, E} of the photon A 2 (see Appendix E of Supplementary  Information for details). The hybrid CNOT gate is implemented on the polarization DoF of the photon A 1 and the spatial DoF {I, E} of the photon A 2 after disentangling the NV center e 1′ . Figure 4(b) is used to implement the CNOT gate on the spatial DoF {l, r} of the photon A 1 and the polarization DoF of the photon A 2 , i.e., In fact, similar to the evolutions as shown in the Fig. 3(b), the controlled phase gate CZ A e ( , ) . And then, after one Hadamard operation H a on the NV center e 2 in the NV 2 , the followed circuit for the photon A 2 from each spatial mode is used to complete the CNOT gate C ap (e 2 , A 2 ) on the NV center e 2 and the polarization DoF of the photon A 2 (see the Fig. 3(a)). The final joint state is is realized, where − I p will be performed on the photon A 1 from each spatial mode r 1 I 1 and r 1 E 1 for the measurement outcome |−〉 e 2 , see Appendix F of Supplementary Information for details. Moreover, if the second part of the present circuit above is applied to the photon A 1 from two spatial modes l 1 I 1 and l 1 E 1 , the CNOT gate is implemented on the spatial DoF {I, E} of the photon A 1 and the polarization DoF of the photon A 2 , see Appendix G of Supplementary Information for details. Figure 4(c) is used to implement the CNOT gate on the spatial DoF {l, r} of the photon A 1 and the spatial DoF {I, E} of the photon A 2 , i.e, . And then, after one Hadamard operation H a on the NV center e 3 , the followed circuit for the photon A 2 from each spatial mode is used to realized the CNOT gate C e A ( , ) Hybrid CNOT gate on different DoFs of one photon. Figure 5 is a schematic circuit to implement the CNOT gate C ps l r , in the equation (13) on the polarization DoF and the spatial DoF {l, r} of the photon A 1 . In detail, similar to the Fig. 3(a), the controlled-phase flip CZ pa in the equation (7) is used to change the photon A 1 and the NV center e 1 from φ + . And then, after one Hadamard operation H a performed on the NV center e 1 , the followed CNOT gate C asl in the equation (11) is performed on the NV center e 1 and the spatial DoF {l, r} of the photon A 1 (similar to the Fig. 3(b)). After disentangling the NV center e 1 using the measurement under the basis {|± 〉 }, the hybrid CNOT gate C ps l r , is realized on the photon A 1 , where Z p will be performed for the photon A 1 from each mode for the measurement outcome − e 1 , see Appendix J of Supplementary Information for details. Moreover, if the CNOT gate C e A ( , ) as 1 1 IE is performed on the NV center e 1 and the photon A 1 after CZ pa (A 1 , e 1 ), the hybrid CNOT gate C ps I E , in the equation (14) is realized on the photon A 1 after properly disentangling the auxiliary NV center, see Appendix K of Supplementary Information for details.
For the hybrid CNOT gate on the spatial DoF {l, r} and the spatial DoF {I, E} of the photon A 1 , the photon A 1 from the spatial modes r 1 I 1 and r 1 E 1 passes through CBS, − I, CBS, sequentially. The photon A 1 evolves as follows ) / 2 6 6 7 , and α α α ′ = ′ − ′ ( ) / 2 7 6 7 . Similar circuit may be used to realize the hybrid CNOT gate on the spatial DoF {I, E} and the spatial DoF {l, r} of the photon A 1 , see Appendix K of Supplementary Information for details. Moreover, the CNOT gates on the spatial mode DoF and the polarization DoF of the photon A 1 are easily realized by two flip waveplates on two spatial modes r 1 I 1 and r 1 E 1 , or l 1 E 1 and r 1 E 1 , respectively.

Discussions
In ideal conditions, one may neglect the cavity side leakage, and the reflection coefficients satisfy |r 0 (ω)| ≈ 1 and |r(ω)| ≈ 1. The corresponding fidelities of the present CNOT gates are nearly 100%. In experiment, the general fidelity is defined by where |Φ 〉 is the ideal final state without side leakage while ρ f is the final state under a real situation with side leakage. In the resonant condition δω e = 0, if the cavity side leakage is considered, the optical selection rule for the NV-cavity system given by the equation (5) becomes Due to the exchangeability of two spatial DoFs of one photon with respect to random initial photons, the fidelities and efficiencies are evaluated for four CNOT gates: CNOT gate on two polarization DoFs, CNOT gate on two spatial DoFs, CNOT gate on the polarization DoF of one photon and the spatial DoF of the other photon, and CNOT gate on the polarization and spatial DoFs of one photon system, as shown in Figs 6 and 7, respectively. Generally, large cooperativity C and low relative detuning δω c /κ are required for high fidelities and efficiencies. For the diamond NV centers, the photoluminescence is partially unpolarized, and the emission with ZPL is only 4% of the total emission. ZPL with zero phonon line is only 4% of γ = 2 × 15 MHz 31 . For the diamond NV center in a MTR with WGM mode system, |r(ω)| ≈ 0.95 when C ≥ 18 34 with small detuning δω c /κ ≈ 0; |r(ω)| ≈ 1 when C ≥ 50 with small detuning δω c /κ ≈ 0 for κ ≈ 1 GHz or κ ≈ 10 GHz. For our CNOT gates, if C ≥ 18 and δω c /κ ≈ 0.1, their fidelities are greater than 82.6% and their efficiencies are greater than 75.4%. If C ≥ 50 and δω c /κ ≈ 0.1, their fidelities are greater than 98.4% and their efficiencies are greater than 94.7%.
In conclusion, we have investigated the possibility of quantum simulations using photon systems with three DoFs. We have constructed fifteen schematic CNOT gates operating on the spatial and polarization DoFs of the two-photon system or one-photon system. Different from previous CNOT gate on the same DoF of the two-photon system 14,15,56,57 , our schemes are based on different DoFs of two photons or one photon. Compared with hybrid implementations on the photon and stationary electron spins in quantum dots [58][59][60][61] , the present CNOT circuits are ultimately realized on the photon system, and the electron spins in NV center are auxiliary resources to build the correlation between photons. The present schemes have shown that two different spatial DoFs may be viewed as independent qubits simultaneously, which has beyond previous independence of the polarization and spatial DoFs 62,63 . Although different DoFs may be easily exchanged in terms of encoding, the schematic operations are inconvenient for photon systems with two different spatial DoFs. The main reason is that the hybrid CNOT gates are not realized in one-shot manners. Thus, it is difficult to exchange these DoFs during applications, where different DoFs may be used as different encoding types such as the quantum Shor algorithm or the quantum search algorithm. Hence, our results are distinct from all previous quantum logic gates on different photons 14,15,56,57 . Our theoretical schemes have shown that three DoFs of photon systems may be independent in quantum information processing. Two thirds of the quantum resources may be saved in quantum simulations. With the recent experiments of the NV-cavity system [33][34][35] , our schemes are expected to be applicable for the entanglement distribution or large-scale quantum computation.

Methods
A diamond NV center coupled to an MTR with a WGM. The master equation of the whole system may be expressed by a Lindblad form as follows mode by applying the rotating wave approximation and dropping the energy nonconserving terms. σ − and σ + are the Pauli raising and lowering operators, respectively. g is the coupling strength between the cavity and X − .
1 2 s accounts for the damping of the input photon puls e. ρ σ ρσ σ σ ρ ρσ σ  accounts for spontaneous emission of the dipole. The input-output optical relation of the NV center system may be calculated from the Heisenberg equations 67 in terms of the cavity field operator â, input pulse field b and dipole operator σ − ,

Measurement of the NV center e in cavity.
To measure the NV center e of an entangled system α|m − 〉 e |Ω 1 〉 + β|m + 〉 e |Ω 2 〉 , an auxiliary photon c in the state + R L ( ) 1 2 may be used as follows. Let the photon c pass through one CPBS to split the circular polarizations |R〉 and |L〉 , and the right-circular polarization |R〉 interact with the cavity system, and its output combine with |L〉 of the photon c using the other CPBS. Thus, the joint system evolves