Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-Q cavities

Fang-Fang Du; Zhen-Rong Shi

doi:10.1364/OE.27.017493

1. Introduction

Quantum information processing (QIP) is powerful with fascinating capability of computing, communication and sensing using quantum mechanical principles [1]. Universal two-qubit operations are the key ingredients in quantum computing. In 1995, Barenco et al. proved that any natural building block for quantum computing can be achieved by combining controlled-NOT (CNOT) gates and single-qubit rotations [2], therefore, precise control operations on well-defined quantum systems become extremely important [3]. Deterministic CNOT gate has been implemented both in theory and in experiment by using various physical systems, such as nuclear magnetic resonance [4,5], quantum dots [6–8], diamond nitrogen-vacancy (NV) centers [9–11], superconducting circuits [12, 13], and atoms [14–17]. In 1998, Cerf et al. presented a CNOT gate with the spatial-mode of the photon as the control qubit and its polarization as the target one [18]. In 2000, Knill et al. put forward a linear optical quantum computing scheme with respect to feedback from photon detectors, which was robust against errors from photon loss and invalid detector [19]. To constitute two-qubit optical gate in a deterministic way, one can resort to Kerr nonlinearities. For instance, Duan and Kimble proposed a scheme for constructing a controlled phase-flip (CPF) gate between an atom trapped in a cavity and a single photon [14]. The strong-coupling interaction between the atom and the cavity can provide the giant Kerr nonlinearity. Quantum logic gates between flying photonic qubits and stationary qubits hold a great promise for quantum communication and computing, especially for quantum repeaters, distributed quantum computing, and blind quantum computing.

In a network of quantum computation [19–21], photons are often regarded as the flying qubits due to flexible manipulation and robustness against the environment noise. Increasing the information-carrying capacity by utilizing multiple degrees of freedoms (DOFs) of a single photon provides us an effective way to improve the realization of the long-distance quantum computing [22–25]. Recently, the concept for hyperparallel photonic quantum computing was proposed with universal quantum operations performed on two or more DOFs of photon systems [26–29]. For example, exploiting the giant optical circular birefringence induced by the double-sided quantum-dot-cavity system, Wang et al. constructed a deterministic hybrid hyper-CNOT gate, in which the spatial-mode and polarization states of a photon acted as the two control qubits, whereas two stationary electron spins in quantum dots confined inside the optical microcavities served as the two target qubits [27]. With hyperparallel photonic quantum gates, hyperentanglement generation [30–32], hyperentanglement purification [33–35], and hyperentanglement concentration [36, 37] can be accomplished in the relatively simple way, which will improve the channel capacity of quantum communication and simplify the process of operation.

Cavity quantum electrodynamics (QED) is a promising physical platform for constructing universal quantum logic gates, which can enhance the interaction between a photon and an atom trapped in a cavity [15–17]. In the past decades, many theoretical and experimental works have been focused on to realize large-scale QIP [14–16,38–47]. Most of them relied on the efficient single-photon input-output process in the high-Q cavity and the strong-coupling interaction between the confined atoms and the high-Q cavity field. However, the high-Q cavity in the strong coupled cases, sensitive to the ratio g/κ between the coupling strength g and cavity decay rate κ, does not facilitate to carry out the input-output process of the photons in experiment. Therefore, it is significant to realize QIP task in the weak-coupling region of the low-Q cavity. In 1995, Turchette et al. completed the measurement of conditional phase shifts for quantum logic in the intermediate atom-cavity coupling regime with the low-Q cavity [48]. In [49], Dayan et al. demonstrated a robust and efficient mechanism for the regulated transport of photons under the same condition [48]. In 2009, An et al. proposed a special scheme with higher fidelity by using low-Q cavities to entangle distant atoms by a single photon, generate entangled photons, and transfer quantum state to a distant qubit [50]. Besides, there are some new progress in fabrication of microcavities [51,52], and control of microcavities [53,54]. Microcavities also play an important role in sensing [55–57].

Motivated by recent works on atom-based quantum logic gates relying on the low-Q cavities, we investigate the possibility to construct hybrid hyper-CNOT gates on the two-photon system in both the polarization and spatial-mode DOFs, resorting to the cavity-assisted interaction, the single-photon input-output process, and the readout on the auxiliary atom as a result of cavity QED. Our proposals have some advantages. Firstly, it is relatively easier to perform quantum logical gates on qubits residing in different DOFs of the same photon, which also reduces the resources consumed based on the hyperparallel feature. Secondly, simultaneous counterpropagation of two photons economize extremely the operation time in the whole process of our schemes. Besides, our gates work in the weak-coupling region of the low-Q cavities, which further lowers the difficulty of the experimental operation. Moreover, the relatively long coherence time allows multi-time operations between the photon and the cavity-atom system. Our calculations show that the average fidelities and efficiencies of our hybrid hyper-CNOT gates are higher with current experimental technology. These features maybe make our proposals have potential applications for high-capacity quantum computation in the future.

2. An atomic-cavity interaction system

Here we consider a system that an atom is trapped in a single-sided optical cavity. The atom is assumed to be a standard three-level ∧-type system as shown in Fig. 1. The degenerate ground states of the atom, i.e., |0〉 and |1〉, are considered to be the qubit states and the excited level |e 〉 to be the ancillary state. The optically allowed transitions |1〉 ↔ |e 〉 (6S_1/2, F = 4, m = 4 → 6P_3/2, F′ = 5, m′ = 5 of cesium atom) can only be excited by the single left (L) polarized photon under the selection rules, while it decouples the transition |0〉 ↔ |e〉 due to large detuning. The top of the cavity is perfectly reflective and the bottom is partially reflective [50]. Under the Jaynes-Commings model, the Hamiltonian of the whole system composed of a single cavity mode (L-polarized) and an atom can be given by

H = \frac{ℏ ω_{0}}{2} σ_{z} + ℏ ω_{c} a^{†} a + i g ℏ (a σ_{+} - a^{†} σ_{-}) .

Here a and a^† are the annihilation and creation operators of the L-polarized cavity mode with the frequency ω_c, respectively. σ_z, σ₊, and σ₋ are the inversion, raising, and lowering operators of the atom, respectively. ω₀ is the frequency difference between the ground level |1〉 and the excited level |e〉 of the atom. g is the atom-cavity coupling strength, which is affected by the trapping position of the atom. The reflection coefficient of a single-photon pulse with the frequency ω_p injected into the optical cavity can be obtained by solving the Heisenberg-Langevin equations of motion for the internal cavity field and the atomic operator in the interaction picture [58]

{\begin{array}{l} \frac{d \hat{a} (t)}{d t} & = & - [i Δ ω_{c} + \frac{κ}{2}] \hat{a} (t) - g {\hat{σ}}_{-} (t) - \sqrt{κ} {\hat{a}}_{in} (t), \\ \frac{d {\hat{σ}}_{-} (t)}{d t} & = & - [i Δ ω_{0} + \frac{γ}{2}] {\hat{σ}}_{-} (t) - g {\hat{σ}}_{z} (t) \hat{a} (t) + \sqrt{γ} σ_{z} (t) {\hat{b}}_{in} (t), \\ {\hat{a}}_{o u t} (t) & = & {\hat{a}}_{in} (t) + \sqrt{κ} \hat{a} (t), \end{array}

where Δω_c = ω_c − ω_p and Δω₀ = ω₀ − ω_p. κ and γ are the cavity damping rate and the atomic decay rate, respectively. â_out(t) is the output operator. Here the one-dimensional field operator â_in(t) is the cavity input operator which satisfies the commutation relation

[{\hat{a}}_{in} (t), {\hat{a}}_{in}^{†} (t^{'}) = δ (t - t^{'})

. The three-level atom can feel the vacuum input field b̂_in(t) with the commutation relation

[{\hat{b}}_{in} (t), {\hat{b}}_{in}^{†} (t^{'})] = δ (t - t^{'})

.

Fig. 1 Schematic setup to implement the interaction between the atom and the single-photon pulse. The top of the low-Q cavity is perfectly reflective and the bottom is partially reflective. CPBS_i (i = 1, 2) is a circularly polarizing beam splitter which transmits the photon in the right-circular polarization |R 〉 and reflects the photon in the left-circular polarization |L 〉, respectively. With CPBS₁, only the L-polarized component of the single-photon pulse is reflected by the cavity. DL is the time-delay device for making the circularly polarized photon pulses R and L reach the CPBS₂ simultaneously. The states |0〉 and |1〉 indicate hyperfine states of the atom in the ground states while |e 〉 is an excited state.

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In fact, since the atom stays in the ground state at most time, a weak excitation by the single-photon pulse (keeping 〈σ_z〉 = −1) throughout our operation, one can get the input-output optical property of the cavity field [50]

r (ω_{p}) = 1 - \frac{κ [i Δ ω_{0} + \frac{γ}{2}]}{[i Δ ω_{0} + \frac{γ}{2}] [i Δ ω_{c} + \frac{κ}{2}] + g^{2}} .

Here

r (ω_{p}) \equiv \frac{a_{out} (t)}{a_{in} (t)}

is the reflection coefficient for the atom-cavity system. By taking the coupling strength g = 0, one get the amplitude of the output pulse r₀ (ω_p) for an uncoupled cavity (or bare cavity) where the atom does not couple to the input filed,

r_{0} (ω_{p}) = 1 - \frac{κ}{i Δ ω_{c} + \frac{κ}{2}} .

Considering the parameters of the atom-cavity system satisfy the resonant condition ω₀ = ω_c = ω_p, the reflection coefficient can be expressed as

r (ω_{p}) = \frac{- 1 + {(2 g / \sqrt{κ γ})}^{2}}{1 + {(2 g / \sqrt{κ γ})}^{2}}, r_{0} (ω_{p}) = - 1 .

If the photon feels a coupled cavity (g ≠ 0), it will get a phase e^iφ after reflection. Otherwise, if the photon feels a bare cavity, it will obtain a phase shift e^iφ₀. Moreover, when the limitation of a low cavity satisfies κ ≫ g²/κ ≫ γ in the atom-cavity intermediate coupling region, phase shifts ϕ = 0 and ϕ₀ = π can be achieved. In detail, when the atom is initially prepared in the state |1〉 with the transition |1〉 ↔ |e〉, the L-polarized photon feels a hot cavity and the corresponding output state of the L-polarized photon can be written as

| L 〉 | 1 〉 \to r (ω_{p}) | L 〉 | 1 〉 = e^{i φ} | r (ω_{p}) | | L 〉 | 1 〉 \frac{ω_{0} = ω_{c} = ω_{p}}{κ ≫ g^{2} / κ ≫ γ} | L 〉 | 1 〉 .

When the atom is initially prepared in the ground state |0〉, the left-circularly polarized single-photon |L〉 will only sense a bare cavity. As a result, the corresponding output state can be written as

| L 〉 | 0 〉 \to r_{0} (ω_{p}) | L 〉 | 0 〉 = e^{i φ_{0}} | r_{0} (ω_{p}) | | L 〉 | 0 〉 \underset{\to}{ω_{c} = ω_{p}} - | L 〉 | 0 〉 .

As for the R-polarized pulse which remains unchanged due to having no access to the atom-cavity system, we suppose that the reflection coefficient equals 1. Therefore, the evolution rule of the photon in the polarized state |L 〉 (or |R 〉) and different atomic states can be summarized as |L 〉|1〉 → |L 〉|1〉, |L 〉|0〉 → −|L 〉|0〉, |R 〉|1〉 → |R 〉|1〉, and |R 〉|0〉 → |R 〉|0〉. Here the conditional phase shift π allows the construction of a universal quantum gate that can be transformed into any two-qubit gate using rotations of the individual qubits, which are implemented with wave plates for the photon and with Raman transitions for the atom.

3. Three types of hybrid hyper-CNOT gates

3.1. Hybrid hyper-CNOT gate I

The principle of our photonic four-qubit hybrid hyper-CNOT gate I is shown in Fig. 2, in which the quantum circuit is constructed with the reflection rule of L-polarized photon interacting with an atom-cavity system. Here Y₁ and Y₂ represent two types of interactions. Y₁ shows the interaction between only the L polarization photon in spatial mode |k₁〉 (|k₂〉, k = a, b) and the atom trapped in the cavity. For Y₂, both the L and R components of the photon interact with the atom trapped in the cavity in sequence, in which before and after the interaction, the R component will be flipped with the polarization bit-flip operation X $(σ_{X_{K}}^{P} = {| L 〉}_{K} 〈 R | + {| R 〉}_{K} 〈 L |, K = A, B)$ .

Suppose that the auxiliary atom j in the cavity j is initially prepared in the state ${| φ 〉}_{j} = \frac{1}{\sqrt{2}} {(| 0 〉 + | 1 〉)}_{j}$ (j = 1, 2) and the initial states of the photons A and B are

\begin{array}{l} {| ϕ 〉}_{A} = {| ϕ 〉}_{A}^{P} \otimes {| ϕ 〉}_{A}^{S} = {(sin α_{1} | R 〉 + cos α_{1} | L 〉)}_{A} \otimes (sin α_{2} | a_{1} 〉 + cos α_{2} | a_{2} 〉), \\ {| ϕ 〉}_{B} = {| ϕ 〉}_{B}^{P} \otimes {| ϕ 〉}_{B}^{S} = {(sin β_{1} | R 〉 + cos β_{1} | L 〉)}_{B} \otimes (sin β_{2} | b_{1} 〉 + cos β_{2} | b_{2} 〉) . \end{array}

The quantum circuit is described in detail as follows. First, the photon A in spatial mode |a₁〉 or |a₂〉 propagates from left to right (passing through Y₁-atom 1, BS₁, Y₂-atom 2, and BS₂ sequentially), while the photon B in spatial mode |b₁〉 or |b₂〉 propagates from right to left (passing through Y₁-atom 2, BS₃, Y₂-atom 1, and BS₄ sequentially). Due to the interaction between Y₁-atom 1 and the photon A is the same as the interaction between Y₁-atom 2 and the photon B, we use the former as an example to discuss in detail. After the Y₁-atom 1 interacts with the photon A, the joint state |φ〉₁ ⊗ |φ〉_A of the system composed of the atom 1 and the photon A is evolved into

\begin{array}{l} {| Ψ 〉}_{1 A} = \frac{1}{\sqrt{2}} [{| 1 〉}_{1} {(sin α_{1} | R 〉 + cos α_{1} | L 〉)}_{A} + {| 0 〉}_{1} {(sin α_{1} | R 〉 - cos α_{1} | L 〉)}_{A}] \\ \otimes (sin α_{2} | a_{1} 〉 + cos α_{2} | a_{2} 〉) . \end{array}

It is obvious that the interaction Y₁-atom j (j = 1, 2) with the photon A (B) can be viewed as the polarization-controlled-Z gate (P-CZ gate). After performing the Hadamard operation on the atom 1 applying a π/2-pulse (that is,

| 1 〉 \to \frac{1}{\sqrt{2}} (| 1 〉 + | 0 〉)

and

| 0 〉 \to \frac{1}{\sqrt{2}} (| 1 〉 - | 0 〉)

in the basis {|1〉, |0〉}), and the Hadamard operation on the spatial-mode DOF of the photon A (namely,

| a_{1} 〉 \to \frac{1}{\sqrt{2}} (| a_{1} 〉 + | a_{2} 〉)

and

| a_{2} 〉 \to \frac{1}{\sqrt{2}} (| a_{1} 〉 - | a_{2} 〉)

) using BS₁, the state |Ψ〉_1A in Eq. (9) becomes

{| Ψ 〉}_{1 A}^{'} = (sin α_{1} {| R 〉}_{A} {| 1 〉}_{1} + cos α_{1} {| L 〉}_{A} {| 0 〉}_{1}) \otimes [sin (α_{2} + \frac{π}{4}) | a_{1} 〉 + sin (α_{2} - \frac{π}{4}) | a_{2} 〉],

where

sin (α_{2} + \frac{π}{4}) = \frac{1}{\sqrt{2}} (sin α_{2} + cos α_{2})

and

sin (α_{2} + \frac{π}{4}) = \frac{1}{\sqrt{2}} (sin α_{2} - cos α_{2})

. Similarly, after the photon B interacts with Y₁-atom 2, and immediately the Hadamard operations are performed on the atom 2 and the spatial-mode DOF of the photon B, the complex state |ϕ〉_B ⊗ |ϕ〉₂ becomes

{| Ψ 〉}_{2 B}^{'} = (sin β_{1} {| R 〉}_{B} {| 1 〉}_{2} + cos β_{1} {| L 〉}_{B} {| 0 〉}_{2}) \otimes [sin (β_{2} + \frac{π}{4}) | b_{1} 〉 + sin (β_{2} - \frac{π}{4}) | b_{2} 〉] .

Subsequently, the photon A interacts with the Y₂-atom 2, meanwhile the photon B interacts with the Y₂-atom 1. The complex state |Ψ′〉_1A ⊗ |Ψ′〉_2B of the system composed of the two photons (A and B) and two atoms (1 and 2) is evolved into

\begin{array}{l} {| Ψ 〉}_{12 A B} = & {sin α_{1} {| R 〉}_{A} {| 1 〉}_{1} [sin (β_{2} + \frac{π}{4}) | b_{1} 〉 + sin (β_{2} - \frac{π}{4}) | b_{2} 〉] \\ + cos α_{1} {| L 〉}_{A} {| 0 〉}_{1} [sin (β_{2} + \frac{π}{4}) | b_{1} 〉 - sin (β_{2} - \frac{π}{4}) | b_{2} 〉]} \\ \otimes {sin β_{1} {| R 〉}_{B} {| 1 〉}_{2} [sin (β_{1} + \frac{π}{4}) | a_{1} 〉 + sin (β_{1} - \frac{π}{4}) | a_{2} 〉] \\ + cos β_{1} {| L 〉}_{B} {| 0 〉}_{2} [sin (β_{1} + \frac{π}{4}) | a_{1} 〉 - sin (β_{1} - \frac{π}{4}) | a_{2} 〉]} . \end{array}

It is clear that the interaction between Y₂-atom 2 and the photon A (Y₂-atom 1 and the photon B) can be viewed as the spatial-mode controlled-Z gate (S-CZ gate). Here DL makes the photon A (B) in spatial mode a₁ (b₁) and a₂ (b₂) reach the second BS₂ (BS₄) simultaneously. That is, fiber loops to storage the photon for enough long time needed by the interaction between the single photon and the atom. After performing the Hadamard operations on the atom j in the cavity j(j = 1, 2) and the Hadamard operations on the spatial-mode DOF of the photon A (B) using BS₂ (BS₄), the state |Ψ〉_12AB becomes

\begin{array}{l} {| Ψ^{'} 〉}_{12 A B} = & \frac{1}{2} {[sin α_{1} {| R 〉}_{A} {| ϕ 〉}_{B}^{S} + cos α_{1} {| L 〉}_{A} σ_{X_{B}}^{S} {| ϕ 〉}_{B}^{S}] {| 1 〉}_{1} \\ + [sin α_{1} {| R 〉}_{A} {| ϕ 〉}_{B}^{S} - cos α_{1} {| L 〉}_{A} σ_{X_{B}}^{S} {| ϕ 〉}_{B}^{S}] {| 0 〉}_{1}} \\ \otimes {[sin β_{1} {| R 〉}_{B} {| ϕ 〉}_{A}^{S} + cos β_{1} {| L 〉}_{B} σ_{X_{A}}^{S} {| ϕ 〉}_{A}^{S}] {| 1 〉}_{2} \\ + [sin β_{1} {| R 〉}_{B} {| ϕ 〉}_{A}^{S} - cos β_{1} {| L 〉}_{B} σ_{X_{A}}^{S} {| ϕ 〉}_{A}^{S}] {| 0 〉}_{2}} . \end{array}

Here,

σ_{X_{A}}^{S} = | a_{1} 〉 〈 a_{2} | + | a_{2} 〉 〈 a_{1} |

and

σ_{X_{B}}^{S} = | b_{1} 〉 〈 b_{2} | + | b_{2} 〉 〈 b_{1} |

represent the spatial-mode bit-flip operations.

Fig. 2 Quantum circuit for implementing the hybrid hyper-CNOT gate I assisted by an input-output process of low-Q cavities. Here Y₁ and Y₂ represent two types of interactions. Y₁ shows the interaction between the L-polarized photon in spatial-mode |k₁〉 (|k₂〉, k = a, b) and the atom trapped in the cavity. For Y₂, both the L and R components of the photon interact with the atom of the cavity in sequence, in which before and after the interaction, the R component will be flipped with the polarization bit-flip operation X $(σ_{X_{K}}^{P} = {| L 〉}_{K} 〈 R | + {| R 〉}_{K} 〈 L |, K = A, B)$ . The 50:50 beam splitters (BS_m, m = 1, 2, 3, 4) are used to perform the Hadamard operation on the spatial-mode DOF of the photon A (or B). DL makes the photon A (B) in spatial mode a₁ (b₁) and a₂ (b₂) simultaneously reach the second BS₂ (BS₄). The black (blue) arrows represent propagating directions of the photon A (B), respectively.

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Finally, after measuring the auxiliary atoms 1 and 2 under the orthogonal basis {|1〉, |0〉}, if two auxiliary atoms are both in the basis |1〉, the joint system of the two photons A and B collapses into

\begin{array}{l} {| Ψ 〉}_{1} = [sin α_{1} {| R 〉}_{A} {| ϕ 〉}_{B}^{S} + cos α_{1} {| L 〉}_{A} σ_{X_{B}}^{S} {| ϕ 〉}_{B}^{S}] \\ \otimes [sin β_{1} {| R 〉}_{B} {| ϕ 〉}_{A}^{S} + cos β_{1} {| L 〉}_{B} σ_{X_{A}}^{S} {| ϕ 〉}_{A}^{S}] . \end{array}

By performing the polarization phase-flip operation

σ_{Z_{K}}^{P} = | R 〉 〈 R | - | L 〉 〈 L | (K = A, B)

on the polarization DOF of the photon A (B) when the outcome |0〉 of the auxiliary atom 1 (atom 2), the hybrid hyper-CNOT gate I is completed.

In a word, the quantum circuit shown in Fig. 2 can be used to accomplish the deterministic hybrid hyper-CNOT gate I on the two photons A and B. In detail, the polarization state of the photon A is the control qubit when the spatial-mode state of the photon B is the target qubit, meanwhile, the polarization state of the photon B is the control qubit when the spatial-mode state of the photon A is the target qubit. That is, the polarization states of the photons A and B simultaneously control the spatial-mode states of the photons B and A, respectively.

3.2. Hybrid hyper-CNOT gate II

The quantum circuit of the hybrid hyper-CNOT gate II in Fig. 3(a) is similar to the one of the hybrid hyper-CNOT gate I in Fig. 2 in many ways. In Fig. 3(a), the Hadamard operation is performed on the polarization DOF rather than the spatial-mode DOF of the photons A and B by BS, which is substituted by H, that is, $| R 〉 \to \frac{1}{\sqrt{2}} (| R 〉 + | L 〉)$ , and $| L 〉 \to \frac{1}{\sqrt{2}} (| R 〉 - | L 〉)$ . Besides, the interaction between Y₂-atom 1 (Y₂-atom 2) and the photon A (B) and the interaction between Y₁-atom 2 (Y₁-atom 1) and the photon A (B) occur in turn, which leads to realizing S-CZ gate and the P-CZ gate sequentially.

Fig. 3 (a) and (b) represent quantum circuit for implementing hybrid hyper-CNOT gate II and III, respectively. H_n (n = 1, 2, ...) is used to perform a Hadamard operation on the polarization DOF of a photon.

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Suppose that the initial states of the photons A and B are shown in Eq. (7). Moreover, |φ〉_j is the initial state of the auxiliary atom j in the cavity j (j = 1, 2). In Fig. 3(a), the photon A in two spatial modes |a₁〉 and |a₂〉 passes through Y₂-atom 1, H₁, Y₁-atom 2, and H₂ sequentially, while the photon B in the two spatial modes |b₁〉 and |b₂〉 passes through Y₂-atom 2, H₃, Y₁-atom 1, and H₄ sequentially. After the interaction Y_i (i = 1, 2)-atom j (j = 1, 2) with the photon A (B), one can perform the Hadamard operation on the atom j in the cavityj (j = 1, 2) every time. The DL guarantees the photon A (B) in its own two spatial modes |a₁〉 (|b₁〉) and |a₂〉 (|b₂〉) to simultaneously reach H₂ (H₄).

Similar to the hybrid hyper-CNOT gate I, the hybrid hyper-CNOT gate II can be achieved by measuring the states of auxiliary atoms 1 and 2 in the orthogonal basis {|1〉, |0〉} and making some feed-forward operations. In detail, if both the auxiliary atoms 1 and 2 are in the state |0〉, the spatial-mode phase-flip operation $σ_{Z_{K}}^{S} = | k_{1} 〉〈 k_{1} | - | k_{2} 〉〈 k_{2} | (k = a, b; K = A, B)$ on the spatial-mode DOF of the photons A and B are performed, respectively. Otherwise, if two auxiliary atoms are both in the basis |1〉, the result of the hybrid hyper-CNOT gate II is obtained as

\begin{array}{l} {| Ψ 〉}_{II} = [sin α_{2} | α_{1} 〉 {| ϕ 〉}_{B}^{P} + cos α_{2} | a_{2} 〉 σ_{X_{B}}^{P} {| ϕ 〉}_{B}^{P}] \\ \otimes [sin β_{2} | b_{1} 〉 {| ϕ 〉}_{A}^{P} + cos β_{2} | b_{2} 〉 σ_{X_{A}}^{P} {| ϕ 〉}_{A}^{P}] . \end{array}

Here,

σ_{X_{K}}^{P} = {| L 〉}_{K} 〈 R | + {| R 〉}_{K} 〈 L | (K = A, B)

represents the polarization bit-flip operation.

In a word, the spatial-mode states of the photons A and B simultaneously control the polarization states of the photons B and A, respectively. It is self-evident that the control qubits swap with the target qubits in both types I and II.

3.3. Hybrid hyper-CNOT gate III and hybrid hyper-CNOT^N gate

The hybrid hyper-CNOT gate III in Fig. 3(b) is used to complete the task in which the polarization state and the spatial-mode state of the photon A simultaneously control the spatial-mode state and the polarization state of the photon B, respectively. The initial states of the two photons A and B and two auxiliary atoms j in the cavity j(j = 1, 2) are the same as the one of the hybrid hyper-CNOT gate I and II. In the first step, let the photon A get through Y₁-atom 1 and Y₂-atom 2 sequentially. After the Y₁-atom 1 and Y₂-atom 2 interact with the photon A, the atoms 1 and 2 are performed the Hadamard operation. Subsequently, let the photon B get through Y₂-atom 2, BS₅, H₅, Y₁-atom 1, Y₂, H₅, and BS₆ sequentially. Then one continues to perform the Hadamard operation on the atoms 1 and 2 again after the interactions Y₂-atom 2 and Y₁-atom 1 with the photon B. Eventually, one measures the states of auxiliary atoms 1 and 2 in the orthogonal basis {|1〉, |0〉}, and makes the feed-forward operations, similar to the discussion of the hybrid hyper-CNOT gate I and II. If the auxiliary atom 1 (atom 2) is in the state |0〉, the polarization (spatial-mode) phase-flip operation $σ_{Z_{A}}^{P} (σ_{Z_{A}}^{S})$ is performed on the polarization (spatial-mode) DOF of the photon A. Therefore, the hybrid hyper-CNOT gate III is obtained as

\begin{array}{l} {| Ψ 〉}_{III} = [sin α_{1} {| R 〉}_{A} {| ϕ 〉}_{B}^{S} + cos α_{1} {| L 〉}_{A} σ_{X_{B}}^{S} {| ϕ 〉}_{B}^{S}] \\ \otimes [sin α_{2} | a_{1} 〉 {| ϕ 〉}_{B}^{P} + cos α_{2} | a_{2} 〉 σ_{X_{B}}^{P} {| ϕ 〉}_{B}^{P}] . \end{array}

The hybrid hyper-CNOT gate III is different from the hybrid hyper-CNOT gate I and II, because the photons A and B do not simultaneously interact with Y₁-atom 1 and Y₂-atom 2, which means the photon B goes through the quantum circuit after the photon A interacts with the atom-cavity (1 or 2). Moreover, our hybrid hyper-CNOT gate III can be easily extended to establish the multi-qubit hybrid hyper-CNOT^N gate, in which one photon as control qubit and the rest N-photon as target qubits. The stationary-qubit systems [59,60] have been used widely to study the CNOT^N gate. We will describe the procedure of the multi-qubit hybrid hyper-CNOT^N gate below, which useful to realize quantum algorithms [61], correct errors [62], and prepare the hyperentanglement for N photons [63].

Suppose that the initial states of the control photon A and two auxiliary atoms j in the cavity j (j = 1, 2) remain |φ〉_A and |φ〉_j(j = 1, 2), respectively, and the N target photons B₁, B₂, · · · , and B_N are in the superposition state of the form ${| ϕ 〉}_{B_{m}} = {| ϕ 〉}_{B_{m}}^{P} \otimes {| ϕ 〉}_{B_{m}}^{S} (m = 1, 2 \dots)$ , where ${| ϕ 〉}_{B_{m}}^{P} = sin θ_{1}^{m} {| R 〉}_{B_{m}} + cos θ_{1}^{m} {| L 〉}_{B_{m}}$ and ${| ϕ 〉}_{B_{m}}^{S} = sin θ_{2}^{m} | b_{1}^{m} 〉 + cos θ_{2}^{m} | b_{2}^{m} 〉$ . To implement the hybrid hyper-CNOT^N gate, we could use the quantum circuit depicted by Fig. 3(b) by substituting the target photon B in the original hybrid hyper-CNOT gate III with the photon string B₁, B₂, · · ·, and B_N. Besides, the target photon B_m+1 should be input into the cavity in turn after B_m goes away the cavity. Meanwhile, the time delay between each two target photons B_m and B_m+1 could be utilized to discriminate their spatial modes. To be detail, after the control photon A interacts with the Y₁-atom 1 and Y₂-atom 2 in turn, the atoms 1 and 2 are performed the Hadamard operations. Subsequently, let the first target photon B₁ with its spatial modes $b_{1}^{1}$ and $b_{2}^{1}$ get through Y₂-atom 2, BS₅, H₅, Y₁-atom 1, Y₂, H₅, and BS₆ sequentially, similar to the target photon B of the hybrid hyper-CNOT gate III. After the photon B₁ passes though the setup and propagates into the output modes $b_{1}^{1}$ and $b_{2}^{1}$ , let the target photon B_m after the photon B_m−1 be emitted by the cavity with the similar procedure. After all the N target photons pass though the setup, the Hadamard operations on the two atoms 1 and 2 are performed. To decouple the two atoms, the measurements on the atoms are performed. When the outcomes are both the states |0〉, the (N + 1)-photon system is projected into the state ${| Ψ 〉}_{III}^{N}$ . Here

\begin{array}{l} {| Ψ 〉}_{III}^{N} = [sin α_{1} {| R 〉}_{A} \prod_{m = 1}^{N} {| ϕ 〉}_{B_{m}}^{S} + cos α_{1} {| L 〉}_{A} \prod_{m = 1}^{N} σ_{X_{B_{m}}}^{S} {| ϕ 〉}_{B_{m}}^{S}] \\ \otimes [sin α_{2} | a_{1} 〉 \prod_{m = 1}^{N} {| ϕ 〉}_{B_{m}}^{P} + cos α_{2} | a_{2} 〉 \prod_{m = 1}^{N} σ_{X_{B_{m}}}^{P} {| ϕ 〉}_{B_{m}}^{P}] . \end{array}

Here

σ_{X_{B_{m}}}^{S} = | b_{1}^{m} 〉 〈 b_{2}^{m} | + | b_{1}^{m} 〉 〈 b_{2}^{m} |

and

σ_{X_{B_{m}}}^{P} = {| R 〉}_{B_{m}} 〈 L | + {| L 〉}_{B_{m}} 〈 R |

represent the spatial-mode and polarization bit-flip operations on the target photon B_m involved, respectively. That is, the multi-qubit hybrid hyper-CNOT^N gates in double DOFs on the N + 1 photons are achievable after successively operating the control photon A and target photon B₁, B₂, · · · , and B_N.

4. Discussion and conclusion

Our hybrid hyper-CNOT gates work in an intermediate coupling region, which is considerably different from the strong coupling cases [14–17]. There are some technical challenges to realize the strong coupling strength in experiment, compared with intermediate coupling strength of the atom-cavity system [48, 49, 64]. Moreover, in order to obtain shorter operation time, it is necessary to achieve the atom-cavity photon scattering with a bad cavity in experiment. In [48], Turchette et al. made a measurement on the conditional phase shifts for quantum logic with the experimental parameters (g, κ, γ)/2π ∼ (20, 75, 2.5) MHz, which satisfy the limitation of a bad cavity κ ≫ g²/κ ≫ γ exactly in an intermediate coupling region (g = 0.27κ). In [49], Dayan et al. presented an intermediate atom-cavity coupling in experiment, where a Cs atom is trapped in a microtoroidal resonator. They provided a set of experimental parameters (g, κ, γ)/2π ∼ (70, 165 ± 15, 2.6) MHz with the atom-cavity detuning ω₀ = ω_c, which still satisfied the requirements of a bad cavity. The Q value in [49] is more than 10⁴ in the low-Q cavity limit, which is also very suitable to our protocol requiring the Q value of larger than 10². Besides, it should be noted that that although the Q can be low, a high Q/V is usually required in cavity QED systems, in which V is the mode volume. In [64], Tiecke et al. experimentally demonstrated a quantum optical switch, in which a single atom switched the phase of a photon and a single photon modified the atom’s phase, by coupling a photon to a single atom trapped in the near field of a nanoscale photonic crystal cavity attached to an optical fibre taper. The atom is trapped about 200 nm from the surface in an optical lattice formed by the interference of an optical tweezer and its reflection from the side of the cavity. The atom utilizing a short (3 ns) pulse of light was excited in the optical trap and resonant with the |5S_1/2, F = 2〉 → |5P_2/3, F′ = 3〉 transition (near 780 nm) accompanied by the experimental parameters (2g, κ, γ)/2π ∼ (1.096 ± 0.03, 25, 0.006) GHz under the limitation of a low cavity.

When the photon interacts with the atom-cavity system, the incident photon may be inevitably lost. Therefore, it becomes particularly important to consider the efficiencies of the hybrid hyper-CNOT gates, η = η_output/η_input, which is defined as the yield of the incident photon, that is, the ratio that takes into account these gates’ output photon number η_output and input photon number η_input. Besides, all the discussions for the constructions of the hybrid hyper-CNOT gates are in the ideal case r (ω_p) ∼ 1 and r₀(ω_p) = −1. The imperfection in phase and amplitude of the reflection photons reduces the performance of our gates, so it is necessary to consider the feasibilities of our gates, which can be evaluated by the fidelity defined as the overlap of the output states of the system in the ideal case |ψ_ideal〉 and the real case |ψ_real〉, F = |〈ψ_ideal|ψ_real〉|². Taking the hybrid hyper-CNOT gate I as an example, where |ψ_ideal〉 can be described by in Eq. (14), and the corresponding output state |ψ_real〉 can be obtain by substituting the real optical transition rules |L〉|1〉 → r (ω_p) |L 〉|1〉, and |L 〉|0〉 → r₀(ω_p)|L〉|0〉 for the ideal case |ψ_ideal〉 during the evolution of the whole system. Here,

\begin{array}{l} F_{C 1} = \prod_{i \neq j \in {1, 2}} \frac{{| \sum_{k = 1}^{4} (ξ_{2 k - 1} sin β_{j} + ξ_{2 k} cos β_{j}) λ_{k} ζ_{k} |}^{2}}{\sum_{k = 1}^{4} {| ξ_{2 k - 1} sin β_{j} + ξ_{2 k} cos β_{j}) λ_{k} |}^{2} \sum_{k = 1}^{4} {| ζ_{k} |}^{2}}, \\ η_{C 1} = \prod_{i \neq j \in {1, 2}} \frac{\sum_{k = 1}^{4} {| ξ_{2 k - 1} sin β_{j} + ξ_{2 k} cos β_{j}) λ_{k} |}^{2}}{256}, \end{array}

where ξ₁ = ξ₄ = 2+2r, ξ₂ = ξ₃ = 2−2r,

ξ_{5} = ξ_{8} = 2 r + 2 r r_{0} + r^{2} - r_{0}^{2}

,

ξ_{6} = ξ_{7} = 2 r - 2 r r_{0} - r^{2} + r_{0}^{2}

, λ₁ = λ₂ = sin α_i, λ₃ = λ₄ = cos α_i, ζ₁ = sin β_j sin α_i, ζ₂ = cos β_j sin α_i, ζ₃ = cos β_j cos α_i, and ζ₄ = sin β_j cos α_i (i ≠ j ∈ {1, 2}). In order to make our discussion more practical and general, we use the average efficiency defined as

{\bar{η}}_{C 1} = \frac{1}{{(2 π)}^{4}} \int_{0}^{2 π} d α_{1} \int_{0}^{2 π} d β_{1} \int_{0}^{2 π} d α_{2} \int_{0}^{2 π} d β_{2} η_{C 1}

and the average fidelity defined as

{\bar{F}}_{C 1} = \frac{1}{{(2 π)}^{4}} \int_{0}^{2 π} d α_{1} \int_{0}^{2 π} d β_{1} \int_{0}^{2 π} d α_{2} \int_{0}^{2 π} d β_{2} F_{C 1}

to characterize the performance of our hybrid hyper-CNOT gate I. The fidelities F_C2 (F_C3) of the hybrid hyper-CNOT gates II (III) is similar to the one F_C1 of the hybrid hyper-CNOT gate I, and can be obtained by transforming the positions of α₁, α₂, β₁, and β₂ in Eq. (18), so they possess the same average fidelity. Therefore, we adopt the average fidelity F̄_C1 to display the feature of three kinds of hybrid hyper-CNOT gates. So does the following discussion of the average efficiency of the hybrid hyper-CNOT gates. Obviously, the average fidelity and efficiency of our hybrid hyper-CNOT gates are assisted by the nonlinear interaction between the photon and the auxiliary atom in the cavity shown in Figs. 4(a) and 4(b), respectively. Based on the experimental parameters mentioned in [48], the average fidelity and efficiency of our hybrid hyper-CNOT gate are F̄_C1 = 0.9943, and η̄_C1 = 0.9061, respectively. The average fidelity and efficiency of our hybrid hyper-CNOT gate are improved to F̄_C1 = 0.9989 and η̄_C1 = 0.9478 under the condition [49], respectively. The average fidelity and efficiency of our hybrid hyper-CNOT gate become F̄_C1 = 0.9999 and η̄_C1 = 0.9898 in the case of [64], respectively. The results above show that the average fidelities and efficiencies of our hybrid hyper-CNOT gates can acquire higher values with low-Q cavities.

Fig. 4 (a) The average fidelity F̄_C1 and (b) the average efficiency η̄_C1 of the hybrid hyper-CNOT gate I vs the ratio of $g / \sqrt{κ γ}$ with ω₀ = ω_c = ω_p, respectively.

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The mechanism of our hybrid hyper-CNOT gates are in principle determinated in experiment if the unavoidable effects of photon loss, non-ideal single-photon sources, imperfect linear-optical elements (BSs and CPBSs), and invalid dark detectors atomic detections are not taken into account. These imperfections reduce the efficiencies and fidelities of our schemes. An ideal single-photon source should be work efficiently, controlled, exactly, but no single-photon sources can meet these requirements with the current technology. However, these technical imperfections will be improved with the further development of linear optical elements.

In this paper, the hybrid hyper-CNOT gate I (II) shows that the polarization (spatial-mode) states of the photons A and B simultaneously control the spatial-mode (polarization) states of the photons B and A, respectively. It is self-evident that the control qubits of the hybrid hyper-CNOT gate I are changed into the target qubits of the hybrid hyper-CNOT gate II and vice versa. Two photons A and B simultaneously interact with atom-cavity system in the opposite direction, which means shorter operation time required in the whole process of our schemes. Moreover, the hybrid hyper-CNOT gate III shows that the polarization state and the spatial-mode state of the photon A simultaneously control the spatial-mode state and the polarization state of the photon B, respectively. Meanwhile, it could be directly generalized into the multi-qubit hyper-CNOT^N gate, which is of great importance when performing the hyperentanglement preparation and redundant hyperencoding procedure [65,66], and could find its potential application in the memory-less quantum communication [67, 68] in the hyper-parallel quantum network. In addition, our multi-qubit hybrid hyper-CNOT^N gate needs only two auxiliary atoms, which makes our scheme easier, compared with the multi-qubit gate constituted with two-qubit gates and single-qubit gates [2]. These hybrid hyper-CNOT gates can be used to set up some interesting devices for high-capacity direct transmission of the information from one quantum communication node to another with a polynomial gain for the efficiency of distributed QIP. Compared with other protocols with input-output process [14–17], this protocol does not need the confined atom strong coupled to a high-Q cavity, and extends the earlier protocols with single-photon to continuous variable regime, which could greatly relax the experimental requirement. Therefore, these hybrid spatial-polarization hyper-CNOT gates are more robust against photonic dissipation noise.

In conclusion, we have constructed three types of flexible hybrid hyper-CNOT gates assisted by low-Q cavities and investigated the possibility of parallel quantum computation without utilizing extra spatial modes or polarization modes. The hybrid spatial-polarization hyper-CNOT gates consume less quantum resource and are more robust against photonic dissipation noise, compared with the integration of two cascaded CNOT gates in one DOF. In contrast to the hybrid hyper-CNOT gate on the one photon and the stationary electron spins in quantum dots [27], our hybrid hyper-CNOT gates are ultimately realized on the two photon systems, and the atoms of the low-Q cavities are only the auxiliary resource. Besides, the operation time of our hybrid hyper-CNOT gates are economized extremely in the whole process of our schemes due to simultaneous counter-propagation of two photons, which are much different from previous hyper-CNOT gate [26]. Moreover, these gates are performed with cavity-assisted photon scattering in the intermediate coupling region, possessing relatively long coherence time, which are much different from strong-coupling cases of the high-Q ones. Therefore, it is more feasible to realize not only fast quantum operations, but also multi-time operations between the photon and the cavity-atom system. In addition, our calculations show that the average fidelities and efficiencies of our gates are high with current experimental technology.

Funding

National Natural Science Foundation of China under Grant (11747024, 51635011, 61571406, 61704158); in part by the Shanxi “1331 Project ” Key Subjects Construction.

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Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-Q cavities

Abstract

1. Introduction

2. An atomic-cavity interaction system

3. Three types of hybrid hyper-CNOT gates

3.1. Hybrid hyper-CNOT gate I

3.2. Hybrid hyper-CNOT gate II

3.3. Hybrid hyper-CNOT gate III and hybrid hyper-CNOT^N gate

4. Discussion and conclusion

Funding

References

Cited By

Figures (4)

Equations (18)

Optics Express

Abstract

1. Introduction

2. An atomic-cavity interaction system

3. Three types of hybrid hyper-CNOT gates

3.1. Hybrid hyper-CNOT gate I

3.2. Hybrid hyper-CNOT gate II

3.3. Hybrid hyper-CNOT gate III and hybrid hyper-CNOTN gate

4. Discussion and conclusion

Funding

References

Cited By

Figures (4)

Equations (18)

Optics Express

3.3. Hybrid hyper-CNOT gate III and hybrid hyper-CNOT^N gate