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The fundamental concept of four-photon decoherence-free states and single logical qubit information

For robust quantum information processing against collective decoherence, due to uncontrolled coupling between a system and the environment, the utilization of logical qubits (left{ {left| {0_{{text{ L}}} } rightrangle , , left| {1_{{text{ L}}} } rightrangle } right}) based on a decoherence-free subspace has been proposed32,33,34. Thus, one of the proposed concepts32,33,34,35,43,44,45,50,51,52,59,60,61 for logical qubits is the design of four-qubit decoherence-free states44,50,51,52, as follows:

$$begin{aligned} left| {0_{{text{ L}}} } rightrangle_{1234} & equiv frac{1}{2}left( {left| {0101} rightrangle – left| {0110} rightrangle – left| {1001} rightrangle + left| {1010} rightrangle } right)_{1234} \, &= frac{1}{sqrt 2 }left( {left| {01} rightrangle – left| {10} rightrangle } right)_{12} otimes frac{1}{sqrt 2 }left( {left| {01} rightrangle – left| {10} rightrangle } right)_{34} , \ left| {1_{{text{ L}}} } rightrangle_{1234} & equiv frac{1}{{sqrt {12} }}left( {2left| {0011} rightrangle + 2left| {1100} rightrangle – left| {0101} rightrangle – left| {1010} rightrangle – left| {0110} rightrangle – left| {1001} rightrangle } right)_{1234} \ , & = frac{1}{sqrt 3 }left[ {left( {left| {0011} rightrangle + left| {1100} rightrangle } right)_{1234} – frac{1}{sqrt 2 }left( {left| {01} rightrangle + left| {10} rightrangle } right)_{12} otimes frac{1}{sqrt 2 }left( {left| {01} rightrangle + left| {10} rightrangle } right)_{34} } right]. \ end{aligned}$$

(1)

Furthermore, in quantum information processing technologies, a flying photon is a feasible resource to manipulate, transfer, and encode quantum information. Four-photon decoherence-free states (left{ {left| {0_{{text{ PL}}} } rightrangle , , left| {1_{{text{ PL}}} } rightrangle } right}), which consist of photonic spins, photons, can be used as logical qubits, (left{ {left| {0_{{text{ L}}} } rightrangle , , left| {1_{{text{ L}}} } rightrangle } right}), to carry quantum information under collective decoherence, as follows:

$$begin{aligned} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}} & equiv frac{1}{2}left( {left| {RLRL} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle + left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}\ , & = frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AB}}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{CD}}}} , , \ left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}} & equiv frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle + 2left| {LLRR} rightrangle – left| {RLRL} rightrangle – left| {LRLR} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle } right)_{{{text{ABCD}}}} \ , & = frac{1}{sqrt 3 }left[ {left( {left| {RRLL} rightrangle + left| {LLRR} rightrangle } right)_{{{text{ABCD}}}} – frac{1}{sqrt 2 }left( {left| {RL} rightrangle + left| {LR} rightrangle } right)_{{{text{AB}}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle + left| {LR} rightrangle } right)_{{{text{CD}}}} } right], \ end{aligned}$$

(2)

where (left{ {left| R rightrangle equiv left| 0 rightrangle , , left| L rightrangle equiv left| 1 rightrangle } right}) and the circularly polarized states ((left| R rightrangle): right and (left| L rightrangle): left) are related to the linearly polarized states ((left| H rightrangle): horizontal and (left| V rightrangle): vertical), using (left| R rightrangle equiv left( {left| H rightrangle + left| V rightrangle } right)/sqrt 2) and (left| L rightrangle equiv left( {left| H rightrangle – left| V rightrangle } right)/sqrt 2). For robustness against collective decoherence, we can encode arbitrary quantum information onto four-photon decoherence-free states, as follows:

$$left| {psi_{{{text{PL}}}} } rightrangle_{{{text{ABCD}}}} = , alpha left| {0_{{{text{PL}}}} } rightrangle_{{{text{ABCD}}}} + beta left| {1_{{{text{PL}}}} } rightrangle_{{{text{ABCD}}}} ,$$

(3)

where (left| alpha right|^{2} + left| beta right|^{2} = 1). Through this encoding process (single logical qubit information onto a decoherence-free subspace), we can conserve the arbitrary quantum information encoded onto logical qubits under collective decoherence.

Quantum dot confined in a single-sided optical cavity

In this section, we introduce the concept of a quantum dot (QD) within a cavity (QD-cavity system)11,20,22,47,58,62,63,64,65,66,67,68,69, which can induce the interaction of a photon and a singly charged QD (a negatively charged exciton:({text{X}}^{ – })). For the coherence of quantum systems in quantum information processing schemes, the systems of micropillar cavities have been widely utilized to construct quantum controlled gates11,20,22,47,58,62,63,64,65,66,67,68,69. Additionally, quantum information in the QD-cavity system can be effectively isolated from the environment for a long electron-spin coherence time (({text{T}}_{{ 2}}^{{text{ e}}}) ~ ({mu s}))70,71,72,73,74,75 and a limited spin relaxation period (({text{T}}_{{ 1}}^{{text{ e}}}) ~ ({text{ms}}))76,77,78,79.

In Fig. 1, the schematic of the QD-cavity system, Fig. 1a, and the spin selection rule, Fig. 1b, in the QD11,20,22,47,58,62,63,64,65,66,67,68,69 are presented with (left| uparrow rightrangle equiv left| { + 1/2} rightrangle {, }left| downarrow rightrangle equiv left| { – 1/2} rightrangle) (the spin states of the excess electron), and (left| Uparrow rightrangle equiv left| { + 3/2} rightrangle , , left| Downarrow rightrangle equiv left| { – 3/2} rightrangle) (heavy-hole spin states). The single-sided cavity consists of two GaAs/Al(Ga)As distributed Bragg reflectors, DBR: the bottom DBR is partially reflective and the top DBR, 100% reflective, and a transverse index guide for the three-dimensional confinement of light. Figure 1a shows that a singly charged electron self-assembled QD is embedded in the center of the single-sided cavity. When an excess electron is injected into the QD, optical excitations can create a negatively charged exciton (({text{X}}^{ – })), as described in Fig. 1b. By the Pauli exclusion principle, if the spin state of the excess electron in the QD is in the state (left| uparrow rightrangle), then a polarization (left| L rightrangle) of a photon can drive the state (left| { uparrow downarrow Uparrow } rightrangle) of ({text{X}}^{ – }). Moreover, if the spin state of the excess electron in the QD is (left| downarrow rightrangle) and the polarization of a photon is (left| R rightrangle), through the interaction, the state (left| { downarrow uparrow Downarrow } rightrangle) of ({text{X}}^{ – }) can be created. The reflection coefficient (Rleft( omega right)), which is induced by the reflected photon from the interaction between the input photon pulse and the QD-cavity system, can be calculated by the Heisenberg equation of motion57 and the spin selection rule, with the ground state in the QD ((leftlangle {hat{sigma }_{Z} } rightrangle = – 1)) for the steady state in the weak excitation approximation. Additionally, we can obtain the reflection coefficient, (R_{{text{h}}}) ((R_{{0}})) of the hot (cold) cavity when the QD is coupled (uncoupled) to the cavity, depending on the spin selection rule of (left| R rightrangle left| downarrow rightrangle , , left| L rightrangle left| uparrow rightrangle) ((left| R rightrangle left| uparrow rightrangle , , left| L rightrangle left| downarrow rightrangle)), with the coupling strength (g) between ({text{X}}^{ – }) and the cavity mode, and the decay rate (kappa) of the cavity mode, as follows:

$$begin{aligned} & left[ {g ne 0} right]: \,&quad Rleft( omega right) = R_{{text{h}}} left( omega right) equiv left| {R_{{text{h}}} left( omega right)} right|exp left[ {ivarphi_{{{text{Rh}}}} left( omega right)} right] , = , 1 – frac{{kappa left[ {ileft( {omega_{{{text{X}}^{ – } }} – omega } right) + gamma /2} right]}}{{left[ {ileft( {omega_{{{text{X}}^{ – } }} – omega } right) + gamma /2} right]left[ {ileft( {omega_{c} – omega } right) + kappa /2 + kappa_{s} /2} right] + g^{2} }}, , \ & left[ {g = 0} right]:\,&quad R_{{0}} left( omega right) equiv left| {R_{{0}} left( omega right)} right|exp left[ {ivarphi_{{{text{R0}}}} left( omega right)} right] , = , 1 – frac{kappa }{{ileft( {omega_{c} – omega } right) + kappa /2 + kappa_{s} /2}}, \ end{aligned}$$

(4)

where (left| {R_{{text{h}}} } right|) ((left| {R_{{0}} } right|)) and (varphi_{{{text{Rh}}}} = arg [R_{{text{h}}} ]) ((varphi_{{{text{R0}}}} = arg [R_{{0}} ])) are the reflectance and phase shift of the hot (cold) cavity, respectively. (omega_{{{text{X}}^{ – } }}), (omega_{c}), and (omega) are the frequencies of ({text{X}}^{ – }), the cavity mode, and the external field (photon), respectively. Thus, after the interaction between a photon and the QD-cavity system, the reflection operator (hat{text{R}}left( omega right)) from Eq. 4 is given by:

$${hat{text{R}}}left( omega right) = left| {R_{text{h}} left( omega right)} right|e^{i{varphi}_{text{Rh}} left( omega right)} left( {left| R rightrangle leftlangle R right| otimes left| downarrow rightrangle leftlangle downarrow right| + left| L rightrangle leftlangle L right| otimes left| uparrow rightrangle leftlangle uparrow right|} right) + left| {R_{0} left( omega right)} right|e^{i{varphi}_{{text{R}}0} left( omega right)} left( {left| R rightrangle leftlangle R right| otimes left| uparrow rightrangle leftlangle uparrow right| + left| L rightrangle leftlangle L right| otimes left| downarrow rightrangle leftlangle downarrow right|} right).$$

(5)

Here, if we take the experimental conditions of (omega_{{{text{X}}^{ – } }} = omega_{c}) (resonant interaction), such as a small side-leakage rate, (kappa_{s} ll kappa), a strong coupling strength, (g gg left( {kappa , , gamma } right)), and a small (gamma) (~ several µeV)80,81,82, the reflection operators, (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) and (hat{text{R}}_{2}^{{{text{Id}}}} left( omega right)), with regard to (omega – omega_{c}), (2left( {omega – omega_{c} } right)/kappa): frequency detuning, in the ideal case (without vacuum noise and leaky modes, such as sideband leakage and absorption) can be calculated as:

$$begin{aligned} & left[ {omega – omega_{c} = kappa /2} right] , Rightarrow \&quad hat{text{R}}_{1}^{{{text{Id}}}} left( omega right) = left( {left| R rightrangle leftlangle R right| otimes left| downarrow rightrangle leftlangle downarrow right| + left| L rightrangle leftlangle L right| otimes left| uparrow rightrangle leftlangle uparrow right|} right) – ileft( {left| R rightrangle leftlangle R right| otimes left| uparrow rightrangle leftlangle uparrow right| + left| L rightrangle leftlangle L right| otimes left| downarrow rightrangle leftlangle downarrow right|} right), \ & left[ {omega – omega_{c} = 0} right] , Rightarrow \&quad hat{text{R}}_{2}^{{{text{Id}}}} left( omega right) = left( {left| R rightrangle leftlangle R right| otimes left| downarrow rightrangle leftlangle downarrow right| + left| L rightrangle leftlangle L right| otimes left| uparrow rightrangle leftlangle uparrow right|} right) – left( {left| R rightrangle leftlangle R right| otimes left| uparrow rightrangle leftlangle uparrow right| + left| L rightrangle leftlangle L right| otimes left| downarrow rightrangle leftlangle downarrow right|} right), \ end{aligned}$$

(6)

where the values of the reflectances and the phase shifts are (left| {R_{{0}} } right| = left| {R_{{text{h}}} } right| approx 1) and (varphi_{{{text{Rh}}}} approx 0 , left( { approx 0} right)), (varphi_{{{text{R0}}}} approx – pi /2 , left( { approx pi } right)) from Eq. 4, according to the adjustment of the frequencies (omega – omega_{c} = kappa /2 , left( { = 0} right)) between the external field and the cavity mode when (kappa_{s}) is negligible, with (g/kappa = 2.4) and (gamma /kappa = 0.1)11,20,22,42,62,63,67.

Figure 1
figure1

(a) Schematic of a singly charged QD inside a single-sided cavity, interacting with a photon (input and output field operators: (hat{b}_{{{text{in}}}}) and (hat{b}_{{{text{out}}}})), with a side-leakage rate ((kappa_{s})) of cavity mode and decay rate ((gamma)) of ({text{X}}^{ – }). (b) By the spin selection rule in the QD, the induced interaction is (left| uparrow rightrangle to left| { uparrow downarrow Uparrow } rightrangle) ((left| downarrow rightrangle to left| { downarrow uparrow Downarrow } rightrangle)), according to the photon polarization of (left| L rightrangle) ((left| R rightrangle)).

Generation of four-photon decoherence-free states and the encoding process for single logical qubit information

In Fig. 2, we present the design of the scheme to encode single logical qubit information onto four-photon decoherence-free states using the QD-cavity systems and linearly optical devices. The scheme is composed of two parts: the generation of four-photon decoherence-free states and the process of encoding arbitrary quantum information. To obtain quantum information that is robust against collective decoherence34,35,59, our scheme can encode arbitrary quantum information onto four-photon decoherence-free states, (single logical qubit information), such as (alpha left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}} + beta left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}) in Eq. 3. To explain the process in detail, we first prepare the initial state as (left| {psi_{1} } rightrangle_{{{text{ABCD}}}} = left| V rightrangle_{{text{A}}}^{1} otimes left| V rightrangle_{{text{B}}}^{1} otimes left( {left| R rightrangle_{{text{C}}}^{1} + ileft| L rightrangle_{{text{C}}}^{1} } right)/sqrt 2 otimes left( {left| R rightrangle_{{text{D}}}^{1} + ileft| L rightrangle_{{text{D}}}^{1} } right)/sqrt 2) (product state of four photons). And, for the convenience, we define the expressions of path and photon, as (left| {{text{state}}} rightrangle_{{{text{photon}}}}^{{{text{path}}}}).

Figure 2
figure2

Encoding scheme for single logical qubit information onto four-photon decoherence-free states. This scheme is comprised of two parts: the generation of four-photon decoherence-free states and the encoding process. In the generation of the four-photon decoherence-free states, the four (1st, 2nd, 3rd, and 4th) gates employ the QD-cavity systems (QD1, 2, 3, and 4). The final gate in the encoding process also utilizes a QD-cavity system (QD5) to encode the single logical qubit information, with minimal collective decoherence.

1st gate [(photons C, D) ↔ QD1]

In the first gate (Fig. 3), two photons (C and D) and an electron spin 1 (left| { +_{{text{e}}} } rightrangle_{{1}}) (the prepared electron 1) in QD1, sequentially interact with each other, according to the reflection operator (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) in Eq. 6 with a frequency detuning of (omega – omega_{c} = kappa /2), where the spin states are defined as (left| { pm_{{text{e}}} } rightrangle = left( {left| uparrow rightrangle pm left| downarrow rightrangle } right)/sqrt 2).

Figure 3
figure3

Details of the first gate (QD1) in Fig. 2. The sequential interactions between two photons (C and D) and an electron spin 1 in QD1 utilize the reflection operator ({hat{text{R}}}_{1}^{{{text{Id}}}} left( omega right)) in Eq. 6 with a frequency of (omega – omega_{c} = kappa /2). The prepared excess electron spin state is in the state (left| { +_{{text{e}}} } rightrangle_{1}). This electron spin 1 is then measured after the interactions, in accordance with time table for switches (S1 and S2).

After the operation of the first gate, which interacts with two photons (C and D) and QD1 in sequence, according to time table (({text{t}}_{{1}} to {text{t}}_{{2}})), the state of pre-measurement is given by:

$$begin{aligned} & left| {psi_{{1}} } rightrangle_{{{text{ABCD}}}} otimes left| { +_{{text{e}}} } rightrangle_{1} {mathop{longrightarrow}limits^{{1{text{st}},{text{gate}}}}} , left| {psi_{{2}}^{{text{i}}} } rightrangle_{{{text{1ABCD}}}} \ &quad= frac{1}{sqrt 2 }left{ {left| { +_{{text{e}}} } rightrangle_{1} otimes left| {VV} rightrangle_{{{text{AB}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {HH} rightrangle – left| {VV} rightrangle } right)_{{{text{CD}}}}^{{{11}}} – left| { -_{{text{e}}} } rightrangle_{1} otimes left| {VV} rightrangle_{{{text{AB}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {HH} rightrangle + left| {VV} rightrangle } right)_{{{text{CD}}}}^{{{11}}} } right}, end{aligned}$$

(7)

where the interactions of the QD-cavity system are expressed as the reflection operator (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) in Eq. 6, with a frequency of (omega – omega_{c} = kappa /2) between the external field and the cavity mode. For example, if we assume that the result of a measurement in QD1 is in the state (left| { -_{{text{e}}} } rightrangle_{1}), the post-measurement state (left| {psi_{{2}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) will be:

$$left| {psi_{{2}}^{{text{i}}} } rightrangle_{{{text{1ABCD}}}} {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { -_{{text{e}}} } rightrangle_{{1}} } right] , to , left| {psi_{{2}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}} = left| {VV} rightrangle_{{{text{AB}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {HH} rightrangle + left| {VV} rightrangle } right)_{{{text{CD}}}}^{{{11}}} .$$

(8)

Subsequently, two polarizing beam splitters (PBSs) and rectangular-polarization flippers (Rs), as described in Fig. 2, operate to affect the state (left| {psi_{{2}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) of the first gate, as follows:

$$left| {psi_{{2}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}} {mathop{longrightarrow}limits^{{text{PBSs, Rs}}}} , left| {psi_{3} } rightrangle_{{{text{ABCD}}}} = left| {VV} rightrangle_{{{text{AB}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {HH} rightrangle_{{{text{CD}}}}^{{{11}}} + left| {HH} rightrangle_{{{text{CD}}}}^{{{22}}} } right).$$

(9)

2nd gate [(photons A, C, D) ↔ QD2] and 3rd gate [(photons B, C, D) ↔ QD3]

In the second (third) gate, depicted in Fig. 4, three photons, A, C, and D (B, C, and D), and an electron spin 2 (3), [(left| { +_{{text{e}}} } rightrangle_{{2}}) ((left| { +_{{text{e}}} } rightrangle_{{3}})): the prepared electron 2 (3)] in QD2 (QD3) sequentially interact with each other, according to the reflection operator (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) in Eq. 6, with a frequency detuning of (omega – omega_{c} = kappa /2). After the operation of the second gate, which interacts with three photons (A, C, and D) and QD2 in sequence, according to time table (({text{t}}_{2}^{prime } to {text{t}}_{3}^{prime })), the state of the pre-measurement is given by:

$$begin{aligned} & left| {psi_{3} } rightrangle_{{{text{ABCD}}}} otimes left| { +_{{text{e}}} } rightrangle_{2} {mathop{longrightarrow}limits^{{2{text{nd}};{text{gate}}}}} , left| {psi_{{4}}^{{text{i}}} } rightrangle_{{{text{2BACD}}}} \&quad = frac{ – i}{{sqrt 2 }}left| { +_{{text{e}}} } rightrangle_{2} otimes left| V rightrangle_{{text{B}}}^{{1}} otimes frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes left| H rightrangle_{{text{D}}}^{{1}}+ frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes left| H rightrangle_{{text{C}}}^{{2}} } right} \ &quadquad+ frac{ – 1}{{sqrt 2 }}left| { -_{{text{e}}} } rightrangle_{2} otimes left| V rightrangle_{{text{B}}}^{{1}} otimes frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RR} rightrangle + left| {LL} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes left| H rightrangle_{{text{D}}}^{{1}} + frac{1}{sqrt 2 }left( {left| {RR} rightrangle + left| {LL} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes left| H rightrangle_{{text{C}}}^{{2}} } right}, \ end{aligned}$$

(10)

Figure 4
figure4

The second and third gate (QD2 and QD3) in Fig. 2. The sequential interactions between three photons, A, C, and D (B, C, and D), and an electron spin 2 (3) in QD2 (QD3) are operated by the reflection operator ({hat{text{R}}}_{1}^{{{text{Id}}}} left( omega right)) in Eq. 6, with a frequency of (omega – omega_{c} = kappa /2). The prepared excess electron spin state in QD2 [QD3] is in the state (left| { +_{{text{e}}} } rightrangle_{2}) ((left| { +_{{text{e}}} } rightrangle_{3})). This electron spin 1 (2) is then measured after the interactions, in accordance with the second [third] time table for switches, S1, S2, S3, and S4 (S5, S6, S7, and S8), in sequence.

where the reflection operator (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) is given by Eq. 6, with (omega – omega_{c} = kappa /2). If the measurement outcome of QD2 is in the state (left| { +_{{text{e}}} } rightrangle_{2}), the post-measurement state (left| {psi_{{4}}^{{text{f}}} } rightrangle_{{{text{BACD}}}}) is then given by:

$$begin{aligned} & left| {psi_{{4}}^{{text{i}}} } rightrangle_{{{text{2BACD}}}} {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { +_{{text{e}}} } rightrangle_{{2}} } right] , to , left| {psi_{{4}}^{{text{f}}} } rightrangle_{{{text{BACD}}}} \ &quad= left| V rightrangle_{{text{B}}}^{{1}} otimes frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes left| H rightrangle_{{text{D}}}^{{1}} + frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes left| H rightrangle_{{text{C}}}^{{2}} } right}. end{aligned}$$

(11)

In the third gate, three photons (B, C, and D) of the state (left| {psi_{{4}}^{{text{f}}} } rightrangle_{{{text{BACD}}}}) interact with an electron spin state (left| { +_{{text{e}}} } rightrangle_{{3}}) in QD3, as follows:

$$begin{aligned} & left| {psi_{{4}}^{{text{f}}} } rightrangle_{{{text{BACD}}}} otimes left| { +_{{text{e}}} } rightrangle_{3} {mathop{longrightarrow}limits^{{text{3rd gate}}}} , left| {psi_{{5}}^{{text{i}}} } rightrangle_{{{text{3ABCD}}}} \&quad = frac{ – i}{{sqrt 2 }}left| { +_{{text{e}}} } rightrangle_{3} otimes frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{BD}}}}^{{{11}}} + frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{BC}}}}^{{{12}}} } right} \ &qquad + frac{ – 1}{{sqrt 2 }}left| { -_{{text{e}}} } rightrangle_{3} otimes frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {RR} rightrangle + left| {LL} rightrangle } right)_{{{text{BD}}}}^{{{11}}} + frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes frac{1}{sqrt 2 }left( {left| {RR} rightrangle + left| {LL} rightrangle } right)_{{{text{BC}}}}^{{{12}}} } right}, \ end{aligned}$$

(12)

where the reflection operator (hat{text{R}}_{1}^{{{text{Id}}}} left( omega right)) is given by Eq. 6, with (omega – omega_{c} = kappa /2), according to time table (({text{t}}_{3}^{prime prime } to {text{t}}_{4}^{prime prime })). For a measurement outcome in the state (left| { +_{{text{e}}} } rightrangle_{{3}}) of QD3, we obtain the output state (left| {psi_{{5}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) from the third gate, as follows:

$$begin{aligned} &left| {psi_{{5}}^{{text{i}}} } rightrangle_{{{text{3ABCD}}}} {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { +_{{text{e}}} } rightrangle_{{3}} } right] ,to , left| {psi_{5}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}\ &qquad = frac{1}{sqrt 2 }left{ {frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AC}}}}^{{{11}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{BD}}}}^{{{11}}} + frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{AD}}}}^{{{12}}} otimes frac{1}{sqrt 2 }left( {left| {RL} rightrangle – left| {LR} rightrangle } right)_{{{text{BC}}}}^{{{12}}} } right}. \ end{aligned}$$

(13)

Subsequently, as described in Fig. 2, two 50:50 beam splitters (BSs) are applied to two photons, C and D, of the output state (left| {psi_{{5}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}), as follows:

$$begin{aligned} & left| {psi_{{5}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}} {mathop{longrightarrow}limits^{{{text{BSs}}}}} , left| {psi_{6} } rightrangle_{{{text{ABCD}}}} \&quad = frac{1}{sqrt 2 }left[ {frac{1}{2}left{ {frac{1}{2}left( { – left| {RLRL} rightrangle + left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}} } right}} right.\&qquad + left. { frac{sqrt 3 }{2}left{ {frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle + 2left| {LLRR} rightrangle – left| {RLRL} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1111}}} } right}} right] \ &quadquad left[ { + frac{1}{sqrt 2 }frac{1}{2}left{ {frac{1}{2}left( { – left| {RLRL} rightrangle + left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1121}}} } right} } right.\&qquad + left. { frac{sqrt 3 }{2}left{ {frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle + 2left| {LLRR} rightrangle – left| {RLRL} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1122}}} } right}} right] \ end{aligned}$$

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4th gate [photon C ↔ QD4] and the operations dependent on the measurements

In the fourth gate (Fig. 5), the reflection operator (hat{text{R}}_{2}^{{{text{Id}}}} left( omega right)), which is given by Eq. 6, with (omega – omega_{c} = 0), performs an operation between a photon C and an electron spin 4 ((left| { +_{{text{e}}} } rightrangle_{{4}}): the prepared electron 4) in QD4. Subsequently, in the operations depending on measurements, diverse operators [circular-polarization flippers (CFs), (left| R rightrangle)– and (left| L rightrangle)-phase flippers (RPs and LPs), phase flippers (PPs), and a path switch] are applied to the output state from the fourth gate, according to the measurement outcomes of QD1, QD2, QD3, and QD4. After the interaction in the fourth gate, between photon C and QD4 of the state (left| {psi_{6} } rightrangle_{{{text{ABCD}}}}), the state of pre-measurement (left| {psi_{7}^{{text{i}}} } rightrangle_{{{text{ABCD}}}}) is given by:

$$begin{aligned} & left| {psi_{6} } rightrangle_{{{text{ABCD}}}} otimes left| { +_{{text{e}}} } rightrangle_{4} {mathop{longrightarrow}limits^{{text{4th gate}}}} , left| {psi_{{7}}^{{text{i}}} } rightrangle_{{{text{4ABCD}}}} \ &quad = frac{1}{sqrt 2 }left| { +_{{text{e}}} } rightrangle_{4} otimes left[ {frac{1}{2}left{ {frac{1}{2}left( { – left| {RLRL} rightrangle + left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}} } right}}right. \&qquad left.{+ frac{sqrt 3 }{2}left{ {frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle + 2left| {LLRR} rightrangle – left| {RLRL} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1111}}} } right}} right] \ &quadquad + frac{1}{sqrt 2 }left| { -_{{text{e}}} } rightrangle_{4} otimes left[ {frac{1}{2}left{ {frac{1}{2}left( {left| {RLRL} rightrangle + left| {RLLR} rightrangle – left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1121}}} } right} }right. \&qquad left.{+ frac{sqrt 3 }{2}left{ {frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle – 2left| {LLRR} rightrangle + left| {RLRL} rightrangle – left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1122}}} } right}} right] \ end{aligned}$$

(15)

Figure 5
figure5

The fourth gate (QD4) and the operations dependent on the measurements in Fig. 2. For the alignment of the path of photon C, the fourth gate utilizes the reflection operator ({hat{text{R}}}_{2}^{{{text{Id}}}} left( omega right)) given by Eq. 6 with a frequency of (omega – omega_{c} = 0) for the interaction between a photon C and an electron spin 4, which is prepared to the state (left| { +_{{text{e}}} } rightrangle_{4}), in QD4. Subsequently, due to the measurement outcomes of QD1, QD2, QD3, and QD4, the operations, depending on measurements, are performed on photons A, B, C, and D by Feed-Forward.

where the reflection operator (hat{text{R}}_{2}^{{{text{Id}}}} left( omega right)) is given by Eq. 6, with (omega – omega_{c} = 0).

Subsequently, if we assume that the result of a measurement in QD4 is in the state (left| { +_{{text{e}}} } rightrangle_{4}), the post-measurement state (left| {psi_{{7}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) will be:

$$begin{aligned} & left| {psi_{{7}}^{{text{i}}} } rightrangle_{{{text{4ABCD}}}} {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { +_{{text{e}}} } rightrangle_{{4}} } right] , to , left| {psi_{{7}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}} \&quad = frac{1}{2}left{ {frac{1}{2}left( { – left| {RLRL} rightrangle + left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}} } right} \ &quadquad+ frac{sqrt 3 }{2}left{ {frac{1}{{sqrt {12} }}left( {2left| {RRLL} rightrangle + 2left| {LLRR} rightrangle – left| {RLRL} rightrangle – left| {RLLR} rightrangle – left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1111}}} } right}. \ end{aligned}$$

(16)

For the encoding of (arbitrary) quantum information onto four-photon decoherence-free states, (single logical qubit information), we require the superposed state of the four-photon decoherence-free states, (left{ {left| {0_{{text{ PL}}} } rightrangle , , left| {1_{{text{ PL}}} } rightrangle } right}{: } approx left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}} + left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}) in Eq. 2 (the superposition of logical qubits). Thus, our scheme utilizes the operations, depending on measurements, using Feed-Forward, as described in Fig. 5, to transform a superposed state four-photon decoherence-free subspace according to the measurement outcomes of four electron spin states in the QD-cavity systems (QD1 ~ QD4). In Table 1, all possible operations, due to the measurement outcomes of electrons spins 1 ~ 4 in QD1 ~ QD4 (polarization flippers, phase flippers, and path switch), are summarized, to apply to the output state (left| {psi_{{7}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) of the fourth gate by Feed-Forward for the generation of a superposition state of four-photon decoherence-free states (logical qubits: (left| {0_{{text{ PL}}} } rightrangle) and (left| {1_{{text{ PL}}} } rightrangle)). Till this point, we have assumed the measurement outcomes of each of the electron spin states in QD1 ~ QD4. From the results of Eqs. 8, 11, 13, and 16, the measurement outcome of QD1 ~ QD4 is in the state (left| { -_{{text{e}}} } rightrangle_{1} left| { +_{{text{e}}} } rightrangle_{2} left| { +_{{text{e}}} } rightrangle_{3} left| { +_{{text{e}}} } rightrangle_{4}). We can subsequently apply a RP ((left| R rightrangle)-phase flipper from part 4 of the operations dependent on measurements) to path 2 of photon D, as (D-2), in the state (left| {psi_{{7}}^{{text{f}}} } rightrangle_{{{text{ABCD}}}}) by Feed-Forward.

Table 1 For a single type of the superposed four-photon decoherence-free states, the operations (circular-polarization flippers (CFs), (left| R rightrangle)– and (left| L rightrangle)-phase flippers (RPs and LPs), phase flippers (PPs), and a path switch] by Feed-Forward in parts (1), (2), (3), (4), and (5) should be applied to the output state of the fourth gate, due to the measurement results of the QDs. Here, we assign “O” and “N” to mean “Operation” and “No operation” of the Feed-Forward.

Finally, as listed in Table 1, we can obtain the output state (left| {psi_{8} } rightrangle_{{{text{ABCD}}}}) (the superposition of logical qubits) from the generation of decoherence-free states, as follows:

$$left| {psi_{7}^{{text{f}}} } rightrangle_{{{text{ABCD}}}} {mathop{longrightarrow}limits^{{text{Feed }{-}text{ Forward}}}} , left| {psi_{8} } rightrangle_{{{text{ABCD}}}} = frac{1}{2}left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} + frac{sqrt 3 }{2}left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} ,$$

(17)

where the four-photon decoherence-free states ((left| {0_{{text{ PL}}} } rightrangle) and (left| {1_{{text{ PL}}} } rightrangle)) are given in Eq. 2. Furthermore, other types of superposed states from the fourth gate can also be transformed to the superposed state of the four-photon decoherence-free states (left{ {left| {0_{{text{ PL}}} } rightrangle , , left| {1_{{text{ PL}}} } rightrangle } right}) in Eq. 2 (the superposition of logical qubits) by the operation of Feed-Forward in Table 1.

Encoding process for single logical qubit information

In the encoding process depicted in Fig. 6, an arbitrary-beam splitter (BS) is utilized to encode arbitrary quantum information onto the superposed state of the four-photon decoherence-free states [the output state (left| {psi_{8} } rightrangle_{{{text{ABCD}}}}) (the superposition of logical qubits) from generation of decoherence-free states]. To determine the path of photon D, the interaction of the final gate, which utilizes the reflection operator (hat{text{R}}_{2}^{{{text{Id}}}} left( omega right)) in Eq. 6, with (omega – omega_{c} = 0), is performed between photon D and QD5, as described in Fig. 6.

Figure 6
figure6

The encoding process (an arbitrary-BS and the final gate) in Fig. 2. To encode the arbitrary quantum information onto four-photon decoherence-free states, (single logical qubit information), the arbitrary-BS, having a transmission ((tau_{1})) and reflection ((tau_{2})) of arbitrary probabilities, is applied to photon D. The final gate then interacts with photon D using QD5, by the reflection operator ({hat{text{R}}}_{2}^{{{text{Id}}}} left( omega right)), with (omega – omega_{c} = 0), and can discriminate the path of photon D, according to the measurement outcome of QD5.

In Fig. 6, an arbitrary-BS has the independent reflectivity and transmissivity, according to polarizations. The operation of an arbitrary-BS is given by

$$a_{{{text{Down}}}}^{ + } to , left( {cos phi } right)a_{{{text{Up}}}}^{ + } – , left( {sin phi } right)a_{{{text{Down}}}}^{ + } , a_{{{text{Up}}}}^{ + } to , left( {sin phi } right)a_{{{text{Up}}}}^{ + } + , left( {cos phi } right)a_{{{text{Down}}}}^{ + } ,$$

(18)

where (cos phi) and (sin phi) are the transmission and reflection coefficients of the arbitrary-BS43,46,47,83 for (cos^{2} phi + sin^{2} phi = 1). Therefore, by controlling the experimental parameter (phi), we can generate the arbitrary encoding values (we want) onto four-photon decoherence-free states. After the transformed state (left| {psi_{8} } rightrangle_{{{text{ABCD}}}} = left( {left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} + sqrt 3 left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} } right)/2) by Feed-Forward (Eq. 17, results: (left| { pm_{{text{e}}} } rightrangle_{1} left| { pm_{{text{e}}} } rightrangle_{2} left| { pm_{{text{e}}} } rightrangle_{3} left| { +_{{text{e}}} } rightrangle_{4}) in Table 1), photon D in the state (left| {psi_{8} } rightrangle_{{{text{ABCD}}}}) passes through the arbitrary-BS in Fig. 6. The arbitrary quantum information is then encoded as the state (left| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}}), as follows:

$$begin{aligned} left| {psi_{8} } rightrangle_{{{text{ABCD}}}}& {mathop{longrightarrow}limits^{{text{Arbitrary -}text{ BS}}}} \ quadleft| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}} & = frac{1}{2}left[ {left( {tau_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} – tau_{2} sqrt 3 left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} } right) + left( {tau_{2} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} + tau_{1} sqrt 3 left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} } right)} right] \ & equiv frac{1}{sqrt 2 }left[ {left( {alpha_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} + beta_{1} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} } right) + left( {alpha_{2} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} + beta_{2} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} } right)} right], \ end{aligned}$$

(19)

where the specific transmission rate ((tau_{1} equiv cos phi)) and reflection rate ((tau_{2} equiv sin phi)) of the arbitrary-BS can be adjusted for our purposes (e.g., communication, information transfer, or computation) by controlling the experimental parameter (phi), as described in Eq. 18. For convenience, we define the arbitrary quantum information as (left{ {alpha_{1} , , beta_{1} } right} equiv left{ {tau_{{1}} /sqrt {left| {tau_{{1}} } right|^{2} + 3left| {tau_{{2}} } right|^{2} } , , – tau_{{2}} sqrt 3 /sqrt {left| {tau_{{1}} } right|^{2} + 3left| {tau_{{2}} } right|^{2} } } right}) and (left{ {alpha_{2} , , beta_{2} } right} equiv left{ {tau_{{2}} /sqrt {left| {tau_{{2}} } right|^{2} + 3left| {tau_{{1}} } right|^{2} } , , tau_{{1}} sqrt 3 /sqrt {left| {tau_{{2}} } right|^{2} + 3left| {tau_{{1}} } right|^{2} } } right}). As an additional example, (other measurement outcomes (left| { pm_{{text{e}}} } rightrangle_{1} left| { pm_{{text{e}}} } rightrangle_{2} left| { pm_{{text{e}}} } rightrangle_{3} left| { -_{{text{e}}} } rightrangle_{4}) in Table 1), if another output state (left| {psi^{prime}_{8} } rightrangle_{{{text{ABCD}}}}) is in the state (left| {psi^{prime}_{8} } rightrangle_{{{text{ABCD}}}} = left( {left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1122} + sqrt 3 left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1121} } right)/2) from the generation of four-photon decoherence-free states, the arbitrary quantum information can be also encoded as the (left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}) state, as follows:

$$left| {psi^{prime}_{8} } rightrangle_{{{text{ABCD}}}} {mathop{longrightarrow}limits^{{text{Arbitrary-} text{ BS}}}} , left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}} = frac{1}{sqrt 2 }left[ {left( {alpha_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1121} + beta_{1} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1121} } right) + left( {alpha_{2} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1122} + beta_{2} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1122} } right)} right].$$

(20)

In Eqs. 19 and 20, the paths of the photon D are still split (are superposed in terms of paths: 1 and 2), despite the arbitrary information can be encoded onto the decoherence-free states (logical qubits). Thus, to merge the split paths, the final gate (an interaction between photon D and QD5 in Fig. 6) should be operated in the encoding process. Subsequently, after the operation of the final gate to the state (left| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}}) in Eq. 19, the output state (left| {psi_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}}) (pre-measurement) is given by:

$$begin{aligned} & left| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}} otimes left| { +_{{text{e}}} } rightrangle_{5} {mathop{longrightarrow}limits^{{{text{Final }};{text{gate}}}}} , left| {psi_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}} \&quad = frac{1}{sqrt 2 }left| { +_{{text{e}}} } rightrangle_{5} otimes left[ {alpha_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} + beta_{1} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} } right] \ &quadquad+ frac{1}{sqrt 2 }left| { -_{{text{e}}} } rightrangle_{5} otimes left[vphantom{{+ frac{{beta_{2} }}{{sqrt {12} }}left( {2left| {RRLL} rightrangle – 2left| {LLRR} rightrangle + left| {RLRL} rightrangle – left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}} } } {frac{{alpha_{2} }}{2}left( { – left| {RLRL} rightrangle + left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}}}right. \&qquadquad left.{+ frac{{beta_{2} }}{{sqrt {12} }}left( {2left| {RRLL} rightrangle – 2left| {LLRR} rightrangle + left| {RLRL} rightrangle – left| {RLLR} rightrangle + left| {LRRL} rightrangle – left| {LRLR} rightrangle } right)_{{{text{ABCD}}}}^{{{1112}}} } right] \ end{aligned}$$

(21)

where the prepared state of electron 5 in QD5 is in the state (left| { +_{{text{e}}} } rightrangle_{{5}}). According to the measurement outcomes of QD5, we can then acquire the final state of the arbitrary quantum information encoded onto the four-photon decoherence-free states (single logical qubit information), as follows:

$$begin{aligned} & left| {psi_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}} , {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { +_{{text{e}}} } rightrangle_{{5}} } right] , {mathop{longrightarrow}limits^{{{text{Nothing}}}}} , left| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} + )}} = alpha_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} + beta_{1} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1111} , \ & left| {psi_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}} , {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { -_{{text{e}}} } rightrangle_{{5}} } right] , {mathop{longrightarrow}limits^{{text{Feed}- text{ Forward}}}} , left| {psi_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} – )}} = alpha_{2} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} + beta_{2} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1112} , \ end{aligned}$$

(22)

where Feed-Forward [(left| R rightrangle)-phase flippers (RP)] is operated if the measurement outcome is in the state (left| { -_{{text{e}}} } rightrangle_{{5}}). Furthermore, for another state (left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}) in Eq. 20, the final state (single logical qubit information) through the final gate can be obtained as:

$$begin{aligned} & left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}} , {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { +_{{text{e}}} } rightrangle_{{5}} } right] , {mathop{longrightarrow}limits^{{{text{Nothing}}}}} , left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} + )}} = alpha_{1} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1121} + beta_{1} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1121} , \ & left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{5ABCD}}}}^{{text{i}}} , {mathop{longrightarrow}limits^{{{text{measurement}}}}} , left[ {{text{result: }}left| { -_{{text{e}}} } rightrangle_{{5}} } right] , {mathop{longrightarrow}limits^{{text{Feed-}text{ Forward}}}} , left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} – )}} = alpha_{2} left| {0_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1122} + beta_{2} left| {1_{{text{ PL}}} } rightrangle_{{{text{ABCD}}}}^{1122} , \ end{aligned}$$

(23)

where the final state, (left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} + )}}) or (left| {psi^{prime}_{{text{E}}} } rightrangle_{{{text{ABCD}}}}^{{{text{F(}} – )}}), shows another path, 2, for photon C, compared with Eq. 22.

We have designed a scheme to encode arbitrary quantum information onto four-photon decoherence-free states (single logical qubit information) using QD-cavity systems and linearly optical devices for immunity against collective decoherence. For the experimental implementation of our scheme, we analyze the interactions between a photon and an excess electron in a QD, within a single-sided cavity.

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