Error-rejecting quantum computing with solid-state spins assisted by low-Q optical microcavities
aa r X i v : . [ qu a n t - ph ] J a n Error-rejecting quantum computing with solid-state spins assisted by low-Q opticalmicrocavities ∗ Tao Li and Fu-Guo Deng † Department of Physics, Applied Optics Beijing Area Major Laboratory,Beijing normal University, Beijing 100875, China (Dated: June 17, 2018)We present an efficient proposal for error-rejecting quantum computing with quantum dots (QD)embedded in single-sided optical microcavities based on the interface between the circularly polarizedphoton and QDs. An almost unity fidelity of the quantum entangling gate (EG) can be implementedwith a detectable error that leads to a recycling EG procedure, which improves further the efficiencyof our proposal along with the robustness to the errors involved in imperfect input-output processes.Meanwhile, we discuss the performance of our proposal for the EG on two solid-state spins withcurrently achieved experiment parameters, showing that it is feasible with current experimentaltechnology. It provides a promising building block for solid-state quantum computing and quantumnetworks.
I. INTRODUCTION
Compared with the traditional computer, quantumcomputing [1] can factor an n -bit integer with the magi-cal Shor algorithm [2], exponentially faster than the best-known classical algorithms. It can also run the famousquantum search algorithm, the Grover algorithm [3] orthe optimal Long algorithm [4], for unsorted databasesearch, which requires O( √ N ) operations only, ratherthan O( N ) operations involved in its classical coun-terpart. Both circuit-based quantum computing andthe measurement-based one require quantum entanglinggates. That is, the ability to entangle the quantum bits(qubits) is an essential building block in the constructionof a quantum computer [1]. Since the early quantum en-tangling gate (EG) for single atoms was designed withthe assistance of a high-Q optical cavity [5], more andmore attention has been paid to the entangling opera-tion between stationary qubits [5–13].The previous EG between two stationary qubits is im-plemented by various methods that resort to differentinteractions, i.e., the coherent control of the direct qubit-qubit interaction, the indirect interaction meditated withhigh-Q optical cavities [5–10], or the controllable ex-change interaction involved in the solid-state spin sys-tems [11–13]. The typical absence of a heralding mea-surement in the EG resulting from the direct or indirectqubit-qubit interaction will lead to some ambiguous er-ror, such as the one originating from the photon loss as aresult of the cavity decay or the radiative deexcitation ofthe stationary dipole. These proposals could work suc-cessfully under the condition that the amount of noiseinvolved in these EGs is less than a small threshold value[14]. It will be more physical-resource consuming andlargely increase the complexity of the target quantumsystem when performing scalable quantum computing ∗ Published in Phys. Rev. A , 062310 (2016) † Corresponding author: [email protected] with EGs of a higher error probability [15, 16]. How-ever, with a layered quantum-computer architecture, theresources required for error correction will become man-ageable when the physical error rate is about an order ofmagnitude below the threshold value of the chosen code[16].An alternative strategy exploits a measurement on theauxiliary photonic qubits that entangle with the corre-sponding stationary qubits to project the target station-ary system into an entangled state, which constitutes aquantum EG of high fidelity. Meanwhile, its success isheralded by the detection of photons [17–25]. Its fidelitydoes not suffer from the photon loss noise, and it is rela-tively robust to the variation of the system parameters.Since these special schemes involve the optical Bell-statemeasurement (BSM) assisted by linear optical elements,they can succeed with the maximal efficiency of 1 / .
99, and it is signaled by thedetection of a photon of orthogonal polarization as a re-sult of cavity QED [51], where only one effective input-output process is involved in single-sided cavities, sim-ilar to the one in a nitrogen-vacancy center coupled totwo-sided cavities [52]. This is the origin of our highefficiency rather than the maximal efficiency 1 / II. ERROR-REJECTING ENTANGLING GATEFOR TWO QDS IN LOW-Q OPTICALMICROCAVITIES
Let us consider a quantum system consisting of asingly charged self-assembled In(Ga)As QD embedded ina single-sided micropillar cavity [34, 42–44]. The quanti-zation axis z is chosen along the growth direction of theQD and is also parallel to the light propagation direc- (cid:1) |↑(cid:3) (cid:1)(cid:1) |↓(cid:3)|↑↓⇑(cid:3) (cid:1) |↑↓⇓(cid:3) (cid:1) |(cid:8) (cid:3) (cid:1)(cid:1) |(cid:9) (cid:3) (cid:1) Dark transitions a (cid:11)(cid:12) (cid:1) a (cid:13)(cid:14)(cid:15) (cid:1) (a) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (b) FIG. 1: The spin-dependent transitions for negatively chargedexciton X − . (a) A singly charged QD inside a single-sidedoptical micropillar cavity. (b) The relative energy levels andthe optical transitions of a QD. tion, shown in Fig. 1(a). The dipole transition associatedwith the negatively charged QD is strictly governed byPauli’s exclusion principle [53], shown in Fig. 1(b). Thesingle electron ground states have J z = ± /
2, denoted | ↑i and | ↓i , respectively, and the optical excited statesare the trion states ( X − = {| ↑↓⇑i or | ↑↓⇓i} ) consist-ing of two antisymmetric electrons in the singlet state1 / √ | ↑↓i − | ↓↑i ) and one hole with J z = ± / | ⇑i and | ⇓i ). The dipole-allowed transitions between theground state and the trion state are | ↑i ↔ | ↑↓⇑i and | ↓i ↔ | ↑↓⇓i , along with the absorbtion of a right-handedcircularly polarized photon | R i and a left-handed one | L i ,respectively, while the crossing transitions are dipole for-bidden [53].When a circularly polarized probe photon is launchedinto the single-sided cavity, it will be reflected by thecavity with a spin-dependent reflection coefficient r j ( ω )[34, 42–44]. The dynamic process can be represented byHeisenberg equations for the cavity field operator ˆ a anddipole operator ˆ σ − in the interaction picture [54], d ˆ adt = − h i ( ω c − ω )+ κ κ s i ˆ a − g ˆ σ − −√ κ ˆ a in + ˆ R,d ˆ σ − dt = − h i ( ω X − − ω )+ γ i ˆ σ − − g ˆ σ z ˆ a + ˆ N, (1)where ω X − , ω c , and ω are the frequencies of the dipoletransition, the cavity resonance, and the probe photon,respectively. ˆ R and ˆ N are noise operators which helpto preserve the desired commutation relations. The pa-rameter g is the coupling strength between X − and thecavity mode. κ describes the coupling to the input andoutput ports, while κ s and γ represent the cavity leakagerate and the trion X − decay rate, respectively. In theweak excitation limit where the QD dominantly occupiesthe ground state, assisted by the standard cavity input-output theory ˆ a out = ˆ a in + √ κ ˆ a [54], one can obtain thespin-dependent reflection coefficient [42, 43, 55, 56]: r j ( ω ) = 1 − κ (cid:2) i ( ω X − − ω ) + γ (cid:3)(cid:2) i ( ω X − − ω ) + γ (cid:3)(cid:2) i ( ω c − ω )+ κ + κ s (cid:3) + jg . (2)Here the subscript j is used to discriminate the case thatthe polarized probe photon agrees with the trion transi-tion ( j = 1) and feels a QD-cavity coupled system andthe case that the polarized photon decouples from thetrion transition ( j = 0) and feels an empty cavity.Suppose the electron spin s of a QD is initialized to | ψ s i = α | ↑i s + β | ↓i s , with | α | + | β | = 1. When the inputphoton is in the polarized state | ψ p i = √ ( | R i p − | L i p ),the photon reflected by the cavity directly due to the mis-matching between the incident probe photonic field andthe cavity mode, or reflected by the desired cavity-QDsystem, together with the QD, evolves into an unnormal-ized state | Φ i H = η in √ h ( r × α | ↑i s + r × β | ↓i s ) ⊗ | R i p − ( r × α | ↑i s + r × β | ↓i s ) ⊗ | L i p i + q − η in | ψ s i ⊗ | ψ p i . (3)Here η in is the probability amplitude of the photon re-flected by the desired cavity-QD system [57]. If onerewrites | Φ i H with the linear-polarization basis {| H i ≡ √ ( | R i + | L i ), | V i ≡ √ ( | R i − | L i ) } , one can get thesystem composed of the photon p and the electron spin s evolving into a partially entangled hybrid state, | Φ i H = (cid:20) η in r + r )+ q − η in (cid:21) ( α | ↑i + β | ↓i ) s ⊗ | V i p + η in ( r − r )2 ( α | ↑i − β | ↓i ) s ⊗ | H i p . (4)Here the photon p is partially entangled with the elec-tron spin s , and one can determine the state of the spinaccording to the outcome of the measurement on photon p . In detail, the detection of an | H i p photon leads toa phase-flip operation on spin s . Alternatively, the de-tection of a | V i p photon signals an error and results inan unchanged electron spin s , no matter where the er-ror originates (the mismatch between the incident fieldand the cavity mode, the low-Q cavity, or the detuning).For simplicity, we can take η in ≡ − η in . Meanwhile, the output state ofthe combined hybrid system composed of the spin s andthe probe photon p only depends on the combined coeffi-cients r − r or r + r of the cavity-QD system, while it isindependent of the particular parameters that affect thereflection coefficients r j , shown in Eq. (2). Therefore, theoutput states of two individual inhomogeneous electronspins embedded in different optical microcavities along FIG. 2: The schematic setup of the EG. BS represents a 50 :50 beam splitter. PBS is the polarizing beam splitter thattransmits | H i photons and reflects | V i photons. BS ai denotesthe beam splitter with adjustable reflection coefficient r ai , i.e., r a = r a = 1 is utilized for two identical cavity-QD systems;otherwise, | r a r a | < D ′ i and restart the recycling procedure beforea phase-flip operation on spin s i . with their respective probe photons could be amendedto be the same one by utilizing an adjustable beam split-ter [51, 58]. The negative effect of the inhomogeneity ofthe solid-state spins could be eliminated formally, whichleads to the same result as in homogeneous cavity-QDsystems [34, 42–44].With the faithful process described above, we can con-struct an error-rejecting EG, shown in Fig. 2, for twoidentical electron spins s and s (the reflection coeffi-cients r ai = 1 of the adjustable beam splitter BS ai areadopted), which will collapse spins s and s into a statewith a deterministic parity after the entangling process.Suppose the electron s i ( i = 1 ,
2) is initially in the state | Φ i s i = α i | ↑i s i + β i | ↓i s i with | α i | + | β i | = 1. One probephoton p in state | Φ i p = √ ( | R i p − | L i p ) launched intothe import of the EG passes through the beam splitter(BS ), and it will be reflected by either the left cavitycontaining the electron spin s or the right one contain-ing s . The unnormalized state of the hybrid systemcomposed of the photon p and the electron spins s and s after being reflected by the cavities evolves into | Φ i H = 1 √ n ( r + r )( α | ↑i + β | ↓i ) s ( α | ↑i + β | ↓i ) s ⊗ ( | V i p + | V i p )+( r − r ) (cid:2) ( α | ↑i− β | ↓i ) s ⊗ ( α | ↑i + β | ↓i ) s | H i p +( α | ↑i + β | ↓i ) s ⊗ ( α | ↑i− β | ↓i ) s | H i p (cid:3)o . (5)Here the subscripts p and p denote photon componentsthat occupy the left path and the right path, respectively.When the photon is in the horizonal polarized state | H i p or | H i p , the two different spatial modes of photon p arecombined on the BS . The interference of | H i p and | H i p modes will collapse the hybrid system into | Φ i H = 12 ( r − r ) (cid:2) ( α α | ↑i s | ↑i s − β β | ↓i s | ↓i s ) | H i p +( α β | ↑i s | ↓i s − β α | ↓i s | ↑i s ) | H i p (cid:3) . (6)Upon a click of the detector D or D , the EG is com-pleted and the electron-spin system s s is projected intoa subspace with a deterministic parity, which is indepen-dent of the reflection coefficients r j , since r j only appearsas a global coefficient in Eq. (6). In detail, when thephoton detector D clicks, the spins s s collapse intothe even-parity entangled state of the form | Φ i E = α α | ↑i s | ↑i s − β β | ↓i s | ↓i s . (7)When the photon detector D clicks, the spins s s areprojected into the odd-parity entangled state of the fol-lowing form | Φ i O = α β | ↑i s | ↓i s − β α | ↓i s | ↑i s . (8)Both states | Φ i E and | Φ i O keep the information of theinitial state. Therefore, the coefficient α i and β j couldbe the state of other QD spins that are entangled with s and s , which makes the EG effective for constructingcluster states in the next section. The total probabilitythat either D or D detects one photon of horizonalpolarization is η H : η H = | r − r | . (9)Here η H equals the efficiency of the EG without recyclingprocedure.The first term on the right-hand side of Eq. (5) con-tains the vertical polarization component | V i p ( | V i p )and it will lead to a click on the photon detector D ( D ).In this time, the state of the electron spins s s is pro-jected into | Φ i s ⊗ | Φ i s , exactly identical to the originalone without any interaction between the spins and thephoton p , which takes place with probability η V : η V = | r + r | . (10)Here η V equals the heralded error efficiency of the EG,and the electron spins s s , in this case, could be directlyused in the recycling EG procedure.In a word, one can obtain two kinds of useful resultswith our EG setup. When only one probe photon is ex-ploited, the probabilities of heralded success or failureof the EG are η H or η V , respectively. When the her-alded error of EG takes place, a | V i polarized photonis detected and the state of the spin subsystem has notbeen changed. One can input another probe photon p ′ in state | Φ i p ′ = √ ( | R i − | L i ) p ′ to repeat the EG pro-cess until a horizonal photon | H i is detected by D or D . This procedure will project the spin system s s into an even-parity subspace or an odd-parity one even-tually. By taking the recycling procedure into account,the total success probability η S of our error-rejecting EGis η S = | r − r | − | r + r | , (11)which is state independent, resulting in a more efficientquantum computing [1]. Note that each recycling pro-cedure is conditioned on a click of either vertical detec-tor D or D , and it should be stopped when photonloss takes place. Subsequently, one has to reinitialize thespins before performing a new EG operation on the spins. III. CLUSTER STATE GENERATION WITHOUR EG FOR MEASUREMENT-BASEDONE-WAY QUANTUM COMPUTING
Our error-rejecting EG can be used directly to imple-ment the one-way quantum computing [30, 59, 60] basedon QDs embedded in optical cavities. In the following,we demonstrate that our EG can be used to construct thetwo-dimensional (2D) QD cluster state [30, 61, 62], whichconstitutes the base of one-way quantum computing onsolid-state spins.Suppose there are j + 1 QD electron spins { s , s , . . . , s j } and s j +1 , and s j +1 is initialized to bethe state √ ( | ↑i j +1 − | ↓i j +1 ) and the first j spins areinitially in the one-dimensional (1D) cluster state of theform | ψ j i = ( | ↑i + | ↓i ˆ Z )( | ↑i + | ↓i ˆ Z ) · · ·⊗ ( | ↑i j − + | ↓i j − ˆ Z j )( | ↑i j + | ↓i j ) , (12)with the phase flip operator ˆ Z i = | ↑i i h ↑ | − | ↓i i h ↓ | .To increase the length of the 1D cluster state, an error-rejecting EG for spins s j and s j +1 is applied. When theEG fails, the state of spin s j is ambiguous and a statemeasurement on s j with basis {| ↑i , | ↓i} will collapse theremaining spins into a 1D cluster state of j − Z j − feedback operation. When theEG succeeds in the case that is heralded by the click ofphoton detector D , the j + 1 spins will be projected into | ψ ′ j +1 i = ( | ↑i + | ↓i ˆ Z )( | ↑i + | ↓i ˆ Z ) · · · ( | ↑i j − + | ↓i j − ˆ Z j )( | ↑i j | ↑i j +1 + | ↓i j | ↓i j +1 ) , (13)which could be transformed into the 1D cluster state sim-ilar to that in Eq. (12) of length j + 1 by a Hadamardoperation ˆ H j [ ˆ H completes the following transformation: | ↑i → √ ( | ↑i + | ↓i ) and | ↓i → √ ( | ↑i − | ↓i )] performedon spin j . If the success of the EG for s j and s j +1 issignaled by the click of D , a local operation ˆ H j +1 ˆ X j +1 (here the spin-flip operator ˆ X = | ↑ih ↓ | + | ↓ih ↑ | ) on spin s j +1 could also evolve the j + 1 spins into the desired 1Dcluster state.This procedure of cluster growth discussed above couldbe used to generate a larger cluster, since the efficiencyof our error-rejecting EG η s > . M and N available are, respectively, of lengths m and n , | ψ m i = ( | ↑ M i + | ↓ M i ˆ Z M ) · · · ( | ↑ M i m − + | ↓ M i m − ˆ Z M m )( | ↑ M i m + | ↓ M i m ) , (14) | ψ n i = ( | ↑ N i + | ↓ N i ˆ Z N ) · · · ( | ↑ N i n − + | ↓ N i n − ˆ Z N n )( | ↑ N i n + | ↓ N i n ) . (15)Before performing EG on M m and N , a phase-flip oper-ation ˆ Z M m is applied on spin M m . The success of the EGheralded by the click of photon detector D will projectthe entire spin system into | ψ nm i = ( | ↑ M i + | ↓ M i ˆ Z M ) · · · ( | ↑ M i m − + | ↓ M i m − ⊗ ˆ Z M m )( | ↑ M i m | ↑ N i + | ↓ M i m | ↓ N i ˆ Z N ) ⊗ ( | ↑ N i + | ↓ N i ˆ Z N ) · · · ( | ↑ N i n − + | ↓ N i n − ˆ Z N n )( | ↑ N i n + | ↓ N i n ) . (16)An additional Hadamard operation ˆ H on M m will evolvethe spin system into a 1D cluster | ψ m + n i of m + n qubits.As for the case that the success of the EG is signaled bya click of detector D , a local single-qubit operation ˆ H ˆ X on M m can also evolve the m + n spins into the 1D cluster | ψ m + n i .The cluster-connecting procedure based on parity mea-surement above is similar to that used in linear opti-cal quantum computing [65], whereas both the outcomeof the EG operation and the feedback operations afterthe EG are quite different. It generates a 1D cluster of m + n qubit, rather than m + n − IV. PERFORMANCE OF OURERROR-REJECTING EG WITH CURRENTEXPERIMENTAL PARAMETERS
The total success probability η S together with η H and η V of our EG are shown in Fig. 3 as a function of theside leakage κ/κ s with the cooperativity C ≡ g /γκ T , E ff i c i e n c y h S h H h V (a)10 -4 -2 h S h H h V E ff i c i e n c y k / k s h S h H h V -4 -2 h S h H h V k / k s FIG. 3: The efficiency of the EG vs different parameters with ω X − /ω c = 1 and γ/κ = 0 .
1: (a) ( ω c − ω ) /κ = 0, C = 1 / ω c − ω ) /κ = 0, C = 1; (c) ( ω c − ω ) /κ = γ/κ , C = 1 / ω c − ω ) /κ = γ/κ , C = 1. κ T ≡ κ s + κ , and γ/κ = 0 . ω X − of the QD to be resonant to thatof the cavity, ω X − /ω c = 1 [67]. When the probe pho-ton is also resonant to cavity [see Figs. 3 (a) and (b)], η rS = 0 .
255 and η pS = 0 .
559 can be achieved in the regimeof resonance scattering with C = 1 / C = 1, respectively, for κ/κ s = 13 [48, 68].When the probe photon detunes from the trion transi-tion by ( ω c − ω ) /κ = γ/κ , shown in Figs. 3 (c) and (d), η rS = 0 .
194 and η pS = 0 .
538 can be achieved for the sameremaining parameters, and the contribution from the re-cycling procedure η V increases. Furthermore, the EGcould enjoy a higher efficiency with a lower side leakageand a higher cooperativity C , which can be achieved byutilizing adiabatic cavities with smaller pillar diameters[66, 68]. In other words, the near -unity efficiency of theerror-rejecting EG can be achieved when the deep Pur-cell regime with low side leakage is available, and we caneasily attribute this improvement of the efficiency to theenhancement of the photon into the cavity mode.In the above discussion, we can get an efficienterror-rejecting EG for QDs with the perfect spin qubitand the monochromatic ( δ -function-like) single photonwavepacket. In fact, every single photon pulse is of fi-nite linewidth, i.e., a polarized photon of pulse shapein Gaussian function f ( ω ) = exp ( − ω/ ∆) / ( √ π ∆) withbandwidth ∆. This finite-linewidth character usually in-troduces some additional infidelity in the previous EGprotocols [42–50], while it has little harmful effect onthe fidelity of our EG. When one constitutes our EGwith a polarized single-photon pulse p of Gaussian shape, | ψ p i = √ R dωf ( ω )[ˆ a † R ( ω ) − ˆ a † L ( ω )] | i , where ˆ a † k ( ω ) is thecreation operator of a k -polarized photon with frequency ω , the state of the hybrid system composed of the pho-ton p and electron spins s and s just before photondetection process, shown in Eq. (6), will be modified to | Φ i H = 12 Z dω [ r ( ω ) − r ( ω )] f ( ω ) n ( α α | ↑i s | ↑i s − β β | ↓i s | ↓i s ) ⊗ ˆ a † H ( ω ) | i p +( α β | ↑i s | ↓i s − β α | ↓i s | ↑i s ) ⊗ ˆ a † H ( ω ) | i p o , (17)where ˆ a † H ( ω ) = √ [ˆ a † R ( ω ) + ˆ a † L ( ω )]. Upon the click ofphoton detector D or D , one can still complete the EGby projecting spins s and s into a subspace of deter-mined parity, as one does with a monochromatic pho-ton wave packet. One can find that the fidelity of ourEG is independent of the finite linewidth of the pho-ton pulse, since the frequency-dependent reflection co-efficients r j ( ω ) appear only in the global coefficient.In fact, the effects of dephasing and decay of electronspins will affect the performance of the EG. The timeneeded for the coherent control of single electron spinin QDs is on the scale of picoseconds [38, 39] and thecavity photon time is tens of picoseconds when the cavity Q -factor is about 1 × − × [66]. Therefore, itis the spontaneous emission lifetime of a QD, which isabout 1 ns, that sets the upper limit for the fidelity ofthe EG. Meanwhile, the electron spin coherence time of10 ns has been achieved at zero magnetic field [69], andit could be extended to several microseconds if the all-optical spin echo technique is exploited [40, 41]. The ratioof the decoherence time of electron spins to the operationtime needed to complete the EG can exceed 1 × , andthus the fidelity of the EG proposal will be larger than0 .
99 when taking into account the dephasing process ofthe electron spin, which suggests the strong promise ofelectron spin in QDs for scalable quantum computing.
V. DISCUSSION AND SUMMARY
Our scheme of error-rejecting EG can work efficientlywith almost unity fidelity in the strong coupling regime, g > κ T , γ , the Purcell regime, C > /
4, or even theresonantly scattering regime, C = 1 /
4. It is robust tothe imperfections involved in the practical input-outputprocess, i.e., the nonzero bandwidth, QD or cavity decay,and the finite coupling g/κ , since the fidelity of our EG isindependent of the reflective coefficients r j ( ω ) and thusindependent of the cooperativity C , which is far differ-ent from other schemes that depends on C [43–48]. Theoriginal low fidelity or error items originating from thepractical input-output process are converted into a rel-atively lower efficiency in our EG. Fortunately, the lowfidelity or error items trigger the single-photon detector D or D , which can be used to improve the efficiency ofthe EG by introducing the recycling procedure. In fact,our recycling procedure can contribute little when perfectcircular birefringence ( C ≫ ω X − = ω c = ω ) is avail-able, since the efficiency η H of the EG without recyclingprocedure approaches unity in this situation. Although our proposal is detailed with the QD-cavity system, itcould also be implemented with solid-state spin coupledto a photonic crystal waveguide [70].The previous EG performed in a resonantly scatteringregime, C = 1 / ω X − = ω c = ω could also be com-pleted with high fidelity, since a reflectivity r = 0 inthe resonantly scattering regime could be automaticallyeliminated, and only the photon that decouples the elec-tron spin could be reflected. Therefore, one can entangletwo spins by subsequently probing the two spin-cavitysystems with a linear polarized photon or entangle twolinear polarized photons by subsequently importing theminto a spin-cavity system, where the even-parity subspaceof the spin system or the photon system could be easilypicked out, since the odd-parity case will inevitably besignaled by photon loss [48], and the corresponding effi-ciency of the EG is 0 .
25. It is quite different from ourEG where the linear polarized photon, after being re-flected by the QD-cavity system, in the ideal case C ≫ ω X − = ω c = ω , is supposed to change its polarizationinto the orthogonal polarization and exert a phase-flipoperation on the spin. The interference of the photonafter being reflected by two cavities in superposition canproject the two spins into either even-parity subspace orodd-parity subspace in a heralded way.The error-rejecting EG only involves one effectiveinput-output process, which makes our scheme more ef-ficient than others since the practical input-output cou-pling η in < | V i polarization and it will trigger the single-photon detector D or D , which signals the restartingof the EG. This makes the EG different from the onebased on a double-sided cavity where the photon is en-coded in its Fock state [52]. The photon loss during theEG process owing to the inefficiency of the single-photondetector or cavity absorbtion will decrease the efficiencyof the EG, but it does not affect the fidelity of our EGsince both the success of the EG and the restarting ofthe EG are signaled by a click of single photon detectors.In conclusion, we have proposed an efficient error-rejecting EG proposal for two electron spins of QDs em-bedded in low- Q optical microcavities. With our error-rejecting EGs, a cluster-state connection scheme could becompleted efficiently. Under the practical experimentalcondition, the EG could be performed well with almostunity fidelity and an efficiency of η s > .
53 for C = 1. Webelieve the EG could provide a promising building blockfor solid-state scalable quantum computing and quantumnetworks in the future. ACKNOWLEDGMENTS
This work is supported by the National Natural Sci-ence Foundation of China under Grants No. 11474026 and No. 11674033, and the Fundamental ResearchFunds for the Central Universities under Grant No.2015KJJCA01. [1] M. A. Nielsen and I. L. Chuang
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