Optically generated 2-dimensional photonic cluster state from coupled quantum dots
OOptically generated 2-dimensional photonic cluster state from coupled quantum dots
Sophia E. Economou , Netanel Lindner , and Terry Rudolph Naval Research Laboratory, Washington, DC 20375, USA Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA Optics Section, Blackett Laboratory, Imperial College London, London SW7 2BZ, United Kingdom (Dated: October 27, 2018)We propose a method to generate a two-dimensional cluster state of polarization encoded photonicqubits from two coupled quantum dot emitters. We combine the recent proposal [5] for generating1-dimensional cluster state strings from a single dot, with a new proposal for an optically inducedconditional phase (CZ) gate between the two quantum dots. The entanglement between the twoquantum dots translates to entanglement between the two photonic cluster state strings. Furtherinter-pair coupling of the quantum dots using cavities and waveguides can lead to a 2-dimensionalcluster sheet. Detailed analysis of errors indicates that our proposal is feasible with current tech-nology. Crucially, the emitted photons need not have identical frequencies, and so there are noconstraints on the resonance energies for the quantum dots, a standard problem for such sources.
Measurement-based quantum computation (MQC) isan alternative to the well-known ‘circuit model’ of quan-tum computation [1]. The main idea in MQC is to ro-bustly create, upfront, a highly entangled state. Oncethis ‘cluster state’ is created, which is the challengingpart of this approach, only single qubit measurementsare necessary to perform the actual computation. Inthe case of photon polarization qubits, performing singlequbit rotations followed by photon number detection iseasily done with high fidelity, which makes them partic-ularly attractive for MQC. In fact this is one of the mostfault-tolerant architectures known for quantum comput-ing [2], and is particularly tolerant to qubit losses [3],of importance for optical architectures. The creation ofthe initial entangled cluster state is, however, a difficultproblem on which much current research efforts are fo-cused. To date the most promising methods have involveoptical interference of nearly identical photons [4]. Bycontrast, our proposal here allows for direct generationof the entangled photons.In Ref. [5] a proposal was developed for generating alinear (one-dimensional) cluster state of polarization en-coded photons from single photon emitters with a certainenergy level structure, such as those found in quantumdots (QDs). The relevant states of the QD are the twospin states |↑(cid:105) , |↓(cid:105) of the electron along the optical axis z and the two optically excited states called trions, whichhave total angular momentum 3 / z -direction of ± / | / (cid:105) , | ¯3 / (cid:105) . The broken symmetry of the QD along the z axis sets a preferred direction, along which the opticalpolarization selection rules are circularly polarized, andenergetically separates the excited trion states with to-tal angular momentum ± / π pulses, to an electron that is in a superposi-tion state | ↑(cid:105) + | ↓(cid:105) , exciting it to a superposition ofthe two trion states | / (cid:105) + | ¯3 / (cid:105) . Because QDs havelarge dipole moments, spontaneous emission is very fast,both compared to atoms and to the other relevant timescales in the QD dynamics, at least for very low mag-netic fields. Therefore the trion will spontaneously de-cay to the electron state almost instantaneously uponexcitation, emitting a photon of either right ( R ) or left( L ) circular polarization, thereby effecting transitions | / (cid:105) → |↑(cid:105)| R (cid:105) , | ¯3 / (cid:105) → |↓(cid:105)| L (cid:105) . The state of the emittedphoton+spin is |↑(cid:105)| R (cid:105) + |↓(cid:105)| L (cid:105) - i.e. they are entangled asboth recombination paths take place simultaneously. Theremaining degrees of freedom of the system are the same,so they are factored out and omitted for brevity. Subse-quent precession of π/ y − direction is performed,denoted R y ( π/ z basis. The entangled emitterstherefore generate photons which are themselves entan-gled. Explicitly, an entangled chain can be created. Thiscircumvents the need for ‘fusion gates’ [4]. Moreover, inour approach, the photons need not be identical in fre- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r (a) (b) (c) (d) (e) (f) (g) (h) FIG. 1: A sequence of diagrams depicting the generation ofthe cluster state using the standard diagrammatic represen-tations of such states. The electronic qubits are depicted asfilled circles, the initial electronic state is taken to be |↑(cid:105)|↑(cid:105) . Atstep (a) both spins precess under R y ( π/ CZ gate is applied, at (c) a pulse excitation followed by triondecay produces photons (open circles). These procedures arethen repeated, leading to the states of (d),...(h). Details ofthe states produced are in the text. Note that to recover thestandard form of cluster states one must use a mapping wherethe logical qubit | (cid:105) state is equivalent to the photonic state −| L (cid:105) , this is because, for practical reasons, R y ( π/
2) gates areused instead of Hadamard gates. quency, so that there are no constraints on the resonanceenergies of the two QDs.The state evolution for the idealized abstract proto-col is depicted in Fig. 1, for a quantum circuit logicallyequivalent to the protocol see Fig. 2. For simplicity weassume that the two QDs are initialized in the spin upstate | ↑(cid:105)| ↑(cid:105) (in fact no initialization is necessary - itcan be effected later via measurements on the photons).First we apply a R y ( π/
2) operation on each spin yielding( | ↑(cid:105) + | ↓(cid:105) )( | ↑(cid:105) + | ↓(cid:105) ), as in Fig. 1(a). This is followedby a C-Z gate entangling the dots, ( |↑(cid:105)|↑(cid:105) + |↑(cid:105)|↓(cid:105) + |↓(cid:105)|↑(cid:105) − |↓(cid:105)|↓(cid:105) ), producing the bond in Fig. 1(b). Imme-diately after this we apply the pump pulse to each dot,and the creation of the subsequent photons yield the state( |↑(cid:105)| R (cid:105)|↑(cid:105)| R (cid:105) + |↑(cid:105)| R (cid:105)|↓(cid:105)| L (cid:105) + |↓(cid:105)| L (cid:105)|↑(cid:105)| R (cid:105) − |↓(cid:105)| L (cid:105)|↓(cid:105)| L (cid:105) ).In the circuit of Fig. 2 this is equivalent to the CNOT gates. This resulting state is equivalent to a 2-qubit clus-ter state, where the logical state | (cid:105) ( | (cid:105) ) is redundantlyencoded [4] in 2 qubits as | ↑(cid:105)| R (cid:105) ( −| ↓(cid:105)| L (cid:105) ). Graphi-cally such a situation is depicted with the circles for eachqubit adjacent to each other, Fig.1(c). A second R y ( π/ C-Z gate is implemented optically bycoupling to trion states which are higher in energy thanthe ones used for the single dot photon emission. Thesehigher-energy trion states are delocalized, i.e. the volt- y R y R y R y R top dot bottom dot photon photon photonphoton FIG. 2: A quantum circuit which is logically equivalent to theidealized evolution of the two QDs. The CZ gates correspondto the interdot coupling, the CNOT gates to photon emission,and the R y to precession by π/ y direction. age bias is such that one of the electrons in these higherenergy states is tunnel-coupled, in contrast to the single-electron ground states (denoted | B (cid:105) and | T (cid:105) for bottomand top QD respectively) and the lower energy trionstates, which are isolated from one another and local-ized to their respective quantum dots. This trion medi-ating the inter-dot interaction has two electrons in the | B (cid:105) and | T (cid:105) (“s”-type) states, and the third electron inthe first excited (“p”-type) orbital, which we take to bethe one that is a delocalized (also called ‘molecular’ or‘extended’) state, denoted | E (cid:105) . It has been shown ex-perimentally that such a regime is feasible [7]. The holeis taken to occupy a single orbital state | H (cid:105) , which forsimplicity we take to be completely confined to one QD(this assumption does not deleteriously affect the overallproposal).With the three electrons in distinct orbital states, thespin configuration can acquire any of its allowed values byadding the three angular momenta. So, for a given orbitalconfiguration there are a total of eight electron states(two S = 1 / S = 3 / S = 3 / | / (cid:105) = | A (cid:105)| ↑↑↑(cid:105) (1) | / (cid:105) = | A (cid:105) ( | ↑↑↓(cid:105) + | ↑↓↑(cid:105) + | ↓↑↑(cid:105) ) / √ | ¯1 / (cid:105) = | A (cid:105) ( | ↓↓↑(cid:105) + | ↓↑↓(cid:105) + | ↑↓↓(cid:105) ) / √ | ¯3 / (cid:105) = | A (cid:105)| ↓↓↓(cid:105) (4)where | A (cid:105) = 1 √ (cid:0) | T BE (cid:105) − |
BT E (cid:105) − |
T EB (cid:105) + | BET (cid:105) − |
EBT (cid:105) + | ET B (cid:105) (cid:1) . (5)The electron and hole in semiconductors are coupled byexchange interactions. In QDs, these are quite strong (onthe order of, or stronger, than typical Zeeman energies),and they are separable into ‘isotropic’ and ‘anisotropic’terms [8]. The isotropic term is much stronger - typicalvalues of this are 0.3-0.5 meV - so it is the leading termin our parameter regime. Its physical origin is the lack ofinversion symmetry (along the growth direction) in theQD. We will ignore the anisotropic term, which originatesfrom in-plane asymmetry (deviation of the QD cross sec-tion from a disk) and is typically small, in the order of µ eV [8, 9]; its effects can be incorporated as standarderrors in the gate.The Hamiltonian is therefore given by H = (cid:80) i α z ( r i , r h ) s iz j z , where α z ( r i , r h ) is an operator act-ing on the envelope wavefunctions of the electrons andthe hole. The index i runs over the three electrons, r h denotes the position of the hole, and z is the growth axis.The operator j acts on the hole spin, which we take to bea pseudospin, j z = ± / H as H = 13 (cid:88) i α z ( r i , r h ) S z j z (6)+ 13 (cid:88) ( ij ) ∈{ (12) , (23) , (31) } ( α z ( r i , r h ) − α z ( r j , r h ))( s zi − s zj ) j z , with S z = (cid:80) i s iz . The first term on the RHS of Eq. (6)conserves the total electron spin, so it only has nonzeromatrix elements within the three total spin subspaces dis-cussed above. The second set of terms has nonzero ma-trix elements only between different total electron spinstates. Since typical values of the electron-electron ex-change are about one order of magnitude more than typi-cal electron-hole exchange interactions, we can ignore thetotal spin mixing terms and focus on the Hamiltonian H (cid:39) (cid:88) i α z ( r i , r h ) S z j z , (7)and only consider the states (1)-(4) tensored with thehole state, which is an 8 × α in state | A (cid:105)| H (cid:105) is (cid:88) i (cid:104) H |(cid:104) A | α ( r i , r h ) | H (cid:105)| A (cid:105) = 2 × (cid:88) K = B,T,E (cid:104) H |(cid:104) K | α ( r, r h ) | H (cid:105)| K (cid:105)≡ δ BH + δ T H + δ EH . (8) Assuming that the hole is localized in one of the twoquantum dots, say the one labeled by B , we have δ T H =0. We will define the sum of the nonzero terms to be δ .Now we have the operator H / = δ S z j z , (9)acting only on the spin states. Clearly, this operatoris already diagonal in the basis we have chosen. Sinceit is invariant under the simultaneous flip of S z and j z ,we expect the states to be doubly degenerate. Then theeigenenergies and corresponding eigenstates are: E = δ | / (cid:105)| ⇑(cid:105) , | ¯3 / (cid:105)| ⇓(cid:105) (10) E = δ
12 with eigenstates | / (cid:105)| ⇑(cid:105) , | ¯1 / (cid:105)| ⇓(cid:105) (11) E = − δ
12 with eigenstates | / (cid:105)| ⇓(cid:105) , | ¯1 / (cid:105)| ⇑(cid:105) (12) E = − δ | / (cid:105)| ⇓(cid:105) , | ¯3 / (cid:105)| ⇑(cid:105) (13)The states with energy E are dark. The remaining onesare optically accessible. We are particularly interested inthe states with energy E . These states are coupled onlyto the two-qubit states | ↑↑(cid:105) and | ↓↓(cid:105) by polarization σ − and σ + respectively. The two-qubit states | ↑↓(cid:105) and | ↓↑(cid:105) couple to the states with E , E with these polarizations.We take advantage of the energy splitting between E and E , E to selectively address only the two-qubit | ↓↓(cid:105) state and realize the C-Z gate.For simplicity we fix the polarization of the pulse to σ + (behaviour for the orthogonal polarization is foundby flipping all the spins). If we label the dipole matrixelement for transition | ↓↓(cid:105) → | ¯3 / (cid:105)| ⇑(cid:105) to be d . Thenonly the triplet state | T + (cid:105) couples to the excited state | ¯1 / (cid:105)| ⇑(cid:105) with dipole strength (cid:113) d .Given these three transitions, we can implement the c-z gate by acting with a resonant 2 π pulse on the | ↓↓(cid:105) state and avoid coupling to the other transitions.We now turn to a consideration of the various sourcesof errors and imperfections. A crucial feature of our pro-posal is the fact that all non-leakage errors in the system localize . By non-leakage errors we refer to any decoher-ence which eventually returns the electrons back into thecomputational subspace - ie back into any state such thatone electron is located in the orbital ground state of eachdot. By localize we refer to the fact that the action of anydecoherence map on the electrons is (mathematically)equivalent to a (different) decoherence map on some ofthe emitted photons, however crucially the number ofaffected photons is at most the four photons emittedaround the time the decoherence event occurs. This en-sures that the final output state takes the form of an idealcluster subject to localized random noise–a noise modelfor which fault tolerant procedures are known to work.In particular we emphasize that this allows for produc-tion of photonic cluster states for arbitrarily longer timesthan the electron decoherence timescales might suggest.The error localization might be seen in quite a generalmanner as follows. Consider the quantum circuit of Fig. 2encoding the generic evolution. Let some decoherenceoccur which is described by a set of Kraus operators { K i } acting on the spin only. If we denote by U the unitaryevolution which corresponds in the figure to the circuitconsisting of four photon emissions (i.e., two photons perdot and including the CZ gate acting between the dots)then an error and subsequent evolution takes the genericform ρ (cid:48) i = U ( I ⊗ I ⊗ K i )( ρ spin ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | )( I ⊗ I ⊗ K † i ) U † . It is a remarkably nice feature of this process that in factwe can find a Kraus operator ˜ K i acting now only on thefour emitted photons, such that ρ (cid:48) i = ( ˜ K i ⊗ I ) U ( ρ spin ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) U † ( I ⊗ ˜ K i † ) . Physically this means that an error occuring on the spinis (mathematically) identical to some different error oc-curring on the photons subsequently emitted. Cruciallyit affects only the next four photons, and no more–hencethe term localization.We now discuss some specific sources of error, and theirexpected impact.
1. Imperfect CZ gate —If we label Ω the Rabi fre-quency of the target transition from | ↓↓(cid:105) , then the othertransitions see a Rabi frequency of Ω = Ω / √ = (cid:113) Ω , with a large detuning. As such some popu-lation is transferred to those excited states and it is notreturned via stimulated emission. Instead, the incoherentprocess of spontaneous emission redistributes that popu-lation. For simplicity we assume that the small popula-tion transferred is equal for the two unwanted transitionsand that spontaneous emission equally redistributes it.The simplest way to express the Kraus operators { K j } describing the generalized quantum evolution in the twospin qubit subspace is by one nearly unitary, CZ opera-tor: K = u | ↑↑(cid:105)(cid:104)↑↑ | + | ψ − (cid:105)(cid:104) ψ − | + u | ψ + (cid:105)(cid:104) ψ + | − | ↓↓(cid:105)(cid:104)↓↓ | , plus eight more operators describing the redistributionof the populations. For a pulse of a total duration of40 ps and for anisotropic exchange δ = 0 . | u | (cid:39) | u | ∼ .
99. Then the remaining operators, { K , K , ..., K } , are √ −| u | | k (cid:105)(cid:104) | and √ −| u | | k (cid:105)(cid:104) | ,with k = 1 , , ,
4. Since the operator sum representa-tion is not unique, we can find a different set of Krausoperators { M j } for which M is proportional to the CZ gate. Setting u = u ≡ u , these are M = α CZ , M = e iφ √ − α K − α √ ( K + K ), and M = √ ( K − K ), with φ = arctan (cid:16) Im ( u )1 − Re ( u ) (cid:17) . For j = 3 , ..., M j = K j .The value of α is a measure of how close the operation isto a unitary CZ . For u = u = 0 .
99 we find α = 0 . − α = 0 .
02. Physically we can thereforeinterpret the action of the gate as follows: With proba-bility α we obtain a perfect CZ gate, with probability(1 − α ) we obtain some other type of evolution.
2. Unequal g factors —In general, the two QDs com-prising the QD molecule will have different g factors, andtherefore different precession frequencies. This meansthat we cannot get both spins to undergo a R y ( π/
2) op-eration solely based on precession. One can correct forthis mismatch by spin-echo type control by applying tothe fast spin at time τ = π ( ω − f − ω − s ) / π rotation about the optical axis to delay it ( ω f , ω s arethe fast and slow Zeeman splittings respectively). Theserotations are by design fast (in the ps regime) [10] andhave been demonstrated experimentally [11].
3. Decay of one of the two resident electrons intothe other QD —When this error occurs it will causeboth quantum dots to stop emitting photons, and thusamounts to a detectable loss error on the cluster state.We want the hole to occupy the dot for which the sin-gle particle energy is lowest so that recombination willnow correct for this error. In general we expect the com-putation to be quite resilient to such loss. Dependingon the height of the tunneling barrier between the dots,eventually the system will decay back into the desiredcomputational basis with one electron in each dot.
4. Precession during CZ gate —This is an error whoseeffect will be essentially the same as the case of the sin-gle dot machine gun[5]. It results in a localizable er-ror, which, provided the magnetic field strength is chosensuitably, can be extremely low.In conclusion, we have developed a scheme for gener-ation of 2 × N dimensional photonic cluster state basedon coupled quantum dots. Analysis of the relevant errorsshows our proposal to be robust and feasible with currentstate of the art systems. This scheme can be generalizedto the generation of a two-dimensional sheet either byconsidering multiple stacked dots, or by employing cavi-ties and waveguides to couple distant dots. Future workwill include specific cavity-waveguide-quantum dot de-signs for generation of a cluster state sheet of arbitrarydimensions.This work was supported in part by the Engineeringand Physical Sciences Research Council and the US Of-fice of Naval Research. [1] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. ,5188 (2001).[2] R. Raussendorf et. al. , Annals of Physics , 2242(2006); [3] M. Varnava et. al , Phys. Rev. Lett. , 120501 (2006);ibid, Phys. Rev. Lett., , 060502 (2008).[4] D. E. Browne and T. Rudolph Phys. Rev. Lett. ,010501 (2005); Q. Zhang, et al, Phys. Rev. A. , 062316(2008).[5] N. H. Lindner and T. Rudolph, Phys. Rev. Lett. ,113602 (2009).[6] S. E. Economou and T. L. Reinecke, Phys. Rev. B ,115306 (2008).[7] E. A. Stinaff, M. Scheibner, A. S. Bracker, V. L. Pono-marev, V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, and D. Gammon, Science , 636 (2006).[8] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gor-bunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer,T. L. Reinecke, et al., Phys. Rev. B , 195315 (2002).[9] M. Scheibner et. al. , Phys. Rev. B , 245318 (2007)[10] S. E. Economou, L. J. Sham, Y. Wu, and D. G. Steel,Phys. Rev. B , 205415 (2006).[11] A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev,D. Reuter, A. D. Wieck, T. L. Reinecke, and M. Bayer,Nat. Phys.5