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Downloaded by guest on September 28, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1711003114 optics them states. characterize network and 2D approach of this states terms using quantum in produced many-body be of can class the that implementation identify the We guided allows feedback. coupled mirror where of mirror, the system while distant atom-like , a the or with represent waveguide atom 1D single a a implemen- to physical consider a we As feedback. tation, emit- quantum quantum time-delayed single 2017) via 18, a June ter using review states for two- cluster (received generate 2017 photonic to 19, protocol dimensional September deterministic a approved analyze and and CA, propose Pasadena, We Technology, of Institute California Kimble, Austria J. Innsbruck, Harry A-6020 by Sciences, Edited of Academy Austrian , Quantum and Optics 02138; Quantum MA for Cambridge, University, Harvard , of a Pichler Hannes feedback time-delayed via quantum photonic Universal esta eeeyrsrcstecaso civbesae.I con- In states. achievable of pro- class emission the sequential restricts is the severely of difference that nature cess The Markovian measurement- (20). the universal in for (MBQC) rooted resource computation a con- quantum as PEPSs based serve classically, simula- simulated that and efficiently states computation be tain can quantum while they for Significantly, as use tion, (25). higher limited (PEPSs) a have states to MPSs pro- pair leads so-called entangled feedback by jected delayed captured (MPSs) structure one-dimensional that states entanglement below product a dimensional show in we so-called only (24), by entangled characterized generic time- are way, a using in process enriched emitted pro- photons qualitatively sequential emission While be feedback. sequential can structure delayed entanglement a resultant the in the that of demonstrate emitter previously We single shown (21–23). be cess a was can by qubits it focus photonic generated where between We entanglement systems, computation. multipartite that optical quantum quantum universal basic on enable most resource its that may a in show already as form and we used entanglement be follows, quantum generate can what to feedback In quantum (7). coherent engineering delayed and science of state-of-the-art current in available experiments. using already simulation resources quantum to minimal and avenues opens computation back and quantum field states photonic entangled genera- photonic deterministic emitted complex for atom-like of allows the or tion approach of Our atomic emitter. part feedback the single reflecting delayed onto a by The cre- with introduced waveguide. to photonic (20) is protocol to state explicit coupled cluster emitter an 2D quantum present a var- We deterministic ate in (14–19). and (11–13) emitters systems 10) and ious photons (5, single photons between operations single of motivated gen- is high-fidelity eration demonstrating approach progress experimental Our recent (7–9). by quan- time-delayed feedback with using combination quantum in computation emitter coherent single quantum we a universal of Here control of tum systems. power optical with full states photonic linear the generate deterministically quantum of can one of scalability that technolog- show nature the and probabilistic limits (3) the gates conceptual (4–6), of breakthroughs number ical a despite However, Q nttt o hoeia tmc oeua n pia hsc,HradSihoinCne o srpyis abig,M 02138; MA Cambridge, Astrophysics, for Center Harvard-Smithsonian Physics, Optical and Molecular Atomic, Theoretical for Institute iedlydfebc sapwru oli eea areas several in tool powerful a is feedback Time-delayed en cieyepoe o h attodcds(,2). (1, decades two past is the photons for optical explored with actively processing being information uantum | eae feedback delayed a,b,1,2 ono Choi Soonwon , | htncqatmcomputation quantum photonic b,1 ee Zoller Peter , c nttt o hoeia hsc,Uiest fInbuk -00Inbuk Austria; Innsbruck, A-6020 Innsbruck, of University Physics, Theoretical for Institute c,d n ihi .Lukin D. Mikhail and , 2 1 the under Published Submission. Direct PNAS a is article This interest. wrote of conflict M.D.L. no and declare .Z., authors S.C., The H.P., and data; analyzed paper. S.C. the and H.P. research; performed encoding waveguide, the into (| pulses absence the of via emis- train qubits an a to a of leading consists of emitter, a protocol sion quantum Our and single, this 1A. waveguide of a Fig. excitation 1D in repeated is a depicted to approach as mirror coupled our distant (Q) of emitter quantum block driven building fundamental The State Cluster a 2D Photonic with already MBQC photonic delayed emitter. enabling that single stress for we allows (22), works emitters feedback previous quantum systems multiple While Markovian using of resource. by limitation this universal for we a compensating that proposed create memory to quantum non- effective harness system an the introducing renders Markovian, feedback quantum time-delayed trast, quency field from classical excited be two a can supporting by 2A, manipulated Fig. coherently in states depicted structure metastable internal an with cluster 2D the of 1C). example Fig. specific also the (see for state now discuss we delay as time ture the by separated pulses τ photon pho- between emitted also subsequently can but emitter between tons the only way, this not In correlations 1B). create (Fig. once than more emitter delay the time a with ter the uhrcnrbtos ...... n ...dsge eerh ..adS.C. and H.P. research; designed M.D.L. and P.Z., S.C., H.P., contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom work. To this to equally contributed S.C. and H.P fetvl hslast w-iesoa nageetstruc- entanglement two-dimensional a to leads this Effectively . htn sn iia xeietlresources. experimental with minimal processing using information photons quantum computation. for avenues quantum of opens for creation This resources for universal are allows that already form states feedback basic experiments most quantum state-of-the-art the delayed emitters, states. of in quantum quantum that atom-like single photonic show with achievable we particular, of dramatically In feedback class quantum the photons. delayed expands that between emit- correlations show quantum we quantum a Here induce using can photons that directly, generate ter sequentially interact approach One to not element. is nonlinear do a requires photons them infor- entangling quantum quantum Since for as processing. challenge photons central mation a with is carriers states information entangled large Creating Significance o oceees efcso neitr(representing emitter an on focus we concreteness, For k t us.Temro ed hs htn akt h emit- the to back photons these feeds mirror The pulse. -th Ω 2 (t olwn ahectto,teao ildcyto decay will atom the excitation, each Following ). b NSlicense. PNAS |g |g 1 2 τ | ≡ i i uhta h htn a neatwith interact can photons the that such , oastate a to 0i 0i k rpeec (|1i presence or ) Q and |e L i |g sn ae ihRb fre- Rabi with a using 2 | ≡ i NSEryEdition Early PNAS 1i Ω Q b 1 k Department (t fapoo in photon a of ) hc a be can which , h emitter The ). d Institute | f6 of 1 Q)

PHYSICS A quantum in state |g1i or the photon mode is empty, there is no interaction. emitter This process implements a controlled σz gate mirror 1D wave guide ˆ 1 z ZQ,k = |0iQh0| ⊗ k + |1iQh1| ⊗ σk [1]

B vaccuum input entangling the atom and the k-th photon. In turn, the subse- quently generated k + N -th photon inherits this entanglement, thereby giving rise to an effective 2D entanglement structure (Fig. 1C). delay line, time delay output: 2D Formally, the protocol can be interpreted as a sequen- tial application of gates XˆQ,k+N ZˆQ,k HˆQ, on the atom and C photonic qubits k and k + N , on the initial trivial state |0i N |0i C D Hˆ = √1 (σz + σx ) Q k k (Fig. 2 and ). Here Q 2 Q Q and ˆ 1 x XQ,k = |0iQh0|⊗ k + |1iQh1| ⊗ σk . One can show that after (M + 1) × N turns, this gives exactly the 2D cluster state on a M × N square lattice with shifted periodic boundary conditions (see Materials and Methods):

N (M +1)  Y ˆ ˆ ˆ O |ψC2D i =  XQ,k+N ZQ,k HQ |0iQ |0ik . [2] k=1 k

2D cluster state Universal quantum computation (27) can be performed by sequentially measuring each photonic directly at the output Fig. 1. Schematic setting. (A) A quantum emitter Q is coupled to a 1D port—for example, using another atom as a high-fidelity mea- waveguide that is terminated on one side by a (distant) mirror. (B) In each surement device. This can be implemented using pulse shaping time step k, Q can emit a phtoton (k + N) toward the mirror—that is, into techniques (Fig. 3A), allowing for a perfect absorption of each a delay line—and interact with a photon k returning from the delay line, photon by the second atom (28, 29). exiting at the output port. (C) of the resulting entanglement structure: Wrapping the photon channel displayed in B around a cylinder Experimental Requirements. Photons generated in this pulsed with the proper circumference (equivalent to the time delay τ), it can be scheme have a finite bandwidth B. To realize the controlled phase seen that in each time step Q interacts with two photons that are neigh- gate given in Eq. 1, this bandwidth must be small—that is, B  γR boring along the artificial, second dimension. Therefore, as time progresses, (26); otherwise, the wave packet is distorted when scattered Q creates entanglement between neighboring photonic qubits, both in the physical and the artificial dimension. by the atom, reducing the gate fidelity, FZ (Fig. 3C). Narrow- bandwidth photons and high-fidelity gates can be obtained by shaping the temporal profiles, eliminating the error to first order

|g2i, emitting a photon into the waveguide. For now, we assume that the atom–photon coupling is chiral (26) such that every such photon is emitted unidirectionally—that is, to the left in Fig. 1A AC (see however below). Finally, another |eRi, degen- erate with |eLi, couples to the right, moving photons reflected from the mirror. We denote the corresponding decay rates by γL and γR.

Protocol. Our protocol starts by first generating 1D cluster states of left-propagating photons (21). To this end, the atom is initially D prepared in the state |0iQ. Then, a rapid π/2 pulse is applied on the atomic qubit, followed by a π pulse on the |g2i → |eLi tran- B sition (Fig. 2 A and B). The subsequent decay from state |eLi to |g2i results in entanglement between the atom and the emit- ted photon—that is, |0iQ|0i1 + |1iQ|1i1. Repeating this pulse sequence for n times leads to a train of photonic qubits in the form of a 1D cluster state. An analogous scheme has been demonstrated in an experiment using a (23), fol- lowing a seminal proposal by Lindner and Rudolph (21). Remarkably, the 2D cluster state is generated from exactly the same sequence if we take into account the effect of the mirror and the scattering of the reflected photons from the atom. We Fig. 2. Protocol to generate the photonic 2D cluster state. With the level are interested in the situation where the time delay τ is so large structure (A) and periodic, pulsed coherent drive (B), the setup in Fig. 1A that the k-th photon interacts for the second time with the atom leads to a 2D cluster state of photons in the output. During each step between the generation of the (k + N − 1)-th and k + N -th pho- k = 1, 2, ... (of duration T), the pulse (B) sequence realizes a unitary involv- τ = (N − 1/2)T ing the emitter Q (A) and two photonic qubits, k and k + N [realized by ton. This is achieved, for example, by setting a right (R) and left (L) moving photon, respectively], with N = 3 in D. The (with T the protocol period; see Fig. 2B). Crucially, if the atom corresponding circuit diagram is shown in C and D. Each step consists of a |g i ˆ ˆ is then in the state 2 , the returning photon is resonantly cou- Hadamard gate HQ and a controlled phase gate ZQ,k between the atom and pled to the |g2i → |eRi transition, picking up a scattering phase the photon qubit k returning from the delay line, followed by a controlled- ˆ shift of π without any reflection (26). In contrast, if the atom is NOT gate XQ,k+N (see Eq. 2).

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1711003114 Pichler et al. 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PHYSICS K ! K Y O |ψ i = |+i | i . AB C1D HQXQ,k Q 0 k [9] k=1 k=1 We note that this representation of the 1D cluster state has been also used in ref. 21. We now proceed to the construction of the 2D cluster state. From its definition, the 2D cluster state can be obtained from the 1D cluster state above by introducing additional entanglement (via phase gates) between qubits k and k + N (27). This gives a 2D cluster state on a square lattice with shifted periodic boundary conditions, where the extend of the (shifted) periodic direction is set by N.

 K  K ! Y Y O |ψ i = |+i |0i C2D  Zk,k−N HQXQ,k Q k Fig. 4. Tensor network representation of the generated states. (A) Graph- k=N+1 k=1 k i ical representation of the rank-5 tensors U[k] . The “physical index” i K ! a,b,c,d Y O denotes the state of the qubit k, the “horizontal bonds” b and d run over = HQZk,k−NXQ,k |+iQ |0ik. [10] internal degrees of freedom of Q, and “vertical bonds” a and c enumerate k=1 k the quantum states of input into and output from the delay line, respec- In the second line, we used the fact that [Zj,j+N, XQ,k] = 0 for j > k, and tively. (B) Representation of the state Eq. 4 in terms of these tensors. Con- Z , |0i = 1 ⊗1 |0i . Now we make use of the identity nected lines indicate contractions, and red open lines denote physical indices k j j k j j

corresponding to the states of qubits in the output. In the top row, the ten- Zn,mXQ,n|ψiQ,m|0in = XQ,nZQ,m|ψiQ,m|0in, [11] sors are contracted with the initial state of Q and the vacuum state for the where |ψi is an arbitrary state of Q and every qubit including m 6= n first N qubits in the string, as indicated by the black circles. Open legs at the Q,m but not n. Again, this relation is not an operator identity but a property of bottom correspond to the state of the qubits in the delay line after step k, a state where qubit n is in the separable state |0i . Using this identity, we and the open line on the bottom right corresponds to the state of Q. n arrive at

K ! Y O |ψ i = |+i |0i C2D HQXQ,kZQ,k−N Q k [12] We finally note that this possibility of creating PEPSs as output k=1 k states of delayed feedback schemes allows for developing vari- ational photonic quantum simulators of strongly correlated 2D which is (up to a shift of the index and a rotation of Q) exactly our protocol given in Eq. 3. systems, extending recently implemented proposals for 1D sys- tems (18, 36) to higher dimensions. This is particularly impor- General Entanglement Structure. In this section, we provide details on the tant, since unlike in 1D, quantum simulators in higher dimen- explicit connection between the class of achievable states from using a sin- sions cannot be efficiently simulated by classical . gle emitter with delayed quantum feedback and 2D tensor network states (PEPS). To this end, we use induction to show that the repeated application Outlook of Uˆ[k] in Eq. 4 results in a 2D tensor network that “grows” by an additional Our work can be extended in several ways, including the addi- tensor in each time step (Fig. 4B). This can be seen by induction: we assume tion of multiple delay lines, leading to tensor networks in higher that the quantum state of the system after k − 1 steps can be written as X iQ,{ij} dimensions. This is of relevance for fault-tolerant implementa- |Ψ(k − 1)i⊗|0i = C[k − 1] |iQ,{ij}i ⊗ |0i, [13] tions of MBQC using 3D cluster states (20). Additionally, on iQ,{ij} can envisage implementing variants of MBQC that can toler- ate up to 50% of counterfactual errors due to photon loss (37). where |0i is the input state of the k + N-th qubit (Fig. 1B) and iQ,{i } Finally, besides nanophotonic setups, our “single-atom quantum C[k − 1] j denotes the contraction of the 2D tensor network formed ” can be implemented in a circuit QED system with by U[`] (` ∈ {1, ··· , k − 1}) in Fig. 4B with open indices iQ, {ij}. The basis state |iQ, {ij}i enumerates configurations of Q and the array of k − 1 out- microwave photons (16, 18), surface-acoustic, or bulk-acoustic put qubits (j ∈ {1, ··· , k − 1}) and N memory qubits in the delay line waves (32, 38). (j ∈ {k, ··· , k +N −1}). The application of Uˆ[k] in the k-th step modifies the quantum state of k-th and k + N-th qubits and Q. From Eqs. 3 and 4, new Materials and Methods quantum amplitude A for a configuration |i , {i }i can be written as iQ,{ij} Q j 2D Cluster State Representation. In this section, we prove that |ψC i in Eq. 2D 0 0 X ik Y i ,{i } 3 represents the 2D cluster state. To this end, we start first with the repre- A = U[k] δ 0 C[k − 1] Q j iQ,{ij} i ,i ,i0 ,i0 i ,i + k+N Q k Q j j sentation of the 1D cluster state on K 1 qubits: i0 ,{i0} j∈Q/ ,k Q j K i ,{i } Y ⊗ + Q j |ψ i = |+i K 1. = C[k] , [14] C1D Zk,k+1 [6] k=1 ik where the second line is achieved by identifying U[k] 0 0 as a ik+N,iQ,i k,i Q Using the swap operator Si,j that exchanges the quantum states of qubits i 0 rank-5 tensor in Fig. 4A and by realizing that the summations over i k and and j, we can rewrite the 1D cluster state as 0 i Q are equivalent to the contraction of vertical and horizontal bonds of K ! K newly introduced tensor U[k], respectively (see Fig. 4B). We note that the Y O |ψ i = |+i |+i generated network in Fig. 4B is contracted with shifted periodic boundary C1D SQ,kZQ,k Q k [7] k=1 k=1 conditions. This inductive construction also shows that the arbitrary 2D tensor net- where we used the relation Zb,c = Sa,bZa,cSa,b and identified the K + 1th work satisfying Eq. 5 can be generated from our method via appro- qubit with the ancilla Q. Note that throughout this paper, we use a priate choice of unitary Uˆ[k]. Interestingly, this class of states includes Qk convention where the ordering in the product is defined via j=1 Mj = exotic states with such as string-net states (34) or the MkMk−1 ... M1. We now use the relation of Kitaev’s Hamiltonian (35). For example, the latter state can be represented by a 2D network of translationally invariant tensors SQ,v ZQ,v |ψiQ ⊗ |+iv = HQXQ,v Hv |ψiQ ⊗ |+iv , [8] i ,i ,i ,i U 1 2 3 4 = δ δ δ δ /4, where i , ..., i denote the states a,b,c,d a+b,i1 d+a,i2 c+d,i3 b+c,i4 1 4 where Hx is the Hadamard gate acting on qubit x. We note that this equality of four physical qubits in a unit cell and {a, b, c, d} run over bond dimen-

is not an operator identity but a property of states of the form |ψiQ ⊗ |+iv , sion 2. This tensor explicitly satisfies Eq. 5 and can immediately be translated where the qubit v must be in the state |+iv while the ancillary system |ψiQ into a protocol similar to the one for generating the 2D cluster state—that may be in an arbitrary state, potentially entangled to other systems. Using is, using only controlled single photon generation and atom–photon phase these relations, our state can be written as: gates between that atom and feedback photons.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1711003114 Pichler et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 6 o C ta.(05 rbn h unu aumwt natfiilao nfotof front in atom artificial an with vacuum quantum the Probing (2015) al. et IC, Hoi 16. crystals. photonic in interactions S Atom–light 15. (2014) al. et atom. A, single a Goban with switch 14. phase quantum Nanophotonic (2014) optical flying al. a et between TG, gate Tiecke quantum A 13. (2014) S Ritter G, Rempe N, Kalb A, Reiserer for 12. platform Nonlinear (2014) nanophotonics A Rauschenbeutel diamond C, Junge integrated M, Scheucher An J, Volz (2016) 11. al. et A, Sipahigil 10. 7 tt ,e l 21)Qatmsaetase rma o oaphoton. a to ion an from transfer Quantum-state (2013) al. et A, Stute 17. a eotie by obtained be can for that, shows calculation Straightforward packet wave time. Gaussian in the example, symmetric are that  gate: phase controlled the of fidelity the obtain we this, With overlap the calculate to straightforward is It ihe tal. et Pichler via coupling the shape we If packet wave the that such state in is atom the transmission: (if the by scattering forward chiral the normalization for a chose we where profile, spectral Lorentzian a has step single is profile a temporal in whose produced photon each Imperfections. .PclrH olrP(06 htnccrut ihtm easadqatmfeedback. quantum and delays time with control. circuits Photonic (2016) feedback P Zoller quantum H, Pichler Time-delayed 9. (2015) AL Grimsmo 8. (2010) GJ Milburn HM, Wiseman emitters. 7. single-photon Solid-state (2016) M Toth D, Englund I, and Aharonovich efficiency 6. extraction high with photons single On-demand (2016) al. et X, Ding 5. . quantum one-way Experimental (2005) computation al. quantum et efficient P, Walther for scheme 4. A (2001) GJ Milburn R, Laflamme E, interferometry. Knill and 3. entanglement Multiphoton (2012) al. et JW, Pan qubits. photonic with 2. computing quantum optical Linear (2007) al. et P, Kok 1. − cuits. 7:219–222. mirror. a 5:3808. Nanotechnol atom. trapped single a and photon atom. resonator-enhanced single a Photon with interacting photons fibre-guided single networks. quantum-optical Lett Rev Phys 115:060402. UK). Cambridge, Press, Univ ton micropillar. a in Lett dot Rev quantum Phys driven resonantly a from indistinguishability near-unity 434:169–176. optics. linear with 84:777–838. Phys R lnrI ta.(05 eemnsi htneitrculn ncia htnccir- photonic chiral in coupling photon–emitter Deterministic (2015) al. et I, ollner ¨ 0 t 10:631–641. dsγ ˜ f 79:135–174. a Nanotechnol Nat (t ) 8:965–970. L f a Phys Nat (s)/2 = = (t F ) i − Z 508:241–244. = Z 116:093601. 116:020401. ihu hpn h aepceso h mte photons, emitted the of packets wave the shaping Without = √  √ us hpn losfrcetn htnwv packets wave photon creating for allows shaping Pulse . d γ 2 11:1045–1049. a Nanotechnol Nat L ω γ γ  γ L R e √ l 5 γ 2π γ γ −γ 10:775–778.   Z f L γ R R (t (t L γ γ γ γ L − + ) ) R dtf R t l ω ω l Science f /2 = = γ γ (t + + − + t Θ(t L L ∗ rnfrsinto transforms ) (ω q √ e 1 (t 3 i i γ γ −γ   ) ) π B unu esrmn n Control and Measurement Quantum ) ˜ f R R 2 = 354:847–850. a Nanotechnol Nat (1 + (t R 409:46–52. = /2 /2 / 2B L √ t ) /2 dt − 5 ω ω γ i = e ω π Z L −B |f − + = (t − erf(B e + − −B (t ) d i i 1 2 2 = 1 γ γ )| i 1 ω 1 (t γ γ R − R 2 2 −t − + (t L /2 /2 R 4Ω (t √ = /2 2π −t γ − 5 4 0 − γ γ γ γ 2 R ) .Tesatrn hs shift phase scattering The 1. 2 L L (t e 0 L γ γ γ /γ /γ t ) ) −i 508:237–240. R L 2 0 ω then , L /2 ))) e ω R R + + −γ t . O 1 i γ R L

t |g /2 /2 γ γ 2 f  sdetermined is i) L R (t e 2 2 −i π Θ(t a Nanotechnol Nat ! ) = hsRvLett Rev Phys . ω hs hf for shift phase e o Phys Mod Rev a Commun Nat t ) p (Cambridge a Photon Nat γ e Mod Rev a Pho- Nat L (t exp ) [21] [19] [18] [20] [17] [16] [15] [22] Nat Nat 4 urcae ,Aud ,VdlG(09 xlcttno ewr ersnainfor representation network tensor Explicit (2009) G Vidal M, Aguado long- O, of Buerschaper demonstration for 34. Proposal (2017) T Rudolph P, Shadbolt A, atom. Milne artificial T, an Nutz to coupled 33. phonons Propagating (2014) al. et trans- MV, induced Gustafsson Electromagnetically (2005) 32. J Marangos A, Imamoglu crystals. M, photonic in Fleischhauer light Slow 31. (2008) T Baba 30. computer. quantum one-way optics. A quantum (2001) HJ Chiral Briegel (2017) R, Raussendorf al. 27. et P, sys- Lodahl body quantum-many 26. for Renormalization (2004) JI Cirac F, Verstraete 25. pho- entangled of state Sch cluster a 24. of generation Deterministic (2016) pho- al. et 2-dimensional I, generated Schwartz Optically (2010) 23. T photonic Rudolph of N, sources Lindner on-demand SE, pulsed Economou for 22. Proposal (2009) T Rudolph NH, Lindner 21. D DE, Browne HJ, quan- Briegel single and 20. photons single Interfacing (2015) S Stobbe S, Mahmoodian P, Lodahl experi- 19. by systems many-body quantum interacting Exploring (2015) al. et C, Eichler 18. 9 ehlM ta.(04 irwv-otoldgnrto fsae igepoosin photons single shaped of generation Microwave-controlled entan- (2014) al. and et transfer M, Pechal state Quantum 29. (1997) H Mabuchi HJ, Kimble P, Zoller JI, Cirac 28. s with is, via the from computed can process this for fidelity gate The the error. quantity before step an even next to the proceed leading otherwise, would photon; protocol emitted our the in of extend temporal the than gate the for order |e X utinSineFn,a ela h uoenRsac oni Synergy Council Research European the Special as Matter. the Ultracold by as the Grant of Fund, supported Science Science Quantum is of Austrian Applications Innsbruck and Foundations at Program Smithsonian Research Work the Kwanjeong from Foundation. and support Atomic, University Educational the Theoretical Harvard acknowledges for S.C. Institute at Observatory. sup- the Astrophysical Physics is for H.P. Optical grant University Fellowship. a and through Multidisciplinary Molecular, Faculty NSF the Bush the Vannevar via by the ported Research and the Scientific Atoms, Initiative, through of Ultracold Research supported for Office Center was Force the work (NSF), Air This Foundation Science discussions. National useful the for Verstraete F. and ACKNOWLEDGMENTS. faster. wave again photon shaped is a approach of this case packet, the in but exponentially, 1 approaches fidelity large For Eq. photon shaped wave symmetric temporally a of fidelity. consequence the a improve shaping significantly is the can without This which case packet, packet. previous wave the a in unlike of expression this in vanishes fidelity the gives with from calculated be can gate phase the of fidelity corresponding The ˆ Q,k L ntepooe mlmnaint raete2 lse tt,tegate the state, cluster 2D the create to implementation proposed the In | → i ag lse tt nageeti h rsneo htnloss. models. string-net of photon states ground of the presence the in entanglement 2:066103. state cluster range ence media. coherent in Optics parency: 480. cond–mat–0407066. . arXiv.org dimensions. higher and two in tems states. multiqubit entangled tons. dots. quantum coupled from state cluster tonic strings. state cluster computation. quantum based nanostructures. photonic with dots tum circuits. superconducting in X states Rev product matrix continuous creating mentally ici unu electrodynamics. network. quantum a in 3221–3224. nodes distant among distribution glement 86:5188–5191. F +N x X nC oaoE esreeF ia I ofM 20)Sqeta eeainof generation Sequential (2005) MM Wolf JI, Cirac F, Verstraete E, Solano C, on ¨ = = 346:207–211. Science g sraie yeiso fapoo soitdwt h transition the with associated photon a of emission by realized is γ 5:041044. 2 B L 1 x i (t /γ = − ) uigtescn afo ahtm tpwt period with step time each of half second the during = R B 354:434–437. 2 3 n h asinerrfunction error Gaussian the and T γ Z  L /4  F + dtf si Eq. in as = X Z ˆ  22 hsRvLett Rev Phys Q,k = 6 1 ∗ Z  ,w have we 1, (t t 1 2 etakJ .Crc .Heea,L in,N Schuch, N. Jiang, L. Haegeman, J. Cirac, I. J. thank We +N 0 (with t ihu hpn h htnwv packet—that wave photon the shaping Without . ) rW asedr ,VndnNs 20)Measurement- (2009) M Nest den Van R, Raussendorf W, ur 0 − ¨ ˜ f +T (t owr,w hsrequire thus we work, to 5 8 hsRvLett Rev Phys ) oegets 15—one /2 x a Phys Nat = = t 2 0 dt + −1 1 hsRvX Rev Phys = 103:113602. γ e o Phys Mod Rev − O L  −T (t + → (x √ 5:19–26. exp ) hsRvB Rev Phys e o Phys Mod Rev )w find we /4) 4x 4 π e .W oeta h ierodrin order linear the that note We ). 95:110503. −x e 2 4x 4:041010. a Photon Nat +   2 1 2 = /( hsRvLett Rev Phys − O √ 77:633–673. e Z 1 (x 79:085119. −γ x 0 π − 4 t NSEryEdition Early PNAS x erf(z ) dsγ L 87:347–400. .I ohcss h gate the cases, both In ). erf a Nanotechnol Nat  T /2 2:465–473. = T L ) hl o h pulse- the for while , (s) ob uhlarger much be to /2 1 = X 2x ˆ 105:093601. 1  − Q,k √  2 π √ N R hsRvLett Rev Phys 2 scompleted, is 0 z √ hsRvLett Rev Phys dte 1+erf(B P Photon APL erf(B −t | 541:473– T 2 /4) This . f6 of 5 T T [24] [23] Phys In . /4) Sci- 78: x .

PHYSICS 35. Kitaev AY (2003) Fault-tolerant quantum computation by . Ann Phys 303: 37. Varnava M, Browne DE, Rudolph T (2006) Loss tolerance in one-way quantum com- 2–30. putation via counterfactual error correction. Phys Rev Lett 97:120501. 36. Barrett S, Hammerer K, Harrison S, Northup TE, Osborne TJ (2013) Simulating quan- 38. Chu Y, et al. (2017) Quantum acoustics with superconducting qubits. arXivorg p. tum fields with cavity QED. Phys Rev Lett 110:090501. 1703.00342.

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