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is 2, correction January error review Quantum for (sent 2021 6, July Zeilinger, Anton by Contributed and Japan; 560-8531, China; ; Austria; 230026, Vienna, Vienna, Anhui A-1090 A-1090 Hefei, Sciences, , China; of Quantum 230026, Academy and Hefei Austrian Information China, Quantum of in Technology Centre and Innovation Science of University Physics, Modern a Lu Chao-Yang ee ainlLbrtr o hsclSine tMcocl,Uiest fSineadTcnlg fCia ee 306 China; 230026, Hefei China, of Technology and Science of University Microscale, at Sciences Physical for Laboratory National Hefei swl nw htqatmmcaispoie new a provides of transmission quantum and manipulation, that creation, the known for well paradigm is t unu ro orcinwrsb noigteinformation the encoding by works correction error Quantum h ainlIsiueo nomtc,Tko1183,Japan 101-8430, Tokyo Informatics, of Institute National 01Vl 1 o 6e2026250118 36 No. 118 Vol. 2021 a,b,c a,b,c,1 iLnWang Xi-Lin , g a,b,c,2 T ai eerhLbrtre,NTRsac etrfrTertclQatmPyis T oprto,Kngw 4-18 Japan; 243-0198, Kanagawa Corporation, NTT Physics, Quantum Theoretical for Center Research NTT Laboratories, Research Basic NTT igCegChen Ming-Cheng , | unu ro correction error quantum no Zeilinger Anton , f iiino dacdEetoisadOtclSine rdaeSho fEgneigSine sk nvriy Toyonaka University, Osaka Science, Engineering of School Graduate Science, Optical and Electronics Advanced of Division a,b,c esk Fujii Keisuke , a,b,c,1 d,e,2 | n inWiPan Jian-Wei and , aulErhard Manuel , unu teleportation quantum f iLi Li , e inaCne o unu cec n ehooy aut fPyis nvriyo Vienna, of University Physics, of Faculty Technology, and Science Quantum for Center Vienna a,b,c a-eLiu Nai-Le , d,e,1 a,b,c,2 | a-e Zhong Han-Sen , a,b,c doi:10.1073/pnas.2026250118/-/DCSupplemental at online information supporting contains article This 2 1 BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This interest.y competing no declare authors Roma The di Studi degli Universita F.S., and Sapienza.y Sciences; La Photonic of Institute H.d.R., Reviewers: Bell a utilizes teleportation the 1C, Fig. in shown As space. subscript the where ulse etme ,2021. 3, September Published and A.Z., C.-Y.L., M.-C.C., W.J.M., Y.-H.L., K.N., and N.-L.L., L.L., data; 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PHYSICS √ A choice of the state |Ai = (|0i + eiπ/4|1i)/ 2, known as a magic state, is the most relevant to quantum computation. It is used to implement the T gate through magic state injection (3, 13)—a crucial approach toward a fault-tolerant non-Clifford gate. The same mechanism holds for a fault-tolerant implementation of nontransversal gates when the offline state preparation achieves the required precision through repeat-until-success strategies. More generally, a recursive application of this protocol allows us to implement a certain class of gates fault tolerantly, includ- ing a (14), which is also indicated in Fig. 1B. It is equally important to note that the quantum teleportation to the logical qubit is an important building block for distributed quan- tum computation and global quantum communications. The B teleportation-based schemes thus have the potential to significantly lower the technical barriers in our pursuit of larger-scale processing (QIP). In stark contrast to theoretical progress, quantum tele- portation and QECC have been developed independently in the experimental regime. We have seen quite a number of remarkable quantum teleportation demonstrations (15–27) and QECC experiments (28–35) performed in a number of physi- cal systems. However, the experimental combination of these operations, quantum teleportation-based quantum error cor- rection, is still to be realized. Given that it is an essen- tial tool for future larger-scale quantum tasks, it will be our focus here. In this work, we report on an experimental realization of the teleportation of information encoded on a physical qubit C into an error-protected logical qubit. This is a key step in the development of quantum teleportation-based error correc- tion. We begin by establishing an Einstein−Podolsky−Rosen channel—the entangled resource state for an error-protected logical qubit. Quantum teleportation involving a physical qubit of the entangled resource state transfers the quantum infor- mation encoded in one single qubit into the error-protected logical qubit. The quality of the entanglement resource state and the performance of the quantum teleportation are then evaluated. Experimental Implementation Fig. 1. Schematic illustration of teleportation-based error correction state encoding. In A and B, we show the fault-tolerant before The scheme shown in Fig. 1B is conceptually very similar to the and after combining with quantum teleportation, where the unreliable original teleportation protocol; however, currently, it is signif- operations, unknown state encoding, and nontransversal gate U2 are icantly more challenging due to the necessity of creating the marked with red blocks. The flow of quantum information is transmitted entangled resource Eq. 1 involving a logical encoded qubit— along the circuit from left to right. In A, errors will be accumulated as the especially when one considers optical implementations. Here our number of unreliable operations grows. In contrast, by introducing quan- logical qubit states tum teleportation, the “fragile nodes” can be replaced with preestablished entanglement states taking a specific form. As shown in B, the encoding 0 U 1 ⊗3 process and nontransversal gate U2 are replaced with states |ψ i and |ψ 2 i. |0i = √ (|000i + |111i) , Upon encountering “fragile nodes,” such as encoding, the circuit is paused L 2 2 0 + 1 [2] until a suitable |ψ i = |Φ i is generated. Then the BSM transforms quan- |1i = √ (|000i − |111i)⊗3 tum information holding by the initial state into the QECC, which can then L 2 2 be further operated by following logical gates. Scheme in C illustrates the teleportation-based QECC encoding where, to encode the unknown initial state, a physical qubit is entangled with logical qubit encoded in a specific are associated with the (9,1,3) Shor code (2), which is a repetition QECC. Then the BSM is performed between initial qubit and the physical of the three- Greenberger−Horne−Zeilinger (GHZ3) qubit with the measurement results fed forward to complete the transfer of state (36). More details concerning Shor code can be found in our quantum information into the QECC. SI Appendix. Now, given the complexity here, it is crucial to design and configure our optical circuit efficiently, remember- ing that, in linear optical systems, most multiple-qubit gates are state measurement (BSM) between the initial state |ψi to be probabilistic (but heralded) in . Only gates including the teleported and the single physical qubit of |Φ+i. Classical feed- controlled NOT (CNOT) gate between different degrees of free- forward of our BSM result ensures the initial is dom (DOFs) on the same single photon can be implemented in teleported into the encoded qubit. All these procedures, includ- a deterministic fashion. ing the generation of |Φ+i together with BSM, can, in principle, Our experiment is divided into three key stages: 1) the cre- be performed in a fault-tolerant manner (2). Quantum telepor- ation of the entangled resource state |Φ+i; 2) the preparation tation allows us to perform nontransversal gates offline, where and teleportation of the initial physical qubit |ϕi into the logical the probabilistic gate preparation can be done, as shown in Fig. qubit |ϕiL; and 3) readout of the logical state |ϕiL and detection 1B. The initial state |ψi could be an arbitrary state; however, the of error syndromes.

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QECC state physical–logical qubits. entangled 10-qubit desired target the generated the thus as have act state DoFs GHZ qubit two as control other acts the the always With polarization and the control, where the applied, The are their respectively. gates in CNOT polarization, initially are antidiagonal DoFs other and diagonal the where to only us elements enables optical also linear (see using but gates required in CNOT photons deterministic efficient of use resource number only the not of is terms DoFs per angular DoFs additional orbital more Using and directed two path (OAM). are use the 3 we with nine- photons, associated and the three photon construct 2, with to 1, code Now, photons Shor circuit. qubit qubit encoding while physical qubit BSM, logical the the the as to in acts used 4 be photon to photons, four these Among transmits which (PBS), splitter (H beam horizontally polarizing a two-photon GHZ on a entangled create combined to order maximally in so, par- a and This state, produces (37). geometry sandwich-like ticular a in by down- (SPDC) parametric begins conversion spontaneous It type-II beam-like 2. using (GHZ (36) Fig. GHZ state in four-photon shown polarization-entangled a circuit generating quantum the using formed only. stage readout qubit logic the for SPDs 24 OAM-to- thus and detectors an photon, single-photon using eight per in measured (SPDs) results is total, by DoF in followed This, OAM converter. measured, polarization the is Finally, DoF. DoF path polarization con- the three the contains First, syndromes stages. error measurement the measure secutive to used box) (purple stage teleport to necessary combina- state BSM in quantum (BS) the the splitter implement beam to a detection is coincidence box with green tion the in Shown trigger. a as state serves qubit arbitrary an cre- PBS with a with combination GHZ in a NLCs Two ate total. in photons six create to (NLCs) 2. Fig. Initial state xeietly h raino h hrcd Fg )begins 2) (Fig. code Shor the of creation the Experimentally, h rtkysaei h raino the of creation the is stage key first The NLC NLC NLC IAppendix SI Qubit 2 Qubit 5 PBS Qubit 1 Qubit 4 Qubit 3 |±i rnil xeietlshm.W mlytrenniercrystals nonlinear three employ We scheme. experimental Principle 4 tt nteplrzto o.Tefit htni programmed is photon fifth The DoF. polarization the in state (|H = ,0 0 0, |0, GHZ |ψ |Φ 04 oaie htn n eet etcly(V vertically reflects and photons polarized ) |ϕi 4 | ± i o details). for i + | ± i (| = ftefit htnit h ECsae h readout The space. QECC the into photon fifth the of i Encoding V (|H = Teleportation ,1 1 1, 1, H i)/ |±i H H H i|+i |ϕ BS √ i|0i i 2 i n agtqubits target and ob eeotd hl h it photon sixth the while teleported, be to ⊗3 and sgnrtdo ahpoo.We photon. each on generated is L + + |0i |−i |V |ψ |V tt.Te w consecutive two Then state. 4 i|−i i|1i 4 (|H = i tt,poos2ad3are 3 and 2 photons state, (|H = L Logic qubit ⊗3 )/ | − i )/ √ i Phase/ flips ◦ ⊗4 three-qubit a |0i, √ nigtefirst the ending 2, hstransforms This . |Φ V 2, + + i)/ |V i √ tt per- state i 2 ⊗4 Readout denote )/ √ [3] 4 2. ) ) eut ntefloiglgclstate: logical following the in results state the of h xetto auso l orosralsaoe requiring matrices Pauli above, the determining of ues to four logi- equivalent all and is of 4 physical values fidelity the expectation the for the Measuring operators qubit. Pauli cal usual the involving of matrix density the The as between qubits. expressed entanglement physical the and evaluate logical logical to The instead inequality level. we Clauser–Horne–Shimony–Holt (CHSH) the so physical and and fidelity the state purpose, the our at measure serves it perfectly evaluation evaluate measurements level this to of eliminate to number task us the allows daunting structure to code due the of unfeasible However, involved. quality is the qubits tomography evaluate 10 state first on quantum physical Typical to state. the important resource entangled is between of this It generation state qubit. the logical quantum is and entangled experiment maximally our for the ingredient crucial The Results Experimental Shor described complete are details the pho- Further to qubits. three access in with physical of nine and OAM measured of consisting DoFs, us space different code qubit, be the give three logical can transfer in complete measurements tons it to the These where to used analyzer. access one is polarization polarization gate another a swap polarization to OAM standard a state the with qubit, polariza- For out with interferometers. encoded read encoded Mach−Zehnder directly and qubit is analyzers The OAM paths destroy- step. and and paths, by or tion (39). polarization, step disturbing DoFs of measured other DoFs without are the the 3 experiment, in our and encoded In information 2, quantum 1, the ing photons for flips DoF phase and bit also of transformations. can unitary combinations arbitrary code (see linear form Shor that of or measurements The flips flip excluded. syndrome be phase bit error can detect a the qubit thus 2, and logical of Appendix) Fig. the change SI encoding in a qubits displayed in physical As results nine example. space; the an code of error-protected as one the 30 within ref. lie see that and results photons on post- extra we select require here Instead, would techniques. the This correction measuring feed-forward syndromes qubit. active before logical error errors the any the of correct measure state to to results those logi- qubits use the ancilla and of use consists should experiment one the of state’s part cal final and third The onto project we HWPs, using symmetrical splitter the beam the before state the state Bell antisymmetric the entangled onto the from qubit physical resource the and beam polar- qubit 50/50 that encoded on a ization measurement with the BSM coincidence out subsequent A into carried and plates. is state splitter teleportation quarter-wave single-qubit the and arbitrary implement half- to an using encode DoF to polarization pair) separate the a in of in prepared 5) depicted (photon as photon qubit, β a logical use we protected Here QECC Fig.1B. the into qubit cal bru oae(B)cytl(eaddb h eodphoton second the by (heralded crystal (BBO) borate -barium × h eodsaeo h xeietcnen h teleportation the concerns experiment the of stage second The ial,w a neednl esr n edoteach out read and measure independently can we Finally, IAppendix. SI 2 8 1,024 = ρ = |Φ |ϕi 4 1 + |ϕi (I sal,ti ehdpoet h w photons two the projects method this Usually, i. L etnsi oa.Fruaey h xetto val- expectation the Fortunately, total. in settings edu n ro ydoe eeto.Ideally, detection. syndromes error and readout ⊗ = (|HH I α|H cs |ϕi + i i X + + L I , = ⊗ |VV β Z |V α| X a eotie iheulsettings. equal with obtained be can L 0i cs i)/ i https://doi.org/10.1073/pnas.2026250118 L − nisonidpnetphysi- independent own its on |ψ √ + Y − 2 β oee,b transforming by however, i; ⊗ |1i tt 3) hc,ideally, which, (38), state Y L L . cs + Z ⊗ Z |Φ PNAS L cs + ), i a be can | f5 of 3 |ϕi [5] [4] L

PHYSICS Further owing to special features of the Shor code stabilizers, the number of settings can be further reduced to 250 in total (see SI Appendix). For each setting, we record fourfold coinci- dences for 10 s, yielding a coincidence rate of ∼150 s−1. We obtain a fidelity for the ideal state |Φ+i as F = 0.703(2). This clearly surpasses the genuine entanglement 0.5 threshold. How- ever, this fidelity F is insufficient to violate a CHSH inequality with hCHSHi = 1.974(3) < 2 experimental determined. Detailed measurement results for the estimation of the fidelity and CHSH inequality are shown in Fig. 3. Next, we exclude the influence of correctable errors by con- fining the state of the logical qubit to the actual code space using the projectors Ics to the code space (see SI Appendix for details). Experimentally, the overlap results in hI ⊗ Icsi = 0.808(2), representing the overlap between the logic qubit pre- pared in our experiment and the code space. This is then used to exclude all errors that can be detected by the stabilizers, yield- ing an error-corrected state fidelity F = 0.870(3) and hCHSHi = 2.443(3) > 2 violation within the code space (Fig. 3). Further- more, the encoded state fidelity F = 0.870 > 0.85 would enable Fig. 4. Experimental teleportation of an arbitrary single-qubit state. Here with error-corrected . Our we show the teleportation results of three representative states |Hi, |+i, results clearly demonstrate the effectiveness of QECC in our and |Ri that are eigenstates of σz, σx, anb σy, respectively, with eigen- approach, but unity fidelity was not achieved, due to multipair value +1. For each state, the fidelity with and without correction is shown emissions and nonfactorizable joint spectral amplitudes within together with the projection . After correction, the averaged / |Φ+i fidelity of the three teleported states is 0.786(17), well exceeding the 2 3 the SPDC process utilized for generating the state. Such shown as a red dashed line. errors cannot be corrected by our encoding, as they sit inside the code space (see SI Appendix for details). With the entangled resource state characterized, we now need 0.05 Hz. The achieved experimental fidelities (with and with- to explore the operation of teleporting a physical qubit into the out correction) and the projection Ics are shown logical qubit space. For such a quantum system, it is necessary in Fig. 4. to show its performance, comprehensively exceeding any classi- The averaged fidelity of the three logic states is 0.520(7), cal methods. Thus, in our experiment, we select eigenstates with while, after projection into the code space, it increases to eigenvalue +1 of three X , Y , and Z , denoted 0.786(17). This is well above the classical limit of 2/3. Fur- as |0i, |+i, and |Ri, respectively, and measure their teleported thermore, in our experimental arrangements, the teleportation√ fidelity. We measure 125 settings for |0i , |Ri and 98 settings fidelity of any state of the form (|0i + eiφ|1i)/ 2 is inde- for |+i. For each setting, we accumulate, on average, ∼60 coin- pendent of the phase φ. For example, the fidelities of φ = cidences in 1,200 s, which corresponds to a count rate of ∼ 0 and φ = π/2 are consistent in 1 SD, as shown in Fig. 4. The obtained results demonstrate the ability of our approach to write via quantum teleportation arbitrary quantum states, including the magic state φ = π/4 for T gate, from a sin- Without correction Without correction A With correction B With correction gle physical qubit into the logical code space consisting of 1.0 nine physical qubits. Moreover, the postselected error correc- tion scheme employed here significantly increases the observed 0.5 average fidelities from ∼ 52 to ∼ 78% limited only by noncor- e 0.5

lu rectable errors stemming from multipair emissions of the SPDC a processes. on v on value ti ti 0.0 0.0 Discussion and Conclusion

xpecta In summary, we have demonstrated the teleportation of a physi- E Expecta -0.5 cal qubit into a logical qubit formed from a QECC. This is a key -0.5 step for optical quantum calculation on a larger scale. Although the results achieved are far from the threshold, -1.0 our work is still far reaching. It demonstrates the ability to intro- XX YY ZZ C1 C2 C3 C4 duce well-developed quantum teleportation to the QIP at the Observables Correlation functions logical level within current technology, and, as such, represents a Fig. 3. Characterization of the entanglement teleportation resource state. crucial step toward fault-tolerant QIP. Such an ability is essential In A, we show the measured expectation values of X ⊗ XL, Y ⊗ YL, and for probabilistic gate operations to be performed on an unknown Z ⊗ ZL without (orange bars) and with (green bars) correction. One can state in a scalable manner. More specifically and importantly, it determine the fidelity of entangled state as F = 0.703(2) before and F = allows for magic state injection, a critical task in error-corrected 0.870(3) after correction. Similarly, B shows the measured correlation func- quantum computation. Our experiment can be further modi- tions required for the CHSH inequality without (orange bars) and with fied to adapt the fault-tolerant manner. Moreover, within the (green bars)√ error correction. The physical qubit is measured in the E1,2 = theoretical scheme, it can be further concatenated with indepen- (Z ± X)/ 2 basis, while the QEEC is measured with XL, ZL, respectively. dently developed modules, such as magic state distillation and The four correlation functions C ,C ,C , and C denote E ⊗ X , E ⊗ X , 1 2 3 4 1 L 2 L transversal logical operation block, that may become a useful E1 ⊗ ZL, and E2 ⊗ ZL, respectively. Then hCHSHi = C1 − C2 + C3 + C4 gives 1.974(3) before and 2.443(3) after correction. All reported measurements part of future implementations of fault-tolerant quantum com- are without background or accidental count subtraction, while the stated puter. For a global quantum internet based on optical fibers, it measurement errors are obtained using Monte Carlo simulation with an will be necessary to employ quantum repeaters to overcome the underlying Poissonian distribution of photon counting statistics. intrinsic losses in the optical fibers. To distribute quantum entan-

4 of 5 | PNAS Luo et al. https://doi.org/10.1073/pnas.2026250118 Quantum teleportation of physical qubits into logical code spaces Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 0 .G Ren G. J. 20. Long-distance Gisin, N. Zbinden, H. Zhang Q. Tittel, 18. W. Riedmatten, de H. Marcikic, I. 17. 9 .L Wang L. X. 19. Furusawa A. 16. Towards codes: Surface Cleland, N. A. Martinis, M. J. Mariantoni, M. Fowler, G. A. 13. Bennett sets. gate H. quantum C. encoded 12. transversal on Restrictions Knill, code. E. correcting Eastin, error quantum B. Perfect Zurek, 11. H. W. Paz, P. J. Miquel, C. Laflamme, R. 10. 4 .Gtemn .Cun,Dmntaigtevaiiyo nvra unu com- Bouwmeester quantum D. universal 15. of viability the Demonstrating Chuang, I. Gottesman, D. 14. unu eeotto fpyia uisit oia oespaces code logical into qubits physical of teleportation Quantum al. et Luo an is boundary the basic in the is qubit which principle, logical holographic simulate the the to of central and structure implementation the basic qubit between a correlation physical also quantum The is gravity. It quantum exponentially (40). in is efficient used which resource strategy realize communication, the enables quantum to to It long-distance similar method tasks. computing quantum information divide-and-conquer deep-depth quantum teleportation-based novel building a versatile many a is for qubit logical block a and qubit physical a between quantum future a in useful internet. be this could In scheme necessary. presented potentially our is sense, QECC network, a such in glement .A .Sen,Errcretn oe nqatmtheory. quantum in codes correcting Error memory. Steane, M. quantum A. in 9. decoherence beginners. reducing for for Scheme correction Shor, error W. 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