Quantum walks on a programmable two-dimensional 62-qubit superconducting processor
Ming Gong, Shiyu Wang, Chen Zha, Ming-Cheng Chen, He-Liang Huang, Yulin Wu, Qingling Zhu, Youwei Zhao, Shaowei Li, Shaojun Guo, Haoran Qian, Yangsen Ye, Fusheng Chen, Chong Ying, Jiale Yu, Daojin Fan, Dachao Wu, Hong Su, Hui Deng, Hao Rong, Kaili Zhang, Sirui Cao, Jin Lin, Yu Xu, Lihua Sun, Cheng Guo, Na Li, Futian Liang, V. M. Bastidas, Kae Nemoto, W. J. Munro, Yong-Heng Huo, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, Jian-Wei Pan
QQuantum walks on a programmable two-dimensional 62-qubit superconductingprocessor
Ming Gong , , , ∗ Shiyu Wang , , , ∗ Chen Zha , , , ∗ Ming-Cheng Chen , , , He-Liang Huang , , , YulinWu , , , Qingling Zhu , , , Youwei Zhao , , , Shaowei Li , , , Shaojun Guo , , , Haoran Qian , , , YangsenYe , , , Fusheng Chen , , , Chong Ying , , , Jiale Yu , , , Daojin Fan , , , Dachao Wu , , , HongSu , , , Hui Deng , , , Hao Rong , , , Kaili Zhang , , , Sirui Cao , , , Jin Lin , , , Yu Xu , , , LihuaSun , , , Cheng Guo , , , Na Li , , , Futian Liang , , , V. M. Bastidas , Kae Nemoto , W. J. Munro , ,Yong-Heng Huo , , , Chao-Yang Lu , , , Cheng-Zhi Peng , , , Xiaobo Zhu , , , and Jian-Wei Pan , , Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei 230026, China Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China NTT Basic Research Laboratories and Research Center for Theoretical Quantum Physics,3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan and National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
Quantum walks are the quantum mechanical analogue of classical random walks andan extremely powerful tool in quantum simulations, quantum search algorithms, andeven for universal quantum computing. In our work, we have designed and fabricatedan 8x8 two-dimensional square superconducting qubit array composed of 62 functionalqubits. We used this device to demonstrate high fidelity single and two particle quan-tum walks. Furthermore, with the high programmability of the quantum processor,we implemented a Mach-Zehnder interferometer where the quantum walker coherentlytraverses in two paths before interfering and exiting at a single port of it. By tuningthe disorders of the sites on the evolution paths, we observed interference fringes withsingle and double walkers. Our successful demonstration of a programmable quan-tum walks on a two-dimensional solid-state system is an essential milestone in thefield, brings future larger scale quantum simulations closer to realization, and alsohighlights the potential of another model of universal quantum computation on theseNISQ processors.
Classical random walks are a stochastic process de-scribing the random hopping of a particle to its neigh-bors, which have been widely used as an important toolboth in the modeling of various physical processes butalso for the development of algorithms [1]. The quan-tum principles of superposition and entanglement allowa new and more powerful form of random walk – termeda quantum walk (QW). Introduced by Aharonov et al. [2]in 1993, these walks have attracted considerable atten-tion with many applications known in quantum trans-port [3], quantum simulation [4, 5], quantum search algo-rithms [6–8], and even universal quantum computing [9–11]. Childs et al. showed that both quantum walksusing a single walker encoded into qubits [9] and mul-tiple interacting quantum walkers without a qubit en-coding [11] are universal for quantum computation andmay give an exponential algorithmic speedup [12]. Onthe other hand, multiple non-interacting quantum walk-ers without the qubit encoding (as seen in boson sam-pling) may not be universal but could still show quan-tum advantage [13, 14]. Motivated by the rich potentialapplications of QWs, numerous proof of principle experi-mental demonstrations have been performed using a widevariety of hardware platforms, including photons [15– 18], trapped ions [19, 20], neutral atoms [21, 22], nu-clear magnetic resonance [23], and even superconductingqubits [24, 25].The current experimental demonstrations were de-signed to showcase the principles of quantum walk; how-ever, they are not ready for real world applicationswhere the performance should be better than what canbe classically achieved. It is well-known [26] that QWbased quantum search algorithms require at least a two-dimensional configuration to achieve a speedup over clas-sical algorithms (a one dimensional QW search cannot).Furthermore, easy circuit programmability where theconfiguration can be changed on a walk-by-walk basis(including the adjustability of tunneling amplitude andgraph structure [27]) is an essential requirement for ap-plications in universal quantum computing. Achievingboth of these simultaneously have proved experimentallychallenging. Superconducting circuits, one of the lead-ing quantum computer approaches, have shown excel-lent scale up potential in terms of the number of func-tional qubits on a chip [28]. Along with the excellent realtime programmability, they are now an excellent candi-date system for the realization of fully configurable two-dimensional QWs. a r X i v : . [ qu a n t - ph ] F e b Q0 Q1 Q0 Q1 Q0 Q1 Q0 Q1Q3 Q2 Q3 Q2 Q3 Q2 Q3 Q2Q0 Q1 Q0 Q1 Q0 Q1 Q0 Q1Q3 Q2 Q3 Q2 Q3 Q2 Q3 Q2Q0 Q1 Q0 Q1 Q0 Q1 Q0 Q1Q3 Q2 Q3 Q2 Q3 Q2 Q3 Q2Q0 Q1 Q0 Q1 Q0 Q1 Q0 Q1Q3 Q2 Q3 Q2 Q3 Q2 Q3 Q2 (cid:56)(cid:19)(cid:19) (cid:56)(cid:19)(cid:20) (cid:56)(cid:19)(cid:21) (cid:56)(cid:19)(cid:22)(cid:56)(cid:20)(cid:19) (cid:56)(cid:20)(cid:20) (cid:56)(cid:20)(cid:21) (cid:56)(cid:20)(cid:22)(cid:56)(cid:21)(cid:19) (cid:56)(cid:21)(cid:20) (cid:56)(cid:21)(cid:21) (cid:56)(cid:21)(cid:22)(cid:56)(cid:22)(cid:19) (cid:56)(cid:22)(cid:20) (cid:56)(cid:22)(cid:21) (cid:56)(cid:22)(cid:22)
Readout resonatorReadout X Y Z X Y Z X Y Z X Y Z Q0 Q1 Q2Q3
Readin FilterCoupling resonator
AB C
FIG. 1.
The layout and architecture of the superconducting quantum processor. ( A ) The schematic diagram of the2D superconducting quantum processor. The orange crosses represent the qubits arranged in an 8 × B ) The circuit diagram ofa unit of the qubit array. Each qubit (orange) has an XY Z control line (black) for microwave and pulse control. The qubitcouples to an individual λ/ λ/ C ) The labels of functional qubits. Two broken qubits, namely U03Q2 and U22Q1, are marked in gray. In our work, we started with the design of a moder-ate scale 2D superconducting qubit array, which is def-initely a non-trivial exercise – especially when movingfrom a one-dimensional design. In fact, it is one of thekey issues in scaling to larger size systems. One needs toconsider not only the numbers of qubits and their con-figurations, but also the control systems that manipulatethem. While the number of control and readout linesscales linearly with the number of qubits N , the bound-ary length of 2D arrays scale from 4( √ N −
1) for squareto N for bi-linear ladder configurations. One immedi-ately notices the problem associated with planar wiring and how it can be realized to control all the qubits as thesize of the 2D array increases, particularly in the caseof the square configurations that are preferable from aquantum algorithmic point of view. One solution hasbeen 3D wiring using techniques like ‘flip chip’ [29, 30]or ‘through-silicon vias’ [31, 32], but these high-end spe-cialized (and expensive) approaches have limited the de-velopment of 2D superconducting processors. In thiswork, we provide an alternative technical-friendly solu-tion based on pass through holes. This is applied to an8 × A ) that is composedof 16 units whose circuit diagram is shown in Fig. 1 B . Time = 0 ns P o s i t i o n j Time = 100 ns Time = 200 ns Time = 300 ns
Position i P o s i t i o n j Position i
Position i
Position i
Time (ns) C d / C d m a x Time (ns) D i s t a n c e L R b o u n d AB CD
FIG. 2.
Quantum walks on a 2D superconducting qubits array . ( A ) The evolution of the measured populations h n j i of all qubits at times t = 0 ns, 100 ns, 200 ns, and 300 ns respectively with the two walkers initialized in the state | ... i .During the initial state preparation, we excite qubits U00Q0 and U33Q2 at their idle frequency before moving all qubits tothe interaction frequency of 5.02 GHz. We note that thermalization on a few qubits causes abnormal occupations, especiallyfor qubit U10Q2. In the Supplementary Materials we show how post-selection using conservation of the total number ofexcitations improves the fidelity of our quantum walk [34]. ( B ) Numerical simulation of the qubits population evolution underthe same conditions as ( A ). ( C ) The correlation function of single-particle QW as a function of time. The blue, red, orange,and purple circles represent the measured data points for the correlation function between the initial excitation site and thesites on the diagonal with distance d = √ d = 2 √ d = 3 √ d = 4 √ p ∆ x + ∆ y , in which ∆ x (∆ y ) is the number of sites between two sites along X(Y) axis.The corresponding Gaussian curves are Gaussian fittings to the data. ( D ) The Lieb-Robinson bounds. From the two-sitecorrelation functions for a single quantum walker starting at U00Q0 (top left) we extracted the time required for a correlationto be established between the initial excitation and sites at certain distances. Those times are represented as blue circles. Usinga linear time fit with distance (blue line) we determine the velocity of the walker propagation as 22 . ± . µs , where the95% confidence interval comes from the fitting. The gray shadow shows Lieb-Robinson bound with v max = 35 . µs , whichexhibits the linear light cone with exponential decay tails. Here each unit contains four frequency tunable transmonqubits which are dispersively coupled to individual read-out resonators but share a common band-pass filter forstate readout. Each qubit couples to its four nearestneighbors via coplaner waveguide resonators. The sep-aration between qubits is approximately 4 mm, leavingsufficient space to cut holes in the chip substrate usinga picosecond laser. The control lines and readout filtersare connected to the fan-out PCB on the bottom of thechip by wire bonding through the holes, from which theycan be further connected to the coaxial cables installed inthe dilution refrigerator. Parasitic slot-line modes [33],which may arise, are then suppressed using air-bridgesapplied over the control lines, readout resonators, filtersand coupling resonators to connect the ground planes.In the fabrication of any advanced quantum proces-sor containing thousands of elements, slight imperfectionwill arise and may have an effect on that device. In our8 × C ), twoof the qubits U03Q2 and U22Q1 and one coupling res-onator (between U10Q0 and U10Q3) are non-functional.It is important to mention that we optimized the qubitfrequency setup to maximize the energy relaxation time T of all qubits while preventing the effect of defects, ZZ coupling, and microwave cross-talk (further details aboutthe qubit parameters are presented in the SupplementaryMaterials[34]). Next, the design of our processor has thequbits large detuned from the coupling resonators (6.0GHz) enabling dispersive coupling between two qubits.This enables us to represent the effective Hamiltonian ofthe qubit system using the Bose-Hubbard model as: b H = X j ∈{ Q i } ~ ω j b n j + ~ U j b n j ( b n j − X i ∈{ Q i } ,j ∈{ C Qi } ~ J i,j eff ( b a † i b a j + b a i b a † j ) , (1)where ω j is the j th qubit frequency, U j the anharmonic-ity, J i,j eff the effective coupling strength between Q i and Q j where j labels the group { C Q i } of coupling qubits to Q i . Next b a † j ( b a j ) are the qubits bosonic creation (annihi-lation) operator with b n j = b a † j b a j being the correspondingnumber operator which has values of zero or one.For the realization of our continuous-time quantumwalks (CTQWs), we prepare our initial walkers beforewe tune all qubits to the same interaction frequency for D R - S L - B S B S Time (ns)
A BC D EF
FIG. 3.
Single-particle Mach-Zehnder interferometer . ( A ) The circuit diagram of the programmable paths for therealization of the single-particle Mach-Zehnder interferometer in a qubit array. Our walker is a qubit excited at the source S .After BS
1, the quantum walker goes into two paths { L } and { R } , in which the phases of the quantum walker can be shifted bythe disorders in two paths. Then these two paths combine at BS D . The qubits in theprocessor not used in the interferometer (marked in gray) are detuned to 4.97 GHz (25 J eff below the interaction frequency) sothey will not contribute to the evolution of the system. On paths { L } and { R } , extra disorders can be added conditionally bycontrolling the qubits detuned away from the interacting frequency. From R ( L ) to R ( L ), the disorder increases graduallyfrom d R ( d L ) to 5 d R (5 d L ), and then decreases gradually from R ( L ) to R ( L ). In ( B ) and ( D ) we illustrate the dynamicsevolution of population h n j i of all relative sites in experiment and simulation, respectively. The red arrow marks the time t = 650 ns when the population of D is maximized. ( C ) The population of D at t = 650 ns under different disorder steps in twopaths, which displays obvious interference fringes. ( E ) The circuit diagram of the interferometer with the { R } path blocked at R and R by detuning these qubits to 4.97 GHz. ( F ) The population of D under different disorder steps in two paths with R path blocked, showing no interference fringes. time-independent system evolution. With all qubit atthe same frequency, our effective evolution Hamiltonianis given by: b H evo = X i ∈{ Q i } ,j ∈{ C Qi } ~ J i,j eff ( b a † i b a j + b a i b a † j ) (2)which forms an interference network. To achieve thisin practice and ensure its stability, three experimentalrequirements need to be realized: (i) the stability of thequbit frequency, (ii) the precise knowledge of the couplingstrength, and (iii) high precision control of the qubit-frequency alignment. This is achieved through a seriesof calibration experiments [34]. We begin with a Z -pulsedistortion calibration [25] to ensure the stability of qubitfrequency while performing detuning operations. Then,by setting the qubits at the interaction frequency of 5.02GHz, we determined the effective coupling strengths J i,j eff by measuring two-qubit swapping oscillations. This al-lows us to establish J eff / π = 2 . ± .
07 MHz. The finalrequirement is achieved through several rounds of qubit-frequency alignment calibrations and corrections whichallows us establish that the disorders of all qubits areno larger than 1.6 MHz (0 . J eff / π ). With the systemnow well calibrated, we can now explore CTQWs in ourquantum processor. We begin by exploring continuous-time quantum walksusing one and two walkers, where we create the initialstates | ... i and | ... i for the single-walkerand | ... i for the two-walker situation, respectively.Here | ... i ( | ... i ) corresponds to the casewhere only qubit U00Q0 (U33Q2) has been initially ex-cited while | ... i has both U00Q0 and U33Q2 ex-cited. Once these initial states are prepared, we tune allthe qubits to the interaction frequency of 5.02 GHz andallow the system to naturally evolve under Eq.(2) for acertain evolution time. We then measure the population h ˆ n j i of all 62 qubits in their σ z basis for evolution timesranging from 0 to 600 ns. For each time point, we per-formed 50,000 single-shot measurements. In Fig. 2 A wepresent the experimental results for the two-walker QWwith a comparison from numerical simulations shown inFig. 2 B (the supplementary material [34] shows the re-sults for the single walker QW). Remarkable agreementis observed, indicating the high accuracy characterizationand high precision control of our system.For quantum walks it is also extremely useful to deter-mine the propagation speed of the walker (s) through thenetwork compared to the Lieb-Robinson (LR) bound [35].Focusing on the single walker situation for simplicity,we can use the two-site correlation function defined by
A BC DE FG H
FIG. 4.
Two walkers in the Mach-Zehnder interferom-eter. ( A, C, E, G ) Circuit diagrams of the programmablepaths for the Mach-Zehnder interferometer. The initial statecomposed of either a single or dual walker is prepared by theexcitation of the sites marked with pink. In ( A ) and ( C ),there are two qubits excited at L and R while in ( E ) and( G ), the qubit is independently excited at L and R , re-spectively. In ( C ), the BS S are removed from theinterferometer by detuning those two qubits to 4.97 GHz.( B,D,F,H ) The population of D at t = 550 ns are shownfor the various programmable detailed above. In ( B ), inter-ference fringes are seen somewhat similar to those of Fig. 3 C .In ( D ) which corresponds to the path situation of ( C ) thereare no interference fringes observed. Finally, for the singlewalker situations ( F ) and ( H ), interference fringes are ob-served. However neither of them nor the sum of them is thesame as that of ( A ). C ij ( t ) = h ˆ σ iz ˆ σ jz i−h ˆ σ iz ih ˆ σ jz i [25] to achieve this. In Fig. 2 C we plot the correlation function of single-particle QW asa function of time. This allows us to estimate the time ittakes to establish correlations between the initial excita-tion site and other sites with different distances. This isshown in Fig. 2 D from which we can determine the prop-agation speed as 22 . ± . µs . It has also been shownthat the maximal group velocity for two-dimensional sys-tems [36, 37] is given by v max = 2 √ J eff / (1 − J / U )which equates to v max = 35 . /µs in our system. Now v < v max clearly shows that our propagation speed islimited by the LR bound and caused by the lack of longrange interactions.The single and multi walker continuous-time quantumwalks demonstration establishes a solid basis for the re-alization of programmable QW devices. Furthermore,our ability to accurately vary the frequency of each qubitin our processor enables us to define propagation paths(even intersecting ones) for the quantum walkers. Thisis critical for QW-based quantum computing where weneed to deal with graph problems with different struc-tures. The simplest non-trivial configuration we couldconsider is probably the Mach-Zehnder (MZ) interferom-eter involving two intersecting paths. In Fig. 3 A , we de-fine two paths in our 62-qubit superconducting processorto demonstrate a MZ interferometer where the qubits inthe path are tuned to the interaction frequency of 5.02GHz, while those not involved are detuned by -50 MHzto 4.97 GHz. With the paths defined we can now ex-plore the evolution of our walkers - beginning with thesingle particle situation starting at U30Q2 (site S ). Asshown in Fig. 3 A , after exciting the site S , the walkerwill propagate to BS L to L ) and( R to R ). These paths are reconnected at BS D . At this stage, it isimportant to mention a key difference between our super-conducting circuit implementation and the more typicalphotonic one. In the former, all sites in the interferometercan be directly measured, which in turn provides valu-able information about the walkers dynamics as it ‘walks’through the interferometer. As such the time evolutionof all sites’ population is measured from t = 0 to 1000ns and presented in Fig. 3 B . It clearly shows the sin-gle walker transversing both the L − L and R − R paths. At t = 650 ns, a refocusing of the QW with thepopulation as high as 0.43 is observed. Again, excellentagreement is found compared with the numerical simula-tions shown in Fig. 3 D .Our flexibility in being able to adjust the qubit fre-quencies provides another freedom we can exploit asso-ciated with the disorders on paths { L } and { R } . Thisallows us to vary the phase in the paths. For the { R } path elements we adjust the disorder of the sites R to R from d R to 5 d R respectively, while for sites R to R we did the opposite changing from 5 d R to d R . Simi-lar disorder changes (scaling as d L ) are made in the { L } path. By controlling the difference in disorder sizes d R , d L on these two paths, we measured the population onsite D at t = 650 ns, and observed interference fringes asshown in Fig. 3 C . To confirm the origin of these fringes,we blocked the path of { R } on R and R as shown inFig. 3 E and found no interference fringes present as illus-trated in Fig. 3 F . Such results show that the disorder notonly changes the tunneling amplitude between neighbor-ing sites, but also provide the quantum walker a differentphase accumulated in propagation that gives rise to theinterference fringes. Moreover, for that interference tohave occurred, our walkers must have maintained coher-ence as they both traverse a superposition of distinct spa-tially separated paths. The generation of those non-localcorrelations is essential for the development of QW baseduniversal quantum computation. On the other hand, al-most no population is observed on the four corners ofboth Figs. 3 C and F , which is associated with a localiza-tion phenomenon due to the relatively large disorders.The natural question that now arises is what occurswhen we have multiple walkers in our MZ interferome-ter. So, let us turn our attention to the two walker case.We create our two walkers as illustrated in Fig. 4 A onsites L and R by exciting these respective qubits, andthen let the system evolve. We measured the populationon site D after t = 550 ns, and observed the interferencepattern shown in Fig. 4 B . This is a similar pattern towhat we observed in the single walker case. To determinethe origin of this interference pattern in the two walkercase, we performed a number of control experiments be-ginning with the removal of sites BS S for the MZinterferometer (see Fig. 4 C ). Removing site BS D , no interference fringes areobserved anymore. This indicates that the pattern comesfrom the interference between the single-particle forwardand back propagation. In our next control experiment,we created a single walker at either site L or R as shownin Figs. 4 E,G respectively, and let it walk through the in-terferometer before measuring the population at D . Theresults are presented in Figs. 4 F,H , and clearly show in-terference fringes. Neither one of them nor their sumare the same as what we observed in Fig. 4 B . This re-enforces our observation that the two walkers present inthe MZ interferometer must have interacted with eachother. Such results agree well with our understanding oftransmon qubit physics in the hard-core boson limit [38],where | U/J | ∼
ACKNOWLEDGMENTS
The authors thank the USTC Center for Micro- andNanoscale Research and Fabrication for supporting thesample fabrication. The authors also thank Quantum-CTek Co., Ltd., for supporting the fabrication and themaintenance of room-temperature electronics.
Funding:
This research was supported by theNational Key R&D Program of China, Grant2017YFA0304300, the Chinese Academy of Sciences,Anhui Initiative in Quantum Information Technolo-gies, Technology Committee of Shanghai Municipality,National Science Foundation of China (Grants No.11574380), Natural Science Foundation of Shanghai(Grant no. 19ZR1462700), and Key-Area Researchand Development Program of Guangdong Provice(Grant No.2020B0303030001). This work was alsosupported in part by the Japanese MEXT QuantumLeap Flagship Program (MEXT Q-LEAP), Grant No.JPMXS0118069605.
Author contributions:
X.Z. and J.-W.P. conceivedthe research. M.G., S.W, C.Z., M.-C.C., and X.Z. de-signed the experiment. S.W., M.G., Q.Z., Y.Z., Y.Y.,F.C., C.Y., and X.Z. designed the sample. S.W., C.Z.,H.R., H.D., K.Z., S.C., and Y.-H.H. prepared the sample.S.G., H.Q., and H.D. prepared the Josephson paramet-ric amplifiers. Y.W. developed the programming plat-form for experiment. M.G., S.W., C.Z., Y.Z., S.L., C.Y.,J.Y., D.F., D.W., and H.S. contributed to the build-ing of the measurement system. M.G. and S.W. carriedout the measurements. C.Z. performed numerical sim-ulations. M.G., S.W., C.Z., M.-C.C., H.-L.H., V.M.B.,K.N., W.M., C.-Y.L., and X.Z. analyzed the results. J.L.,Y.X., F.L., C.G., L.S., N.L., and C.-Z.P. developed theroom-temperature electronics. M.G., C.Z., S.W., M.-C.C., H.-L.H., K.N., W.M., and X.Z. contributed to thedevelopment of the manuscript. All authors contributedto discussions of the results. X.Z. and J.-W.P. supervisedthe whole project.
Competing interests:
None declared.
Data and materials availability:
All data neededto evaluate the conclusions in the paper are present inthe paper or the supplementary materials.
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Klimov, et al., A method for building low loss multi-layer wiring for su-perconducting microwave devices, Applied Physics Let-ters , 063502 (2018).[34] See supplementary materials for more details .[35] E. H. Lieb and D. W. Robinson, The finite group ve-locity of quantum spin systems, in Statistical mechanics(Springer, 1972) pp. 425–431.[36] M. Cheneau, P. Barmettler, D. Poletti, M. Endres,P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath,and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature , 484 (2012).[37] Y. Takasu, T. Yagami, H. Asaka, Y. Fukushima, K. Na-gao, S. Goto, I. Danshita, and Y. Takahashi, Energy re-distribution and spatio-temporal evolution of correlationsafter a sudden quench of the bose-hubbard model, arXivpreprint arXiv:2002.12025 (2020).[38] Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pu-gatch, and Y. Silberberg, Quantum walk of two interact-ing bosons, Physical Review A , 011603 (2012). upplementary materials for “Quantum walks on a programmable two-dimensional62-qubit superconducting processor” Ming Gong , , , ∗ Shiyu Wang , , , ∗ Chen Zha , , , ∗ Ming-Cheng Chen , , , He-Liang Huang , , , YulinWu , , , Qingling Zhu , , , Youwei Zhao , , , Shaowei Li , , , Shaojun Guo , , , Haoran Qian , , , YangsenYe , , , Fusheng Chen , , , Chong Ying , , , Jiale Yu , , , Daojin Fan , , , Dachao Wu , , , HongSu , , , Hui Deng , , , Hao Rong , , , Kaili Zhang , , , Sirui Cao , , , Jin Lin , , , Yu Xu , , , LihuaSun , , , Cheng Guo , , , Na Li , , , Futian Liang , , , V. M. Bastidas , Kae Nemoto , W. J. Munro , ,Yong-Heng Huo , , , Chao-Yang Lu , , , Cheng-Zhi Peng , , , Xiaobo Zhu , , , and Jian-Wei Pan , , Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei 230026, China Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China NTT Basic Research Laboratories and Research Center for Theoretical Quantum Physics,3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan and National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
CONTENTS
I. Experimental wiring setup 1II. Parameters of the superconducting quantumdevice 2III. System calibration 4A. Idle frequency setup 4B. Frequency-alignment optimization 5C. Optimization of the interferometer 6IV. Thermalization and Post-selection 6V. Numerical simulation method 7VI. Extended data 7References 7
I. EXPERIMENTAL WIRING SETUP
As shown in Fig.S1, the 62-qubit processor is installedat the base temperature stage of the dilution refrigerator(DR), which is cooled down to 10 mK. In the DR, weused totally 186 control lines for qubits, 32 control linesfor Josephson parametric amplifiers (JPAs), 16 readoutinput lines and 16 readout output lines. For each qubit,there are three control lines, the XY line for qubit driv-ing, the fast Z control line to apply the Z pulse control,and the DC line to bias the qubit to its idle point. Thefour qubits in a unit share one readout input line and onereadout output line.In the DR, attenuators and filters are installed at dif-ferent stages to reduce noise. To reduce the thermal noise ∗ These authors contributed equally to this work. from higher-temperature stages, there are totally 41 dB,34 dB and 61 dB attenuations for XY control line, fast Z control line and readout input line, respectively. Inaddition, at the base temperature stage, we installed 8GHz low pass filters for all XY , fast Z , DC control lines,readout input and readout output lines, 500 MHz lowpass filters for fast Z control lines and 80MHz low passfilters for DC control lines, to further reduce the high-frequency noise. At the 4 K stage, we installed a RCfilter of 10 KHz cut-off frequency for each DC controlline. After passing through the attenuators and filters,the XY , fast Z and DC signals are combined togetherby a bias tee at base temperature and then reach thequantum processor. For qubit state readout, the readoutinput signal passes through the attenuators and the 8GHz low pass filter and then reaches the quantum proces-sor. The output signal firstly passes through the 8 GHzlow pass filter and two circulators, then amplified by aJosephson parametric amplifier (JPA), of which the noiseas well as the noise from higher temperature stages hasbeen blocked by the preceding of two circulators. Thenthe signal passes through a third circulator, amplified bya high electron mobility transistor (HEMT) amplifier atthe 4K stage and a low noise amplifier at room temper-ature respectively, and finally captured and analyzed bythe room temperature electronics. To reduce the noise incontrolling JPA, we installed 31dB attenuators and a 18GHz low pass filter for each JPA pump line, a RC filter,an 8 GHz and an 80 MHz low pass filter for each JPAbias line.At room temperature, we use two digital to analogconverters (DAC) channels to generate Gaussian shapedpulses for each XY control. These two intermediate fre-quency pulses are up-converted to the driving frequencyby an IQ mixer with a carrier frequency of 5.72 GHzgenerated by the microwave source. We use one DACchannel for the Z pulse control and one DC source for theDC bias. For the readout of qubit state, we use two DACchannels and one microwave source to generate a multi- a r X i v : . [ qu a n t - ph ] F e b MW DAC DAC DC MW DAC ADC MW DC d B d B d B d B d B G H z d B d B d B d B d B G H z M H z d B d B d B d B G H z M H z R C fi l t e r d B d B d B d B d B G H z d B d B d B d B d B d B d B d B d B d B G H z d B d B d B d B M H z G H z R C fi l t e r Qubit 50 K4 KStill Cold PlateMCXY Z DC RI RO JPApump JPA DC: 0dB att. d B : 1dB att. d B : 3dB att. d B : 10dB att. d B : 20dB att. d B : RC filter: JPA: RC filter : HEMT: Circulator: Room-temperature amplifier: Bias Tee : I-Q mixer MW : Microwave source DAC : Digital-analog converter DC : DC voltage source ADC : Analog-digital converter G H z Dilution refrigerator
FIG. S1. The schematic diagram of control electronics and wiring. For each qubit, there are individual XY , Z and DC controllines, which are combined together via bias tees before connected to the quantum processor. In the dilution refrigerator,attenuators and filters are installed at various stages to reduce noise. The Josephson parametric amplifiers (JPAs) and highelectron mobility transistors (HEMTs) are used to amplify the readout signals. At room temperature, digital to analogconverters (DAC) and microwave sources are used to generate pulses for qubit XY control and readout. Qubit Z control pulsesare generated by DACs. DC voltage sources are used to bias the qubits to their idle points and to bias the JPAs. The readoutsignals amplified by the room-temperature amplifiers can be digitized and demodulated by analog to digital converters (ADCs). tone readout input signal by side-band mixing with an IQmixer. The readout output signal is down-converted intotwo signals, which are digitized and demodulated by thetwo ADC channels to extract the qubit state information. II. PARAMETERS OF THESUPERCONDUCTING QUANTUM DEVICE
Our experiments are implemented on a 62-qubit quan-tum processor arranged in a 8 × T . The qubits can be tuned to their workingpoints by Z pulses to realize interaction. The anhar-monicity η/ π of qubit, which is equal to the nonlinearon-site interaction U/ π in the Bose-Hubbard Hamilto-nian (1) in the main text, has the mean value of -248.9MHz. Each qubit couples to its four nearest neighborsvia coplaner waveguide resonators, whose frequencies are Parameters Median Mean Stdev.Qubit maximum frequency (GHz) 5.434 5.442 0.116Qubit idle frequency (GHz) 5.148 5.200 0.198Qubit anharmonicity η/ π (MHz) -251.0 -248.9 7.0 T at idle frequency ( µ s) 12.48 13.56 5.53 T at working point ( µ s) 11.24 12.26 5.45 T ∗ at idle frequency ( µ s) 1.61 1.63 0.67Coupling strength between qubit and readout resonator (MHz) 95.49 94.95 4.60Effective coupling strength between neighboring qubits (MHz) 1.99 2.01 0.07Dispersive shift χ/ π (MHz) 1.05 1.14 0.34Resonator linewidth κ/ π (MHz) 4.91 5.06 1.63Readout fidelity of | i | i Qubit maximum frequency (GHz)
Qubit anharmonicity /2 � (MHz) -251 -255 -255 -257-241 -226 -244 -246-240 -247 -238 -245-252 -244 -237 -243-246 -239 -254 -249-250 -245 -243 -255-242 -253 -251-239 -253 -240 -249-254 -247 -249-239 -240 -254 -255-254 -254 -252 -256-254 -255 -235 -255-255 -255 -256 -255-251 -263 -254 -255-254 -250 -255 -244-253 -244 -253 -254 T at working point ( s) T * at idle frequency ( s) Qubit idle frequency (GHz) T at idle frequency ( s) FIG. S2. Qubit parameters distributions, including qubit maximum frequency, qubit anharmonicity, qubit idle frequency, qubitenergy relaxation time T at the idle and working points, and qubit dephasing time T ∗ at idle frequency. Each square in thediagrams represents a qubit, the number and color in the square show the value of the corresponding parameter. designed to be 6 GHz, and the coupling strength be-tween the qubit and coupling resonator is designed to bearound 44 MHz. The mean value of the measured effec-tive coupling strength between neighboring qubits is 2.01MHz at the interaction point of 5.02 GHz, which gives | U/J | = 124. This means our system can be describedby hard-core bosons [S2]. For qubit state readout, the mean value of dispersive shift is 1.14 MHz, and the meanvalue of readout resonator linewidth is 5.06 MHz. Withthese parameters, we achieve an average readout fidelityof 96.6% for state | i and 91.9% for state | i . The effec-tive qubit temperature extracted from the excited stateprobability in measuring | i state is determined as 66 mKin average. More details of the parameters for each func- Readout drive frequency (GHz)
Qubit coupling to readout resonator (MHz)
Dispersive shift /2 � (MHz) Resonator linewidth /2 � (MHz) Readout fidelity of |0>
Readout fidelity of |1>
Effective qubit temperature (mK)
54 76 65 8456 62 74 7874 62 73 7462 61 82 6256 48 59 5958 78 67 6068 66 5854 75 75 4973 71 8476 68 86 8665 63 59 5165 69 79 4964 82 63 6756 62 86 8458 71 55 5257 57 65 50
FIG. S3. Qubit readout parameters distributions, including readout drive frequency, the coupling strength between the qubitand its individual readout resonator, the dispersive shift, the resonator linewidth, the readout fidelity of state | i , the readoutfidelity of state | i and the effective qubit temperature. tional qubit are shown in the Fig.S2, Fig.S3 and Fig.S4,and their statistical values are shown in table S1. III. SYSTEM CALIBRATIONA. Idle frequency setup
In our experimental setup, the qubits are biased attheir idle frequencies for state preparation and readout.In calibrating and optimizing qubit idle frequency, thereare several key elements we need to focus on. They are:
Coupling strength between neighboring qubits (MHz)
U03Q2U22Q1U00Q0
U01Q0
U02Q0
U03Q0
U10Q0
U11Q0
U12Q0
U13Q0
U20Q0
U21Q0
U22Q0
U23Q0
U30Q0
U31Q0
U32Q0
U33Q0
U00Q1
U01Q1
U02Q1
U03Q1U10Q1
U11Q1
U12Q1
U13Q1
U20Q1
U21Q1
U23Q1
U30Q1
U31Q1
U32Q1
U33Q1
U00Q2
U01Q2
U02Q2
U10Q2
U11Q2
U12Q2
U13Q2
U20Q2
U21Q2
U22Q2
U23Q2
U30Q2
U31Q2
U32Q2
U33Q2U00Q3
U01Q3
U02Q3
U03Q3
U10Q3
U11Q3
U12Q3
U13Q3
U20Q3
U21Q3
U22Q3
U23Q3
U30Q3
U31Q3
U32Q3
U33Q3
FIG. S4. Distribution of the coupling strengths betweenneighboring qubits. The square connecting two qubits showsthe effective coupling strength between them when these twoqubits are tuned to the interaction frequency of 5.02 GHz. Energy relaxation.
It is well known that theenergy relaxation time of the qubit varies stronglywith frequency. Defects including two-level-systems(TLSs) [S3], slot-line modes [S4], and some other mi-crowave modes, induce significantly short energy re-laxation times at certain frequencies. This will fur-ther limit the readout and single-qubit operation fi-delities. Those frequencies affected by defects shouldbe avoided.2)
Microwave crosstalk.
In our superconductingquantum device, though the microwave crosstalk hasbeen suppressed to below -40 dB for next-nearest-neighboring qubits, for the nearest-neighboring qubitsit is still not negligible as being around -25 dB. Con-sidering the maximal driving strength to be around 25MHz, the minimal frequency gap between two neigh-boring qubits is set to be 50 MHz. Meanwhile, toavoid the two-photon excitation by crosstalk, the fre-quency of exact match with f / f is the frequencydifference between the ground state and the secondexcited state.3) ZZ coupling. For the coupled qubits, the ZZ cou-pling strength [S5] is given by Ω ZZ = − g ( η + η ) / [(∆ − η )(∆ + η )], where η and η are the qubitanharmonicities, g is the coupling strength betweenthese two qubits, and ∆ is the difference in qubit fre- quencies. From the above expression, it is noticedthat if one wants to limit the ZZ coupling to be be-low 0.2 MHz, the frequency gap between f of thetarget qubit and f of the coupling qubit should belarger than 45 MHz, where f is the frequency differ-ence between the first and second excited states.4) Readout fidelity.
We perform the readout opera-tion at the idle frequency. The large detuning fromthe readout resonator reduces the dispersive couplingstrength of the qubit, and further affects the readoutfidelity. However, we also need to balance it with thefrequency crowding issues. We set the minimal idlefrequency to be 4.9 GHz.We construct an optimization procedure based onthese principles, containing the three following steps.1. First, we measure the frequency-dependent T ofall qubits to determine the defect-affected frequen-cies. We then generate tables of available frequen-cies { f iavl } for all qubits by removing those bad-performance points. The frequency step in the ta-bles is 1 MHz.2. Second, based on { f iavl } , we search for a solutionthat all qubits are initialized in their available fre-quencies. To be more specific, we randomly chooseone frequency from f qavl as the initial frequencyfor the qubit q with the shortest length of f qavl .Then, we update { f iavl } with the frequencies of de-termined qubits, following the principles we listedpreviously. We repeat this progress until all qubitsare set. However, if there is no available frequenciesfor one of the qubits, we restart the search again.We note that we use T as the weight of each fre-quency point, thus those frequencies with betterperformance have more chances to be chosen.3. Last, we optimize all single-qubit performance inparallel, and then measure T of all qubits at theiridle frequencies. It is still possible that when thequbits are biased to different idle points, some mi-crowave modes and TLSs may be changed, result-ing in different available frequencies. The changescan be identified from the relatively low values andthe large variations in fitting T . We then opti-mize those qubits again by measuring T at differ-ent available frequencies to find a frequency withbetter performance. This step is repeated for sev-eral rounds until the performance of all qubits areacceptable. B. Frequency-alignment optimization
In previous experiments [S6], we optimize thefrequency-alignment by tuning all qubits to the same in-teraction frequency for evolution, and then use the pop-ulation propagation to fit the disorders on each site forfurther corrections. However, when the system size growsto be more than 60 qubits, such a strategy becomes un-achievable because of the non-negligible time cost in thenumerical simulations.In this work, we optimize the frequency-alignment withthe following procedure:1. Set the initial alignment frequency for all qubits.In this work, it is 5.02 GHz.2. For each qubit, we measure the population prop-agation between the qubit and its coupled qubits,which is defined as ‘multi-qubit swapping’ experi-ment. We begin by exciting a single qubit. Thenwe tune the qubit and its coupled qubits to thealignment frequency for system evolution. All otherqubits are tuned to 4.97 GHz to prevent any un-wanted state leakage. After an evolution time rang-ing from 0 to 1 µs , we tune these qubits back totheir idle points to readout the population of eachqubit jointly.3. With the data sets for all qubits, we use the ‘Nelder-Mead’ algorithm to search for a disorder map whichhas the best fit to the data. To be more specific, foreach data set, we can calculate the distance fromthe experimental data to the numerically simulateddata with the disorders given by the disorder map.By defining the cost function as the sum of thesquare of the distances, we optimize the disordermap to minimize the cost function. The disor-der map is the final frequency differences we needto correct the alignment frequency for each qubit.Meanwhile, we define and calculate the overall dis-tance using the sum of the square of the distances,which comes from the data and the numerical sim-ulations with no disorders.4. We then determine the sign of correction by addingor subtracting the disorder map for the alignmentfrequency. Then, there are two alignment fre-quency setups, of which one setup is worse and willinduce larger overall distance. By running ‘multi-qubit swapping’ experiments for both setups, wecompare the overall distances. The smaller over-all distance indicates the better setup, and can beused to further updating.5. By repeating steps 2 to 4 for several rounds, theoverall distance will saturate. In our case the fi-nal maximal disorder determined is smaller than0.8J eff . C. Optimization of the interferometer
Though the disorders of each site have been suppressedto be below 0.8J eff , the residual disorders in the interfer-ometer are still non-negligible. As the number of effective coupling is reduced from 4 to 2 or 3 in the interferom-eter, the disorder causes the reflection in spreading andreduces the effective coupling strength. As a result, themaximal population at site D at t = 650 ns is only 0.12before optimization.The ultimate objective of the optimization is to en-hance the population of site D at t = 650 ns when singleexcitation is involved in site S . However, the direct opti-mization of all sites in the interferometer may result in adifferent local minimum, which corresponds to the block-ade of one path. Therefore, we use a procedure utilizinga two-step optimization to search for the best correction.1. First, we optimize the alignment frequency of theinterferometer except for sites BS D . Afterexciting the qubit at site S , we tune all qubits inthe interferometer except for sites BS D tothe alignment frequency. Note that in this step,sites BS D are not in interaction. We usethe product of the populations of sites L and R at t = 550 ns as the cost function, and use‘Nelder-Mead’ algorithm to optimize the alignmentfrequencies of the corresponding qubits.2. Second, we optimize the alignment frequency for allsites. After exciting the qubit at site S , we tune allsites in interaction and then use the population ofsite D at t = 650 ns as the cost function. Again,we use the ‘Nelder-Mead’ algorithm to optimize thealignment frequencies for all sites. Now based onthe first step optimization, the two paths have beenin balance, thus the local minimum with one pathblocked will not occur.Following this two-step optimizations, the maximalpopulation at site D for t = 650 ns can be optimizedto be above 0.43. IV. THERMALIZATION ANDPOST-SELECTION
We note that for several qubits, thermal noise is non-negligible when performing detuning operations, whichis possibly caused by the heating of the bonding wiresto the electrodes of the control lines. These affects thequality of our quantum walks.To suppress the thermal excitation, one solution is toutilize post-selection [S7], a technique commonly used inlinear optical quantum computation [S8]. Post-selectioninvolves selecting data events where the total number ofrecorded excitations are the same as the total number ofinitial excitations. This technique allows us to partiallyremove the effects of circuit loss, thermalization and de-tector inefficiency. All of these individually change thetotal excitation number and so if there is only one errorwe can eliminate it. However we do not remove the caseswhere two errors occurs which maintain the excitation
Time = 0 ns
Position i P o s i t i o n j Time = 100 ns
Position i Time = 200 ns
Position i Time = 300 ns
FIG. S5.
The two-particle quantum walks after post-selection.
For the same set of raw data in Fig. 2 A in the maintext, we performed post-selection with the conservation of the total excitation number. After post-selection, the behavior ismuch closer to ideal as the Thermalization is suppressed. number. Examples of these two error processes for in-stance include when an excitation is loss in the circuitwhile one detector reports | i when it was actually | i .Another is when one detector reports | i (but was | i )while the second reports | i (but was | i ). Both these ex-amples maintain the total excitation number and so cannot be distinguished from the ideal case. However twoerror events occur with a much lower probability thansingle error events. The improved quantum walks afterpost-selection with two walkers are shown in Fig. S5. Wenote that for the other results, no post-selection is used. V. NUMERICAL SIMULATION METHOD
To obtain the theoretical results as the comparison ofthe experiment results, we numerically simulate the evo-lution of the system with 62 functional qubits under thespin model [S9]. This is a reasonable approximation ofthe Bose-Hubbard model when | U | (cid:29) | J | along with lowfilling factor. However, the full Hamiltonian space of the62-qubit system consists of 2 dimensions, whose cor-responding matrix has 2 bases. The space has beenbeyond a state-of-the-art classical supercomputer. So wetruncate the full Hamiltonian space to a subspace whosebases have the same number of excitations as the initialstates. For example, if we have a single excitation in thequbit array, we truncate the full space to the subspacecomposed of {| · · · i , | · · · i , · · · · · · , | · · · i} . Inthis way, we reduce the number of dimensions from 2 to 1891 for two excitations and 62 for a single excitation.The corresponding matrix of the truncated Hamiltonian is small enough to be easily handled with a laptop.Due to the time-independent Hamiltonian, we cantransform the Schrodinger equation to | Ψ( t ) i = e − iHt/ ~ | Ψ(0) i , where | Ψ(0) i is our initial state while | Ψ( t ) i is the state of system at time t . In this way,with the matrix of truncated Hamiltonian and the vectorof the truncated initial state, we can numerically simu-late the evolution of the initial state at any time underthe given Hamiltonian. And then we can obtain the ex-pectation of observables associated with the system as afunction of time, such as the population h n j ( t ) i , the cor-relation function C ij ( t ), etc. VI. EXTENDED DATA
In Fig. S6 and Fig. S7, we illustrate the quantum walksof single particle excited at U00Q0 and U33Q2, respec-tively. The time evolution of population with one path( { R } ) blocked in the interferometer is shown in Fig. S8.In Fig. S9, we show the simulated interference fringes incomparison with that of Fig. 3 C and 3 F . For the caseof two particles in the MZ interferometer, the numericalsimulated time evolution of all sites are shown in Fig. S10.In Fig. S11, we show the numerical simulated interferencephenomenon in comparison with that in Fig. 4 B , D , F , H in the main text.Moreover, the extended movies (supplied online only)show the time-resolved population h n j i of the two-dimensional quantum walk (Movie S1, S2) and the evo-lution of population h n j i in the MZ interferometer withsingle walker (Movie S3) and two walkers (Movie S4). [S1] J. Koch, M. Y. Terri, J. Gambetta, A. A. Houck,D. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M.Girvin, and R. J. Schoelkopf, Charge-insensitive qubitdesign derived from the cooper pair box, Physical Re-view A , 042319 (2007).[S2] Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pu-gatch, and Y. Silberberg, Quantum walk of two interact-ing bosons, Physical Review A , 011603 (2012). [S3] P. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant,B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, et al. , Fluctuations of energy-relaxation times in super-conducting qubits, Physical review letters , 090502(2018).[S4] J. M. Martinis and A. Megrant, Ucsb final report forthe csq program: Review of decoherence and materi-als physics for superconducting qubits, arXiv preprint Time = 0 ns A P o s i t i o n j Time = 100 ns Time = 200 ns Time = 300 ns
Position i
FIG. S6.
Quantum walks in a 2D superconducting qubits array with excitation at U00Q0 . ( A ) In the statepreparation, qubit U00Q0 at the top left corner is excited. Then all qubit are tuned in interaction and finally measured jointlyat t = 0 ns, 100 ns, 200 ns, and 300 ns, respectively. ( B ) The numerical simulation with the same condition as in ( A ). Time = 0 ns A P o s i t i o n j Time = 100 ns Time = 200 ns Time = 300 ns
Position i
FIG. S7.
Quantum walks in a 2D superconducting qubits array with excitation at U33Q2 . ( A ) We excite qubitU33Q2 at the bottom right corner in the state preparation and then measure all qubit jointly after an interaction time of t = 0ns, 100 ns, 200 ns, and 300 ns, respectively. ( B ) The numerical simulation with the same condition as in ( A ).arXiv:1410.5793 (2014).[S5] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank,E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler,B. Campbell, et al. , Superconducting quantum circuits at the surface code threshold for fault tolerance, Nature , 500 (2014).[S6] M. Gong, G. D. Neto, C. Zha, Y. Wu, H. Rong, Y. Ye,S. Li, Q. Zhu, S. Wang, Y. Zhao, et al. , Experimen- D R - S L - B S B S
FIG. S8.
The evolution of population h n j i in single-particle Mach-Zehnder interferometer with one path blocked. The circuit diagram is the same as that in Fig. 3 E in the main text with { R } path blocked. The qubit is excited at the source S . In A and B , we illustrate the dynamical evolution of population h n j i of all site in experiment and simulation, respectively.
The numerical simulated interference fringes in single-particle Mach-Zehnder interferometer . In ( A )and B , we show the numerical simulated interference fringes with all parameters the same as that in Fig. 3 C and Fig. 3 F inthe main text respectively for comparison.tal characterization of quantum many-body localizationtransition, arXiv preprint arXiv:2012.11521 (2020).[S7] Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang,W. Ren, H. Dong, D. Zheng, Y.-R. Zhang, et al. , Obser-vation of energy-resolved many-body localization, NaturePhysics , 1 (2020). [S8] E. Knill, R. Laflamme, and G. J. Milburn, A scheme forefficient quantum computation with linear optics, nature , 46 (2001).[S9] T. Onogi and Y. Murayama, Two-dimensional superflu-idity and localization in the hard-core boson model: Aquantum monte carlo study, Physical Review B , 9009(1994). D R - S L - B S B S
Experiment Simulation
FIG. S10.
The evolution of population h n j i with two particles excited at L and R in two different situations. The circuit diagram for ( A and B ) and ( C and D ) is the same as that of Fig. 4 A and 4 C in the main text, respectively. Thedifference is in C and D , sites BS S are removed from the interferometer. B and D are the numerical simulated resultswith the conditions the same as that of A and C , respectively.