Deterministic Bidirectional Communication and Remote Entanglement Generation Between Superconducting Quantum Processors
N. Leung, Y. Lu, S. Chakram, R.K. Naik, N. Earnest, R. Ma, K. Jacobs, A. N. Cleland, D. I. Schuster
DDeterministic Bidirectional Communication and Remote Entanglement Generation BetweenSuperconducting Quantum Processors
N. Leung, ∗ Y. Lu, ∗ S. Chakram, R. K. Naik, N. Earnest, R. Ma, K. Jacobs,
2, 3
A. N. Cleland, and D. I. Schuster † The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA U.S. Army Research Laboratory, Computational and Information Sciences Directorate, Adelphi, Maryland 20783, USA Department of Physics, University of Massachusetts at Boston, Boston, MA 02125, USA Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
Abstract
We propose and experimentally demonstrate a simple andefficient scheme for photonic communication between tworemote superconducting modules. Each module consists ofa random access quantum information processor with eight-qubit multimode memory and a single flux tunable transmon.The two processor chips are connected through a one-meterlong coaxial cable that is coupled to a dedicated “communi-cation” resonator on each chip. The two communication res-onators hybridize with a mode of the cable to form a dark“communication mode” that is highly immune to decay inthe coaxial cable. We modulate the transmon frequency viaa parametric drive to generate sideband interactions betweenthe transmon and the communication mode. We demonstratebidirectional single-photon transfer with a success probabilityexceeding 60%, and generate an entangled Bell pair with afidelity of 79.3 ± Introduction
A practical quantum computer requires a large numberof qubits working in cooperation [1], a challenging task forany quantum hardware platform. For superconducting qubits,there is an ongoing effort to integrate increasing numbers ofqubits on a single chip [2–8]. A promising approach to scalingup superconducting quantum computing hardware is to adopta modular architecture [9–11] in which modules are connectedtogether via communication channels to form a quantum net-work. This reduces the number of qubits required on a singlechip, and allows greater flexibility in reconfiguring and ex-tending the resulting information processing system. In suchan architecture, each module is capable of performing univer-sal operations on multiple-bits, and neighboring modules areconnected through photonic channels, allowing communica-tion and entanglement generation between remote modules.Remote entanglement between superconducting qubits hasbeen realized probabilistically [12–14]. Conversely, realiz-ing deterministic photonic communication requires releasinga single photon from one qubit and catching it with the remotequbit. In the long-distance limit, the photon emission and ab-sorption are from a continuum density of states. In this limit,static coupling limits the maximum transfer fidelity to only54% [15, 16]. This limit is exceeded by dynamically tailoring ∗ These authors contributed equally to this [email protected]@uchicago.edu † [email protected] the emission and absorption profiles [17–20]. These capabil-ities are presently being used to perform photonic communi-cation between superconducting qubits connected by a trans-mission line within a cryostat [21–24]. In these experiments,the use of a circulator enables the finite-length transmissionline to be modeled as a long line with a continuum density ofstates, at the cost of added transmission loss.Here, we establish bidirectional photonic communicationbetween two multi-qubit superconducting quantum proces-sors through a multimodal communication channel. Ratherthan inserting a circulator, the multimode nature of the fi-nite length transmission line is made manifest and exploited[25]. For intra-cryostat communication, the required connec-tion coaxial cable length of 1 m or less results in a free spec-tral range on the order of hundreds of MHz. In this setting,the resonances of the coaxial cable form hybridized normalmodes with on-chip communication resonators, and photonsare transferred coherently through the discrete modes of thechannel in contrast to emission/absorption through a contin-uum. We use parametric flux modulation of the qubit fre-quency to generate resonant sideband interactions between thequbit and the communication channel [26–29]. This approachavoids the loss due to the circulator that significantly limitsthe communication fidelity, and enables bidirectional quan-tum communication. Results
Network of two multimode modules
We extend the random access quantum processor mod-ule presented in Ref. [30] to allow photonic communicationbetween two remote modules, thereby realizing a two-nodequantum network. Each processor consists of an eight-qubitmultimode memory comprised of two chains of four identi-cal and strongly coupled superconducting resonators, a sin-gle flux-tunable transmon, and two additional resonators [31].The first of these resonators is used for readout, and the sec-ond is coupled to the coaxial cable to enable the inter-modulecommunication. The transmon can resonantly couple to all theresonators (readout, multimode and communication) throughparametric flux modulation to realize intra-module gate oper-ations and inter-module photonic communications. Figure 1shows a schematic of our two modules. The readout res-onators have the lowest frequencies [module 1: 5.7463 GHz;module 2: 5.7405 GHz], the communication resonators havethe highest frequencies [ ≈ ≈
200 MHz. For the circuit design, we arranged the mul- a r X i v : . [ qu a n t - ph ] A p r S i d e b a n d f r e q u e n c y ( G H z ) S i d e b a n d f r e q u e n c y ( G H z ) transmon |e> population transmon |e> population Figure 1 | Device schematic and stimulated vacuum Rabi oscillations.
Each chip consists of a frequency-tunable transmonand two chains of four identically designed, lumped-element resonators. In addition, a resonator is included for readout, and asecond resonator is coupled to the coaxial cable ( ∼ ≈
200 MHz. timode resonators to be spatially separated from the readoutand communication resonators by placing the high impedancetransmon in-between, preventing Purcell loss of the multi-mode resonators through the low Q readout and communi-cation resonators [32–34]. We operate the transmons at thestatic frequency of [1: 4.7685 GHz; 2: 4.7420 GHz] with ananharmonicity of [1: 109.8 MHz; 2: 109.9 MHz].We induce resonant interactions between the transmon andan individual mode by modulating the transmon frequency viaits flux bias. The modulation creates sidebands of the trans-mon excited state, detuned from the original resonance by thefrequency of the applied flux tone. When one of these side-bands is resonant with a mode, the system experiences stim-ulated vacuum Rabi oscillations [30]. This process is simi-lar to resonant vacuum Rabi oscillations [35], but occur at arate that is controlled by the modulation amplitude [26, 27].To illustrate the application of parametric control, we employthe following experimental sequence. First, the transmon isexcited via its charge bias. Subsequently, we modulate theflux bias to create sidebands of the transmon excited stateat the modulation frequency. This is repeated for differentflux pulse durations and frequencies, with the population ofthe transmon excited state measured at the end of each se-quence. When the frequency matches the detuning betweenthe transmon and a given eigenmode, we observe full-contraststimulated vacuum Rabi oscillations. Figure 1 shows that thetransmon can selectively interact with each of the eigenmodesby choosing the appropriate modulation frequency. As pre- viously demonstrated, this sideband interaction and rotationsof the transmon are sufficient for universal operations on eachset of multimode resonators [30]. Similarly, the photon trans-fer process between two remote qubits is initiated by switch-ing on the sideband interactions targeting the communicationresonator on each chip. As the bare frequencies of the trans-mon and the communication resonator are far detuned ( ∆ ≈ g ≈
50 MHz), the sideband coupling scheme for pho-tonic communication achieves a high on/off ratio.
Multimode communication channel
The two communication resonators are designed to haveidentical frequencies. They are chosen to be coplanar waveg-uide resonators with a large center pin and gap width to makethe frequency insensitive to fabrication variations [36]. Theseresonators are coupled via the one-meter long coaxial cable,where the cable can be thought of as a multimode resonatorwith a free spectral range of around 200 MHz. The couplingstrength between the cable and the communication resonatorsis g c ≈ MHz. The cable mode that we use for communi-cation has a frequency that is within g c of the frequencies ofthe communication modes. Since the free spectral range ofthe coaxial cable is an order of magnitude larger than g c , weconsider the cable as a single mode nearly resonant with thecommunication resonators. The cable and the communica-tion resonators thus together produce three hybridized normalmodes which are depicted in Figure 2. The near-degeneracy ofthe coaxial cable with the two communication resonators give Normal modes wavefunctions a b on-chipresonator on-chipresonatorcoaxialcable S i d e b a n d f r e q u e n c y ( G H z ) transmon |e> population Figure 2 | Hybridized normal modes. a.
The schematicshowing the wavefunctions of the coupled system involvingthe communication resonators and the coaxial cable. Thethree degenerate modes hybridize and form three normalmodes with distinct signatures. The center normal mode hasminimal participation in the lossy cable mode and has highquality factor. b. Stimulated vacuum Rabi oscillations aroundthe communication modes. The near-degeneracy of thecoaxial cable with the two communication resonators give riseto this almost equally-spaced three-mode structure. Being thetwo bright modes that include the lossy cable mode, and thedark “communication” mode of the two resonators. The lattercouples more strongly to both qubits, and has a lifetime that isideally only limited by the internal quality factors of thecommunication resonators. By fitting the simulation toexperimental data, we found that the coaxial cable has aslightly higher frequency than the on-chip communicationresonators [see Appendix]. rise to this almost equally-spaced three-mode structure, whichcan be seen from the three stimulated vacuum Rabi chevronsin Fig. 2 b. The center normal mode used for communica-tion ideally has no participation in the cable mode, and as aresult, its loss rate is limited by the internal quality factors ofthe communication resonators and small Purcell losses fromneighboring cable modes. In comparison to the neighboringmodes, the center normal mode couples more strongly to bothqubits due to higher wavefunction participation at the commu-nication resonators. Thus, this communication mode has boththe advantages of high quality factor and high coupling rate.For any practical device, the center normal mode does havea non-zero participation in the lossy coaxial cable due to afrequency mismatch between the two on-chip communicationresonators. From the measurements of previous individual testchips, the detuning between these two resonators is expectedto be less than 3 MHz ( < g c ), an assumption that is validatedby the simulation shown in the appendix, resulting in a lessthan 5% of cable mode participation in the communicationmode.The coherence time of the communication mode can becharacterized using protocols analogous to those for the trans-mon; the qubit pulses are merely sandwiched between a pairof transmon-mode iSWAP pulses to transfer the quantum statebetween the transmon and the mode [30]. We find T = P o p u l a t i o n P o p u l a t i o n a c1f1c2f2 c1f1c2f2 b eggg ge eeegggge ee Figure 3 | Bidirectional excitation transfer.
The inset attop right shows the pulse sequence used to implementexcitation transfer. The labels c , c denote the charge driveson qubits 1 and 2, respectively, and f , f the respective fluxdrives. We first apply a π pulse to excite one of the qubits,then simultaneously switch on the sideband flux pulse to drivethe transfer process. Using the same sideband sequence, butinstead applying the π pulse to the other qubit, we can send asingle photon in the opposite direction. The transfer fidelity islimited by qubit dephasing and photon decay in thecommunication mode. Described in the following, the transferprocess in different directions have slightly different lossmechanisms. a. Excitation transfer from qubit 1 to qubit 2.Notice that in this transfer process the sender qubit is not ableto fully receive its excitation (population of | eg (cid:105) does not reachzero). As confirmed by the master equation simulation, this isdue to the dephasing of qubit 1. The remaining errors arisefrom communication cavity loss and dephasing of qubit 2,which is less than that of qubit 1. b. Excitation transfer fromqubit 2 to qubit 1. In this process, while qubit 2 releases mostof its excitations (population of | ge (cid:105) comes close to zero), thedephasing of qubit 1 prevents it from capturing all theexcitations in the communication mode, resulting in a slightlyhigher final population in | gg (cid:105) . The resulting fidelities for thetransfer in the two directions are similar: { P | ge (cid:105) , P | eg (cid:105) } ≈ ns and T ∗ = 1 µ s, corresponding to a quality factor ofabout 4000. This quality factor is reasonably high, consid-ering that it involves losses from the long lossy cable, wire-bonds, solder of the SMA connector, and the copper leads ofthe sample holder. The two neighboring normal modes havemuch lower coherence times due to the higher participation ofthe lossy cable mode. From fitting to fig. 2b we estimate anupper bound of T for these modes to be ∼
200 ns.
Bidirectional communication
To demonstrate photonic communication between the twochips, we send a single photon from one chip to the other.First, we excite the sender qubit, then we switch on side-band interactions simultaneously on both qubits, targeting thecommunication channel. We send a photon in the reverse di-rection using the same sideband sequence but instead excit-ing the other qubit, thus demonstrating bidirectional photontransfer. Figure 3 shows the transmon population plotted asa function of the sideband pulse length. The master equationsimulation results (solid lines) are shown along with the ex-perimental data (dots). We are able to obtain photon transferwith a success rate of { P | ge (cid:105) , P | eg (cid:105) } ≈ T ∗ ≈
700 ns) than qubit 2 ( T ∗ ≈ µ s). Thedephasing rate of qubit 1 is comparable to the sideband cou-pling rate, with the result that this qubit is not able to fullyrelease its excitation during the transfer process. Conversely,for transfer in the other direction qubit 1 is not able to receiveall of the excitations. This transfer infidelity can be largelymitigated by using a fixed-frequency qubit less susceptible tothe flux noise, with its coupling strength to the communicationmode parametrically controlled via a tunable coupler circuit[37–41]. The remaining loss of transfer fidelity comes fromthe loss in the communication mode. From our numerical sim-ulations detailed in the appendix, we estimate that the overallphoton loss in both the qubits and the communication modecontribute to an infidelity of 24%, while the dephasing errorof the two qubits accounts for an infidelity of 15%. The side-band coupling rate of the transmon is limited by the range overwhich its frequency can be parametrically tuned, resulting ina maximum effective sideband coupling to the communica-tion resonator of ≈ Bell state entanglement
We now entangle two qubits by creating a Bell state be-tween the transmons on the respective chips [44]. We cancreate such a state by first applying the √ i SWAP gate be-tween the excited qubit 1 and the communication mode, whichgenerates the Bell state ( | g (cid:105) + | e (cid:105) ) / √ between them. Weimplement the √ i SWAP by applying a sideband modulationpulse to qubit 1 to perform a π/ rotation. Subsequently,we transfer the state of the communication mode to qubit 2through the i SWAP gate by applying a sideband modulationpulse to the latter to perform a π rotation. Ideally this se-quence prepares the Bell state | Ψ + (cid:105) = ( | ge (cid:105) + | eg (cid:105) ) / √ shared between the two remote qubits. To minimize deco-herence the sender and receiver pulses can be applied simul-taneously, so long as the lengths and amplitudes of the pulsesare adjusted appropriately. Choosing qubit 1 as the sender gg ge eg ee gggeegee 0.00.10.20.30.40.5 ba IX IY IZ XI XX XY XZ YI YX YY YZ ZI ZX ZY ZZ1.00.50.00.51.0 E x p e c t a t i o n v a l u e c1f1c2f2 Figure 4 | Bell pair creation. a.
Real component of thedensity matrix. b. Expectation values of two-qubit Paulioperators. We create a Bell state between two remote qubits,one on each module. This is achieved by first applying the √ i SWAP gate between the excited qubit 1 and thecommunication mode, which is implemented by a sidebandmodulation pulse to qubit 1 to perform a π/ rotation. Asimilar pulse, this time a π rotation, applied to the secondqubit performs an i SWAP that transfers the entanglement fromthe communimation mode to the second qubit. As shown inthe inset, we implement the two pulses simultaneously toreduce decoherence. We obtain the resulting Bell state withfidelity (cid:104) Ψ + | ρ exp | Ψ + (cid:105) = ± and using square pulses, we found — both in our simula-tion and in the experiment — that maximal fidelity was ob-tained by setting both pulses at the same coupling rate andthe length of the receiver pulse to be slightly longer thantwice that of the sender. The resulting Bell state has a fi-delity of (cid:104) Ψ + | ρ exp | Ψ + (cid:105) = ± ρ exp using quantum state tomography with anover-complete set of measurements complemented with themaximum likelihood method [45]. It can be inferred fromthe data that the fidelity is almost equally limited by photondecay in the cable and the qubit dephasing errors. We alsonote that the Bell state fidelity is significantly higher than thesuccess probability we achieved for photon transfer. Likelyexplanations for this is that qubit 1 is actively involved in theprocess for only half the duration of the protocol and there isless excitation in the cable over the duration of the protocol.The process is thus less sensitive to the dephasing noise in thequbit and decay loss in the cable. Conclusion
We have built upon the random access quantum informationprocessor previously presented in Ref. [30], so as to realizephotonic communication between two remote modules, a firststep in realizing a modular network. The sideband modulationof the transmon qubit in each module can be applied to im-plement local operations on the multimode resonators and toperform photon transfer between the two modules. The multi-mode characteristic of the communication channel (a coaxialcable) is enabled by the absence of a circulator. This modestructure results in normal modes that are superpositions of amode of the inter-module communication cable and the on-chip resonators. One of the these normal modes is “dark”to the coaxial cable mode, thus avoiding much of the cableloss and allowing for high fidelity photon transfer. We charac-terized our system by performing single photon transfer with61% fidelity and Bell-state preparation with 79.3% fidelity.These fidelities can be increased by improving the qubit co-herence time and the strength of the coupling to the com-munication channel. Future work will include implementingmore sophisticated photon transfer protocols (e.g. STIRAP),applying heralding protocols to protect against photon trans-mission error, implementing local gates on memory modes inconjunction with photonic communication to facilitate large-scale computation, and integrating the present architecturewith high-quality-factor 3D superconducting cavities [46].
Data availability
Data available on request from authors.
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The authors thank R. Cook, Y. Zhong, and A. A. Clerkfor useful discussions, and A. Oriani for support with cryo-genic facilities. This material is based upon work supportedby the Army Research Office under (W911NF-15-2-0058).The views and conclusions contained in this document arethose of the authors and should not be interpreted as repre-senting the official policies, either expressed or implied, ofthe Army Research Laboratory or the U.S. Government. TheU.S. Government is authorized to reproduce and distributereprints for Government purposes notwithstanding any copy-right notation herein. Research was also supported by the U.S. Department of Defense under DOD contract H98230-15-C0453. This work was partially supported by the Universityof Chicago Materials Research Science and Engineering Cen-ter, which is funded by the National Science Foundation un-der Award No. DMR-1420709. This work made use of thePritzker Nanofabrication Facility of the Institute for Molecu-lar Engineering at the University of Chicago, which receivessupport from SHyNE, a node of the National Science Foun-dation’s National Nanotechnology Coordinated Infrastructure(NSF NNCI-1542205). We gratefully acknowledge supportfrom the David and Lucile Packard Foundation.
Author contributions
N.L., Y.L designed and fabricated the device, designed theexperimental protocols, performed the experiments, and ana-lyzed the data. K.J. provided theoretical support. S.C., R.N.,N.E., R.M provided fabrication and experimental support. Allauthors co-wrote the paper.
Competing financial interests
The authors declare no competing financial interests.
Cryogenic setup and control instrumentation d B d B Still (700 mK) MC (10-30 mK) 4 K 50 K 300 K
Filter
HEMT Miteq S R S P r ea m p li f i e r A L A Z A R T e c h A c qu i z i t i on c a r d ( G S a / s ) A g ill en t E D Minicircuits amplifiers ZX60-83LN-S+ A g ill en t E D T e k t r on i x A W G C ( . G S a / s ) B NC I(t) Q(t) K e ys i gh t M A ( G S a / s ) Y O K OG A W A G S A ( DC c u rr en t s ou r c e ) L L R R QI I Charge + Readout driveRF flux driveDC flux drive trig clk D i g i t a l M a r k e r s A na l og O u t MultimodeModule 1 E cc o s o r b RFDC H P - . G H z L P - . G H z L P - G H z L P - G H z L P - M H z Flux in
Out
Bias tee 50 Ω Ω DC Block C i r c u l a t o r s clk trig trig clk trig clk trig clk LP- 6 GHz - K&L FilterLP- 2 GHz - Minicircuits VLF - 1800 +LP- 3 GHz - Minicircuits VLF - 3400 +HP- 0.4 GHz - Minicircuits SHP - 400 +Bias-T - Minicircuits ZFBT -4R2GW+Eccosorb - Lossy IR filterBandpass - Fairview FMFL002 d B µ metal shield Output S R S F S R b a t o m i c c l o ck +30 dB+25 dB+16 dB+16 dB Charge in d B d B d B d B HEMT Miteq Charge + Readout driveRF flux driveDC flux drive
MultimodeModule 2 E cc o s o r b RFDC H P - . G H z L P - . G H z L P - G H z L P - G H z L P - M H z Flux in
Out
Bias tee 50 Ω Ω DC Block C i r c u l a t o r s d B µ metal shield Output +30 dB+25 dBCharge in d B d B FilterBandpass
Minicircuits amplifiers ZX60-83LN-S+
L RI +16 dB+16 dB
Communication (~1m coaxial cable)
I(t) Q(t)
LR QI Y O K OG A W A G S A ( DC c u rr en t s ou r c e ) Bandpass
Figure 5 |
Detailed schematic of the cryogenic setup, control instrumentation, and the wiring of microwave and DC connectionsto the device.
The device is heat sunk via an OFHC copper post to the base stage of a Bluefors dilution refrigerator (10-30 mK). The sampleis surrounded by a can containing two layers of µ -metal shielding and a layer of lead shielding, thermally anchored using aninner close fit copper shim sheet, attached to the copper can lid. The schematic of the cryogenic setup, control instrumentation,and the wiring of the device is shown if Supplementary Figure 5. Each device is connected to the rest of the setup through threeports: a charge port that applies qubit and readout drive tones, a flux port for shifting the qubit frequency using a DC-flux biascurrent and for applying RF sideband flux pulses, and an output port for measuring the transmission from the readout resonator.The readout pulses are generated by mixing a local oscillator tone (generated from an Agilent 8257D RF signal generator), withpulses generated by a Tektronix AWG5014C arbitrary waveform generator (TEK) with a sampling rate of 1.2 GSa/s, using anIQ-Mixer (MARQI MLIQ0218). The charge drive pulses are generated with Keysight M8195A arbitrary waveform generatorby direct synthesis, and subsequently combined with the readout drive pulse. The combined signals are sent to the device, afterbeing attenuated a total of 60 dB in the dilution fridge, using attenuators thermalized to the 4K (20 dB), still (20 dB) and basestages (20 dB). The charge drive line also includes a lossy ECCOSORB CR-117 filter to block IR radiation, and a low-pass filterwith a sharp roll-off at 6 GHz, both thermalized to the base stage. The flux-modulation pulses are also directly synthesized bythe Keysight M8195A arbitrary waveform generator and attenuated by dB at the 4 K stage, and bandpass filtered to within aband of 400 MHz - 3.4 GHz at the base stage, using the filters indicated in the schematic. The DC flux bias current is generatedby a YOKOGAWA GS200 low-noise current source, attenuated by 20 dB at the 4 K stage, and low-pass filtered down to abandwidth of 1.9 MHz. The DC flux bias current is combined with the flux-modulation pulses at a bias tee thermalized at thebase stage. The state of the transmon is measured using the transmission of the readout resonator, through the dispersive circuitQED readout scheme [47]. The transmitted signal from the readout resonator is passed through a set of cryogenic circulators(thermalized at the base stage) and amplified using a HEMT amplifier (thermalized at the 4 K stage). Once out of the fridge, thesignal is filtered (narrow bandpass filter around the readout frequency) and further amplified. The amplitude and phase of theresonator transmission signal are obtained through a heterodyne measurement, with the transmitted signal demodulated usingan IQ mixer and a local oscillator at the readout resonator frequency. The heterodyne signal is amplified (SRS preamplifier) andrecorded using a fast ADC card (ALAZARtech). Device Hamiltonian
Without connecting to the coaxial cable, the Hamiltonian of the i-th (i=1,2) circuit can be modeled by ˆ H = hν i,q ( t )ˆ a † i ˆ a i + 12 α i ˆ a † i ˆ a i (ˆ a † i ˆ a i −
1) + hν i,r ˆ b † i,r ˆ b i,r + hν i,c ˆ b † i,c ˆ b i,c + (cid:88) m =1 hν i,m ˆ b † i,m ˆ b i,m + hg i,r (ˆ b i,r + ˆ b † i,r )(ˆ a i + ˆ a † i ) + hg i,c (ˆ b i,c + ˆ b † i,c )(ˆ a i + ˆ a † i ) + (cid:88) m =1 hg i,m (ˆ b i,m + ˆ b † i,m )(ˆ a i + ˆ a † i ) (B1)where ˆ a i , ˆ b i,r , ˆ b i,c and ˆ b i,m stand for the annihilation operators of the flux-tunable qubit, the readout resonator, the com-munication cavity and the m-th multimode on the i-th chip. The communication cavities of the two chips are of identicalcoplanar waveguide resonator design with large center pin and gap width, leading to approximately the same resonant frequency ν ,c ≈ ν ,c = ν c and the same coupling strength g l to the coaxial cable mode ˆ b l , ˆ H int = (cid:88) i =1 hν c ˆ b † i,c ˆ b i,c + hν l ˆ b † l ˆ b l + (cid:88) i =1 hg l (ˆ b l ˆ b † i,c + ˆ b † l ˆ b i,c ) . (B2)B2 can be directly diagonalized, yielding three normal modes ˜ˆ b , ˜ˆ b and ˜ˆ b c , ˜ˆ H int = hν c ˜ˆ b † c ˜ˆ b c + hν ˜ˆ b † ˜ˆ b + hν ˜ˆ b † ˜ˆ b , (B3)where ν = ν c + δ (cid:113) g l + δ ,ν = ν c + δ − (cid:113) g l + δ , (B4)and ˜ˆ b c = 1 √ b ,c − ˆ b ,c ) , ˜ˆ b = 1 (cid:113) r + √ r ) (ˆ b ,c + ˆ b ,c + ( r + (cid:112) r )ˆ b l ) , ˜ˆ b = 1 (cid:113) r − √ r ) (ˆ b ,c + ˆ b ,c + ( r − (cid:112) r )ˆ b l ) . (B5)Here δ stands for the deviation of the cable mode frequency from the communication resonator frequency, i.e. δ = ν l − ν c , and r = δ/ g l . The normal mode frequencies relative to the qubit frequency can be readily obtained from Fig. 2.b, so that δ and g l can be calculated from Eq. B3 and B4. Eq. B1 and B5 together give the renormalized coupling strengths between the qubit andthese normal modes, ˜ˆ g c = g c √ , ˜ˆ g = g c (cid:113) h + √ h ) , ˜ˆ g = g c (cid:113) h − √ h ) . (B6)It is worth noting that the center normal mode, ˜ˆ b c , is selected to be our communication channel mode in the experiment, fortwo obvious reasons: it contains only the two resonator modes with no convolution with the cable mode, as seen in Eq. B5, thushighly immune to the photon loss of the cable. Eq. B6 shows that it also couples more strongly to the qubit comparing to theother two normal modes, which also agrees well with Fig. 2.b where the center chevron has the fastest oscillation. S i d e b a n d f r e q u e n c y ( G H z ) S i d e b a n d f r e q u e n c y ( G H z ) a b Figure 6 | Stimulated vacuum Rabi oscillation between the qubit and the communication mode.
By fitting to theexperimental data (a) using our analytical model, we extracted the deviation of the cable mode frequency from the twocommunication resonators to be 4.25 MHz, while the coupling between the cable mode and the communication resonator is . MHz. Plugging these along with other circuit parameters obtained from the experiment into a master equation, we cansimulate the experimental result with decent agreement (b).
Fitting Eq. B4 to Fig. 2b, we obtain δ = 4 . MHz and g l = 6 . MHz, from which we can numerically reproduce the chevronpatterns observed in the experiment (Fig.6).Here we list the relevant circuit parameters in the following table:sample 1 sample 2 ν q static: 4.7685 GHz; range: ≈ ≈ α ν r ν c ≈ ≈ ν m T µ s 7.9 µ s T ∗ µ s 1.4 µ s0 Sideband interaction and calibrations S i d e b a n d f r e q u e n c y ( G H z ) qubit 1 S i d e b a n d f r e q u e n c y ( G H z ) qubit 2 transmon |e> population readoutmultimodememorycommunication Figure 7 | Full sideband Rabi spectrum of each qubit.
Stimulated vacuum Rabi oscillation with sideband frequency scancovering the band of all resonance frequencies of the resonators. The clean chevron patterns indicate that our transmons arefree from spurious crosstalks. We can clearly identify ten chevron patterns corresponding to one readout resonator (lowestfrequency), eight multimode memory resonators and one communication resonator (highest frequency).
In these scans, we can clearly identify ten chevron patterns corresponding to one readout resonator, eight multimode memoryresonators, and one communication resonator. The crosstalk at sideband frequency ≈ S i d e b a n d a m p li t u d e ( a r b ) qubit 1 S i d e b a n d a m p li t u d e ( a r b ) qubit 2 Figure 8 | DC offset scan.
There is a shift (DC-offset) of the qubit frequency during the flux modulation, arising from thenon-linear flux-frequency relation of the transmon. To calibrate this effect, we sweep sideband transition frequency at differentflux amplitudes and obtain the calibration with linear interpolation. The black dots on the figures show the tracked resonancesideband frequency for the considered range of amplitude. The pattern of three normal modes persisted for the consideredrange of sideband amplitudes. In this experiment, we set the sideband length to be inversely proportional to the sidebandamplitude. This ensures high contrast features even for small sideband amplitude which the coupling is weak.
With the calibrated frequencies, we sweep the sideband length with a range of sideband amplitudes and obtain stimulatedvacuum Rabi oscillation. The experimental data is displayed in figure 9. As expected, a higher sideband amplitude impliesa higher effective coupling rate. Using this data, we obtained the effective qubit dissipation parameters during the sidebandcoupling. These dissipation parameters are subsequently being applied in master equation simulation of photon transfer and Bellentanglement generation. S i d e b a n d a m p li t u d e ( a r b ) qubit 1 S i d e b a n d a m p li t u d e ( a r b ) qubit 2 Figure 9 | Sideband Rabi sweep.
Using the calibrated DC offset, we obtained sideband rabi data between transmon andcommunication resonator for different sideband amplitude. Notice that the contrast of qubit 1 is much smaller than qubit 2. Thisis because qubit 2 has a higher coherence time. The trajectories of these scans are used for fitting the effective qubit decayparameters during the sideband coupling. These decay parameters are subsequently being applied in master equationsimulation of photon transfer and Bell entanglement generation.
Lastly, we calibrated the timing of the two flux sideband pulses. Due to slightly different travel path length of flux line controlfrom AWG to sample, we expect a slightly different timing between the two flux sideband pulses. Since the simultaneity of twoflux sideband pulses is essential for high fidelity transfer, it is important to calibrate this systematic error. The experiment wasconducted with two equal length sideband pulses but sweeping the software delay between two pulses. Here, a negative receiverdelay means the sender qubit (qubit 1) sideband pulse starts before the receiver qubit (qubit 2) sideband pulse. Figure 10 showsthe population of the sender qubit with sweeping parameters of two sideband length and receiver delay. The center of the “K"pattern corresponds to the scenario where the photon is maximally captured by the receiver qubit. We obtained the “K" patternas symmetric around receiver delay time of ≈ -10 ns, indicating the flux sideband pulse of the receiver qubit (qubit 2) lags theflux sideband pulse of the sender qubit (qubit 1). As a sanity check, we switched the role of sender and receiver qubit, such thatsender is qubit 2 and receiver is qubit 1. In such case, we found that the pattern is symmetric around receiver delay time of ≈ +10 ns. This confirms our conclusion that indeed the qubit 2 lags the flux sideband pulse of qubit 1 due to a delay in the lines.2 R e c e i v e r d e l a y ( n s ) Figure 10 | Delay calibration.
This figure shows the population of the sender qubit with sweeping parameters of two sidebandlength and receiver delay. Ideally, the two flux sidebands during photon transfer should start simultaneously. However due toexperimental conditions (e.g. different travel path length of flux line control from AWG to sample) causes the sideband pulsesstart at a different time on the devices, even the AWG is programmed to initiate two pulses simultaneously. To calibrate thiseffect, we sweep the delay between two sideband pulses and found that the flux control of qubit 2 is delayed by 10 ns.Throughout the experiment we time-advanced the control of qubit 2 flux by 10 ns in our pulse generation software.
Master equation simulation
In order to calculate the communication processes between the remote qubits using master equation simulations, we first writeout the circuit Hamiltonian under flux modulations, based on Eq. B1 ∼ B6, as ˆ H = (cid:88) i =1 3 (cid:88) j =1 h ( ν i,q + (cid:15) i cos ω i t )ˆ a † i ˆ a i + hα i ˆ a † i ˆ a i + hν j ˆ b † j ˆ b j + hg j (ˆ b j + ˆ b † j )(ˆ a i + ˆ a † i ) , (D1)where ˆ b j stand for the three normal mode and g j their coupling strengths to the two transmon qubits. Assuming weak flux mod-ulation with ω i ≈ ν c − ν i,q , and under the rotating frame transformation U = exp [ − ih (cid:80) i =1 (cid:80) j =1 (( ν i,q t − (cid:15) i ω i cos ω i t )ˆ a † i ˆ a i + ν c ˆ b † j ˆ b j t )] , Eq. D1 can be rewritten as ˆ H = (cid:88) i =1 3 (cid:88) j =1 (cid:26) hα i ˆ a † i ˆ a i + h ( ν j − ν c )ˆ b † j ˆ b j − ihg j J (cid:18) (cid:15) i ω i (cid:19) (cid:104) ˆ b j ˆ a † i e i ( ω i − ν j − ν c ) t − ˆ b † j ˆ a i e − i ( ω i − ν j − ν c ) t (cid:105)(cid:27) . (D2)Here J ( x ) stands for the Bessel function of the first kind of the first order, and all the fast-oscillating terms have beenabandoned. With the flux-modulation frequencies being ω i = ν c − ν i,q , and applying the two-level-approximation for thequbits, we find the "transfer Hamiltonian" as ˆ H = (cid:88) i =1 2 (cid:88) j =1 h ( ν l,j − ν c )ˆ b † l,j ˆ b l,j − ihJ (cid:18) (cid:15) i ω i (cid:19) (cid:104) g l,j (cid:16) ˆ b l,j ˆ σ + i − ˆ b † l,j ˆ σ − i (cid:17) + g c (cid:16) ˆ b c ˆ σ + i − ˆ b † c ˆ σ − i (cid:17)(cid:105) , (D3)where ˆ b l, and ˆ b l, are the two lossy “bright” normal mode, and b c is the “dark" communication channel mode. Plugging thisinto the master equation, ˙ ρ = − i [ ˆ H, ρ ] + κ l,j D [ˆ b l,j ] ρ + κ c D [ˆ b c ] ρ + γ D [ˆ σ − ] ρ + γ φ D [ˆ σ z ] ρ, (D4)we are able to simulate the bidirectional photon transfer experiment (Fig. 3) and the remote entanglement experiment (Fig. 4).Simultaneous square sideband pulses are adopted in both the photon transfer and Bel state creation experiment to achieve theshortest pulse time possible, as is shown in fig. 3 and 4. However, there is a possibility that better fidelities could be acquiredthrough further minimizing the photon loss in the communication mode, by making use of adiabatic protocols in a manner akinto the stimulated Raman adiabatic passage (STIRAP). A typical STIRAP protocol has a pulse sequence shown in fig. 11a, whereafter the excitation of the sender qubit, the receiving pulse turns on first, and slowly ramps down together with the ramping up ofthe sending pulse. When the ramping of the pulses are done adiabatically w.r.t the gap between the communication mode and the3qubit modes, the transfer could be completed without inducing the communication mode population, and is therefore immuneto the photon loss in the communication mode. However, this comes at the cost of much longer transfer time, which introducesmore loss from the qubits.For simplicity we model the sender and receiver pulses as two Gaussian pulses with the same maximum amplitude as thesquare pulse scheme used in our experiment. In the time domain, the two pulses are set to be f s ( t ) = (cid:40) Ae − ( t − t σ , | t − t | (cid:54) σ , | t − t | > σ , f r ( t ) = (cid:40) Ae − ( t − t − ∆ t )22 σ , | t − t − ∆ t | (cid:54) σ , | t − t − ∆ t | > σ . (D5)The fidelity yielded by this protocol is calculated as a function of both the pulse width σ and the delay time ∆ t , via masterequation simulation with real circuit parameters. Fig. 11b shows that a maximum fidelity of 56% is achieved when two Gaussianpulses with σ = 120 us overlap each other, which indicates that non-adiabatic transfer with shortest time is favorable in ourcurrent parameter regime. This also justifies our choice of the simultaneous square pulse scheme which is the fastest in allnon-adiabatic schemes. In contrast, if the coherence of the qubit is improved to T = 20 us and T = 20 us, the same simulationresults in a maximum fidelity of 85% at delay time ∆ t = (fig. 11c) that is higher than the simultaneous square pulse fidelity of82%, proving the usefulness of the adiabatic protocol for future improvements.
100 150 200 250 (ns) t ( n s ) a b c
100 150 200 250 (ns) t ( n s ) Figure 11 | STIRAP-like protocol. (a) Pulse sequence of the STIRAP protocol for photon transfer. After initializing of the senderqubit state in the excited state, two Gaussian pulses with same duration and amplitude (set to be the maximum amplitudeachievable in the experiment) are applied to the flux channels of the two qubits, with the receiver pulse turned on ahead of thesending pulse by a time of ∆ t . (b) Calculation of the transfer fidelity as a function of the Gaussian RMS width, σ , as well as thedelay time ∆ t . A maximum fidelity of occurs at { σ = 120 us, ∆ t = 0 us } (labeled by the yellow dot), which is worse than the60% fidelity achieved by the simultaneous square pulse scheme. This indicates that, in our current parameter regime, the fidelityis optimal with simultaneous square pulse scheme which has the shortest pulse length. (c) With better qubit coherenceproperties of T , T us, the STIRAP protocol promises 85% maximum fidelity at { σ = 145 us, ∆ t = 95 us } (labeled by theyellow dot), which is higher than the maximum fidelity of 82% yielded by the simultaneous square pulse scheme under the sameparameters. Readout and state tomography
To measure the two-qubit state, we record the homodyne voltage for each qubit from every run. For example, run i of theexperiment would result in a 4D heterodyne voltage values ( V I ,i , V Q ,i , V I ,i , V Q ,i ). These voltages are random numbersgenerated from a specific distribution corresponding to state projection and experimental noise. To measure the population inthe four two-qubit basis states: | gg (cid:105) , | ge (cid:105) , | eg (cid:105) , | ee (cid:105) we construct the histograms for these states by applying π pulses to thequbits. These histograms approximate the probability distribution for measuring a given voltage pair when the system is in agiven basis state.We employed logistic regression for classification of the two-qubit states. By setting decision thresholds for maximizingthe classification accuracy for the two-qubit basis states according to the voltage distribution, we obtain a confusion matrixrepresenting the correct and incorrect identification of basis state. For an unknown density matrix ρ we construct the classificationdistribution for ρ from N measurements, and project onto the basis states by applying the inverse of the calculated confusionmatrix.We perform state tomography using the standard method by calculating the linear estimator, ρ est = (cid:88) i,j T r [( σ i ⊗ σ j ) ρ ]( σ i ⊗ σ j )4 (E1)4To calculate the term T r [( σ i ⊗ σ j ) ρ we apply a unitary operator U to ρ prior to measurement. For two-qubits, there are ninerequired measurements corresponding to the following unitary operators, ( I, R Y ( π/ , R X ( π/ ⊗ ( I, R Y ( π/ , R X ( π/ .This simple linear estimator method can return unphysical results because it projects onto the space of all Hermitian matriceswith Trace 1. However a physical density matrix must also be positive semi-definite. Following the maximum likelihood protocoloutlined in [31, 45], we estimate the most likely physical density matrix by minimizing the function, F [ ρ est ] = N, (cid:88) i =1 ,j =1 ( (cid:104) j | U † i ρ est U i | j (cid:105) − P i,j ) (E2), where U i are the set of N applied tomography pulses, | j (cid:105) is the j th basis state, P i,j is the measured probability, and ρ est is aphysical density matrix satisfying the physical constraints. The starting guess for the minimization is the density matrix estimatedfrom the linear estimator with all negative eigenvalues set to zero. To form a over-complete set for a total of 17 tomographymeasurements, we also measure the negative pulse set [48] ( I, R Y ( − π/ , R X ( − π/ ⊗ ( I, R Y ( − π/ , R X ( − π/ . Online Gaussian process for Bell state optimization B e ll f i d e li t y Figure 12 | Optimization of Bell state creation with an online Gaussian process.
We employed an online optimizationdirectly applying on the experimental device. In each iteration, the Gaussian process model proposes 8 candidate solution (1obtained from L-BFGS-B optimization on the Gaussian model, and 7 obtained from random sampling filtered with the best modelprediction), and we also test two candidate solutions from pure random sampling to improve parameter space exploration. Therandom samplings lead to the apparent spikes of low fidelity Bell state during the optimization iterations. The model quicklystarts to converge, and after some time we obtained Bell state with a fidelity close to 80%.
For two square pulses, there are in total 6 parameters (amplitude, frequency, and duration of each square pulse). The linearinterpolation calibration of the DC offsets relates the amplitude and frequency parameters, thus resulting in 4 parameters tobe optimized. All the parameters are fairly dependent on each other in the process of simultaneous transfer, meaning all 4parameters have to be optimized together. Exhaustive search is quite forbidden even with just 4 parameters. Therefore, weemployed optimization techniques, in particular, the Gaussian process to assist in optimizing Bell state creation. We employedan online optimization directly applied to the experimental device. In each iteration, the Gaussian process model proposes 8candidate solution (1 obtained from L-BFGS-B optimization on the Gaussian model, and 7 obtained from random samplingfiltered with the best model prediction), and we also test 2 candidate solution from pure random sampling to improve parameterspace exploration. Figure 12 shows the optimization trajectory of Bell state creation. The model quickly starts to converge, andafter some time we obtained Bell state with a fidelity close to 80%. Since only half of the excitation is being transmitted in theprocess, the transmission is less likely to be lost. We are able to obtain bell state creation with a fidelity higher than single photontransfer. During the optimization, we clipped the value of density matrix to a maximum of 0.5 for the calculation of fidelity.Without doing so, we found our numerical optimization results bias towards a higher excited population ( > > ±±