Independent, extensible control of same-frequency superconducting qubits byselective broadcasting

S. Asaad,1, 2 C. Dickel,1, 2 S. Poletto,1, 2 A. Bruno,1, 2 N. K. Langford,1, 2 M. A. Rol,1, 2 D. Deurloo,1, 3 and L. DiCarlo1, 2

1QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands2Kavli Institute of Nanoscience, Delft University of Technology,

P.O. Box 5046, 2600 GA Delft, The Netherlands3Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 155, 2600 AD Delft, Netherlands

(Dated: August 28, 2015)

A critical ingredient for realizing large-scale quantum information processors will be the ability tomake economical use of qubit control hardware. We demonstrate an extensible strategy for reusingcontrol hardware on same-frequency transmon qubits in a circuit QED chip with surface-code-compatible connectivity. A vector switch matrix enables selective broadcasting of input pulses tomultiple transmons with individual tailoring of pulse quadratures for each, as required to minimizethe effects of leakage on weakly anharmonic qubits. Using randomized benchmarking, we comparemultiple broadcasting strategies that each pass the surface-code error threshold for single-qubit gates.In particular, we introduce a selective-broadcasting control strategy using five pulse primitives, whichallows independent, simultaneous Clifford gates on arbitrary numbers of qubits.

Building a fault-tolerant quantum computer requiresthe ability to efficiently address and control individualqubits in a large-scale system. Many leading exper-imental quantum information platforms, among themtrapped ions [1], electronic spins in impurities and quan-tum dots [2] and superconducting circuits [3], employqubits with level transitions in the microwave frequencydomain. Addressing these transitions often involves ex-pensive microwave electronics scaling linearly with thenumber of qubits. To move beyond the state of the art inmicrowave-frequency quantum processors, such as thoserecently used for small-scale quantum error correction insuperconducting circuits [4–6], it will already be bene-ficial to have a hardware-efficient control strategy thatharnesses economies of scale. One approach is to use mi-crowave pulses from a single control source for multiplequbits [7], requiring frequency-matched qubits and high-speed routing of pulses to separate control lines. The lin-ear scaling of control equipment could then be reducedto a constant overhead for the most expensive resources.

Using control equipment for multiple qubits has previ-ously been demonstrated for optical addressing in atomicsystems, where qubits naturally have the same fre-quency [8–11]. Such frequency reuse also becomes pos-sible in circuit quantum electrodynamics (cQED) [12] inthe context of fault-tolerant computation strategies [13–15] which rely only on local interactions between qubitsmediated by bus resonators. The natural isolation be-tween different lattice sites allows the use of repeatingpatterns of qubit frequencies with selectivity provided byspatial separation. A tileable unit cell with a handful ofqubit frequencies [16] could therefore provide a promis-ing route towards scalability. Crucially, this also solvesthe frequency-crowding problem which arises when try-ing to fit many distinct-frequency qubits within the finiteuseful bandwith of the circuit-based devices, particularlyfor designs based on weakly anharmonic qubits wherehigher levels must also be avoided [17, 18]. While noqubit experiments have yet shown the viability of this

approach, Hornibrook et al. have recently demonstrateda cryogenic switching matrix for pulse distribution op-erating at 20 mK, triggered by a field-programmablegate array at 4 K [7]. Cryogenic control equipment mayshorten feedback latency and reduce wiring complexityacross temperature stages, but the isolation and opera-tional frequency range achieved are currently insufficientfor typical cQED experiments.

In this Letter, we demonstrate frequency reuse in anextensible solid-state multiqubit architecture. Specif-ically, we show independent simultaneous control oftwo same-frequency qubits with a home-built room-temperature vector switch matrix (VSM). The VSM al-lows tailoring of control pulses to individual qubit prop-erties, and routing of the pulses to either one or both ofthe qubits using fast digital markers. We develop sev-eral different approaches to selective pulse broadcasting,including a simple scheme for implementing independentClifford control on an arbitrary number of qubits witha constant overhead in time. The device for this ex-periment is designed to allow testing in a circuit withthe correct connectivity of a relevant surface-code lat-tice [19, 20]. Using randomized benchmarking (RB), weshow that all control schemes exceed the fidelity thresh-old for surface code and are dominated by qubit relax-ation. We also develop a method for measuring leakageto the second excited state directly within the context ofRB [21, 22]. We characterize the limitations of our sys-tem and find no major obstacles to scaling up to largerimplementations.

Demonstrating frequency reuse in a context relevantfor increasing system sizes requires two key elements: amethod for distributing control pulses to multiple qubitsusing a single qubit-control source, and a multiqubit de-vice containing same-frequency qubits with relevant con-nectivity. Here, we focus on a particular implemen-tation of the surface code, where the connectivity be-tween nearest-neighbor qubits is achieved via bus res-onators [13]. Figure 1 illustrates a conceptual design

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FIG. 1. (color online). Schematic of independent control ofsame-frequency qubits using a vector switch matrix (VSM).The device (gold-colored box on right) connects two trans-mons with matched frequency (QT and QB: 6.220 GHz) indi-rectly through coupling buses and a third non-matched trans-mon (QM: 6.550 GHz). This provides the smallest relevantsubunit of the four-frequency surface-code fabric illustratedabove right. The VSM (blue box) allows independent, si-multaneous transmon control with tailored DRAG pulsing bycombining and directing Gaussian and derivative-of-Gaussianinput pulses to separate outputs T and B connected to ded-icated drive lines for each qubit. The link between inputsand outputs can be switched on nanosecond timescales usingthe digital marker inputs MT and MB (orange lines). DRAGpulses for the targeted qubit are independently tuned in bothamplitude and phase for each input-output pair (top left).

based on repeated tiling of a unit cell consisting of fourqubits with unique frequencies that are coupled via busresonators.

Our device contains a small block of this design,consisting of two same-frequency transmon qubits (QTand QB), which are connected to a third qubit (QM)via separate bus resonators (Fig. 1). Each qubit hasa capacitively-coupled drive line for individual qubitcontrol [23], a readout resonator coupled to a com-mon feedline for frequency-division multiplexing read-out [24, 25], and a flux-bias line for individual fre-quency tuning (Fig. 1). While QT and QB were designedto be identical, fabrication uncertainties resulted in asweet-spot (maximum) frequency of QT 57 MHz higherthan that of QB. With QB and QM kept at their re-spective sweet-spots (6.220 GHz and 6.550 GHz, respec-tively), QT was then flux tuned to match QB with anaccuracy of 50 kHz, determined using Ramsey measure-ments. The coherence times at the operating frequencycan be found in the Supplemental Material [26]. Becauseof the transmon’s weak anharmonicity [27], high-fidelityfast single-qubit control is achieved using the methodof derivative-removal-via-adiabatic-gate (DRAG) puls-ing, where the in-phase Gaussian pulse is combined withan in-quadrature derivative-of-Gaussian pulse [28, 29].For each qubit, this requires independent amplitude con-trol of the two constituent quadrature pulses.

The VSM was designed to accept multiple input pulsesand selectively fan them out to multiple qubits with in-dividual pulse tuning for each qubit (Fig. 1). Our home-built room-temperature 4 × 2 (four input, two output)VSM allows independent control of amplitude and phasefor each of its input-output combinations. Fast marker-controlled digital switches enable routing of pulses tothe qubits at nanosecond timescale, with approximately50 dB isolation in the frequency range from 4 to 8 GHz(see [26] for VSM specifications). By directing the twoconsistuent pulses of DRAG control through separate in-puts of the VSM, this allows independent, in-situ DRAGtuning for both same-frequency qubits using four AWGchannels [26].

The first critical test of our control architecture is toassess the VSM’s ability to implement high-precision con-trol of one qubit while leaving the other qubit idle. Todo this, we use the standard technique of single-qubitRB based on Clifford gates [30–32], which allows us tocharacterize control performance independently of statepreparation and measurement errors. After initializingall qubits in the ground state by relaxation, we use theVSM to selectively apply random sequences of Cliffordsgates to only one of the same-frequency qubits, withlengths m ranging from 1 to 800 gates and the resultsaveraged for 50 different random sequences. We decom-pose each gate into the standard minimal sequence of πand ±π/2 pulses around the x and y axes (16 ns pulsesseparated by a 4 ns buffer; tp = 20 ns) [22], requiringon average 〈Np〉 = 1.875 pulses per Clifford. This is incontrast to atomic pulses, where the 24 single-qubit Clif-fords can each be implemented with a single pulse [33].After applying a final Clifford that inverts the cumula-tive effect of all m previous Cliffords, the driven qubitis ideally returned to the ground state, but as a resultof imperfections such as gate errors and decoherence,the final ground-state population decays as a functionof m. The decay rate can be related to the average fi-delity per Clifford FC [30, 31]. In a strictly two-level sys-tem, the measured ground- and excited-state populationsaveraged over many sequences (〈P0〉 and 〈P1〉) both con-verge to 0.5 for large m. For weakly anharmonic trans-mon qubits, leakage to the second-excited state can be animportant additional source of gate error, which can leadto a shift of the asymptotic value away from 0.5. We ad-dress this issue by performing the RB protocol both withand without an additional final π pulse [34], which allowsus to explicitly estimate the populations of the first threetransmon states (see [26] for details).

We implement the above characterization for one qubitat a time by switching off the marker for the undrivenqubit at the VSM [Fig. 2(a,b)], in each case measuringthe effect on both qubits simultaneously via multiplexedreadout. From the results in Fig. 2(c,d), we calculatethe average Clifford fidelities for the two individually-driven qubits to be 0.9982(2) (QT) and 0.9986(2) (QB), inboth cases surpassing the best known surface-code fault-tolerance threshold for single-qubit gates of ∼ 0.99 [35–

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FIG. 2. (color online). Single-qubit control of same-frequencyqubits using the VSM. (a,b) Schematics showing DRAGpulses routed exclusively to either QT or QB using the corre-sponding markers (always on or off). Details of the tune-upprocedure for optimizing pulse amplitude and phase parame-ters for each qubit are provided in Ref. 26. (c,d) Character-ization of single-qubit control by randomized benchmarking(RB) of Clifford gates. Average populations of QT and QB inthe ground, first- and second-excited states (〈P0〉, 〈P1〉 and〈P2〉, respectively) as a function of the number of Cliffordgates applied. Curves are best fits of single exponentials withoffsets. Single-qubit Clifford fidelities for each qubit are ex-tracted from the decay of ground-state populations. For bothqubits, the fidelity surpasses the surface-code fault-tolerancethreshold and lies close to the theoretical value expected forT1-only relaxation-limited performance. (e,f) Expanded plotsof second-excited state leakage during RB. Curves are best fitsaccording to Eq. (2). (g,h) Cross-excitation of the undrivenqubit resulting from control pulses applied to the driven qubit.Further measurements indicate that this effect can be at-tributed to microwave leakage.

37]. We compare these values with the expected averageClifford fidelities assuming only T1 decay [38]:

FT1C '[

1

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(3 + 2e−tp/2T1 + e−tp/T1

)]〈Np〉. (1)

This shows our results are predominately limited by re-laxation effects, the difference in performance being con-sistent with the different T1 times for the two qubits.Additional measurements in the Supplemental Materialfurthermore show that there is no difference in perfor-mance when both qubits are driven simultaneously bythe same pulse sequence (both markers on). From themeasured leakage populations 〈P2〉, we also extract esti-mated per Clifford leakage rates κ of 4.1(2)× 10−6 (QT)and 1.3(4) × 10−6 (QB) by fitting the following simple

model to the data (see [26] for details):

〈P2[m]〉 ' κT2→1(

1− e−m〈Np〉tp/T2→1), (2)

where T2→1 is the second- to first-excited-state relaxationtime. As these leakage rates are much smaller than thegate errors (1−FC), it is reasonable to neglect them whenestimating the Clifford fidelity.

We next explore the effect of the single-qubit controlpulses on the undriven qubit [Fig. 2(g,h)], which shouldideally remain in the ground state. We fabricated thequbits as close to each other as possible in order to studythe worst-case scenario for such cross-excitation effects.While QT remains largely unaffected when driving QB,a substantial deviation from the ground state is mea-sured in QB when driving QT. There are several possi-ble mechanisms for cross-excitation effects in our system:cross-coupling (higher-order quantum coupling mediatedby the bus resonators and QM) and cross-driving (spu-rious driving of the idle channel by microwave leakageresulting from imperfect isolation either in the VSM oron chip). From single-excitation swap experiments (seeRef. 26), we observe a residual exchange interaction [12]between QT and QB with strength J/2π ≤ 36 ± 1 kHz.This symmetric swapping of excitation is therefore un-likely to explain the strong asymmetry in the amount ofcross-excitation measured for the different qubits. Fur-thermore, in RB experiments, where the state of thedriven qubit is moved randomly around the Bloch sphereduring each run of the experiment, we expect the effectof cross-coupling to be dramatically reduced, because theperiod π/J is far longer than the average Clifford gatetime. Direct, independent measurements of microwaveisolation in both the VSM and on chip indicate that theon-chip cross-driving significantly dominates [26]. Thiswas also confirmed in-situ by performing the same RBmeasurements with the undriven qubit physically dis-connected from the VSM. This showed no significantreduction in cross-excitation. Furthermore, numericalsimulations show that the observed effects are consistentwith cross-driving alone at levels similar to the measuredvalues [26]. We note, however, that while the plots inFig. 2(g,h) are useful diagnostics for identifying the pres-ence of the cross-driving effect, they should not be inter-preted in the same way as the RB plots for the drivenqubit. A more comparable way to study this effect is touse interleaved RB [39], where pulses on the driven qubit(ideally identity operations on the undriven qubit) areinterleaved with a random sequence of Cliffords appliedto the undriven qubit [26]. From this, we estimate theaverage idling fidelity for QB to be 0.9986(5), which isconsistent with the error due to T1 decay during idling.This confirms that cross-excitation effects do not domi-nate the error per Clifford, as characterized by RB.

The defining test of extensibility in our control archi-tecture is to demonstrate the simultaneous, independent,single-qubit control over same-frequency qubits that isenabled by selective broadcasting using the VSM. We ex-

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FIG. 3. (color online). Selective broadcasting schemes forsimultaneous single-qubit control of multiple qubits. Mainfigure: Example of a single Clifford round for n=2 qubitstargeting C2 (C13) in QT (QB) (Cj defined in Ref. 26). Inthe sequential scheme, the pulses implementing C2 are di-rected to QT, after which the pulses implementing C13 aredirected to QB. In the compiled scheme, the two Cliffords arerealized concurrently using a pre-determined pulse sequence,with appropriate markers, which minimizes the total numberof pulses, NP (see [26] for the compilation algorithm). Finally,in the 5-primitives scheme, a fixed sequence of five pulses is re-peated in each round (Np=5). The targeted Cliffords are thenapplied simultaneously by selecting the appropriate subset ofpulses for each qubit (see [26] for the 5-primitives marker ta-ble). Top-right: Scaling of the average pulses per multiqubitcombination of Cliffords, 〈Np〉, versus qubit number n. Theconstant scaling achieved by the 5-primitives scheme providesa dramatic improvement over the linear scaling of the sequen-tial scheme. While 〈Np〉 is always lowest for the compiledscheme, pre-compiling the optimal pulse and marker combi-nations is impractical for n & 5, and the improvement overthe simpler 5-primitives scheme is negligible by n ∼ 10.

plore three paradigmatic schemes for implementing selec-tive broadcasting of Cliffords on an arbitrary number ofqubits n (Fig. 3). In the most straightforward selective-broadcasting scheme, the individual qubits are driven se-quentially, with each pulse being directed to one qubitat a time. This results in a linear scaling of the averagenumber of pulses per Clifford round (〈Np〉 = 1.875× n).By contrast, the second paradigm takes best advantage ofthe VSM’s capability to broadcast simultaneously to mul-tiple qubits by compiling the constituent Clifford pulsesto minimize Np for each Clifford combination in the se-quence. The compilation is performed by searching allpossible combinations of single-qubit Clifford decompo-sitions and finding the one that minimizes Np (see [26] forfurther information). In Fig. 3, we show exact values andestimates for 〈Np〉 for the compiled scheme up to n = 10.Unfortunately, the compilation run-time increases expo-nentially with the number of qubits. This motivates ourfinal broadcasting paradigm, where all Clifford gates canbe implemented using the same fixed, ordered sequence offive pulse primitives (Fig. 3). Independent Cliffords canbe applied to all qubits, irrespective of n, by selectively

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FIG. 4. (color online). Characterization of sequential, com-piled and 5-primitives selective-broadcasting schemes by RB.(a,b) Evolution of the average transmon populations for QT(a) and QB (b) as a function of the number of Clifford rounds.Curves are the best fits of single exponentials with offsets.(c,d) Expanded plots of second-excited-state leakage duringRB. (e) Average single-qubit Clifford-gate fidelities for QTand QB in each scheme, extracted from the decay of the cor-responding ground-state population. All fidelities surpass thesurface-code fault-tolerance threshold and closely track thoseexpected for T1-relaxation-limited performance. Compiled se-lective broadcasting performs best, as expected, having thelowest total number of pulses (see Fig. 3).

directing the appropriate subset of pulses to each qubit,achieving a constant overhead in time for single-qubitClifford control. While the compiled scheme by defini-tion always provides the minimum sequence length, ourestimates of 〈Np〉 for the compiled scheme suggest thatit asymptotes to the same value of 5 achieved by thesimple, prescriptive 5-primitives scheme. We note thatthe specific choice of pulse primitives is not unique, butat least five primitives are required (four pulses allow amaximum of 16 unique gate decompositions, comparedwith the 24 single-qubit Cliffords).

To demonstrate the full functionality of this control ar-chitecture, we implement all three selective-broadcastingschemes and measure their performance using parallelsingle-qubit RB with independent Clifford sequences foreach qubit. Figure 4 shows that the compiled schemeperforms best, followed by the sequential and then 5-primitives schemes, consistent with the average numberof pulses required for each (Fig. 3). In all cases, the av-erage fidelity per Clifford still surpasses the surface-codefault-tolerance threshold, and the average error is againdominated by relaxation. The results are completely con-

5

sistent with the values obtained in the test for isolatedsingle-qubit control, indicating no substantial decrease ingate performance using selective broadcasting schemes.

Our VSM allows efficient reuse of qubit control equip-ment on same-frequency qubits. It enables high-precisionsingle-qubit control of multiple qubits with a perfor-mance that is mainly limited by relaxation. We havedemonstrated three selective broadcasting schemes, allof which achieve a performance that surpasses the fault-tolerance threshold for the surface code. In particular,the 5-primitives scheme implements arbitrary Cliffordcontrol with a fixed five-pulse sequence, where the targetClifford is selected by routing a subset of the pulses usingdigital markers. By adding a sixth, non-Clifford gate tothe five pulse primitives, this can be extended to achieveuniversal single-qubit control. Combining the connectiv-ity of our device, the VSM-based control, and the fixedpulse overhead of the 5-primitives broadcasting strategy,our experiment realizes the simplest element of an exten-

sible qubit control architecture. While we do not yet seeexplicit savings in control hardware for two qubits, thisdesign can be expanded to more same-frequency qubitswithout any further increase in microwave sources or ar-bitrary waveform generators. This experiment suggeststhat surface-code tiling with frequency reuse is a viablepath towards large-scale quantum processors.

ACKNOWLEDGMENTS

We thank R. N. Schouten, W. Vlothuizen, andP. Koobs de Hartog for experimental contributions andB. Criger, T. Chasseur, and D. J. Reilly for discussions.We acknowledge funding by the Dutch Organization forFundamental Research on Matter (FOM), the Nether-lands Organisation for Scientific Research (NWO/OCWand Vidi scheme), the EU FP7 project ScaleQIT, and anERC Synergy Grant.

[1] C. Monroe and J. Kim, Science 339, 1164 (2013).[2] D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu,

and J. R. Petta, Science 339, 1174 (2013).[3] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169

(2013).[4] J. Kelly, R. Barends, A. Fowler, A. Megrant, E. Jeffrey,

T. White, D. Sank, J. Mutus, B. Campbell, Y. Chen,et al., Nature 519, 66 (2015).

[5] A. D. Corcoles, E. Magesan, S. J. Srinivasan, A. W.Cross, M. Steffen, J. M. Gambetta, and J. M. Chow,Nat. Commun. 6 (2015).

[6] D. Ristè, S. Poletto, M. Z. Huang, A. Bruno, V. Vester-inen, O. P. Saira, and L. DiCarlo, Nat. Commun. 6(2015).

[7] J. M. Hornibrook, J. I. Colless, I. D. Conway Lamb, S. J.Pauka, H. Lu, A. C. Gossard, J. D. Watson, G. C. Gard-ner, S. Fallahi, M. J. Manfra, and D. J. Reilly, Phys.Rev. Appl. 3, 024010 (2015).

[8] C. Knoernschild, X. L. Zhang, L. Isenhower, A. T. Gill,F. P. Lu, M. Saffman, and J. Kim, Appl. Phys. Lett. 97,134101 (2010).

[9] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau,P. Schauss, T. Fukuhara, I. Bloch, and S. Kuhr, Nature471, 319 (2011).

[10] S. Crain, E. Mount, S. Baek, and J. Kim, Appl. Phys.Lett. 105, 181115 (2014).

[11] T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Pi-otrowicz, L. Isenhower, and M. Saffman, Phys. Rev. Lett.114, 100503 (2015).

[12] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).

[13] D. P. DiVincenzo, Physica Scripta 2009, 014020 (2009).[14] F. Helmer, M. Mariantoni, A. Fowler, J. Von Delft,

E. Solano, and F. Marquardt, Europhys. Lett. 85, 50007(2009).

[15] J. Ghosh, A. G. Fowler, and M. R. Geller, Phys. Rev. A86, 062318 (2012).

[16] J. Gambetta and J. Smolin, “Frequency arrangement forsurface code on a superconducting lattice,” (2014), US

Patent App. 13/827,326.[17] R. Schutjens, F. A. Dagga, D. J. Egger, and F. K. Wil-

helm, Phys. Rev. A 88, 052330 (2013).[18] V. Vesterinen, O.-P. Saira, A. Bruno, and L. DiCarlo,

arXiv:cond-mat/1405.0450 (2014).[19] S. B. Bravyi and A. Y. Kitaev, arXiv:quant-ph/9811052

(1998).[20] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.

Cleland, Phys. Rev. A 86, 032324 (2012).[21] T. Chasseur and F. Wilhelm, arXiv:1505.00580 (2015).[22] J. M. Epstein, A. W. Cross, E. Magesan, and J. M.

Gambetta, Phys. Rev. A 89, 062321 (2014).[23] A. Fragner, M. Göppl, J. M. Fink, M. Baur,

R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff,Science 322, 1357 (2008).

[24] J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C.de Groot, T. Picot, G. Johansson, and L. DiCarlo, Phys.Rev. Lett. 111, 090506 (2013).

[25] M. Jerger, S. Poletto, P. Macha, U. Hübner, E. Il’ichev,and A. V. Ustinov, Appl. Phys. Lett. 101, 042604 (2012).

[26] See supplemental material below for additional data.[27] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I.

Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin,and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).

[28] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.Wilhelm, Phys. Rev. Lett. 103, 110501 (2009).

[29] J. M. Chow, L. DiCarlo, J. M. Gambetta, F. Motzoi,L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Phys.Rev. A 82, 040305 (2010).

[30] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B.Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin,and D. J. Wineland, Phys. Rev. A 77, 012307 (2008).

[31] E. Magesan, J. M. Gambetta, and J. Emerson, Phys.Rev. Lett. 106, 180504 (2011).

[32] E. Magesan, J. M. Gambetta, and J. Emerson, Phys.Rev. A 85, 042311 (2012).

[33] B. R. Johnson, M. P. da Silva, C. A. Ryan, S. Kimmel,J. M. Chow, and T. A. Ohki, arXiv:1505.06686 (2015).

[34] D. Ristè, C. C. Bultink, K. W. Lehnert, and L. DiCarlo,

6

Phys. Rev. Lett. 109, 240502 (2012).[35] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98,

190504 (2007).[36] A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys.

Rev. A 80, 052312 (2009).[37] D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Phys.

Rev. A 83, 020302 (2011).[38] E. Magesan, private communication.[39] E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan,

J. M. Chow, S. T. Merkel, M. P. da Silva, G. A. Keefe,M. B. Rothwell, T. A. Ohki, M. B. Ketchen, and M. Stef-fen, Phys. Rev. Lett. 109, 080505 (2012).

7

SUPPLEMENTAL MATERIAL

This supplement provides experimental details and ad-ditional data supporting the claims in the main text.The device and experimental setup, including imagesof the device and a full wiring diagram, are describedin Section I. The microwave performance of the vec-tor switch matrix (VSM) and its use for independentcontrol of two same-frequency qubits are demonstratedin Sec. II A. The techniques employed for tuning qubitpulses are discussed in Sec. III. Section IV contains theresults of randomized benchmarking (RB) experimentsin the two-qubit global broadcasting context. Our tech-nique for assessing the effects of leakage is detailed inSec. V. Cross-coupling and cross-driving effects are char-acterized in Sec. VI, along with numerical simulations ofthe effects of cross-excitations on RB. In Section VI D, weintroduce a method for generating selective-broadcastingpulse sequences that are robust to cross-excitation effects.The decompositions of the 24 single-qubit Cliffords intoa minimal set of pulses and the 5-primitive pulses aregiven in Sec. VII. Finally, the algorithm used to compileoptimal pulse sequences for implementing independentsingle-qubit Clifford gates on multiple qubits is explainedin Sec. VIII.

I. QUANTUM CHIP AND EXPERIMENTALSETUP

A. Chip design and fabrication

Our quantum chip consists of three transmons(top: QT, middle (ancilla): QM, and bottom: QB) withdedicated voltage drive lines (DT, DM and DB, respec-tively), flux-bias lines, and readout resonators. All read-out resonators are capacitively coupled to one commonfeedline which crosses various on-chip components us-ing airbridge crossovers. Qubits QT and QM are cou-pled by one bus resonator, and QM and QB by another(fundamental frequencies 4.9 and 5.0 GHz, respectively).All resonators are open-ended on the coupling side, andshort-circuited at the other.

The chip fabrication method is similar to that in Ref. 1,but with some important differences which we now ex-plain. Rather than sapphire, we use a high-resistivity in-trinsic silicon substrate prepared by HF dip and HMDSsurface passivation before sputtering a 300-nm-thick filmof NbTiN, as introduced in Ref. 2. This change aimsto improve the substrate-metal interface and thereby in-trinsic quality factors for both resonators and qubits.After sputtering, the patterns are etched into the su-perconducting layer using reactive-ion etching with anSF6/O2 plasma. In contrast with the Al transmon ca-pacitor plates commonly used, in this experiment wemake them also from NbTiN, with an aim to improvethe substrate-metal interface and avoid large AlOx sur-faces which may house unwanted two-level systems. Only

FIG. S1. Scanning electron micrograph (with added falsecolour) of a device twin to the one used in the experiment andfabricated in the same batch. Notably, the feedline crosses thethree readout resonators.

the Josephson junctions are made by the standard tech-nique of Al-AlOx-Al double-angle evaporation. A furtherHF dip just prior to evaporation also helps to contactthe junctions directly to the NbTiN capacitor plates. InRef. 1, air bridges were already used to cross the feed-line over flux-bias lines on chip. Here, we extend thistechnique to allow the feedline to cross three readout res-onators.

A key requirement for this experiment was the abilityto match qubit frequencies without sacrificing coherence.Flux-bias lines allow easy compensation for mismatch,but at the cost of reduced coherence in the qubit detunedfrom its maximal frequency (coherence sweet spot [3]).We aimed for identical maximum frequencies of QT andQB, as determined by capacitor and junction geometries.The capacitors were easily matched in fabrication. Wethen selected the chip with the closest matching room-temperature resistance values for the relevant qubit junc-tions.

B. Experimental Setup

The chip is anchored to a copper cold-finger connectedto the mixing chamber of a Leiden Cryogenics CS813He/4He dilution refrigerator with 7 mK base temper-ature. A copper can seals the sample space, with aninner surface that is coated with a mixture of Stycast2850 and silicon carbide granules (15 to 1000 nm diam-eter) used for infrared absorption [4]. The copper can isin turn magnetically shielded by an aluminum enclosureand two outer Cryophy enclosures (1 mm thick) [2].

A complete wiring schematic showing all cryogenic

8

and room-temperature components is shown in Fig. S2.The four analog channels of the Tektronix AWG5014Ccreate the in-phase and in-quadrature pulses for QTand QB by single-sideband modulation of a commoncarrier. Because single-sideband modulation requirestwo AWG channels to modulate an IQ mixer, indepen-dent derivative-removal-via-adiabatic-gate (DRAG) tun-ing with the VSM therefore requires four AWG channels,irrespective of the number of qubits. The VSM can bescaled up to many output channels, and direct hardwaresavings can be realized as soon as three or more same-frequency qubits are driven by a single set of AWG in-puts. These pulses are input at ports 1 and 2 of the VSM.The VSM combines these pulses with individually tunedinsertion loss and phase to each of two outputs (labelledT and B). Input-output combinations can be switched onnanosecond timescales using the gate inputs of the VSM,provided by digital markers of the AWG5014C. A sec-ond AWG5014C with the appropriate carrier frequencyis used to excite transmons QT and QB to the second-excited state (Sec. V), to pulse measurement tones, andto trigger the AlazarTech ATS9870 acquisition card.

FIG. S2. Detailed schematic of the experimental setup andoptical image of the chip (installed in a Leiden CryogenicsCS81 dilution refrigerator). Note that while our VSM has twopairs of analog inputs [(1,2) and (3,4)] and four gate inputs(activating links from each pair to each output), the pair (3,4)and its associated gate inputs are not used throughout thisexperiment.

9

C. Device frequencies

Qubit fmax (GHz) fbias (GHz) fres (GHz)QT 6.277 6.220 6.700QM 6.551 6.551 6.733QB 6.220 6.220 6.800

TABLE S1. Table of sweet-spot frequencies fmax and bias-point frequencies fbias of the three qubits, as well as the funda-mental frequencies fres of their dedicated readout resonatorsat the bias point. The qubits are tuned into the bias point bya combination of spectroscopy and standard Ramsey experi-ments.

D. Qubit coherence times

0 5 10 15 200.00

0.25

0.50

0.75

1.00 T1 =8.7 µs

QT

0 5 10 15 20

T1 =7.4 µs

QM

0 5 10 15 20

T1 =11.1 µs

QB

0 2 4 6 8 100.00

0.25

0.50

0.75

1.00

P1

T ∗2 =4.7 µs

0 2 4 6 8 10

T ∗2 =9.7 µs

0 2 4 6 8 10

T ∗2 =12.5 µs

0 5 10 15 20 250.00

0.25

0.50

0.75

1.00 TE2 =8.9 µs

0 5 10 15 20 25Wait time (µs)

TE2 =14.8 µs

0 5 10 15 20 25

TE2 =17.0 µs

FIG. S3. Measurements of relaxation (T1, top), Ramsey de-phasing (T ∗2 , middle) and echo dephasing (T

E2 , bottom) times

for the three qubits at the bias point. When measuring QTor QB, the other qubit is detuned by −50 MHz to suppresscross-coupling effects. Ramsey fringes for QT (middle, leftpanel) fit better to a Gaussian (shown) than an exponentialdecay, reflecting the susceptibility of QT to low-frequency fluxnoise away from its sweet-spot. P1 denotes excited-state pop-ulation.

II. VECTOR SWITCH MATRIX

A. Measured isolation

To characterize the isolation of the VSM in the range4 to 8 GHz, we have measured the insertion loss betweenall input (1 and 2) and output ports (T and B) with staticsettings at the two gate inputs (Fig. S4). Ideally, eachgate activates (on state) and deactivates (off state) thelink of both inputs 1 and 2 to one output, independent ofthe other gate. As shown in Fig. S4, the typical relativeisolation with the relevant gate in the off state is ∼ 50 dB.

100

80

60

40

20

0

20(a) |ST1|

offToffBoffTonBonToffBonTonBnoise

(b) |SB1|

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0Frequency (GHz)

100

80

60

40

20

0

20

Inse

rtion

loss

(dB)

(c) |ST2|

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

(d) |SB2|

FIG. S4. Insertion loss between inputs and outputs of theVSM for four static combinations of the gate inputs. The in-sertion loss is measured relative to the level with both gateinputs activated (onTonB). The black curves indicate thenoise background in our scalar network analyzer measure-ment. The dashed vertical lines indicate the common fre-quency (6.220 GHz) of QT and QB at the bias point.

B. Individual qubit tune-up

The VSM enables independent control of the on/offstate, insertion loss and phase for every input-outputcombination. We exploit this feature to perform DRAG-compensated pulses on QT and QB, that are individuallytailored for each qubit. Different types of gate errors,such as non-ideal in-phase and in-quadrature amplitudes,can be distinguished using an AllXY sequence [5], con-sisting of 21 combinations of two pulses drawn from theset

{I,Xπ, Yπ, Xπ/2, Yπ/2

}(Table S2). Figure S5 shows

AllXY sequence results for QT and QB as the amplitudeof each quadrature on QT is varied independently. Whilethe AllXY signature of QT reveals changing levels of am-plitude and phase errors, there is no noticeable change inthe AllXY signature of QB. This demonstrates the useof the VSM for individual tune-up of pulses for same-frequency qubits.

10

0.00

0.25

0.50

0.75

1.00P

1(a) QT

High drive amplitudeChosen drive amplitudeLow drive amplitude

(b) QBHigh drive amplitudeChosen drive amplitudeLow drive amplitude

0 5 10 15 20AllXY ID

0.00

0.25

0.50

0.75

1.00(c)

High DRAG parameterChosen DRAG parameterLow DRAG parameter

0 5 10 15 20

(d)High DRAG parameterChosen DRAG parameterLow DRAG parameter

FIG. S5. AllXY results for QT (left) and QB (right), as the in-phase amplitude (top) or in-quadrature amplitude (bottom)of QT pulses is varied. During each measurement, both qubitsare driven simultaneously, but read out individually. TheAllXY results of QB all overlap independent of the setting onQA, showing independent control of pulse parameters for eachqubit.

ID Ideal P1Pulses

ID Ideal P1Pulses

First Second First Second1 0 I I 12 0.5 Yπ Yπ/22 0 Xπ Xπ 13 0.5 Xπ Xπ/23 0 Yπ Yπ 14 0.5 Xπ/2 Xπ4 0 Xπ Yπ 15 0.5 Xπ Xπ/25 0 Yπ Xπ 16 0.5 Yπ/2 Yπ6 0.5 I Xπ/2 17 0.5 Yπ Yπ/27 0.5 I Yπ/2 18 1 I Xπ8 0.5 Xπ/2 Yπ/2 19 1 I Yπ9 0.5 Yπ/2 Xπ/2 20 1 Xπ/2 Xπ/210 0.5 Xπ/2 Yπ 21 1 Yπ/2 Yπ/211 0.5 Yπ/2 Xπ

TABLE S2. The 21 two-pulse combinations comprising theAllXY pulse sequence [5].

III. PULSE-CALIBRATION ROUTINES

We tune up qubit pulses by alternating the calibra-tion of in-phase and in-quadrature pulse amplitudes untila simultaneous optimum is found. The two calibrationroutines are discussed below.

A. Accurate in-phase pulse amplitude calibration

The in-phase quadrature amplitude is calibrated byfirst applying a π/2 pulse to the qubit, followed by a

train of π pulses. The pulse sequence is (Xπ)2N

Xπ/2 |g〉,where |g〉 is the qubit ground state and N ∈ [0, 49]. Inthe absence of gate errors and decoherence, the drivenqubit would end on the equator of its Bloch sphere forall N . However, over- or under-driving produces a posi-

0 10 20 30 40 50Pulse repetitions 2N

0.00

0.25

0.50

0.75

1.00

P1

Over-drivingChosen drivingUnder-driving

FIG. S6. Fine calibration of pulse amplitude by initial π/2pulse, followed by 2N repeated π pulses. The initial slope de-termines if the qubit is over- (positive slope) or under-driven(negative slope).

tive or negative initial slope on P1 versus N , respectively(Fig. S6). We choose the in-quadrature amplitude thatminimizes the absolute slope.

B. DRAG-parameter calibration

To minimize phase errors resulting from the presenceof the second- and higher-excited states, we optimizethe scaling of the in-quadrature pulse. As in conven-tional DRAG [6, 7], we choose as the envelope of thein-quadrature pulse the derivative of the Gaussian enve-lope on the in-phase pulse. The DRAG scaling param-eter is calibrated using the method detailed in Ref. [5].Specifically, we measure the difference in excited-statepopulation produced by the YπXπ/2 and XπYπ/2 pulsecombinations (AllXY ID 10 and 11). Ideally, for both,the final qubit state would lie on the equator. However,any phase error shifts the final excited-state population inopposite directions in these cases. We choose the DRAGscaling parameter minimizing this shift.

IV. GLOBAL BROADCASTING

Aside from single-qubit control and selective broad-casting, the VSM also allows global broadcasting ofpulses to all qubits simultaneously by keeping the mark-ers for both qubits on (Fig. S7). While this does pro-vide simultaneous control of QT and QB, marker controlis needed to achieve independent control. Using RB, wemeasure the performance of both qubits when broadcast-ing pulses to both qubits, and compare the results withthose obtained from single-qubit control (Fig. S7). Theglobal broadcasting RB measurements were alternatedwith the single-qubit RB measurements, and aside frommarker settings all other settings were identical. Com-parison of the results in Fig. S7 show that the qubit gateperformance does not depend on whether a single qubit iscontrolled, or both are controlled simultaneously throughglobal broadcasting.

11

Global broadcasting QMQB

QT

(a) MT on

MB on

0 200 400 600 800

Number of Clifford gates m

0.00

0.25

0.50

0.75

1.00

Pop

ulat

ions〈P

i〉

(b)QT

FC 0.9981(2)

F T1C 0.9986

〈P0〉〈P1〉〈P2〉

Single qubitBoth qubits

0 200 400 600 800

(c)QB

FC 0.9986(2)

F T1C 0.9989

FIG. S7. Global broadcasting of DRAG pulses to same-frequency qubits. (a) Illustration of global broadcasting. Twosimultaneous pulses, one with Gaussian envelope at input 1,and another with derivative-of-Gaussian envelope at input 2,are simultaneously directed to QT and QB (both markers al-ways on). The insertion loss and phase shift of each pulseis separately optimized for each output to produce precisionDRAG pulses for each qubit. (b,c) Comparison of single-qubitdriving versus driving both qubits (broadcasting) by RB ofClifford gates composed from π/2 and π pulses [8]. Averagepopulation of QT and QB in the ground, first- and second-excited state (〈P0〉, 〈P1〉 and 〈P2〉, resp.) as a function ofthe number of Clifford gates applied. Curves are the best fitsof single exponentials with offsets to the populations. Thesingle-qubit Clifford-gate fidelity for each qubit is extractedfrom the decay of the corresponding ground-state populationwhen using global broadcasting.

V. LEAKAGE TO SECOND EXCITED STATE

Leakage is fundamentally different from unitary qubiterrors. To quantify leakage, we monitor the populationsPi of the three lowest energy states (i ∈ {0, 1, 2}) duringRB and calculate the average values 〈Pi〉 over all seeds.To do this, we calibrate the average signal levels Vi for thetransmons in level i, and perform each RB measurementtwice, the second time with an added final π pulse onthe 0–1 transition. This final π pulse swaps P0 and P1,leaving P2 unaffected. Under the assumption that higherlevels are unpopulated (P0 + P1 + P2 = 1),[

V0 − V2 V1 − V2V1 − V2 V0 − V2

] [P0P1

]=

[S − V2S′ − V2

], (S1)

where S (S′) is the measured signal level without (with)final π pulse. The populations are extracted by matrixinversion.

Measuring 〈P2〉 as a function of the number of Cliffordgates allows us to estimate an average leakage per Clif-ford, κ. Because the populations are ensemble averagesover different random seeds, we assume that leakage ofthe average qubit-space populations to 〈P2〉 is incoher-ent, and, provided 〈P2〉 remains small (κ small), we also

assume that leakage is irreversible. We therefore modelleakage using the following difference equation for 〈P2〉:

〈P2[m+1]〉−〈P2[m]〉 ' tp〈NP〉κ−tp〈NP〉T2→1

〈P2[m]〉, (S2)

where T2→1 is the second- to first-excited-state relax-ation time. Assuming no initial population in the second-excited state, the solution is Eq. (2), which shows goodagreement with measured data. We extract κ by fittingEq. (2) to 〈P2〉 data. T2→1 is obtained from the best-fitdecay constant (not directly measured) and κ from thebest-fit prefactor.

VI. CROSS-COUPLING AND CROSS-DRIVINGEFFECTS

There are several sources of spurious cross-qubit inter-actions [see Fig. 2(g,h) of the main text] which play arole in our experiments. We divide these into two mainclasses: cross-coupling, where the qubits themselves arecoupled via a direct or indirect quantum interaction, andcross-driving, where input microwaves directly drive theuntargeted qubit with a reduced amplitude as a resultof imperfect isolation either on or off chip. These cross-excitation effects depend strongly on specific chip design,in our case chosen according to our primary aim to studythe potential of frequency reuse in a circuit fully compati-ble with a larger surface-code lattice. This governed bothhow qubits and resonators were connected, as well as theselection and arrangement of frequencies. In addition,in order to fully explore the limitations of such tech-niques, the qubits were positioned on the chip as closeto each other as possible to provide a worst-case scenariofor cross-excitation effects.

A. Cross-coupling

In our device, the same-frequency qubits QT and QBare connected through a linear chain of coupled quan-tum elements: the two bus resonators and intermedi-ate ancilla qubit QM. They are also coupled capaci-tively through the ground plane. When the intermedi-ate elements are detuned in frequency away from thenear-resonant qubits, such geometries typically give riseto higher-order exchange-type interactions between thequbits of the form J(σ+Tσ

−B + σ

−Tσ

+B ) [9, 10]. Consistent

with this picture, we observe coherent swapping of exci-tations between QT and QB at a rate strongly dependenton the qubit detuning (Fig. S8).

In order to characterise the cross-coupling between QTand QB, we measure the evolution of excited-state pop-ulations after a single excitation is injected at one of thequbits with a π pulse. To place a tight upper bound onthe interaction strength J , the qubit frequencies must bematched as closely as possible. We achieve an accuracyof around 50 kHz using Ramsey experiments, limited by

12

0 5 10 15 20 25wait time (us)

0.00

0.25

0.50

0.75

1.00P

1(a) Initially exciting QT

QTQBSum

0 5 10 15 20 25

(b) Initially exciting QB

FIG. S8. Temporal evolution of a single excitation after ini-tially exciting QT (a) or QB (b). The oscillations of populationin both qubits are out of phase and have a common frequency.The sum of both populations shows approximately exponen-tial decay, with time constant 10.2(2) µs, intermediate be-tween the relaxation times of QT and QB.

a combination of factors: the resolution of the flux tun-ing, the fitting resolution limit imposed by qubit T2 de-phasing times, and also the frequency shifting inducedby the qubit-qubit exchange interaction itself. As shownin Fig. S8, the total excitation number exhibits a typicalexponential T1 decay, while the individual populationsshow a symmetric oscillation of the excitation betweenthe two qubits. The measured common frequency of de-caying oscillations for both qubits sets an upper boundon the coupling strength of J/2π ≤ 36± 1 kHz.

B. Cross-driving

When driving one qubit, the signal applied to its ded-icated voltage line residually drives the distant qubitson the chip. This results from microwave leakage in theVSM, near the sample, or from direct on-chip couplingof the drive line to the distant qubits. In Section II A, wefully characterise the isolation in the VSM. This leadsto leakage of around −57 dB and −54 dB on QT andQB, respectively, at the qubit operating frequency forthe conditions used in the main experiments. To char-acterize the residual cross-driving at the device, we dis-connect the VSM and compare the amplitude requiredfor pulses applied on one drive line (either DT or DB) toperform a π rotation on both QT and QB. We define thecross-driving strength of each drive line as the ratio rcof the π-pulse amplitude for the directly-driven to thatfor the cross-driven qubit. For this test, pulses are firstamplified and then attenuated using a step attenuator toallow the large amplitude range required. Results shownin Fig. S9 demonstrate on-chip cross-driving strengthsbelow 1% (−53 dB and −45 dB on QT and QB, respec-tively), but still somewhat larger than cross-driving ef-fects resulting from finite isolation in the VSM. Duringthe main experiments, both VSM and on-chip leakage areable to contribute to cross-driving effects.

In the excitation-swap experiments of the previous sec-tion, it is clear the effects result from cross-couplingrather than cross-driving, because the exchange dynam-

6.00 6.05 6.10 6.15 6.20 6.25QT frequency (GHz)

0.00

0.25

0.50

0.75

1.00

Cro

ss-d

rivin

g (%

)

(c) (d)

QubitDriving strength

DT DB

QT 1.00000.0023

(-53 dB)

QB0.0057

1.0000(-45 dB)

FIG. S9. Rabi oscillations of QT (a) and QB (b) induced bypulses on DT at the bias point. The pulse amplitude is nor-malized to the π-pulse amplitude for QT. (c) Resonant cross-driving of QB via DT for a range of QT frequencies. The weakdependence of the cross-driving strength shows that this ef-fect is most likely due to direct coupling of DT to QB. (d)Summary of cross-driving for DT and DB at the bias point.

ics occur while the drive pulses are off. Similarly, we canrule out these residual driving effects being related tocross-coupling, because the 16-ns Rabi pulses take onlya fraction of the time required for a cross-coupling ex-change oscillation. As a further check that the effectsobserved are indeed due to cross-driving, we measure thecross-driving on QB through DT as QT is tuned throughresonance with QB [Fig. S9(c)]. The cross-driving ratiois essentially constant, showing no significant dependenceon the detuning between QT and QB.

C. Cross-excitation in randomized benchmarking

The isolated single-qubit control experiments in themain text [Fig. 2(g,h)] show that significant spurious ex-citations can build up in the idling qubit over the courseof the long gate sequences tested in RB (particularly inthe case of idling QB while driving QT). It may there-fore initially be somewhat surprising that virtually thesame individual qubit-control performance is achievedin both selective-broadcasting (Fig. 4, main text) andglobal-broadcasting (Fig. S7) scenarios. As discussed inthe main text, the observed cross-excitation is unlikelyto result from cross-coupling, primarily because a sym-metrical quantum coupling should not result in stronglyasymmetric effects on the different qubits. We now showthe results are, however, consistent with the effects ofcross-driving by numerically simulating RB with cross-driving under experimentally realistic conditions (usingindependently measured qubit and cross-driving param-eters). Simulations are performed using QuTiP [11, 12].

13

0 200 400 600 800Number of Clifford rounds m

0.00

0.25

0.50

0.75

1.00〈P

0〉

QT simulation

Single qubitAsymmetric 5-primitivesSymmetric 5-primitives

0 200 400 600 800

QB simulation

FIG. S10. Simulation of cross-driving effects during RB.The results shown are averaged over ten runs, using thesingle-qubit minimal-set decomposition (red), and usingthe selective-broadcasting asymmetric and symmetric 5-primitives schemes (green and blue, respectively). Cross-driving effects are largely suppressed in the 5-primitivesschemes by choosing the five pulse primitives such that con-stituent pulses largely cancel out. The symmetric 5-primitivesscheme further reduces cross-driving effects by alternating be-tween the five pulse primitives and the inverse pulses.

We model our system as two uncoupled qubits, QTand QB, subject to T1 relaxation (with corresponding re-laxation times) and cross-driving. We approximate thesystem dynamics using instantaneous unitary pulse oper-ators from the standard Pauli set {Xπ, Yπ, X±π/2, Y±π/2}with 20 ns delays of T1-only qubit relaxation betweenpulses implemented using a master equation. When ap-plying a pulse to one qubit, cross-driving of the otherqubit is implemented by applying a pulse with the samerotation axis, but with the original rotation angle mul-tiplied by the relevant cross-driving ratio. We note thatalso trying to model the effects of qubit dephasing usinga simple master equation does not produce RB data con-sistent with the experimental observations (e.g., Figs 2and 4 of the main text). This reflects the non-uniformphase noise spectrum which affects the transmon qubit.The long RB pulse sequences consisting of π and π/2pulses around the X and Y axes seem to provide someform of dynamical decoupling which makes the RB mea-surements robust to qubit dephasing. As will be seen,the experimental results can be well modelled using onlyT1-type noise processes.

Randomized benchmarking is implemented by gen-erating independent Clifford sequences for each qubit.We decompose Clifford gates using either the minimalset decomposition or one of the selective-broadcastingschemes. Figure S10 shows simulated results of cross-driving for the isolated single-qubit control scenario re-ported in Fig. 2 of the main text. In this section, we areonly concerned with the red curves, which correspond toimplementing single-qubit RB with the standard set ofpulse decompositions [8]. These simulations can be com-pared directly with the curves in Fig. 2(g,h). While themaximum excitation population observed in the simula-tions is larger than the value observed in the experiments,the simulations for both qubits show the same qualita-tive behaviour as the measured data. The quantitative

0 200 400 600 800Number of Clifford rounds m

0.00

0.25

0.50

0.75

1.00

〈P0〉

QT simulation

T1 limitrc =0.0037rc =0.03rc =0.05

0 200 400 600 800

QB simulation

T1 limitrc =0.0076rc =0.03rc =0.05

FIG. S11. Simulation of sequential (interleaved) RB for sev-eral cross-driving ratios. For each cross-driving ratio, 50simulation runs were performed, using sequential RB up tom = 800 Clifford rounds. Under the cross-driving levels per-taining in our experiments (0.0037 and 0.0076 for QT and QB,respectively), the error per Clifford for the idling operation isdominated by the effects of T1 relaxation as calculated fromEq. (1).

difference may be explained by the fact that the directmeasurements of cross-driving were made at a differenttime from the main measurement run and we observedsome small fluctuations in cross-driving levels over time.

As discussed in the main text, while the plots of cross-excitation during RB are useful diagnostics of the pres-ence of a spurious cross-driving effect, they may give amisleading impression when presented in parallel withRB results. Although the decay curves look superfi-cially similar, they should not be interpreted in the sameway. By contrast, the technique of interleaved RB (IRB),which was introduced to enable rigorous quantification ofthe performance of individual gates, allows us to calcu-late a meaningful error per Clifford for the idling opera-tion [13]. In IRB, the usual random sequence of Cliffordsis alternated with identical repetitions of an individualgate. By comparing the interleaved decay rate with thedecay rate for a standard RB measurement, it is possi-ble to calculate a robust error per gate for the individ-ual gate in question. In this context, the target gate isthe nominal identity operation on one qubit which re-sults from a random Clifford being applied to the otherqubit. The IRB pulse sequence is therefore identicalto the sequence implemented in the sequential selective-broadcasting scheme (see Fig. 4 of the main text). In themain text, we use the formulas in Ref. [13] to calculatethe idling error per Clifford, but for these simulations,the performance of sequential selective-broadcasting al-ready provides a simple way to assess the performanceof cross-excitation during idling. Figure S11 shows thatidling performance as quantified by RB is limited mainlyby T1 relaxation. Finally, when identical gate sequencesare being applied to both qubits, cross-driving will resultin a small amount of over-driving on each qubit (over-driving ratio ro), which would also look like an error inpulse rotation angle. Figure S12 shows that the Clifforderror is insensitive to both cross-driving and over-drivingto first order.

14

0.000 0.025 0.050 0.075Rotation error

0.985

0.990

0.995

1.000F

CQT simulation

FT1C

Cross-drivingOver-driving

0.000 0.025 0.050 0.075

QB simulation

FIG. S12. Simulations of Clifford fidelity FC as a function ofthe cross-driving ratio (rc: red) and the relative over-drivingrotation error (ro−1: green). For each error type, 50 sim-ulation runs were performed, using sequential RB on bothqubits simultaneously up to m = 800 Clifford rounds, afterwhich FC was extracted from an exponential fit to averageddata. The T1 limit [Eq. (1)] is given by a horizontal dashedline. The data shows that FC is first-order insensitive to bothcross-driving and to over-rotations.

D. Making pulse sequences robust to cross-driving

We have already shown that cross-excitation does nothave a dominant effect on single-qubit control in bothglobal and selective broadcasting. We show here thatany residual effect can be largely eliminated also whilea qubit is idling by choosing robust pulse sequences fordecomposing the Clifford gates.

If a qubit is idle, every pulse that is applied to thedriven qubit rotates the idle qubit by an amount de-pending on the cross-driving ratio. The random applica-tion of successive pulses to the driven qubit can there-fore be viewed as a random walk for the idle qubit.As we will discuss in more detail in Sec. VIII, thereare many ways to compose a given Clifford gate froma small set of standard rotations. By choosing the con-stituent pulses in such a way that their combined ap-plication largely cancels out, cross-driving effects can begreatly reduced. In the standard set of Clifford decom-positions [8], the decompositions involve a majority ofpulses rotating in the positive direction, biasing the ran-dom walk and producing a pronounced net cross-drivingeffect. This effect can be countered by choosing de-compositions which minimize the bias. We have imple-mented this in the 5-primitives scheme, by choosing thefirst three pulses, {Xπ/2, Yπ/2, Xπ/2}, to be positive ro-tations, and the last two, {X−π, Y−π}, to move in thenegative direction. Even though the pulse subset that isapplied depends on the Clifford chosen, the pulses stilllargely cancel out after applying many Cliffords. Fur-thermore, as the single-qubit Clifford operations form agroup, the inverse of all Cliffords also form the Cliffordgroup. The complete inverse of the five pulse primitives,{Xπ, Yπ, X−π/2, Y−π/2, X−π/2}, can therefore also gener-ate each of the 24 Cliffords using an appropriate sub-set of the pulses. By alternating between the normalfive-pulse primitives and the inverted five pulses, cross-driving effects can be further reduced. (In fact, we note

0 100 200 300 400Number of Clifford rounds m

0.50

0.75

1.00

〈P0〉

QT

SequentialCompiledSymmetric 5-primitives

0 100 200 300 400

QB

FIG. S13. Measured cross-driving results when performingselective broadcasting. Even though the selective broadcast-ing schemes are meant for multiple qubits, the markers ofthe measured qubit are turned off to measure cross-drivingeffects. The cross-driving effects of QB are stronger thanfor QT, in agreement with single-qubit results. With pulsedecompositions whose cumulative effect largely cancels out,cross-driving is strongly reduced in the 5-primitives method.

that this exactly eliminates all cross-driving that occursvia leakage in the VSM, because all pulses are alwayspresent at that distribution stage.) We refer to this asthe symmetric 5-primitives technique and this is the tech-nique we implement in the main experiments describedin Fig. 4 of the main text. Our simulations in Fig. S10show that the asymmetric 5-primitives technique alreadydramatically reduces the effect of cross-driving, and inthe case of isolated single-qubit control, cross-driving iseffectively eliminated completely using the symmetric 5-primitives scheme. This is also confirmed by measure-ments of cross-driving for the three selective-broadcastingschemes (Fig. S13), where we implement the symmet-ric 5-primitives technique. While we have only demon-strated this technique for the 5-primitives scheme, itcould also be relatively straightforwardly applied to thesequential scheme, but the far better scaling of the 5-primitives scheme make it more interesting for scalingup to larger system sizes.

15

VII. CLIFFORD PULSE DECOMPOSITION

The decompositions of the 24 single-qubit Cliffordgates into a minimal set of π/2 and π pulses and intothe 5-primitives scheme are shown in Table S3.

Clifford ID Minimal set decomposition 5-primitives decompositionFirst Second Third Xπ/2 Yπ/2 Xπ/2 X−π Y−π

1 I 0 0 0 0 02 Yπ/2 Xπ/2 0 1 1 0 03 X−π/2 Y−π/2 1 1 0 1 04 Xπ 0 0 0 1 05 Y−π/2 X−π/2 0 1 1 0 16 Xπ/2 Y−π/2 1 1 0 0 17 Yπ 0 0 0 0 18 Y−π/2 Xπ/2 0 1 1 1 19 Xπ/2 Yπ/2 1 1 0 0 010 Xπ Yπ 0 0 0 1 111 Yπ/2 X−π/2 0 1 1 1 012 X−π/2 Yπ/2 1 1 0 1 113 Yπ/2 Xπ 0 1 0 1 014 X−π/2 0 0 1 1 015 Xπ/2 Y−π/2 X−π/2 1 1 1 0 116 Y−π/2 0 1 0 0 117 Xπ/2 0 0 1 0 018 Xπ/2 Yπ/2 Xπ/2 1 1 1 0 019 Y−π/2 Xπ 0 1 0 1 120 Xπ/2 Yπ 1 0 0 0 121 Xπ/2 Y−π/2 Xπ/2 1 1 1 1 122 Yπ/2 0 1 0 0 023 X−π/2 Yπ 1 0 0 1 124 Xπ/2 Yπ/2 X−π/2 1 1 1 1 0

TABLE S3. Two decompositions of the 24 single-qubit Clif-ford gates. The first, taken from Ref. [8], minimizes the num-ber of π/2 and π pulses around the ±x and ±y axes. Thesecond is our decomposition into 5 primitives. Pulses are ap-plied from left to right.

VIII. COMPILED SELECTIVEBROADCASTING ALGORITHM

A. Finding the optimal pulse sequence

When using a selective broadcasting architecture tosend pulse sequences to multiple qubits, pulses can bedirected to any subset of the qubits, but distinct pulsesmay not be applied simultaneously. In compiled selectivebroadcasting, the total number of pulses required to im-plement single-qubit gates on all qubits of a multi-qubitsystem is minimized by searching all possible combina-tions of single-qubit Clifford decompositions and group-ing together like pulses where possible. In this section,we introduce an algorithm for determining the short-est compiled pulse sequence implementing independentsingle-qubit Clifford gates on n qubits.

On average, there are approximately 38 distinct de-compositions for each single-qubit Clifford gate, given thebasis set of X and Y pulses: {I,Xπ, Yπ, X±π/2, Y±π/2},resulting in approximately 38n different decompositionsfor a given n-qubit combination of Cliffords. Here, weonly consider sequences of up to four pulses, because the5-primitives decomposition identified in Table S3 already

provides a recipe for decomposing an arbitrary n-qubitClifford combination into five pulses. We do not includetrivial decompositions where sequential pulses cancel out.

Given a particular choice of n Cliffords (Cα1 , . . . , Cαn),where αi is the Clifford ID for qubit i, we write a spe-cific decomposition as

((P 11 , ..., P

1m1

), ...,

(Pn1 , ..., P

nmn

)),

where P ij is the jth of mi pulses which implement Cαi .While this already fixes the order in which pulses mustbe applied to individual qubits, we still have the freedomto choose in which order the distinct pulses are applied todifferent qubits. For each possible decomposition, we usethe following recursive algorithm to search and minimizeover all possible pulse orderings.

We first define an empty broadcasting sequence Pseq tostore the compiled multi-qubit pulse sequence. In orderto convert from parallel single-qubit pulse sequences tothe single broadcasting sequence Pseq, we define a vectorof indices β = (β1, . . . , βn) to store the current positionin each single-qubit sequence. Initially, β = (1, ..., 1).At each instant, Pβ = (P

1β1, . . . , Pnβn) contains the next

pulses to be applied to each qubit. When βi = mi + 1,the pulse sequence for that qubit is completed, and sothere is no Pβi to be added to Pβ. The recursive part ofthe algorithm then proceeds as follows:

1. Define P ′β to be the set of distinct pulses in Pβ.

If:: P ′β is empty, store the number of pulses in Pseqand abort this recursion branch (Pseq is a com-pleted pulse sequence such that all Cliffordsare applied to the corresponding qubits).

Else:: Continue.

2. For each pulse P in P ′β, perform the following steps:

(i) Append P to Pseq;

(ii) Copy indices β to βnew = (βnew1 , ..., βnewn );

(iii) For all indices i for which P iβi = P , increasethe index βnewi = βi + 1;

(iv) Recursively loop to step 1 using the new in-dices βnew for β.

After considering all possible pulse sequences and loop-ing over all possible decompositions, choose the sequencewith the minimum number of pulses NP in Pseq.

This algorithm determines the minimum number ofpulses NP required to implement a given n-qubit com-bination of single-qubit Clifford gates. However, this be-comes prohibitively resource intensive as the number ofqubits increases. It is therefore important to assess howthe performance of the optimal compiled decompositioncompares with the 5-primitives decomposition, which canbe applied to any number of qubits without any extraoverhead in resources (in neither calculation time nor se-quence length). To do this, we use the average number ofpulses 〈NP〉 required per n-qubit combination of Cliffords(per Clifford).

In the case of compiled selective broadcasting, finding〈NP〉 requires minimizing the sequence length for all 24n

16

Exact calculation Random samplingn 〈NP〉 n 〈NP〉1 1.875 5 4.139 (2)2 2.925 6 4.380 (12)3 3.521 7 4.570 (15)4 3.874 8 4.721 (10)5 4.137 9 4.808 (14)

10 4.857 (24)

TABLE S4. The average number of pulses 〈NP〉 requiredto perform one Clifford gate on each of n qubits in com-piled selective broadcasting. Exact values were obtained for1 ≤ n ≤ 5 in under two hours, using the outlined optimiza-tions. Approximate values were obtained for 5 ≤ n ≤ 10using random sampling. These results are plotted in Fig. 4 ofmain text.

possible Clifford combinations. This problem again scalesexponentially with n. For example, for n = 5 qubits, thisrequires 245 · 385 ≈ 6.3 · 1014 repetitions of the completerecursive search described above. Nevertheless, by em-ploying a number of optimizations desribed in the nextsection, we have exactly calculated 〈NP〉 for 1 ≤ n ≤ 5qubits in under 2 hours. Using random sampling (find-ing the shortest pulse sequence for a random sample ofClifford combinations), we also approximated 〈NP〉 for5 ≤ n ≤ 10. Exact and approximate results for n = 5agree. As shown in Table S4, the improvement offeredby compiled selective broadcasting over the 5-primitivesmethod is already less than one pulse per Clifford andcontinues to decrease rapidly. Considering how badlythe resource overhead scales for increasing numbers ofqubits for finding a compiled sequence, it is questionablewhether compiling offers any significant benefit over us-ing the prescriptive 5-primitives approach when scalingup to larger system sizes.

B. Optimizations for the Clifford compilationalgorithm

In the first optimization, we place an upper bound Nub

on the pulse sequence length. The upper bound Nub isgiven by the minimum number of pulses found so far thatcan compile a given Clifford combination. At each stage,the algorithm checks if the sum of the pulses in Pseq andall distinct pulses left is equal to or greater than Nub.If this is the case, a shorter combination of pulses usingPseq is not possible, so we stop considering this sequenceand proceed to the next one. Initially Nub = 5, as the5-primitives method proves that there is always a decom-position of an arbitrary number of Cliffords into 5 pulses.Note that, as the limit Nub moves down, the frequencyat which the algorithm stops considering sequences in-creases.

The second optimization relies on decompositions withfewer pulses being more likely to result in an optimal Clif-ford compilation. The decompositions of every Cliffordare therefore arranged in ascending number of pulses.The first decompositions compared are then those withthe minimum number of pulses; these have the highestprobability of finding an optimal Clifford compilation.Even if an optimal Clifford compilation is not found, itis more likely that Nub will be low. This optimizationis especially effective in combination with the first opti-mization.

The third optimization places a lower bound N lb onthe number of pulses. For a given Clifford combination(Cα1 , . . . , Cαn), N

lb is found by looking at the minimumnumber of pulsesNP previously found for all n−1 Cliffordsubsets. Since NP for the n Cliffords can never be lessthan NP for any of the n− 1 Clifford subsets, the maxi-mum length of the n− 1 Clifford subsets therefore placesa lower bound N lb on NP for the n Cliffords. This meansthat if a pulse sequence is found whose length is equal toN lb, it is an optimal Clifford compilation, and all furthersearch is aborted. This is in contrast to the first opti-mization, where only the particular sequence of pulses isaborted upon reaching Nub. Furthermore, as n increases,it becomes increasingly likely that the lower bound is 5.In this case, N lb = Nub, and so the 5-primitives methodis an optimal Clifford compilation. This optimization re-sults in the largest gain in computation time, by severalorders of magnitude.

In the fourth and most complicated optimization, alldecompositions composed of three pulses or less are sepa-rated from those composed of four pulses. First, all com-binations of Clifford decompositions composed of threepulses or less are compared. This reduces the averagenumber of decompositions per Clifford, from 38 to 7, re-sulting in an exponentially reduced number of total de-composition combinations. It is, however, not alwaysthe case that the optimal Clifford compilation is foundusing only up to three pulses per decomposition; some-times optimal Clifford compilations requires that one ofthe decompositions is composed of four pulses. However,

17

after comparing decompositions of three pulses or less,these four-pulse decompositions only need to be consid-ered when N lb ≤ 4 and Nub = 5. If there is a sequencecontaining a four-pulse decomposition that outperformsany found using up to three-pulse decompositions andthe 5-primitives method, the sequence must consist offour pulses. Only one Clifford then has a four-pulse de-composition, while all other Cliffords are subsets of thesefour pulses. We therefore loop, for every Clifford, overeach of the four-pulse decompositions, and test whetherevery other Cliffords can be decomposed into a subset of

these four pulses. This changes the comparison of four-pulse decompositions from scaling exponentially with nto scaling linearly.

The fifth and final optimization is only of use whenall different Clifford combinations need to be consideredto determine 〈NP〉. It stems from the observation thatan optimal compilation for a certain Clifford combina-tion (Cα1 , . . . , Cαn) is the same as for any permutation ofthose Cliffords. We therefore only determine an optimalClifford compilation when β1 ≤ · · · ≤ βn. This reducesthe number of calculations exponentially (81 times fewercomputations for n = 5).

[1] D. Ristè, S. Poletto, M. Z. Huang, A. Bruno, V. Vester-inen, O. P. Saira, and L. DiCarlo, Nat. Commun. 6(2015).

[2] A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden,N. K. Langford, and L. DiCarlo, Appl. Phys. Lett. 106,182601 (2015).

[3] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster,B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Ma-jer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J.Schoelkopf, Phys. Rev. B 77, 180502 (2008).

[4] R. Barends, J. Wenner, M. Lenander, Y. Chen,R. C. Bialczak, J. Kelly, E. Lucero, P. O’Malley,M. Mariantoni, D. Sank, H. Wang, T. C. White, Y. Yin,J. Zhao, A. N. Cleland, J. M. Martinis, and J. J. A.Baselmans, Appl. Phys. Lett. 99, 113507 (2011).

[5] M. Reed, Entanglement and quantum error correctionwith superconducting qubits, PhD Dissertation, Yale Uni-versity (2013).

[6] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.Wilhelm, Phys. Rev. Lett. 103, 110501 (2009).

[7] J. M. Chow, L. DiCarlo, J. M. Gambetta, F. Motzoi,L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Phys.Rev. A 82, 040305 (2010).

[8] J. M. Epstein, A. W. Cross, E. Magesan, and J. M.Gambetta, Phys. Rev. A 89, 062321 (2014).

[9] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M.Girvin, M. H. Devoret, and R. J. Schoelkopf, Phys. Rev.A 75, 032329 (2007).

[10] J. Majer, J. M. Chow, J. M. Gambetta, B. R. Johnson,J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck,A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, andR. J. Schoelkopf, Nature 449, 443 (2007).

[11] J. Johansson, P. Nation, and F. Nori, Computer PhysicsCommunications 183, 1760 (2012).

[12] J. Johansson, P. Nation, and F. Nori, Computer PhysicsCommunications 184, 1234 (2013).

[13] E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan,J. M. Chow, S. T. Merkel, M. P. da Silva, G. A. Keefe,M. B. Rothwell, T. A. Ohki, M. B. Ketchen, and M. Stef-fen, Phys. Rev. Lett. 109, 080505 (2012).

Independent, extensible control of same-frequency superconducting qubits by selective broadcastingAbstract Acknowledgments References Supplemental MaterialI Quantum chip and experimental setupA Chip design and fabricationB Experimental SetupC Device frequenciesD Qubit coherence times

II Vector switch matrixA Measured isolationB Individual qubit tune-up

III Pulse-calibration routinesA Accurate in-phase pulse amplitude calibrationB DRAG-parameter calibration

IV Global broadcastingV Leakage to second excited stateVI Cross-coupling and cross-driving effectsA Cross-couplingB Cross-drivingC Cross-excitation in randomized benchmarkingD Making pulse sequences robust to cross-driving

VII Clifford pulse decompositionVIII Compiled selective broadcasting algorithmA Finding the optimal pulse sequenceB Optimizations for the Clifford compilation algorithm

References