Fast Logic with Slow Qubits: Microwave-Activated Controlled-Z Gateon Low-Frequency Fluxoniums

Quentin Ficheux ,1,* Long B. Nguyen ,1,* Aaron Somoroff,1 Haonan Xiong ,1 Konstantin N. Nesterov ,2

Maxim G. Vavilov,2 and Vladimir E. Manucharyan11Department of Physics, Joint Quantum Institute, and Center for Nanophysics and Advanced Materials,

University of Maryland, College Park, Maryland 20742, USA2Department of Physics and Wisconsin Quantum Institute, University of Wisconsin–Madison,

Madison, Wisconsin 53706, USA

(Received 4 November 2020; accepted 17 March 2021; published 3 May 2021)

We demonstrate a controlled-Z gate between capacitively coupled fluxonium qubits with transitionfrequencies 72.3 and 136.3 MHz. The gate is activated by a 61.6-ns-long pulse at a frequency betweennoncomputational transitions j10i − j20i and j11i − j21i, during which the qubits complete only four andeight Larmor periods, respectively. The measured gate error of ð8� 1Þ × 10−3 is limited by decoherence inthe noncomputational subspace, which will likely improve in the next-generation devices. Although ourqubits are about 50 times slower than transmons, the two-qubit gate is faster than microwave-activatedgates on transmons, and the gate error is on par with the lowest reported. Architectural advantages of low-frequency fluxoniums include long qubit coherence time, weak hybridization in the computationalsubspace, suppressed residual ZZ-coupling rate (here 46 kHz), and the absence of either excessiveparameter-matching or complex pulse-shaping requirements.

DOI: 10.1103/PhysRevX.11.021026 Subject Areas: Quantum Physics,Quantum Information,Superconductivity

I. INTRODUCTION

Macroscopic superconducting circuits have emerged as aleading platform for implementing a quantum computer[1]. Currently available small-scale quantum processors[2–9] have achieved a number of important milestones,including the break-even point in quantum error correctionof a single logical qubit [10], digital quantum simulation[11–16], nontrivial optimization algorithms [17–19], andan example demonstration of quantum supremacy with 53qubits [20]. This progress is even more spectacular as it issolely based on a single qubit type, the transmon, whichis essentially a weakly anharmonic electromagnetic oscil-lator [21]. Although the transmon’s simplicity makesit a remarkably robust quantum system, its weak anhar-monicity has become a major limiting factor for thecurrent performance and scaling of quantum processors.Irrespective of implementation details, the weak anharmo-nicity leads to slower two-qubit gates, which makes them

prone to decoherence errors. Therefore, a motivation hasbuilt up for exploring strongly anharmonic alternatives totransmons that would ideally have higher intrinsic coher-ence and be compatible with the transmon-based scalingarchitectures.In recent years, coherence times in the 100 μs range were

repeatedly observed in superconducting fluxonium qubits[22,23]. In this paper, we describe the first logical operationon a pair of capacitively coupled fluxoniums [Fig. 1(a)].Each fluxonium can be viewed as a transmon with the weakJosephson junction additionally shunted by a large-valueinductance, provided by an array of about 100 strongerjunctions [24]. The inductive shunting makes the chargeson the capacitors continuous, and, hence, fluxoniums canhave highly anharmonic spectra insensitive to the offsetcharge noise. We operate fluxoniums near the half-integerflux bias (“sweet spot”), where the qubit transition fre-quency is first-order insensitive to flux noise and belongs toa 100–1000 MHz range. Such an order of magnitude qubitslowdown, as compared to transmons, the frequency ofwhich is typically constrained to the 5–6 GHz range,dramatically reduces the rate of energy relaxation due todielectric loss and leads to long coherence times [22].However, a fundamental question remains to be answered:How can one physically slow down qubits without endingup with slower two-qubit gates?

*These authors contributed equally to this work.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 11, 021026 (2021)

2160-3308=21=11(2)=021026(16) 021026-1 Published by the American Physical Society

Let us consider the simplest form of circuit coupling bymeans of a mutual capacitance. In analogy with transmons,there is a coupling term proportional to nAnB, where nA andnB are the charge operators of qubits A and B, respectively,normalized to the Cooper pair charge 2e. Indeed, such aterm produces little effect on the computational states j00i,j01i, j10i, and j11i, because the transition matrix elementsof nAðBÞ vanish with the transition frequency. For the samereason, though, the much higher-energy noncomputationalstates j12i and j21i can experience a noticeable repulsion,while the only nearby states j20i and j02i remain unaf-fected due to the parity selection rule; see Fig. 1(b).Therefore, connecting the two subspaces with radiationcan induce an on-demand qubit-qubit interaction. Forexample, Ref. [25] describes a controlled-Z (CZ) gateobtained by applying a 2π pulse to transition j11i − j21iwhile the closest transition j10i − j20i stays unexcited.This condition can always be met if the gate pulse is muchlonger than 1=Δ, whereΔ is the shift of level j21i due to thenAnB term, shown in Fig. 1(b). In fact, the CZ gate can becompleted in a time close to 1=Δ by choosing thecombination of drive detuning and amplitude that synchro-nizes the Rabi rotations of both noncomputational tran-sitions. For our specific device parameters (see Tables Iand II), we get Δ ¼ 22 MHz (1=Δ ¼ 45.5 ns), and theoptimal gate time is 61.6 ns. Remarkably, since the value ofΔ is not directly tied to the qubit frequencies (here, 72.3 and136.3 MHz), the logical operation takes just a few qubitLarmor periods.

Population transit through noncomputational states iscommon for gates realized with transmons [26–31]. Forexample, repulsion of states j11i and j20i enables a CZ gatevia adiabatic flux tuning of these states in and out ofresonance [32,33]. Recent high-fidelity versions of this gaterely on diabatic flux pulses [28,31], resulting in a signifi-cant population of state j20i for a short time, which drawsparallels to our microwave-controlled scheme. In the caseof fixed-frequency qubits, repulsion between states j03iand j12i is used in Ref. [34] to implement a CZ gateactivated by a microwave pulse at a frequency near thetransition j11i − j12i. However, the insufficient transmonanharmonicity introduces many nearby transitions (there isonly one relevant transition j10i − j20i for fluxoniums),and in the end such a scheme proves impractical. Even ingates designed to operate entirely within the computationalsubspace, e.g., the flux-activated j10i − j01i swap gate[35–37] or the cross-resonance (CR) gate [38,39], uncon-trolled population leakage to noncomputational statesremains an important factor limiting gate speed [40].Yet, such coherent errors can be practically eliminated influxoniums owing to their highly anharmonic spectra, asexemplified by the gate scheme reported here.Another noteworthy property of fluxoniums is that the

static ZZ shift, coming from the repulsion of computationaland noncomputational states, is relatively small (here,about 46 kHz), largely thanks to the low qubit frequencies.For transmons, the static ZZ shift is an important sourceof gate error, the mitigation of which draws additionalresources. Thus, in the case of the cross-resonance gate, theZZ term is suppressed by a combination of circuit para-meter matching, additional echo pulse sequences incorpo-rated into the gate protocol [40], and additional qubitrotations [41]. An alternative strategy to eliminate the ZZshift is to use flux-tunable couplers [20,42,43], which inpractice act as separate quantum systems and, hence,increase the circuit complexity. More recently, the ZZ shiftwas suppressed in capacitively shunted flux qubits using an

TABLE I. Parameters of the Hamiltonian given by Eq. (1)extracted by its fitting to the two-tone spectroscopy data in Fig. 2.

EC;α (GHz) EL;α (GHz) EJ;α (GHz) JC (GHz)

Qubit A 0.973 0.457 5.899 0.224Qubit B 1.027 0.684 5.768

FIG. 1. (a) Optical image of two capacitively coupled fluxo-niums fabricated on a silicon chip along with their minimalcircuit model. Devices are similar to those reported in Ref. [22]except that the antennas are intentionally made asymmetric foroptimal coupling to the readout cavity [not shown]. (b) Diagramof the lowest-energy states of the interacting two-fluxoniumsystem. Capacitive coupling induces a shift of level j21i by Δdue to repulsion from level j12i. The shift Δ enables a CZoperation in time approximately given by 1=Δ when eitherqubit is driven at a frequency in between the transitionsj10i − j20i and j11i − j21i.

TABLE II. Frequencies and coherence times of transitionsparticipating in the gate operation.

Transition Freq (GHz) T1 ðμsÞ TR2 ðμsÞ TE

2 ðμsÞj00i − j10i 0.07233 347 5.6 31j00i − j01i 0.13641 282 25.4 64j10i − j20i 5.1766 8.9 2.5 9.3j11i − j21i 5.1986 6.1 1.7 4.3

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additional drive, but at the expense of operating away fromthe flux sweet spot [44].The gate error in our scheme is largely limited by

decoherence outside the computational subspace.Randomized benchmarking yields the gate error of8 × 10−3, which is consistent with the measured fewmicroseconds coherence times of transitions j10i − j20iand j11i − j21i (see Table II). Because these transitions aretransmonlike, we expect their coherence to improve by anorder of magnitude in the next-generation experimentswith improved fabrication and noise-filtering procedures.This step would lower the gate error into the 10−4 range.The presently achieved infidelity is on par with the lowestreported values in microwave-activated gates [41,45](cross-resonance gate by IBM). Very recently, the CR gateerror was reduced even further [46]. Additionally, our gateis considerably faster than cross-resonance-type gates withcomparable errors [40,47]. Our result clearly illustrates thepotential of highly anharmonic circuits for quantum infor-mation processing, and it motivates the exploration oflarge-scale quantum processors based on fluxoniums.The paper is organized as follows: In Sec. II, we describe

the details of our experimental setup, including spectros-copy of the two-fluxonium device, the joint single-shotreadout of fluxoniums, and state initialization procedures.Note that the details of single-qubit gate characterizationare described in the Appendix D. In Sec. III, we detail theconcepts behind our fast microwave-activated CZ gate.Section IV presents the results of the CZ gate characteri-zation, including quantum process tomography and ran-domized benchmarking. In Sec. V, we review the technicallimitations of the present experiment and project the near-term improvements. Section VI concludes the work.

II. TWO-FLUXONIUM SYSTEMCHARACTERIZATION

Our device is composed of two fluxonium artificialatoms with a circuit design introduced in Ref. [22] coupledvia a shared capacitance [see Fig. 1(a)]. The system obeysthe Hamiltonian [25]

Hh¼

Xα¼A;B

�4EC;αn2α þ

EL;α

2φ2α − EJ;α cosðφα − ϕext;αÞ

�

þ JCnAnB; ð1Þ

where EC;α, EL;α, and EJ;α are the charging energy, theinductive energy, and the Josephson energy of fluxoniumindexed by α ¼ A, B, respectively. The operators φα and nαare the phase twist across the inductance Lα and thecharge on the capacitor Cα, respectively, and they commuteaccording to ½φα; nα� ¼ i. We use spectroscopy data inorder to accurately extract the parameters of theHamiltonian (1) (see Fig. 2 and Table I).

When biased at ϕext;α ¼ π, the fluxoniums are at theirsweet spots with respect to external flux. The qubitfrequencies are fA ¼ 72.3 MHz and fB ¼ 136.3 MHz.These relatively low-frequency transitions exhibit longenergy relaxation times T1;A¼347 μs and T1;B ¼ 282 μsowing to their decoupling from dielectric loss mechanisms[22]. A slight nonuniformity in the magnetic field, providedby a single external coil, prevents biasing qubits preciselyat their sweet spots simultaneously, with the offset beingabout 0.14% of the flux quantum. We operate at a bias coilcurrent of 33.7 μA, approximately halfway between thetwo sweet spots, which reduces the spin echo coherencetimes to TE

2;A¼31 μs and TE2;B¼64 μs (see Table II) com-

pared to their sweet-spot values of 47 and 67 μs, respec-tively. The sweet-spot values of coherence times are likelylimited by photon shot noise due to imperfect thermal-ization of the readout cavity [48]. This common issue can

FIG. 2. Two-tone spectroscopy as a function of probe tonefrequency and flux threading the common biasing coil. Theparameters of the system (Table I) are extracted by fitting thespectroscopy data to the diagonalization of the Hamiltonian (1).Dashed lines represent transitions allowed at half-integer flux,while dotted lines represent transitions that are forbidden at thesweet spot. The shared capacitance lifts the degeneracy betweenthe two transitions j11i − j21i and j10i − j20i by an amount Δ(top) that limits the gate speed. The sweet-spot misalignment(bottom) is attributed to a local flux imbalance through the twofluxonium loops ϕext;A ≠ ϕext;B.

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be improved with better filtering of the measurement lines.We note in advance that the small deviation from the sweetspots does not modify appreciably the operation of our two-qubit gates, and the qubit coherence times are not limitingthe gate error.Higher-energy states are separated from the computa-

tional states by an approximately 4.5 GHz gap (Fig. 2, top).The capacitive coupling term JcnAnB in Eq. (1), withJC ¼ 224 MHz, corresponds to the coupling capacitanceof about 1.2 fF. Note that similar value coupling capaci-tances are used in transmon-based quantum processors. Therelevant interaction energy scale Δ (see Fig. 1) is given byΔ ¼ jE20 þ E11 − E10 − E21j=h ¼ 22 MHz, and the staticZZ shift is given by ξZZ ¼ jE00 þ E11 − E01 − E10j=h ¼46 kHz, where Eij is the energy of state jiji.The device is embedded in a three-dimensional copper

cavity with a resonant frequency of fc ¼ 7.4806 GHz.Prior to each experiment, the qubits are initialized byapplying a strong microwave pulse on the cavity (seeAppendix C 3). This procedure prepares each qubit in astate close to the excited state. After the initialization pulse,each qubit is in a mixed state with excited states proba-bilities of 86% and 88% for qubits A and B, respectively,which is sufficient for metrology of gate operations. Theremaining qubit entropies correspond to temperatures ofTA ¼ 3.7 mK and TB ¼ 5.9 mK well below the fridgetemperature of 14 mK. Potentially more efficient initial-ization procedures have been demonstrated on single-fluxonium devices [23,49,50]. We achieve a single-shotjoint readout of the two qubits by sending a cavity tone ofoptimal frequency, power, and duration. The outgoingsignal is further amplified by a Josephson traveling waveparametric amplifier (JTWPA) [51] and commercialamplifiers before down-conversion and numericaldemodulation. We finally correct for readout imperfec-tions by a procedure described in Appendix C 2.

III. FAST CZ GATE BY EXACT LEAKAGECANCELLATION

In this section, we describe how to implement a CZ gatein the case of a two-qubit spectrum shown in Fig. 1(b) in theshortest possible time. Let us start by considering the gatetransitions j10i − j20i and j11i − j21i as two-level sys-tems. Our gate exploits the geometric-phase accumulationsduring round-trips in these two systems. The accumulatedphase can be divided into two parts: the dynamical phase,which is proportional to the evolution time and the energyof the system, and the geometric phase, which dependsonly on the trajectory followed in the Hilbert space.Applying a microwave tone drives the system with theHamiltonian

Hdrive

h¼ ðϵAnA þ ϵBnBÞ cosð2πfdtÞ: ð2Þ

When the drive frequency fd is nearly resonant with oneof the two gate transitions, we observe Rabi oscillations[Fig. 3(a)] with the resonance Rabi frequencies Ω10−20¼jh10jεAnAþεBnBj20ij and Ω11−21¼jh11jεAnAþεBnBj21ij,respectively. A strong hybridization of the j12i and j21istates creates an imbalance between the rotation speedsgiven by the ratio r ¼ Ω11−21=Ω10−20 ≃ 1.36.In order to eliminate leakage to higher states, one needs

to synchronize off-resonance Rabi oscillations determinedby the two transitions to ensure that the state vector alwayscomes back to the computational subspace. This synchro-nization is achieved by matching the generalized Rabifrequencies

Ω ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩ2

11−21 þ δ2q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΩ2

10−20 þ ðδ − ΔÞ2q

; ð3Þ

where δ ¼ f11−21 − fd is the detuning between thej11i − j21i transition and the drive frequency [Fig. 3(a)].A full rotation is then performed in the shortest gate timetgate ¼ 1=Ω. During the gate operation, the state vectortrajectory can be depicted in a Bloch sphere representation[Fig. 3(b)] when the system starts in j10i or j11i. Thesecircular trajectories, which travel in opposite directionswith respect to the centers of the Bloch spheres, define twocones inside the spheres. The cones and directions of traveldefine the solid angles Θ10 ¼ 2π½1 − ðΔ − δÞ=Ω� andΘ11 ¼ 2πð1þ δ=ΩÞ, which correspond to a geometricphase accumulation φij ¼ −Θij=2 on state jiji. Our gate,thus, implements a unitary operation U ¼ diagð1; 1; eiφ10 ;eiφ11Þ. Using virtual Z rotations [52], the phase differencecan be assigned to any computational state such as j11i torealize a controlled-phase operation U ¼ diagð1; 1; 1; eiΔφÞwith Δφ ¼ φ11 − φ10. A CZ gate is obtained whenΔφ ¼ −ðΘ11 − Θ10Þ=2 ¼ −πΔ=Ω ¼ �π. Using this con-dition in Eq. (3), we obtain the optimal drive frequency[brown arrow in Fig. 3(a)]

δ

Δ¼ r2 −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2 − 1Þ2 þ r2

pr2 − 1

≃ 0.29: ð4Þ

For δ given by Eq. (4) and Ω ¼ Δ, a CZ gate with zeroleakage is achieved in time exactly tgate ¼ 1=Δ.Our understanding of the gate process is validated by

simulating the Hamiltonian Eq. (1) in the presence of thedrive term, given by Eq. (2). The full Hamiltonian takes intoaccount the dynamical phase ϕdyn ¼ 2πξZZtgate ≃ 10−2 ≪π, which is negligible after one gate operation thanksto the small ZZ-interaction term. Starting with thedriving frequency given by Eq. (4), we minimize the finalpopulation leakage out of the computational subspace byadjusting the drive amplitude and frequency for everygate duration. Because of the rising and lowering edgesof the pulse in the simulation, we find a slightly longeroptimal gate duration of approximately 62 ns (compared to

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1=Δ ¼ 45.5 ns) for which coherent errors on the gatefall below < 10−3; see Fig. 3(c). Note that, even at the errorlevel of 10−4, the optimal point does not require an excessivefine-tuning of the gate pulse parameters: It corresponds to afraction of a percent variation in terms of the gate time, gatepulse amplitude, and frequency (see Appendix F).

IV. GATE CALIBRATION AND METROLOGY

We start with the following initial gate parameters: Weuse a 60 ns flattop Gaussian pulse with a driving amplitudecorresponding to a full rotation at the gate frequency givenby Eq. (4). To measure the accumulated phase differenceΔφ, we perform a Ramsey-type experiment on qubit B tocompare the phases of the superpositions j11i þ j10i andj01i þ j00i after the application of a CZ gate. Thiscomparison is achieved by inserting a CZ gate betweentwo successive π=2 pulses with a relative phase β createdby a virtual Z rotation [52]. When the angle β is varied,we observe Ramsey fringes with an initial phase encodingthe excited state probability of the control qubit A. Thephase difference when qubit A is prepared in the ground orexcited state yields the relative phase accumulationΔφ=π ≃ 0.95 [in Fig. 4(a)] close to the phase requiredfor the CZ gate.In order to gain more insight in the physics of our gate,

we perform quantum process tomography by preparing 16

independent input states, applying the CZ gate to each ofthem before performing their state tomography. The statetomography is obtained by a maximum likelihood estima-tion similar to the one in Ref. [53] using an overcompleteset of 36 tomography pulses. We represent the quantumprocess [54] in the Pauli basis through the processtomography matrix χ [47,55] [Fig. 4(b)] and adjust thesingle-qubit phases with virtual Z rotations to attributethe relative phase accumulation to the j11i state only. Weobtain process fidelity FQPT ¼ 0.97. Note that, althoughour input states are not pure quantum states, our procedureis valid, since we prepare a set of independent inputstates. This result shows that our scheme can be used toimplement a CZ gate, but the value of the fidelity isprobably dominated by state preparation and measurement(SPAM) errors (Appendixes C 2 and D) and cannot be usedto optimize the gate parameters further.We turn to randomized benchmarking (RB) to obtain a

finer estimate of the gate error. The RB sequence iscomposed of m randomly chosen Clifford operationsfollowed by a recovery gate aimed at bringing the quantumstate back to the initial state. The average population of j11istate decays as aþ bpm þ cðm − 1Þpm−2, where m is thenumber of Clifford operations, p is the depolarizingparameter, and a, b, and c are fitting parameters usedto absorb state preparation and measurement errors [56].The average fidelity of a Clifford operation is given by

(a) (b)

(c)

FIG. 3. Gate principle. (a) Rabi oscillations near the j10i − j20i and j11i − j21i transitions versus driving frequency with a 330-ns-long pulse. The two transitions display different resonance Rabi frequencies characterized by the ratio r ¼ Ω11−21=Ω10−20 ≃ 1.36. Thedrive frequency fd indicated in brown given by Eq. (4) is used to synchronize the oscillations on the two transitions. (b) Bloch sphererepresentations of the trajectories in the fj10i; j20ig and fj11i; j21ig manifolds in the frame rotating at the drive frequency. Thequantum state follows a closed path in the Hilbert space leading to a relative phase accumulation given by the difference of the solidangles spanned by the two trajectories. (c) Gate error versus gate duration simulated using the Hamiltonian given by Eq. (1) in thepresence of a Gaussian flat-topped pulse and without decoherence. For every gate duration, drive frequency and amplitude are optimizedto minimize the coherent leakage error (dashed line). The dots represent the gate error caused by incorrect phase accumulationΔφ ¼ φ11 − φ10 ≠ π. With the optimal pulse duration around approximately 62 ns, the infidelity due to coherent errors is wellbelow 10−3.

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FClifford ¼ 1 − ðd − 1Þð1 − pÞ=d, where d ¼ 2n is thedimension of the Hilbert space with n ¼ 2 the numberof qubits. Interleaving a target gate yields a decay withdepolarizing parameter pgate and a gate error of 1 − F ¼ðd − 1Þð1 − pgate=pÞ=d.To reach the optimal gate fidelity, we perform a

parameter search optimizing the sequence fidelity of fixedlength interleaved randomized sequences [56] using amethod known as “optimized randomized benchmarkingfor immediate tune-up” (ORBIT) [57] with a covariancematrix adaptation evolution strategy (CMA-ES) [58]. Inpractice, we first measure an interleaved randomizedbenchmarking curve [see Fig. 5(b)] before fixing thenumber of gates to n ¼ floor½−1=logðpÞ� ¼ 5, providingthe optimal sensitivity. The survival probability pðj11iÞ ismaximized by adjusting the six gate parameters [Fig. 5(a)]:

amplitude, duration, width of the edges, frequency, andsingle-qubit rotation angles. Figures 5(c)–5(h) show thestochastic evolution of gate parameters during 1700search steps leading to an improvement of gate fidelityfrom 0.967 to 0.992 for final gate duration of30.0ðplateauÞ þ 2 × 15.8ðedgesÞ ¼ 61.6 ns. In our experi-ment, CMA evolution strategy leads systematically to thesame (global) minimum contrary to a nonstochastic algo-rithm such as the Nelder-Mead method that gets moreeasily trapped in local minima.We assess the quality of the optimized CZ gate by

iterative interleaved randomized benchmarking [56].We obtain a reference fidelity FClifford2 ¼ ð96.0� 0.1Þ%.Interleaving the gate CZn, obtained by concatenating npulses corresponding to a single CZ gate, yields the CZgate fidelity of FðCZÞ ¼ ð99.2� 0.1Þ% for n ¼ 1 [yellowcurve in Fig. 6(a)]. The error grows linearly with n (we tryn ¼ 2; 3;…; 10) [Fig. 6(b)], which is a solid evidence thatthe gate error is due to incoherent processes [59].

(a)

(b)

FIG. 4. Controlled-Z gate. (a) Pulse sequences of a Ramsey-typeexperiment measuring the phase of qubit B after a CZ operation.Ramsey fringes are obtained by varying the rotation angle β. Theexcited state population of qubit B, pB

1 ¼ pðj01iÞ þ pðj11iÞ,oscillates with β with a phase that depends on the state of thecontrol qubit A. The blue and brown fringes are obtained byflipping the state of qubit A at the beginning of the sequence,revealing the conditional phase accumulated during the gateΔφ ≃ 0.95π. (b) Quantum process tomography of the CZ gate.The experimentally extracted process tomography matrix χ re-produces a CZ operation with a fidelity of 0.97.

FIG. 5. Optimizing the CZ gate. (a) The gate is optimized oversix parameters: pulse width, plateau, amplitude, and frequency aswell as single-qubit Z rotation angles. (b) We use an approachbased on ORBIT to optimize the sequence fidelity at fixed lengthwith a CMA ES optimization. The number of Clifford gates [here,5 ≃ −1=logðpÞ] is chosen to maximize the sensitivity of the tune-up procedure. (c)–(h) Stochastic evolution of the gate parametersversus the number of function evaluation. The fidelity of the CZgate is improved from 0.967 to 0.992.

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V. OUTLOOK

Although the benchmarked fidelity of our CZ gate isalready high, the current experimental setup contains anumber of imperfections, most of which can be eliminatedin the next-generation experiments. Let us start the dis-cussion by briefly summarizing these imperfections.Taking into account the Ramsey coherence times of

transitions j10i − j20i and j11i − j21i (see Table II), oneexpects the gate error bounded by 7.5 × 10−3 (seeAppendix F 2). This estimate falls close to the measuredgate error ð8� 1Þ × 10−3. We believe that the limitedcoherence time of transitions involving the second excitedstate originates in the first-order sensitivity to external fluxnoise caused by the sweet-spot misalignment. This effectcan be corrected by either improving magnetic fielduniformity, likely through better magnetic shielding, orby using two independent coils. Importantly, fluxonium’s

j1i − j2i transition is very similar in terms of decoherencemechanisms to a transmon qubit transition [22], and we,therefore, expect no fundamental obstacles in reachingcoherence times around 50–100 μs in future work. In thiscase, the Hamiltonian simulations presented in Fig. 3 showthat coherent gate errors in the low 10−4 range are possiblefor our device parameters and without sophisticated pulseoptimization.In the computational subspace, coherence times can

potentially exceed 500 μs, given the measured energyrelaxation times of 250–350 μs of states j10i and j01i.Indeed, coherence time in the few hundred microsecondsrange was previously observed in single-fluxonium experi-ments. However, here, the qubit coherence time is only50–60 μs, even at their flux sweet spots, which can beexplained by the presence of about 5 × 10−3 thermalphotons, on average, in the readout resonator. Coherencetime of the gating transitions j10i − j20i and j11i − j21iwill also be eventually limited by the shot noise of thermalphotons in the readout mode. This dephasing source isgeneric to superconducting circuit experiments, and it canbe mitigated in the future with improved cryogenic filteringof the measurement lines [60,61].Turning to our microwave packaging choice, we use a

single input port in a single 3D-box resonator to performall the gates, for the sake of technical simplicity (seeAppendix B). In Appendix D, we characterize the reductionof addressability in the system and find that simultaneoussingle-qubit gates have an about 3% error rate dominatedby classical cross talk. Using a dedicated driving port perqubit would enable selective addressing of each qubit and,hence, improve significantly the single-qubit gate fidelity[62]. More generally, individual addressing and readout isessential for scaling beyond the two-qubit experiments. Webelieve that scaling of our gate scheme can be done bymoving to the 2D-circuit designs, in complete analogy withprocessors based on capacitively coupled transmons [2–4].Note that our experiment does not benefit from the usuallyhigh-quality factors of 3D resonators, and, hence, it iscompatible with a traditional 2D-circuit technology withoutconceptual modifications. The only foreseen price ofreplacing fixed-frequency transmons by optimally biasedfluxoniums would be the requirement of either a highlyhomogeneous global magnetic field or a static flux-bias lineper qubit.A final remark is that our CZ gate does not require

excessive parameter matching, unlike the majority ofmicrowave-activated gates [34,63]. For example, the leveldiagram in Fig. 1 has no fine-tuned transitions, and thedescribed gate protocol would work for a large class offluxonium spectra and interaction strengths. Essentially, inorder to obtain Δ in the 10–20 MHz range, it is sufficientto arrange the energy of states j12i and j21i not far fromeach other (around 300 MHz difference here) and thatqubit frequencies are well resolved (about 70 MHz here).

(a)

(b)

FIG. 6. Interleaved randomized benchmarking. (a) Averageprobability pðj11iÞ as a function of the sequence length. Foreach sequence length, we average over 100 random sequencesrepeated 1500 times. We insert a variable number n ¼ 0; 1;…; 10(encoded in the color) of CZ gates between the randomly chosenClifford gates. (b) Fidelity of CZn versus n. The gate error growslinearly with n, indicating that the errors are incoherent.Error bars are calculated according to the procedure given inAppendix E.

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This property can mitigate the effects of fabricationimperfections and improve qubit connectivity in thelarge-scale quantum processors.

VI. CONCLUSIONS

We demonstrated that fluxonium qubits are not onlygood at storing quantum information, but they also allowfor fast and high-fidelity logical operations with a minimalengineering overhead. Our implementation of the micro-wave-activated CZ gate can be decomposed into two ideasapplicable to other quantum systems, each harnessing thestrong anharmonicity of fluxoniums. First, the energy scaleΔ limiting the gate speed comes from the repulsion ofnoncomputational states (here, the repulsion of states j21iand j12i), and, hence, the gate time of approximately 1=Δcan be made, in principle, shorter than the qubit’s Larmorperiods. Second, by synchronizing Rabi rotations of theonly two relevant noncomputational transitions (here,j10i − j20i and j11i − j21i), a conditional geometric phaseis accumulated at the end of each Rabi period with zeroleakage outside the computational manifold. Notably, thesynchronization is possible with a single microwave pulseof proper amplitude, frequency, and duration, applied toeither qubit.While the currently achieved combination of the CZ gate

time (61.6 ns) and infidelity (8 × 10−3) is already com-petitive with gates on transmons, our analysis indicates thatthe error can be reduced by an order of magnitude onimproving the fabrication procedures, magnetic shielding,and line filtering. Given the general compatibility of ourfluxonium capacitive coupling scheme with the transmon-based scaling technology, we believe that all necessarydemonstrations have been made to start exploring larger-scale fluxonium-based processors.

ACKNOWLEDGMENTS

We thank Chen Wang, Benjamin Huard, and IvanPechenezhskiy for useful discussions and acknowledgethe support from NSF PFC at JQI (No. 1430094) and fromARO-LPS HiPS program (No. W911NF-18-1-0146).V. E. M. and M. G. V. acknowledge the Faculty ResearchAward from Google and fruitful conversations with themembers of the Google Quantum AI team.

APPENDIX A: DEVICE DESIGN ANDFABRICATION

The device is fabricated by standard e-beam lithographyof a resist bilayer (MAA/PMMA) deposited on a high-resistivity silicon substrate. Two layers of aluminum (20and 40 nm) [22] separated by a thin barrier of oxide aredeposited by double-angle evaporation. Each inductance iscomposed of 310 (qubit A) and 206 (qubit B) identicalJosephson junctions (in agreement with the extractedparameters EL;B=EL;A ≃ 206=310). The loops are closed

by a Josephson junctionwhose insulating barrier parametersdetermine EJ;AðBÞ. The antennas are designed to provide acharging energy EC;A; EC;B ≃ 1 GHz required to bring thesecond excited state transitions in the range 4–5 GHz. Theasymmetry in the antenna design enables us to independ-ently adjust the charging energy and the coupling to thecavity mode.Table II gives the value of the energy relaxation and

coherence times of the selected transitions of the system.All the lifetimes and coherence times are measured byenergy relaxation, Ramsey, and spin echo sequences. Thereduced coherence time of the j10i − j20i and j11i − j21iis limiting the gate fidelity (see the main text).

APPENDIX B: MEASUREMENT SETUPAND WIRING

Qubit pulses are directly generated by an arbitrarywaveform generator (see Fig. 7) instead of modulating aradio frequency tone. Our ability to multiplex and synthe-size qubit pulses with a single digital-to-analog convertersignificantly reduces the hardware cost of the experiment.We use the internal mixing and pulse modulation capabil-ities of two Rhode and Schwarz® SMB100A sources togenerate the readout pulse and the two qubit operations.The output signal is amplified by a JTWPA provided byLincoln Labs [51] followed by commercial amplifiersbefore heterodyne detection.

APPENDIX C: INITIALIZATION AND READOUT

1. Single-shot readout

Both fluxoniums are strongly coupled to a 3D cavity. Weperform a joint readout [33,64] of the qubit states. In thedispersive regime [65], the readout operator can be written

M ¼ βIZI ⊗ σZ þ βZIσZ ⊗ I þ βZZσZ ⊗ σZ; ðC1Þwhere βij are complex coefficients and σi are Paulimatrices. In the single-shot limit, the integrated heterodynesignal distribution can be modeled as the sum of fourGaussian distributions associated with the computationalstates j00i, j10i, j01i, and j11i (see Fig. 8).We observe that the populations (Fig. 8) extracted from

fitting the readout distribution by four Gaussians areaffected by the readout amplitude and duration, indicatingthat our readout scheme is not quantum nondemolition[66–68]. Figure 9 represents the evolution of the populationwith the duration of a cavity pulse preceding the readoutpulse. The origin of this effect will be investigated in moredetail in future studies.

2. Readout cross-talk compensation

We adopt an empirical readout cross-talk compensa-tion introduced in Ref. [37]. This approach is used tocompensate for incorrect state mapping during a bifurcation

QUENTIN FICHEUX et al. PHYS. REV. X 11, 021026 (2021)

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readout, but we believe that it is relevant in our case. First,we compensate for an incorrect mapping j0i → j1i orj1i → j0i of qubit α by correcting the qubit populationof states jijiwith a tensorial products of two 2 × 2matricesCA ⊗ CB where

Cα ¼�

aα 1 − bα1 − aα bα

�;

where aAðBÞ ¼ 0.98ð0.96Þ and bAðBÞ ¼ 0.96ð0.87Þ.In addition, we use a pure cross-talk matrix which takes

into account the possibility to swap excitations betweenthe qubits during the readout process. The total readoutcorrection applied to the qubit populations reads

0BBB@

p000

p010

p001

p011

1CCCA ¼

0BBB@

1 0 0 0

0 1 − b c 0

0 b 1 − c 0

0 0 0 1

1CCCAðCA ⊗ CBÞ

0BBB@

p00

p10

p01

p11

1CCCA;

ðC2Þ

where p0ij are the corrected qubit populations, b ¼ 7%, and

c ¼ 3.5%. The impact of this calibration is best exemplifiedwhen performing Rabi experiments. After the calibration(Fig. 10, right column), we observe that the oscillationsare centered around 0.5 (center of the Bloch sphere) andonly the targeted qubit displays oscillations. Finally, weremind the reader that the two-qubit gate fidelities quotedin the main text are not affected by this procedure, sincerandomized benchmarking is not sensitive to readouterrors [69].

FIG. 8. Experimental (a) and fitted (b) single-shot histogramsof the readout signal when the two qubits are in thermalequilibrium. Histograms in (a) are calculated from 1.5 × 105

experimental realizations with a 10-μs-long readout pulse. Thefitting function is the sum of four Gaussian distributions. Thismeasurement is used to calibrate the position of the center of theGaussian distributions.

3 K

53 K

RT

80 mK

780 mK

14 mK

in

-20d

B

LPF

Eccosorb

-20d

B-1

0dB

LNF

+40dB 4-16 GHz

out

50

50

50coil

-20d

B

Splitter

Cavity

CH2

R&S SMB100A

IPM CH1MK1

RF

IF

AlazarCHB

SplitterLO

RF

AlazarCHA

IFLO

HP 8672A

Local oscillator

Splitter

MinicircuitsZFRSC-183-s+

DC-18 GHz

LNC4_16B

MinicircuitsZX05-153-s+3.4-15 GHz

MinicircuitsZX60-183A-s+

6-18 GHz

4-8 GHz

4-8 GHz

4-8 GHz

Cryoperm + aluminum foil

Copper cavity

50

50

DCDC

JTWPA

4-8 GHz

4-8 GHz

YokogawaGS200

Gate

CH3

R&S SMB100ACH4

CH3MK1IQ

PM

DC

DC

MinicircuitsBLK-18W-S+

PumpR&S SMB100A

50

50

HP 6.6 -11.5 GHz

HP 6.3 -15 GHz

LP 0.25 -12 GHz

LP 0.25 -10 GHz

DC

-20d

B-2

0dB

CH1LP 0-1 GHz

(x2)

(x3)

(x2)

LP 0-1.3 GHz

LP 0-0.14 GHz

LNF

+40dB

50

QubitDrive

Pasternack20 dB coupler4-12.4 GHz

-10d

B

LP 0.25 -12 GHz

FIG. 7. Schematics of the experimental setup. Single-qubitpulses are generated directly at the qubit frequencies using oneanalog output port of a Tektronix® arbitrary waveform generatorAWG5014C (not represented). The signal is combined with CZgate pulses and readout pulses before reaching the input port of the3D cavity. The outgoing signal is amplified using a traveling waveparametric amplifier followed by cryogenic and room temperatureamplifiers before down-conversion by a local oscillator, digitiza-tion by an Alazar® acquisition board (not represented), andnumerical demodulation.

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3. Initialization

One may be concerned in working with qubits whosefrequency is lower than the temperature because of the largeinitial excited state population as well as a modest reductionof the energy relaxation time. However, this situationroutinely takes place in other systems, e.g., NMR. In fact,even high-frequency transmon qubits require an efficientreset procedure [70,71]. Recent experiments show a high-fidelity reset of single-fluxonium qubits using measurement-based feedback [50] and sideband transitions [23,49].We describe how we turn the qubit transitions induced

by cavity photons observed in Fig. 9 to our advantage toinitialize the system. This shortcut is convenient for acoarse initialization procedure for a two-qubit systemwithout any additional hardware. The rate of these tran-sitions is known to generally increase with the number ofcirculating photons in the cavity [72,73]. We, thus, expectto be able to induce incoherent transition rates between thequbit states.We model the photon-induced relaxation using the

incoherent rate equations

dpα0

dt¼ −Γα

↑pα0 þ Γα

↓pα1;

dpα1

dt¼ Γα

↑pα0 − Γα

↓pα1; ðC3Þ

where the total excitation and deexcitation rates are ΓA↑ ¼

38.6� 0.6 kHz, ΓA↓ ¼ 6.4� 0.6 kHz and ΓB

↑ ¼ 53.8�0.3 kHz, ΓB

↓ ¼ 7.4� 0.3 kHz, respectively. Notably,increasing the number of circulating photons by a factorof 2.32 ≃ 5 (from Fig. 9 to Fig. 11) increases the energy

relaxation rates ΓA↓ þ ΓA

↑ and ΓB↓ þ ΓB

↑ by about a factor ofapproximately 5 mainly by increasing the excitation rates.The amplitude of the initialization pulse is chosen tomaximize the steady state purity while keeping the qubitpopulations in the computational space. This procedureleads to the preparation of the qubit states in a statisticalmixture with the excited state probabilities pA

1 ¼ 0.86 andpB1 ¼ 0.88 (see Fig. 11).

FIG. 9. Evolution of the qubit populations after a cavity pulseas a function of the pulse duration. The solid lines correspondto the solution of rate equation (C3) for each qubit with thetransition rate ΓA

↑ ¼ 4.4� 0.1 kHz, ΓA↓ ¼ 2.4� 0.1 kHz and

ΓB↑ ¼ 7.83� 0.05 kHz, ΓB

↓ ¼ 4.64� 0.05 kHz, respectively.The dashed line indicates the duration of the readout pulse. Notethat the impact of the readout pulse in this figure is alreadycorrected by the method described in Appendix C 2.

FIG. 10. Rabi oscillations of qubit A (first row) or qubit B(second row) without (left column) and with (right column)readout error correction. After the calibration, only the targetqubit displays oscillations.

FIG. 11. Initialization of the two fluxoniums. We apply a large(2.3 times the readout amplitude) drive on the cavity in order toreduce the entropy of the system. Populations are extracted by themethod described in Appendix C 3. The solid lines correspondto a fit to rate equations for each qubit with the followingtransition rates: ΓA

↑ ¼ 38.6� 0.6 kHz, ΓA↓ ¼ 6.4� 0.6 kHz and

ΓB↑ ¼ 53.8� 0.3 kHz, ΓB

↓ ¼ 7.4� 0.3 kHz. In all the measure-ments shown in this paper, we apply a 200 μs initializationpulse that leaves the qubits in mixed states with excitationprobability pA

1 ¼ 0.86 and pB1 ¼ 0.88 that correspond to

reversed thermal states with temperatures TA ¼ 3.7 mK andTB ¼ 5.9 mK. The entropy extracted by the initialization pulse isΔS=lnð2Þ ¼ 0.81 bits.

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APPENDIX D: SINGLE-QUBIT GATEFIDELITIES

Single-qubit gate fidelities are calibrated by single-qubitrandomized benchmarking (see Fig. 12). Qubit A (B) gatesare generated using Gaussian edge pulses with a totalduration of 150 ns (75 ns). Reducing the gate duration ofsingle-qubit gates deteriorates the simultaneous random-ized benchmarking fidelity because of cross talk.The addressability of each qubit can be characterized

by the numbers in Table III using the definitions inRef. [62]. The reduction of addressability in our system isrelated to the fact that our control field aimed at one targetqubit is also influencing the other qubit (classical crosstalk) due to our choice of a sample design with a singleinput port.

The use of a first-order model [69] to fit the randomizedbenchmarking decay can be justified by the presence of atime-dependent gate-dependent noise (flux noise) [74].Table IV summarizes the fidelity of various gates usedto generate the Clifford group.

APPENDIX E: ERROR BARS

The error bars displayed in Fig. 6 are the standarddeviations on the population obtained from the populationextracted from the method described in Appendix C.All the curves in Fig. 6(a) are fitted simultaneously tothe first-order model aþ bpðnÞm þ cðm − 1ÞpðnÞm−2,where a, b, and c are fitting parameters used to absorbSPAM errors,m is the number of Clifford gates, and n is thenumber of interleaved CZ gates. The fidelity of an niteration of a CZ gate is given by FðCZnÞ ¼ 1 − ðd − 1Þ½1 − pðnÞ=pðn ¼ 0Þ�=d with d ¼ 4.The error on the gate fidelity is then estimated by the

propagation of error formula

ΔFðCZnÞ ¼ d− 1

d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ΔpðnÞ

pðn¼ 0Þ�

2

þ�pðnÞΔpðn¼ 0Þ

pðn¼ 0Þ2�

2

s

ðE1Þ

to obtain the error bars in Fig. 6(b). Similar procedures areused to obtain all the error bars given in the paper.

APPENDIX F: THEORY AND SIMULATIONSOF GATE ERRORS

1. Unitary errors

To study the unitary dynamics during the gate operation,we solve the Schrödinger equation numerically for the

FIG. 12. Randomized benchmarking for single-qubit gates onqubit A, qubit B, and both. The solid lines are fit by a first-ordermodel [69] aþ bpm þ cðm − 1Þpm−2, where p is the depolariz-ing parameter used to calculate the fidelity F ¼ 1 − ðd − 1Þð1 − pÞ=d, where d is the dimension of the Hilbert space. Errorbars on the fidelity are calculated according to the proceduredescribed in Appendix E.

TABLE III. Table of addressability metrics using the definitionof Ref. [62]. The error rates rα are extracted from the independentrandomized benchmarking experiments in Fig. 12. The error ratesrαjβ are extracted by looking at the depolarization of qubit αduring a simultaneous randomized benchmarking experiment.The error metric of error on α due to the unwanted control of β isgiven by δrαjβ ¼ jrα − rαjβj. The correlations in errors are flaggedby δp ¼ pAB − pAjBpBjA, where pAB is obtained from fitting thedecay of qubit-qubit correlations p00 þ p11.

Parameter Twirl group Experimental value

rA Clifford1 ⊗ I 6.3 × 10−3

rB I ⊗ Clifford1 1.07 × 10−2

rAjB Clifford1 ⊗ Clifford1 5.3 × 10−3

rBjA Clifford1 ⊗ Clifford1 1.4 × 10−2

δrAjB 8.1 × 10−3

δrBjA 5.4 × 10−3

δp 2.2 × 10−2

TABLE IV. Fidelities of various Clifford gates obtained by two-qubit interleaved randomized benchmarking.

Gate Fidelity (%)

Xπ=2 − I 97.9� 0.1Yπ=2 − I 97.9� 0.1Xπ − I 97.6� 0.1Yπ − I 97.6� 0.09I − Xπ=2 98.95� 0.08I − Yπ=2 98.96� 0.08I − Xπ 99.20� 0.08I − Yπ 99.20� 0.08Xπ=2 − Xπ=2 97.8� 0.1Xπ=2 − Yπ=2 97.8� 0.1Yπ=2 − Xπ=2 97.8� 0.1Yπ=2 − Yπ=2 97.8� 0.1Xπ − Xπ 93.9� 0.2Xπ − Yπ 94.0� 0.2Yπ − Xπ 93.8� 0.2Yπ − Yπ 94.2� 0.2

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time-dependent Hamiltonian given by Eqs. (1) and (2) withthe additional account for a Gaussian flat-topped pulse inthe drive term (2). Thus, we multiply the drive term by apulse-shaping function with the rising edge given by

fðtÞ ∝ exp ½−ðt − twidthÞ2=2σ2� − exp ½−t2width=2σ2� ðF1Þ

at 0 < t < twidth, where σ ¼ twidth=ffiffiffiffiffiffi2π

p. The lowering edge

of the pulse is given by a symmetric expression attwidth þ tplateau < t < tgate, where tplateau is the duration ofthe flat part and tgate ¼ 2twidth þ tplateau is the total gateduration. In all the simulations, we keep twidth ¼ 15 ns andvary tplateau in order to vary tgate. To match the measuredratio between resonance Rabi frequencies Ω10−20 andΩ11−21, we choose ϵA=ϵB ¼ 0.9 in the drive term (2).We find the evolution operator U by calculating the

evolution of four basis computational states, projecting theresult into the computational subspace, and performingsingle-qubit Z rotations as described in Ref. [25], whichensures that the only diagonal matrix element that has anonzero phase is h11jUj11i. We then calculate the averagegate fidelity as [75]

F ¼ TrðU†UÞ þ jTrðU†CZUÞj2

20; ðF2Þ

where UCZ ¼ diagð1; 1; 1;−1Þ is the diagonal operatorfor the ideal CZ gate. This expression corresponds tothe randomized benchmarking fidelity [76]. When the realgate operator is U ¼ diagð1; 1; 1; eiΔφÞ, we find that1 − F ¼ ð3=10Þð1þ cosΔφÞ. For a generic gate operator,we define Δφ ¼ argh11jUj11i and use the last expressionto calculate the phase error, which is shown by dots inFig. 3(c). Once projected into the computational subspace,the operator U in Eq. (F2) is generally nonunitary, whichdescribes coherent leakage to noncomputational levels. Theaverage probability of such leakage is given by

Pleak ¼ 1 −1

4TrðU†UÞ: ðF3Þ

We show this leakage error in Fig. 13(a) as well as by thedashed line in Fig. 3(c). The total gate error calculatedusing Eq. (F2) is shown in Fig. 13(b) and by the solid line inFig. 3(c). Results shown in these figures illustrate that wecan achieve coherent gate and leakage errors below 10−4

with the current amplitude and frequency stability of theexperiment.

2. Decoherence effects

We estimate incoherent errors resulting from driving thej10i − j20i and j11i − j21i transitions as follows. We addthe errors coming from cycling each transition independ-ently. We first estimate the state fidelity for an arbitrary

initial state when driving a full rotation on one of thetransitions and average this fidelity over a set of 36 initialstates before summing the two contributions. We solve themaster equation by treating incoherent processes as aperturbation and write the density matrix as an expansionρ ¼ ρð0Þ þ ρð1Þ þ � � �. For the initial state jψð0Þi ¼jψ⊥i þ cj11i, where jψ⊥i describes the amplitudes ofthe remaining computational basis states, we considerthe Hilbert space with only three levels jψ⊥i, j11i, andj21i. Here, hψ⊥jψ⊥i ¼ 1 − jcj2. We consider the driveHamiltonian on the j11i − j21i transition in the interactionpicture and under the rotating wave approximation

HΩ

h¼ Ω

2ðj11ih21j þ j21ih11jÞ

¼ Ω2jþihþj −Ω

2j−ih−j; ðF4Þ

where we introduce the basis j�i ¼ ðj11i � j21iÞ= ffiffiffi2

pthat diagonalizes HΩ. The unitary evolution under thisHamiltonian results in

jψðtÞi ¼ jψ⊥i þ c½cosðπΩtÞj11i − i sinðπΩtÞj21i�

¼ jψ⊥i þ c�e−iπΩtffiffiffi

2p jþi þ eiπΩtffiffiffi

2p j−i

�; ðF5Þ

which gives us the zeroth-order approximation to thedensity matrix ρð0ÞðtÞ ¼ jψðtÞihψðtÞj. We find the first-order correction ρð1ÞðtÞ from the iterative master equation

dρð1Þ

dt¼ −

iℏ½HΩ; ρð1ÞðtÞ� þ Lρð0ÞðtÞ; ðF6Þ

where the Lindblad superoperator Lρ ¼ Pk ½LkρL

†k −

ð1=2ÞðL†kLkρþ ρL†

kLk� describes nonunitary processes.We use the following collapse operators:

(a) (b)

FIG. 13. Simulated leakage error (a) and gate error (b) for thetotal gate duration of tgate ¼ 61 ns versus drive-frequency detun-ing δ and drive amplitude Ωd in units of its optimal value Ωopt.Dashed lines show parameter values satisfying the correct phaseaccumulation Δφ ¼ π.

QUENTIN FICHEUX et al. PHYS. REV. X 11, 021026 (2021)

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L1 ¼ffiffiffiffiffiΓ1

pj11ih21j; ðF7aÞ

Lφ ¼ ffiffiffiffiffiffiffiffi2Γφ

p j21ih21j: ðF7bÞ

The first operator describes relaxation of the j11i − j21itransition with a rate Γ1 ¼ 1=T1ð11 − 21Þ, and thesecond operator describes pure dephasing with a rate Γφ ¼1=TR

2 ð11 − 21Þ − 1=2T1ð11 − 21Þ (see Table II).This result gives the matrix element of the correction in

the basis diagonalizing HΩ:

hmjρð1ÞðtÞjni ¼Z

t

0

hmjLρð0ÞðtÞjnie−2πiνmnðt−t0Þdt0; ðF8Þ

where hνmn ¼ Em − En is the difference between eigen-values of HΩ corresponding to jmi and jni. Theseeigenvalues belong to the set f�hΩ=2; 0g.Using these operators and equations above, we find

the following expression for the state-preparation error(tgate ¼ 1=Ω):

1 − Fψ ¼ −Tr½ρð0ÞðtgateÞρð1ÞðtgateÞ�

¼ ðΓ1 þ 2ΓφÞtgate2

jcj2ð1 − jcj2Þ

þ ð3Γ1 þ 2ΓφÞtgate8

jcj4: ðF9Þ

To find the gate error, we average Eq. (F9) over 36 initialtwo-qubit states generated from single-qubit states j0i, j1i,ðj0i � j1iÞ= ffiffiffi

2p

, and ðj0i � ij1iÞ= ffiffiffi2

p. This averaging gives

the average values hjcj2i ¼ 1=4 and hjcj4i ¼ 1=9, whichresults in the gate error

1 − F ¼ ð2Γ1 þ 3ΓφÞtgate18

¼ 1

36

tgateT1

þ 1

6

tgateT2

; ðF10Þ

where we use the relations Γ1 ¼ 1=T1 and Γφ ¼ 1=T2 −1=2T1. The upper bound for the contribution coming fromthe j10i − j20i has the same form.Comparing the values of T1 and Ramsey T2 in Table II,

we find that the main contribution is coming from T2,so 1 − F ≈ tgate=6T2. Then, for tgate ¼ 1=Δ ¼ 45.5 ns (ouranalysis is valid for a square pulse), we find that 1 − F≈tgate½1=6TR

2 ð10 − 20Þ þ 1=6TR2 ð11 − 21Þ� ≈ 0.75%.

We also integrate numerically the master equation in thesix-level Hilbert space that consists of the computationalsubspace and two upper levels j20i and j21i. We considerthe Hamiltonian generalized from Eq. (F4) to include thedrive of the j10i − j20i transition and to account fordetunings between the drive frequency and two transitionfrequencies as described in Sec. III. Using collapse oper-ators given in Eqs. (F7a) and (F7b) and similar operators forthe j10i − j20i transition, we find the incoherent error to be0.62%, which is slightly smaller than the analytic estimate

of 0.75% because of the proper account for the drive-frequency detuning. Including additional single-qubitrelaxation and dephasing channels with Ramsey T2, wefind the upper numerical bound for the incoherent error tobe 0.95%.

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