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Application Note Superconducting Qubit Characterization Zurich Instruments Applications: Quantum Computing, Circuit QED Products: UHFLI, UHF-AWG, UHF-DIG, UHF-MF Release date: October 2016 Introduction Quantum Computing Quantum computing promises an advance in numeri- cal processing power over classical computing by ex- ploiting quantum physical phenomena [1, 2]. Present- day experimental work in this field focuses on improv- ing the devices that ultimately would take over the function of transistors in classical processors: the quantum bits (qubits). Superconducting qubits are among the most promising candidates explored today [3, 4]. Being based on microfabrication technology and purely electrical control, superconducting qubits come with a clear roadmap for large-scale integration of hundreds of qubits in a single processor. The steady improvement of coherence times achieved over recent years inspires confidence that these devices can be en- gineered to a quality that enables practical numerical algorithms. Growing Complexity Controlling superconducting qubits requires complex pulsed microwave signals on multiple channels with precise synchronization. As the number of qubits and channels increases, there is a need for more inte- grated, standardized, and more reliable electronic con- trol solutions. The Zurich Instruments UHFLI instru- ment addresses this demand as it combines an arbi- trary waveform generator (AWG) for synthesis of pre- cisely shaped pulsed signals, and a 600 MHz lock-in amplifier with digitizer for detection. Combining gen- eration and detection in one instrument reduces setup complexity, improves synchronization, reduces rack space, and enables advanced control methods based on feed-forward from detection to generation, e.g. se- quence branching. Here we describe how to use the UHFLI instrument to characterize superconducting qubits in terms of their frequency, qubit state lifetime and coherence time. The measurements presented also provide tuning of the pulse shapes used to compose quantum comput- ing sequences. The single-qubit device used in the experiments presented here is designed according to the circuit quantum electrodynamics architecture [5] which is extendable towards multi-qubit experiments. The measurements were carried out in Prof. A. Wall- raff’s Quantum Device Lab at ETH Zurich, Switzerland. Setup and Sample Quantum computing devices are designed to enable control and readout of a qubit’s quantum state. The quantum state is generally a quantum superposition of two physical configurations of the qubit device which encodes the information processed in a quan- tum computer. The sample shown in Fig. 1 (a) consists of niobium- and aluminium-based resonant circuits on a sapphire chip. The actual qubit is a nonlinear resonator formed of a small superconducting quan- tum interference device (SQUID) and an on-chip capac- itor (orange in the image). The SQUID design makes the qubit frequency tunable over a frequency range be- tween about 4 and 7.5 GHz by adjusting a small mag- netic field applied to the device. A coplanar waveguide (green) directs external signals directly to the qubit for its control. The qubit is cou- pled to a coplanar-waveguide resonator (blue) [5] at 4.78 GHz. This readout resonator is connected through an on-chip filter to a pair of coaxial cables that pro- vide the interface for reading out the quantum state of the qubit. Figure 1 (b) depicts a schematic of the measurement setup in which the sample is installed in a dilution refrigerator to provide a well isolated low-temperature environment protecting the quan-

Transcript of SuperconductingQubit Characterization - zhinst.com · UHFLI UHF-A WG Arbitrary W aveform Generator...

Page 1: SuperconductingQubit Characterization - zhinst.com · UHFLI UHF-A WG Arbitrary W aveform Generator Ch1 Ch2 Mkr Mkr OUT IN UHF-DIG Digitizer REF/ TRIGGER Cross-domain Trigger fIF fres

Application Note

Superconducting QubitCharacterization

ZurichInstruments

Applications: Quantum Computing, Circuit QEDProducts: UHFLI, UHF-AWG, UHF-DIG, UHF-MF

Release date: October 2016

Introduction

Quantum Computing

Quantum computing promises an advance in numeri-cal processing power over classical computing by ex-ploiting quantum physical phenomena [1, 2]. Present-day experimental work in this field focuses on improv-ing the devices that ultimately would take over thefunction of transistors in classical processors: thequantum bits (qubits). Superconducting qubits areamong the most promising candidates explored today[3, 4]. Being based on microfabrication technologyand purely electrical control, superconducting qubitscome with a clear roadmap for large-scale integrationof hundreds of qubits in a single processor. The steadyimprovement of coherence times achieved over recentyears inspiresconfidence that thesedevicescanbeen-gineered to a quality that enables practical numericalalgorithms.

Growing Complexity

Controlling superconducting qubits requires complexpulsed microwave signals on multiple channels withprecise synchronization. As the number of qubits andchannels increases, there is a need for more inte-grated, standardized, andmore reliableelectronic con-trol solutions. The Zurich Instruments UHFLI instru-ment addresses this demand as it combines an arbi-trary waveform generator (AWG) for synthesis of pre-cisely shaped pulsed signals, and a 600 MHz lock-inamplifier with digitizer for detection. Combining gen-eration anddetection in one instrument reduces setupcomplexity, improves synchronization, reduces rackspace, and enables advanced control methods basedon feed-forward from detection to generation, e.g. se-quence branching.Here we describe how to use the UHFLI instrument to

characterize superconducting qubits in terms of theirfrequency, qubit state lifetime and coherence time.The measurements presented also provide tuning ofthe pulse shapes used to compose quantum comput-ing sequences. The single-qubit device used in theexperiments presented here is designed according tothe circuit quantum electrodynamics architecture [5]which is extendable towards multi-qubit experiments.The measurements were carried out in Prof. A. Wall-raff’s Quantum Device Lab at ETH Zurich, Switzerland.

Setup and Sample

Quantum computing devices are designed to enablecontrol and readout of a qubit’s quantum state. Thequantum state is generally a quantum superpositionof two physical configurations of the qubit devicewhich encodes the information processed in a quan-tum computer. The sample shown in Fig. 1 (a) consistsof niobium- and aluminium-based resonant circuitson a sapphire chip. The actual qubit is a nonlinearresonator formed of a small superconducting quan-tum interferencedevice (SQUID) andanon-chip capac-itor (orange in the image). The SQUID design makesthe qubit frequency tunable over a frequency range be-tween about 4 and 7.5 GHz by adjusting a small mag-netic field applied to the device.A coplanar waveguide (green) directs external signalsdirectly to the qubit for its control. The qubit is cou-pled to a coplanar-waveguide resonator (blue) [5] at4.78GHz. This readout resonator is connected throughan on-chip filter to a pair of coaxial cables that pro-vide the interface for reading out the quantum stateof the qubit. Figure 1 (b) depicts a schematic of themeasurement setup in which the sample is installedin a dilution refrigerator to provide a well isolatedlow-temperature environment protecting the quan-

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Figure 1. (a) Optical micrograph of the sample showing the read-out resonator (blue) and the qubit capacitor plates (orange). Theresonator is coupled to input and output lines via a Purcell filter[6] (cyan). Insets show magnified views of the qubit and the cou-pler between resonator and Purcell filter. (b) Simplified diagram ofthe experimental setup based on the UHFLI instrument integratinga lock-in amplifier and an AWG. The superconducting qubit sampleis cooled in a dilution refrigerator.

tum properties of the sample.The control pulses are generated by quadraturemodu-lation of a microwave signal using two AWG channels.The pulses for qubit readout are generated by feedingone of the AWGmarker channels to the gate input of asignal generator. After transmission through the read-out resonator, the pulses are amplified and the read-out signal is down-converted to an intermediate fre-quency fIF andmeasuredwith one of the UHFLI lock-inamplifier channels.

Measurements

Qubit Spectroscopy

We start with spectroscopic measurements of the fre-quency and linewidth of the qubit and of the readoutresonator. Spectroscopy also allows us to find the op-timummagnetic field bias, set by using a small coil atthe sample.For the measurement of the readout resonator, we ap-ply a continuous microwave tone to the resonator andmeasure the amplitude of the downconverted trans-mitted signal with the lock-in amplifier. The frequencyνres of that tone is swept across the resonator fre-quency fres. Such ameasurement is shown in Fig. 2 (b).The transmission lineshape of the resonator is a nar-row dip at fres on top of a broader peak which is due tothe on-chip Purcell filter [6].For themeasurement of thequbit transition frequency,

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Figure 2. Spectra of the qubit (a) and of the readout resonator (b).The measurement in (a) is obtained by measuring the transmissionat a fixed frequency νrf close to the cavity resonance (b) and sweep-ing thequbit drive frequency νqb. (c) Thearc-shapeddependenceonmagnetic flux Φ is characteristic for this type of qubit. As the qubitfrequency crosses the frequency of the readout resonator along thehorizontal axis in (d), the two exhibit an avoided crossing. The colorscale shows the amplitude of the signal transmitted through thereadout resonator after amplification.

we use a two-tone microwave technique. The first mi-crowave tone is applied to the readout resonator ata fixed frequency close to resonance fres. While mea-suring the transmission amplitude, a second tone isapplied to the qubit gate line and its frequency νqb isswept across a wide range. Owing to the dispersivecoupling between the qubit and the readout resonator,a change in the signal is detected when νqb becomesresonant with the qubit transition at fqb [5]. This leadsto a feature at fqb as seen in Fig. 2 (a).Figure2 (c) and (d) shows thequbit and resonator spec-tra as a function of magnetic field. The qubit transi-tion frequency shows a characteristic, arc-shaped de-pendence on magnetic field [7]. At specific magneticfields, the qubit and readout resonator frequencies ex-hibit an avoided crossing seen in (d). This so-calledvacuum Rabi splitting is an effect of the strong cou-pling between qubit and resonator.All subsequent measurements are pulsed rather thancontinuous to probe the dynamic properties of thequbit. The basis of these measurements is a pulsesequence consisting of a control part and a readoutpart. During the control part we apply pulses to thequbit gate line at a frequency close to fqb. These reso-nant pulses lead to a coherent oscillation of the qubitbetween its ground and excited state known as Rabioscillations. After the control part we apply a pulseto the readout resonator at a frequency close to fresand perform a triggered measurement of the signalafter transmission through the sample. To that end,the return signal is first down-converted in the ana-log domain to fIF =28.125MHz, and then demodulatedin the digital domain using the lock-in amplifier. Us-ing digital lock-in technology instead of direct down-conversion to DC protects the measurement from am-plifier offset voltage drift and 1/f noise [8].The entire cycle is repeated several thousand times in

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Figure 3. Rabi oscillation measurement. As a function of the widthof the control pulse, the qubit evolves periodically fromground stateto excited state and back. This results in a sinusoidal readout signalamplitude as a function of the pulse width (horizontal axis) as seenin the color plot in (a). (b) Time-dependent readout signal at two val-ues of the pulse widthmarked in (a). The time between about 1.2 μsand5.8 μs corresponds to the readout pulse. (c) Rabi oscillation plotobtained by averaging the data in (a) and normalizing the result toobtain a mean excited-state population. The measurement allowsus to determine the parameters for π- and π/2-pulses (red marks).

order to increase signal-to-noise ratio and to averageout quantum fluctuations. Conventionally, the down-conversion is performed after averaging the raw data.The UHFLI’s digital demodulators enable downconver-sion before averaging. This method eliminates onedata post-processing step from the experiment, andit eliminates the neccessity to choose an intermediatefrequency that is commensurate with the repetitionrate. Furthermore it is essential when implementingfeed-forward protocols where the demodulated signalneeds to be available rapidly.

Rabi Oscillations

Applyingacontrolpulse leads toawell-definedchangeof the qubit state that can be used as a single-qubitgate operation. Such operations are required to imple-ment quantum computing algorithms, in a similar wayas for instance the logical inversion is required for clas-sical computing algorithms. A Rabi oscillation mea-surement allows us to determine the pulse parame-ters for a given gate operation.For this measurement we generate a control pulsewith an envelope shaped by smooth edges (rise/falltime 3.9 ns) and a flat plateau with a variable width.The control pulse is followed by a readout pulse whichis a microwave burst with an approximately rectangu-lar envelope. The qubit’s probability for being in the ex-cited state follows a sinusoidal evolution as a functionof thecontrol pulsearea,which isagainproportional to

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Figure 4. (a) Qubit lifetime T1 measurement. The qubit is initial-ized in the excited state with a π-pulse. The subsequent exponen-tial decay is then measured for increasing delays before the read-out pulse. (b) Ramsey fringe measurement. As the delay betweentwoπ/2-pulses is varied, the qubit readout signal follows adecayingoscillatory evolution reflecting the gradual loss of phase coherence.Themeasurement isused todetermine thequbit coherence timeT2*.Bothmeasurements were averaged 40’000 times at a 27 kHz repeti-tion rate.

its width at half maximum. This evolution is reflectedin the measured signal during the readout phase plot-ted in Fig. 3. The curve allows us to identify the widthof the qubit gate pulses relevant for the next mea-surements: the so-called π-pulse width correspondsto the first maximum, and the π/2-pulse width corre-sponds to the first quarter period. The π-pulse bringsthe qubit from its ground to the excited state, whereasthe π/2-pulse brings the qubit to an equal coherentsuperposition of ground and excited state. For thismeasurement, every control pulse was repeated 4000times at a rate of 13 kHz.

Qubit LifetimeDecay from the excited qubit state to the ground statelimits the duration of a computation algorithm. Mea-suring theaverage lifetimeT1 is thereforean importantpart of the device characterization. For a T1 measure-ment we apply a π-pulse at the start of a sequence,then wait for a variable time t, and finally measure thequbit state. The resulting data is plotted in Fig. 4 (a).Fromthe fit toanexponential exp(–t/T1)weextract thelifetime T1 = 8.8 μs.TheUHFLI’s control software LabOne® is able to gener-ate waveforms and sequences directly. For this mea-surement, we generated a series of 200 patterns withthe UHF-AWG using a sequencer loop incrementingthe readout pulse delay from 0 μs to about 26 μs us-ing run-time sequencer variables anddynamicwaitingtimes. This means that all qubit control pulses canbe generated with a single π-pulse envelope with lessthan 400 samples of waveform data for both channels.This is in contrast to conventional techniques where

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patterns for all pulsedelaysneed tobeuploaded to theAWG sample-by-sample, which would require wave-form data of about 200×(26 μs)×(2 channels)×(1.8GSa/s)=18.7 MSa and therefore a much longer wave-form transfer time.

Qubit Coherence TimeTheRamsey fringemeasurementshows the freecoher-ent evolution of the qubit. It provides a measurementof the qubit coherence time. The Ramsey sequenceconsists of two π/2-pulses separated by a varying de-lay time followed by a qubit readout. In the absenceof decoherence, the two π/2-pulses add up to a full π-pulse and the qubit ends up in the excited state. De-phasing and relaxation during the delay time causesthe pulse addition to be imperfect which means theprobability for being in the excited state decays withthe delay time. Furthermore, a slight detuning of thepulse frequency relative to fqb causes a beating effectbetween the qubit phase and the phase of the controlpulse carrier, resulting in an oscillatory contribution inthe readout signal. From themeasurement in Fig. 4 (b),we extract a dephasing time of T2*=1.2 μs.The pulses generated by the dual-channel AWG haveintermediate-frequency carriers at 112.5MHz that aregenerated using the UHF-AWG modulation mode op-erating in the digital domain. The intermediate fre-quency is subsequently upconverted to the qubit tran-sition frequency using an analog mixer. Using ampli-tudemodulationmeans that the carrier frequency andphase can be adjusted or swept without reprogram-ming the AWG.

ConclusionThis series of essential qubit measurements demon-strates thecapabilitiesof theZurich InstrumentUHFLIlock-in amplifier and AWG in an experimental domainrequiringhigh speed, timingprecision, andsignal qual-ity. The measurements provide an extensive charac-terization of a qubit device in terms of its operating fre-quency, lifetime, and coherence time. Based on the el-ementary pulses used for these measurements, morecomplex sequences for experiments such as spin echoor randomized benchmarking are set up in a straight-forward manner.

AcknowledgementsZurich Instruments would like to thank Prof. Wallraffand his team for sharing experimental results and fortheir strong commitment to developing qubit experi-ments with Zurich Instruments technology. Specialthanks go to Michele Collodo for carrying out the mea-surements and providing an excellent framework forthe integration of the UHFLI into the existing labora-tory infrastructure. We thank Theodore Walter for pro-viding the qubit sample and Yves Salathe, SimoneGas-parinetti, and Philipp Kurpiers for support with themeasurements and for discussions.This work was supported by the Swiss Federal Depart-ment of Economic Affairs, Education and Researchthrough the Commision for Technology and Innovation(CTI).

References[1] R. P. Feynman. Simulating physics with comput-

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[2] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura,C.Monroe, and J. L. O’Brien. Quantumcomputers.Nature, 464:45, 2010.

[3] Y. Nakamura, Pashkin, Yu. A., and Tsai J. S. Co-herent control ofmacroscopic quantum states ina single-Cooper-pair box. Nature, 398:786, 1999.

[4] John Clarke and Frank K. Wilhelm. Supercon-ducting quantum bits. Nature, 453:1031, 2008.

[5] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J.Majer, S. Kumar, S.M. Girvin, andR. J.Schoelkopf. Strong coupling of a single photontoasuperconductingqubitusingcircuitquantumelectrodynamics. Nature, 431:1627, 2004.

[6] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun,D. I. Schuster, L. Frunzio, and R. J. Schoelkopf.High-fidelity readout in circuit quantum electro-dynamics using the Jaynes-Cummings nonlin-earity. Phys. Rev. Lett., 105:173601, Oct 2010.

[7] JensKoch, TerriM. Yu, JayGambetta, A. A.Houck,D. I. Schuster, J. Majer, Alexandre Blais, M. H. De-voret, S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit design derived from the Cooperpair box. Phys. Rev. A, 76:042319, Oct 2007.

[8] David Schuster. Circuit Quantum Electrodynam-ics. PhD thesis, Yale University, 2007.

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