PHYSICAL REVIEW A 96, 042339 (2017)

Single qubit operations using microwave hyperbolic secant pulses

H. S. Ku,1,* J. L. Long,1,2 X. Wu,1 M. Bal,1 R. E. Lake,1 Edwin Barnes,3 Sophia E. Economou,3 and D. P. Pappas1

1National Institute of Standards and Technology, Boulder, Colorado 80305, USA2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

3Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA(Received 5 April 2017; published 30 October 2017)

It has been known since the early days of quantum mechanics that hyperbolic secant pulses possess the uniqueproperty that they can perform full-cycle Rabi oscillations on two-level quantum systems independently of thepulse detuning. More recently, it was realized that they induce detuning-controlled phases without changing statepopulations. Here, we experimentally demonstrate the properties of hyperbolic secant pulses on superconductingtransmon qubits and contrast them with the more commonly used Gaussian and square waves. We further showthat these properties can be exploited to implement phase gates, nominally without exiting the computationalsubspace. This enables us to demonstrate a microwave-driven Z rotation with a single control parameter, thedetuning.

DOI: 10.1103/PhysRevA.96.042339

I. INTRODUCTION

Controlled rotations of two-level systems were among thefirst examples of time-dependent quantum phenomena everstudied and continue to be a very active area of researchowing to the central role they play in numerous quantum-basedtechnologies currently being pursued [1–4]. Early investiga-tions of two-level quantum dynamics were conducted usinga double Stern-Gerlach experiment in which spins traverse aregion of rotating magnetic field [5]. In the rotating frame, thisproblem can be mapped onto the familiar Rabi oscillations ofatoms in a field [6], where the time-dependent drive amplitude�(t) and the detuning � from the two-level energy splittingcorrespond to the magnetic-field strength and precession rateof the spins in the field, respectively. It was recognized earlyon by Rosen and Zener [7] that there is an exact solution tothis problem, with �(t) = �0 sech(ρt). More importantly, forspecific values of �0, a spin in an arbitrary superposition of |0〉and |1〉 will always return back to that same initial state, i.e.,cyclic evolutions, independently of the value of the detuning.This surprising result has been leveraged extensively in fieldssuch as spatial solitons, quantum optics, and self-inducedtransparency [8–11]. The cyclic evolution is accompanied bythe acquisition of opposite phases by states |0〉 and |1〉, whichhas been suggested [12–16] as a means to implement phasegates or multiqubit entangling gates in various qubit systemsthrough the use of states outside the computational subspace.

Single-qubit gates are required for quantum computingand simulations. In the case of rotations about an axis inthe XY plane, the control design is rather straightforward:a resonant pulse of any shape will implement such a rotation,with the pulse area (the time integral of the pulse envelope)determining the angle of rotation. The theoretical simplicity ofthis concept has made qubit rotations about XY axes routinein many laboratories [17–20]. On the other hand, Z rotationshave been implemented to date via tuning [21], combiningrotations about X and Y using multiple pulses, or just bykeeping an accounting of all phases on the system [22]. In

*Hsiang-Sheng.Ku@nist.gov

general, these methods can result in increased decoherence insystems of multiple qubits, as it takes the parameters awayfrom the high coherence regime. The other alternatives canresult in increased overhead in pulse time or accounting. Thereis thus a need for a gate that achieves a rotation around the Z

axis using a single pulse to simplify and reduce overheads.One of the difficulties in implementing Z rotations (i.e.,

phase gates) is that, unlike in the case of rotations about axesin the XY plane, there is no generic analytical solution for theevolution operator corresponding to a Z rotation. One generalrequirement for phase gates is that the qubit undergoes a fullRabi cycle, with all populations restored to their initial values.This can be achieved with resonant 2π pulses, however suchpulses induce the same phase factor (equal to −1) to both the|0〉 and the |1〉 state, resulting in a trivial qubit operation thatdoes not change the phase.

In the current work, we overcome this challenge and de-velop an implementation of a microwave-mediated Z rotation,Zsech(�). By using a sech-function microwave pulse, we areable to achieve a phase gate in qubits using only a singleparameter, the detuning �, and nominally driving the lowertwo levels of a transmon qubit. After fixing the peak strength ofthe pulse to satisfy the full Rabi cycle condition, the detuningis used to adjust the angle of rotation. Here we demonstratethe general result that this property is unique to the sechpulse, in contrast with other pulse shapes such as square andGaussian, and demonstrate that sech pulses can be used togenerate microwave-based phase gates that are intrinsicallyhigh fidelity, F > 99%.

II. DRIVING QUBITS WITH SECH PULSES

To demonstrate a Z rotation with superconducting qubits,we use a microwave pulse with a sech envelope to rotatebetween the lowest two energy levels of a transmon. Weconsider a two-level system coupled by an external drive field,where the time-dependent Rabi frequency �(t) of the drivepulse is defined as [12]

�(t) = �0 sech (ρt) cos (ωDt), (1)

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H. S. KU et al. PHYSICAL REVIEW A 96, 042339 (2017)

where �0 is the peak drive amplitude, ρ is the pulse bandwidth,and ωD is the drive frequency. After the rotating waveapproximation, the full Rabi cycle condition is satisfied bychoosing

�0/ρ = n, (2)

where n is an integer [7,10]. The salient feature is that thiscyclic evolution condition is independent of ωD. This enablesus to devise a single-control microwave Z rotation.

The Z rotation is achieved as follows. Suppose the initialstate of a qubit is in a superposition state �0 = a|0〉 + b|1〉.After the qubit undergoes a cyclic evolution, the state |0〉 (|1〉)acquires a phase ξ (−ξ ), i.e., the state is driven to � = a|0〉 +beiφ|1〉 with φ = 2ξ . For a 2π pulse (n = 1), the induced phaseφ is given by

φ = 4 arctan (ρ/�), (3)

where the detuning is � = ωD − ω10 and ω10 is the frequencydifference between the lowest two transmon levels. By fixingthe peak drive amplitude �0 and the bandwidth ρ, we canconstruct a single-control Z gate, Zsech(�), by adjustingonly �.

III. EXPERIMENTAL RESULTS

A. Device

The specific device used for the experimental test ofthis gate is a concentric transmon [23,24]. A transmon isessentially a nonlinear electrical LC oscillator that is readout using capacitive coupling, in the dispersive regime, toa linear resonator. The particular qubit used for this workhad a transition frequency of ω10 = 2π × 5.18 GHz and ananharmonicity of ω10 − ω21 = 2π × 200 MHz where ω21 isthe transition frequency between the first and second excitedstates of the transmon. Further details of the qubit and theheterodyne readout method are given in Appendixes A and B.

B. Rabi oscillation

In order to demonstrate the uniqueness of the sech drive,Rabi oscillations were driven with sech, Gaussian, and squarepulse envelopes as shown in Fig. 1. The synthesized microwavedrive is generated using modulation signals from an arbitrarypulse sequencer to IQ modulate a local oscillator. The fulllength of the pulse extends over ±4σ , where σ is the standarddeviation and can be related to the bandwidth ρ by σ = π/(2ρ)for the sech pulse. This range fully utilizes the full digitizationrange of the arbitrary pulse sequencer, thereby minimizingsharp cutoffs of sech and Gaussian envelopes. As seen in theinset of Fig. 1, the sech pulse is slightly broader with a longertail than the Gaussian. While improvements to the pulse shapecould be made by either creating a hard on-off transition [25]or reducing leakage with derivative removal via adiabatic gate(DRAG) [22,26,27], in this work we have chosen to use asimple sech pulse shape for direct comparison to the theoryand general purpose applications.

A comparison of experimental and theoretical Rabi os-cillations versus the detuning � and the pulse amplitude isshown in Fig. 2 for σ = 25 ns pulses. The theory is simulatedusing the empirical result that our qubit is typically initialized

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Am

plitu

de

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.)

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ns

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Sech

FIG. 1. Three different excitation profiles for sech (solid red),Gaussian (dotted green), and square (dashed blue) pulse shapes. Thesech and Gaussian pulses shown here are truncated at t = ±4σ , whereσ is the standard deviation of the pulse. For the sech pulse, thestandard deviation σ is related to its bandwidth ρ by σ = π/(2ρ).

in an incoherently mixed state of 90% |0〉 and 10% |1〉due to heating in the dilution refrigerator. This results in acorresponding drop of the maximum excited-state probability.The excited-state ellipses obtained from the sech pulses are

Experiment

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FIG. 2. The excited-state probability as a function of pulseamplitude (vertical axis) and detuning (horizontal) for (a) sech,(b) Gaussian, and (c) square pulses. The left and right panels compareexperimental and theoretical simulations. The simulations include thefour lowest energy levels of the transmon and decoherence processT1 = T2 = 10 μs.

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FIG. 3. Line cuts at various detunings vs pulse amplitude fromFig. 2.

qualitatively and quantitatively different compared to thechevron-shaped response exhibited by the Gaussian and squarepulses. The first feature to note is that the widths of the maxima,in the detuning axis, for the sech pulse are approximatelythe same for subsequent oscillation maxima [Fig. 2(a)]. If wetake cuts of the images along constant detunings, this leads touniform periodic oscillations in the excited-state probability asa function of pulse amplitude in the case of the sech, as shownin the one-dimensional (1D) plots of Fig. 3(a). On the otherhand, the Gaussian maxima in Fig. 2(b) progressively widenand curve further downward with each oscillation period. The1D slices shown in Fig. 3(b) for the Gaussian illustrate thatthe points where the population returns to the ground stateshift toward lower pulse amplitude, while the Rabi oscillationcontrast grows with increasing drive strength. This behavioris further exaggerated for the square pulses, as is evident inFigs. 2(c) and 3(c). The uniformity of the oscillations in thesech pulse case is a direct reflection of the fact that, even ifthe drive amplitude is fixed, one can still achieve the cyclicevolution condition. To see this effect quantitatively, the pulseamplitude required for a full Rabi cycle versus the drivedetuning is plotted in Fig. 4 for all three pulse shapes. Thepopulation return amplitudes are found by quadratic fits nearthe minimum region which correspond to 2π pulses in Fig. 2.The sech pulse only has a 8.2% variation in the pulse amplitudeover the |�| � 10 MHz range, while the Gaussian pulse has a43.5% change. This low detuning-dependence behavior allowsus to vary only one parameter, the detuning �, to obtain anarbitrary Z gate.

Although the experimental and theoretical results inFig. 2(a) are in very good agreement, we do expect nonideality

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FIG. 4. The pulse amplitude for the first full Rabi cycle plottedvs the drive detuning for sech (solid red), Gaussian (dotted green),and square (dashed blue) pulse shapes.

in the sech data from two sources. First, the finite bit resolutioncutoff at ±4σ results in a slight flattening of the third Rabiexcitation ellipse in the experimental plot of Fig. 2(a). Thesecond possible source is the existence of higher levels of thetransmon system, resulting in a slight tilt of the theoreticalRabi ellipses. This behavior is illustrated in the simulation formultiple Rabi oscillations. This effect is most likely masked inthe experiment by the finite bit resolution of the arbitrary pulsesequencer. However, note that only the first oscillation [about1/3 of the Y -axis range in Fig. 2(a)] is used to implementZ gates, and for this region there is negligible discrepancybetween theory and experiment. This high degree of frequency

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FIG. 5. Tomography of a single-qubit phase gate, Zsech(�), for astate prepared by a π/2 rotation. Panel (a) shows the pulse sequenceand panel (b) shows the theory predicted θ (dashed cyan line) andφ (solid magenta line) and the measured θ (blue square) and φ (redcircle), where θ and φ are the polar and azimuthal angles of theresulting Bloch vector of the qubit state.

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independence explains the very high fidelities we obtained, aswe discuss below.

C. Z rotation

The preparation and tomography pulse sequence for theZsech(�) phase gate is shown in Fig. 5(a). In this example,the state is initially prepared by a π/2 rotation around the−Y axis. A subsequent 2π sech pulse is then applied withthe drive frequency given by ωD = � + ω10, and in eachexperimental run, either the X, Y , or Z projection of the finalstate is measured to complete the single-qubit tomography.From this, the resulting φ(�) and θ (�) are determined, whereθ is the angle of the Bloch vector from the Z axis [solid pointsin Fig. 5(b)]. For −10 MHz � � � 10 MHz, these data showthat θ remains constant at π/2. This demonstrates that therotation of the Bloch vector is around the Z axis with rotationangle φ. Both are in excellent agreement with the prediction,Eq. (3) [the solid lines in Fig. 5(b)].

To assess the performance of our Zsech(�) gate, we considerthe fidelity of the rotations for the six input states obtainedfrom π/2 rotations about the ±X, ±Y directions, the identityoperator, and a π rotation. To account for the existence ofmixed states in the initial state preparation, we calculate thefidelity of the gate operation, Zsech(�) as

F = Tr[√√

ρ1ρ2√

ρ1], (4)

for each of the initial states. The density matrix ρ1 is recon-structed from tomography measurements and ρ2 is calculatedfrom the theory, Eq. (3), with a 9% excited state before the statepreparation pulse. Complete tomography results for all sixpossible initial states are presented in Appendix C. As shownin Fig. 6, the fidelity averaged over the six different initial statesis Favg(�) > 99% for |�| � 10 MHz. This range correspondsto a ∓π rotation around the Z axis. For |�| � 10 MHz,there is a slight dropoff of Favg for the negative detuning,presumably due to the existence of higher energy leveltransitions at lower frequencies due to the anharmonicity of thetransmon.

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FIG. 6. The average fidelity for the sequence in Fig. 5(a) averagedover the six different initial states prepared as described in the text.

IV. CONCLUSION

In conclusion, we have shown that ideal sech shapedpulses can be used to implement a fast, high-fidelity phasegate with a single control knob, the detuning. The uniqueproperties of the sech drive allow us to achieve this gate whilestaying in the computational subspace throughout the durationof the gate. Our work paves the way toward high-fidelitytwo- and three-qubit entangling phase gates, which havebeen theoretically proposed based on the sech pulse [15,16].Moreover, the sech drive can be advantageous when thelowest energy levels have some spread, as is intrinsic to thesuperconducting devices manufactured with lithographicallydefined and thermally oxidized components. Our results alsolay the ground work for the superconducting circuit-basedexperimental demonstration of self-induced transparency [10],which occurs when, in addition to the temporal profile, thespatial distribution of the pulse is also a sech function. Such anexperiment would be relevant for microwave-based logic withthese circuits, slow-light demonstrations, and the developmentof larger scale circuits.

ACKNOWLEDGMENTS

The authors acknowledge support of the NIST QuantumBased Metrology Initiative. E.B. acknowledges support fromthe Office of Naval Research (N00014-17-1-2971). Contribu-tions to this article by workers at NIST, an agency of the U.S.Government, are not subject to U.S. copyright.

APPENDIX A: DEVICE CHARACTERIZATION

The image of the device is shown in Fig. 7. It consists ofa concentric transmon [24] coupled to a microstrip resonator.The bare resonator frequency is ωr = 2π × 8.5745 GHz witha coupling Qc = 5000 to the transmission line. The resonator-qubit coupling strength is g = 2π × 71.4 MHz which resultsin a 1.5-MHz resonator frequency shift when probed inthe dispersive regime (Fig. 8). The transmon frequency ismeasured as ω10 = 5.18 GHz with an anharmonicity of α =200 MHz (Fig. 9) [28]. The device is cooled down in a dilutionrefrigerator to a base temperature of T = 25 mK. The qubitrelaxation T1 and decoherence T ∗

2 are measured to be 8.6 μs(Fig. 10) and 10.25 μs (Fig. 11), respectively.

input /

output

qubit

resonator

Josephson junction

FIG. 7. Optical microscope image of the device.

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FIG. 8. Color scale of readout resonator response [(a) magnitude(b) phase] vs frequency (x axis) and power (y axis).

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FIG. 9. Readout resonator response [(a) magnitude (b) phase] vsqubit drive frequency showing the qubit energy spectra for high (red,lower line) and low (blue, upper line) driver powers.

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FIG. 10. (a) Energy decay, T1 of qubit. (b) Histogram of multipleT1 measurements over 1 h.

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multiple T ∗2 measurements over 1 h.

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FIG. 12. (a) Schematic for the generation of the pulse-shapedmicrowave drive. Drive 1 is the Zsech(�) gate and drive 2 is forthe state preparation and tomography rotations. (b) Schematic forthe generation of the pulse-shaped microwave drive. Drive 1 is theZsech(�) gate and drive 2 is for the state preparation and tomographyrotations.

APPENDIX B: QUBIT CONTROL ANDMEASUREMENT SYSTEM

The synthesized microwave drive is generated by IQ

modulating a local oscillator (LO) as shown in Fig. 12(a).The two modulation signals, I and Q, are produced from anarbitrary pulse sequencer with preprogrammed pulse shapessuch as sech, Gaussian, or square functions. The amplituderatio between I and Q allows for an arbitrary phase of theoutput pulse relative to the LO, which defines the rotation axis.The pulse amplitudes are calibrated to π/2 and π rotationsby applying repeated pulses to amplify small errors [29]. dcoffsets are added in I and Q signals to suppress LO leakage.The IQ modulator is followed by a variable attenuator anda fast microwave switch for an additional power control andbetter on-off ratio.

Our two-channel heterodyne microwave measurement sys-tem is shown in Fig. 12(b) [30]. This scheme is typicallyreferred to as interferometric readout. Each of the measure-ment channel uses two microwave synthesizers, a LO and areadout, with a frequency difference ωIF between them. Thetwo generator are both split into two paths. In one path, the LOand the readout are directly mixed down to generate a referencetone. In the other path the readout is passed through the device

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alu

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FIG. 13. Tomographies of the Zsech(�) gate for states initiallyprepared by a (a) Y−π/2, (b) Yπ/2, (c) Xπ/2, (d) X−π/2, (e) Xπ , (f) I

rotation.

under test and then mixed with the LO to generate the signalencoded with the qubit state information. The signal and thereference are both digitized in the computer. The amplitude andphase of the signal relative to the reference are then extractedin the control program. The advantage of the interferometricreadout is that it is not necessary to calibrate an IQ mixer vsfrequency for single-sideband modulation. This comes at thecost of an extra synthesizer in the setup, but makes searchingfor readouts much easier.

APPENDIX C: SINGLE-QUBIT TOMOGRAPHY

Tomographies for our Zsech(�) gate operating on sixdifferent initial states [sequence illustrated in Fig. 5(a) ofmain text] are shown in Fig. 13. The measurements (solidpoints) and theory (solid lines) agree with each other wellfor |�| � 10 MHz, which corresponds to a full 2π rotationaround the Z axis. For |�| � 10 MHz, the increased deviationsbetween them are due to the pulse cutoff at ±4σ and theexistence of higher levels of the transmon qubit.

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