High-fidelity and Robust Geometric Quantum Gates that Outperform Dynamical Ones

Tao Chen1 and Zheng-Yuan Xue1, 2, ∗

1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, and School of Physicsand Telecommunication Engineering, South China Normal University, Guangzhou 510006, China

2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics,South China Normal University, Guangzhou 510006, China

(Dated: December 8, 2020)

Geometric phase is a promising element to induce high-fidelity and robust quantum operations due to itsbuilt-in noise-resilience feature. Unfortunately, its practical applications are usually circumscribed by requiringcomplex interactions among multiple levels/qubits and the longer gate-time than the corresponding dynamicalones. Here, we propose a general framework of geometric quantum computation with the integration of the time-optimal control technique, where the shortest smooth geometric path is found to realize accelerated geometricquantum gates, and thus greatly decreases the gate errors induced by both the decoherence effect and operationalimperfections. Meanwhile, we faithfully implement our idea on a scalable platform of a two-dimensional su-perconducting transmon-qubit lattice, with simple and experimental accessible interactions. In addition, numer-ical simulations show that our implemented geometric gates possess higher fidelities and stronger robustness,which outperform the best performance of the corresponding dynamical ones. Therefore, our scheme providesa promising alternative way towards scalable fault-tolerant solid-state quantum computation.

I. INTRODUCTION

Based on fundamental quantum-mechanical principles,quantum computation (QC) can effectively deal with certaincomplex tasks that are hard for classical computers [1], dueto the intrinsic quantum parallelism. Therefore, various sys-tems have been suggested for the physical implementation ofQC, e.g., trapped ions [2], cavity quantum electrodynamics(QED) [3], neutral atoms in optical lattices [4, 5], etc. How-ever, considering the requirements of large-scale integrabilityand flexibility, one of the promising candidates is the super-conducting nanocircuit system [6–8], as it is compatible withmodern ultrafast optoelectronics as well as with nanostructurefabrication and characterization. Besides, recent experimen-tal advances in controlling the coherent evolution of quantumstates [9–12] in larger lattices confirm the superconductingcircuit as an interesting candidate with which to implementscalable QC, which requires at least a two-dimensional (2D)lattice of coupled qubits. Currently, control-induced crosstalk(frequency drift) among adjacent qubits in large qubit latticesis a main source of error for the implementation of quantumgates in that environment. Meanwhile, due to the inevitableinteraction with the surrounding environment, the coherenceof the quantum system is very fragile. Therefore, the questionof how to suppress the effects of quantum operational imper-fections and decoherence is the main challenge in realizingscalable QC.

To suppress quantum operational imperfections, quantumgates induced by geometric phases are promising [13], dueto their built-in noise-resilience features. Explicitly, geomet-ric QC (GQC) [13–15] has been proposed through the use ofadiabatic geometric phases. However, due to the long gatetime required by the adiabatic condition, the decoherence ef-fect causes considerable gate infidelity. To overcome such

∗ zyxue83@163.com

limitations, nonadiabatic GQC has been proposed to achievehigh-fidelity quantum gates based on both Abelian [16–20]and non-Abelian geometric phases [21–23]. Remarkably, ex-perimental demonstrations of elementary gates for GQC havebeen made on various systems [24–33]. However, due to theneed for additional auxiliary energy levels beyond the qubitstates and/or additional auxiliary coupling elements, the im-plementations of high-fidelity universal geometric gates arestill experimentally challenging. Meanwhile, as for the deco-herence effect, the time needed for a nonadiabatic geometricgate is still greater than for the corresponding dynamical one,leading to more gate infidelities: thus, this is the other ma-jor drawback of GQC. Finally, geometric evolution is usuallybased on nonsmooth evolution paths, which further weakensthe intrinsic robustness of the gate .

Here, we propose a general framework for nonadiabaticGQC in a simple setup, which integrates the time-optimal con-trol (TOC) technique [34, 35] to find the shortest geometricevolution path for accelerated geometric quantum gates withsmooth evolution paths, and thus can overcome the above-mentioned disadvantages of GQC perfectly. Meanwhile, ourprotocol can be realized on a 2D square lattice of supercon-ducting qubits, where adjacent qubits are capacitively cou-pled, without increasing the circuit complexity by adding anyadditional auxiliary levels and qubits. In addition, we onlyutilize experimentally accessible two-body interaction by theparametrically tunable coupling [10, 11, 36–38]. Our numer-ical simulations show that, compared with dynamical gates,our implemented geometric gates can perform with highergate fidelities and stronger robustness. Therefore, our proto-col provides a promising route towards scalable fault-tolerantsolid-state quantum computation.

II. GENERAL FRAMEWORK

We first illustrate how to implement nonadiabatic evolutionfor a general two-state system. In the rotating framework with

arX

iv:2

001.

0578

9v3

[qu

ant-

ph]

6 D

ec 2

020

2

respect to the driving frequency, assuming hereafter that ~ =1, a general Hamiltonian for a driven two-level system is

H(t) =1

2

(−∆(t) Ω(t)e−iφ(t)

Ω(t)eiφ(t) ∆(t)

), (1)

where the basis consists of a ground state |0〉 = (1, 0)† andan excited state |1〉 = (0, 1)†; Ω(t) and φ(t) are the amplitudeand phase of the driving microwave field, respectively; ∆(t)is the time-dependent detuning between the qubit transitionfrequency and the frequency of the microwave field. As iswell known, there is a pair of state vectors, |Ψ±(t)〉, thatsatisfy the Schrodinger equation ofH(t) and, in this basis, thetime-evolution operator is

U(t) = Tei∫H(t)dt=

∑j∈+,−

|Ψj(t)〉〈Ψj(0)|,

where T is the time-ordering operator. Here, to induce ourtarget geometric phases, we introduce a set of auxiliary basis|ψj(t)〉 defined by |ψj(t)〉 = e−ifj(t)|Ψj(t)〉 with fj(0) = 0,which does not need to satisfy the Schrodinger equation. Forthe selection of the auxiliary basis vectors, we start from thedynamic Lewis-Riesenfeld invariant I(t) of H(t), satisfying∂I(t)/∂t+ i[H(t), I(t)] = 0, which is [39–41]

I(t) =µ

2

(cosχ(t) sinχ(t)e−iξ(t)

sinχ(t)eiξ(t) − cosχ(t)

), (2)

where ξ(t) = −∆(t) − Ω(t) cotχ(t) cos[φ(t) − ξ(t)] andχ(t) = Ω(t) sin[φ(t) − ξ(t)], in which µ is an arbitrary con-stant. As a matter of fact, we can select its eigenvectors, |ψ+(t)〉 = cos χ(t)

2 |0〉+ sin χ(t)2 eiξ(t)|1〉,

|ψ−(t)〉 = sin χ(t)2 e−iξ(t)|0〉 − cos χ(t)

2 |1〉,(3)

as a set of the auxiliary basis: their evolutionary details ofthem in a Bloch sphere are visualized by the time-dependentpolar angle χ(t) and azimuthal angle ξ(t), as shown inFig. 1(a). That is, by determining the target evolutionpath dominated by χ(t) and ξ(t), the Hamiltonian parame-ters Ω(t),∆(t), φ(t) can then be reverse engineered [41].Therefore, at a final time τ , the two dressed states areU(τ)|ψ±(0)〉 = e±iγ |ψ±(τ)〉 with ±γ= f±(τ), and the cor-responding evolution operator is

U(τ) = eiγ |ψ+(τ)〉〈ψ+(0)|+ e−iγ |ψ−(τ)〉〈ψ−(0)|, (4)

where γ =∫ τ

0〈ψ+(t)|(i ∂∂t−H(t))|ψ+(t)〉dt = γg+γd is the

Lewis-Riesenfeld phase, in which

γd =1

2

∫ τ

0

[ξ(t) sin2 χ(t) + ∆(t)]/ cosχ(t)dt (5)

and

γg = −1

2

∫ τ

0

ξ(t) [1− cosχ(t)] dt (6)

are the dynamical and global geometric parts, respectively.The geometric nature of γg can be verified by the fact that

(a) (b)

FIG. 1. A comparison between TOC and conventional GQC. (a) Ageometric illustration of the evolution paths of the TOC (red line)and conventional (blue line) geometric gates in a Bloch sphere. (b)The X- and Y-axis-rotation gate time of TOC and conventional GQCas a function of rotation angles, with the same time-dependent pulseshape of Ω(t) = Ωm sin(πt/τ).

it is half of the solid angle enclosed by the evolution path andthe geodesic line that connects the initial point [χ(0), ξ(0)]and the final point [χ(τ), ξ(τ)] (for details, see Appendix A).However, the existence of the dynamical phase γd will leadto the loss of the geometric noise-resilient feature: thus, ef-fectively, the ways to deal with the dynamical phase includeeliminating it or transforming it to hold geometric proper-ties. Note that the elimination option requires multiple and/ornonsmooth evolution paths: it is the main culprit in limit-ing the geometric gate time and weakening the robustness ofthe geometric gate. Thus, here, we devote ourselves to ap-plying the latter strategy, i.e., letting γd to meet the form ofγd = αg+`γg , where ` is a gate-independent constant and αgis a coefficient that is dependent only on the geometric featureof the quantum evolution path during gate operation and thusfinally makes the phase γ an unconventional geometric phase[18, 26].

Next, we focus on the dynamical phase γd. By further set-ting

∫ τ0

[ξ(t) + ∆(t)]/cosχ(t)dt = −∫ τ

0ξ(t)dt, the dynam-

ical phase will be γd = [ξ(0) − ξ(τ)] − γg , thus satisfyingthe unconventional geometric condition [26], with ` = −1.Moreover, in this case, αg = ξ(0)− ξ(τ) is just the differencein azimuthal angle between the initial and final points and isindependent of the polar angle χ(t), as shown in Fig. 1(a),which also solely depends on the essential geometric featureof the overall evolution path. In addition, we retain the un-changed polar angle as χ(t) = χ, that is, making the dressedstates evolve along the latitude line of the Bloch sphere. Withthese settings, the constraints for the dressed-state parametersreduce to

ξ(t) = φ(t) + π, χ = tan−1(Ω(t)/[ξ(t) + ∆(t)]), (7)

and the resulting geometric evolution operator from Eq. (4) is

U(τ) =

((cγ′ + isγ′cχ)e−iξ− isγ′sχe

−iξ+

isγ′sχeiξ+ (cγ′ − isγ′cχ)eiξ−

), (8)

where ci = cos i, si = sin i, and γ′ = γ + ξ− = −ξ−with ξ± = [ξ(τ) ± ξ(0)]/2. In this way, the target control ofthe geometric X- and Y-axis rotation operations for arbitraryangles [0, π] can all be done by determining ξ+ = 0 and π/2

3

(a) (b)

(c) (d)

FIG. 2. An illustration of our scheme. (a) A scalable 2D qubit latticeconsisting of capacitively coupled superconducting transmons. (b)The energy levels and coupling configuration of a transmon, wherean external microwave field is meant to couple the two lowest levelsbut will also stimulate transitions among the higher excited states, ina dispersive way. (c) The circuit details and (d) the energy spectrumof two capacitively coupled transmons, where the parametrically tun-able coupling within the single-excitation subspace |01〉, |10〉 canbe used to implement two-qubit geometric gates with TOC.

with the same ξ− = −π/2, respectively, in a single evolutionpath. Note that, up to now, parameters ξ(t) [or φ(t)] and Ω(t)can take arbitrary shapes, provided that the boundary valuesof ξ(t) are fixed to realize different rotation operations.

Furthermore, to pursue higher gate fidelity, we need to min-imize the gate time to reduce the gate infidelity induced by thedecoherence. Therefore, we can incorporate the TOC tech-nique into our framework of GQC, i.e., by engineering theshape of φ(t) and Ω(t), to accelerate the target geometric gate.As for the quantum dynamics under the driving HamiltonianH(t), the different selection of Ω(t) and φ(t) makes the quan-tum system evolve along different paths. The motivation ofTOC is to find the path with the shortest time. And then, inthe realistic physical implementation, the considered interac-tion Hamiltonian asHc(t) = 1

2Ω(t)[cosφ(t)σx + sinφ(t)σy]in Eq. (1) needs to satisfy the following certain constraints:(i) the driving amplitude of the microwave field cannot beinfinite, i.e., f1[Hc(t)] = 1

2 [Tr(Hc(t)2) − 12Ω2(t)] = 0,

and (ii) the form of the interaction Hamiltonian Hc(t) can-not be arbitrary, i.e., f2[Hc(t)] = Tr(Hc(t)σz) = 0, inwhich σx,y,z are the Pauli operators in the computationalsubspace |0〉, |1〉. Then, by solving the quantum brachis-tochrone equation [42] ∂F/∂t = −i[H(t),F], with F =∂(∑j=1,2 λjfj [Hc(t)])/∂H(t) = λ1Hc(t) + λ2σz , in which

λj is the Lagrange multiplier, we can determine the restrictedparameter as follows:

φ(t) = φ0 + φ1(t), φ1(t) =

∫ t

0

[C0Ω(t′)−∆(t)]dt′, (9)

by defining λ1 = 1/Ω(t) and λ2 = −C0/2, where Ω(t) canbe an arbitrary pulse shape, and the coefficientC0 is a constantthat depends only on the type of target gate. It is worth em-

-0.05

0

0.05

0.1

0.94

0.95

0.96

0.97

0.98

0.99

1

/4 /2 3 /40 /4 /2 3 /4

Rotation Angle (x,y

)

-0.1

-0.05

0

0.05

0.1

(c)(a)

(b) (d)

FIG. 3. A comparison of the robustness for the X- and Y-axis ro-tation operations. The results of the gate fidelities versus the qubitfrequency drift δ and the deviation ε of the driving amplitude for the(a),(b) geometric and (c),(d) dynamical X- and Y-axis rotation oper-ations, respectively.

phasizing that the time-dependent Ω(t) allows the incorpora-tion of the pulse-shaping technique into the gate construction,which is essential in eliminating various gate errors. However,the fastest gate needs to correspond to a square pulse shape,i.e., Ω(t) needs to be time independent. Without loss of gen-erality, here we present our framework with a time-dependentΩ(t). Then, the time-optimal form of the Hamiltonian H(t)in Eq. (1) can be determined to realize geometric gates, theresulting evolution path under which is shorter than for thecorresponding conventional ones [19], as shown in Fig. 1(a).

III. UNIVERSAL SINGLE-QUBIT GEOMETRIC GATES

We now proceed to presenting our implementation of nona-diabatic GQC with the TOC technique on a 2D square su-perconducting transmon-qubit lattice, as shown in Fig. 2(a),starting from a single transmon qubit, where the computa-tional subspace |0〉, |1〉 consists of the ground and first ex-cited states of the transmon. Conventionally, as shown in Fig.2(b), arbitrary control over the transmon qubit can be realizedby applying a microwave field driven with the time-dependentamplitude Ω(t) and phase φ(t) on its two lowest levels, whichleads to Eq. (1) with a constant detuning ∆. Note that a fixed∆ is more preferable experimentally, as it will not affect thequbit’s coherent properties. To implement universal single-qubit geometric gates with TOC, we set the restrictions on theparameters in Eq. (9) by defining C0 = cot(θ/2). In this way,the geometric X- and Y-axis rotation operations with TOC, de-noted asRx(θx) andRy(θy), for arbitrary angles θx,y ∈ [0, π]

4

0 20 40 60

m/2 (MHz)

0.996

0.997

0.998

0.999

1G

ate

Fide

lity

20 40 60 80

m/2 (MHz)

0 2 4 6 8 10/2 (kHz)

0.9995

0.9996

0.9997

0.9998

0.9999

1

Gat

e Fi

delit

y

FGx

FGy

(b)

(c)

(a)> 99.90% > 99.90%

FIG. 4. The performance of the single-qubit geometric gates. Theresults of the gate fidelities of the (a) Rx(π/2) and (b) Ry(−π/2)geometric operations as functions of the tunable parameter Ωm.

can all be realized by setting

φ1(τx) = −π, θ = θx, φ0 = −π2

;

φ1(τy) = −π, θ = θy, φ0 = 0, (10)

with the same φ1(0) = 0 and the minimum pulse area of

1

2

∫ τx,y

0

Ω(t)dt =π

2/

√1 + cot2 θ

2, (11)

which are all less than π, required for conventional geometricoperations [19, 21]. As an explicit proof, we take the sim-ple pulse Ω(t) = Ωm sin(πt/τ) as an example: the time-acceleration results are shown in Fig. 1(b). In addition, wetest the robustness of our geometric gates by utilizing thegate-fidelity formula F δ,ε = Tr(R†x,yR

δ,εx,y)/Tr(R†x,yRx,y), in

which Rδ,εx,y are the affected rotation operations. Our simula-tion results in Fig. 3 show that, for both the qubit frequencydrift δ and the deviation ε of the driving amplitude in the formof ∆+δΩm and Ω(t)+εΩm, our implemented geometric gatespossess better noise-resilient features than for the correspond-ing dynamical gates (for details, see Appendix B).

In realistic physical implementations, due to the weak an-harmonicity of the transmon qubit, a target driving on qubitstates will also simultaneously stimulate the sequential tran-sitions among the higher excited states, in a dispersive way.Targeting such an obstacle, we also apply the recent theo-retical exploration of derivative removal via an adiabatic gate(DRAG) [43, 44] to suppress this type of leakage error, to ob-tain the precise qubit manipulation (for details, see AppendixC). To further analyze the performance of the single-qubittime-optimal geometric gates, we take the geometric opera-tionsRx(π/2) andRy(−π/2) as two typical examples. In oursimulation, from the state-of-art experiment [9], we choosethe relaxation and dephasing rates of the transmon to be iden-tical, as κ = κ1

− = κ1z = 2π × 4 kHz, and the anharmonicity

to be α1 = 2π × 220 MHz ( for the details of the simulation,see Appendix D). In Figs. 4(a) and 4(b), we plot the gate fi-delities ofRx(π/2) andRy(−π/2) as functions of tunable pa-rameters, where we find that, when Ωm = 2π × 45 MHz withthe corresponding restricted detuning parameter ∆ ≈ 2π×69MHz forRx(π/2), and Ωm = 2π×40 MHz with ∆ ≈ 2π×11MHz for Ry(−π/2), the fidelities of these two gates are both

close to 99.98%, which compares favorably with the best per-formance of the reported experiments for the same type ofgate [9].

IV. NONTRIVIAL TWO-QUBIT GEOMETRIC GATE

We next work on implementing the nontrivial two-qubit ge-ometric gate with the TOC technique on the 2D square super-conducting transmon-qubit lattice in Fig. 2(a). For the capaci-tively coupled qubit lattice, the coupling strength of two adja-cent transmons, e.g., T1 and T2 in the same row (or T1 and T2′

in the same column) is fixed. Meanwhile, the frequency dif-ference of the two adjacent transmons, ∆1 = ω2 − ω1, is alsogenerally hard to adjust, so that it is difficult to achieve reso-nant coupling and/or off-resonant coupling without changinga qubit frequency to deviate from its optimal working point.To deal with these difficulties, here we introduce an additionalqubit-frequency driving for the transmon T1, which can beexperimentally realized by a longitudinal driving field, in theform of ε(t) = F (t) [38], where F (t) = β sin[νt + ϕ(t)],with ν and ϕ(t) indicating the frequency and phase of thelongitudinal field, respectively: the circuit details are shownin Fig. 2(c). Moving to the interaction picture, the couplingHamiltonian reads

H12(t) = g12

|01〉

12〈10|ei∆1t +

√2|11〉

12〈20|ei(∆1+α1)t

+√

2|02〉12〈11|ei(∆1−α2)t

e−iβ sin[νt+ϕ(t)]

+H.c., (12)

where g12

is the coupling strength between transmons T1

and T2 and αj is the intrinsic anharmonicity of the trans-mon Tj . Utilizing the Jacobi-Anger identity and thenneglecting the high-order oscillating terms, we find thatthe parametrically tunable coupling in the single-excitationsubspace |01〉

12, |10〉

12 and the two-excitation subspace

|02〉12, |11〉

12, |20〉

12 can all be selectively addressed by ad-

justing the frequency ν. The corresponding energy spectraof these two capacitively coupled transmons, T1 and T2, areshown in Fig. 2(d).

However, the use of the interaction of high-excitation sub-spaces tends to cause more decoherence factors than that ofthe single-excitation subspace. Thus, here, we purposely pickthe interactions of the single-excitation subspace. By settingthe frequency ν to satisfy ∆t = ν−∆1, with |∆t| ∆1, ν,and applying the unitary transformation, we can obtain the ef-fective Hamiltonian as

Ht(t) =1

2

(−∆t g′

12e−iϕ(t)

g′12eiϕ(t) ∆t

), (13)

in the single-excitation subspace |01〉12, |10〉12, where theeffective coupling strength g′

12= 2J1(β)g

12, in which J1(β)

is a Bessel function of the first kind. As for the leakagefrom the computational basis |11〉12 to the higher excitationsubspaces, we can further restrain it by optimizing the sys-tem parameters. In the same way, in the above equivalenttwo-level system, to achieve integration with the TOC tech-nique our restricted result in Eq. (9) is denoted by ϕ(t) =

5

1.26 1.28 1.3 1.32 1.34316

318

320

322

324

326

1/2 (

MH

z)

0.998

0.99

8

0.998

0.998

0.998

0.992

0.994

0.996

0.998

1

> 99.80%

FIG. 5. The performance of the implemented time-optimal two-qubitgeometric gate in terms of the gate fidelities as functions of tunableparameters.

ϕ0 + ϕ1(t), based on the effective square-pulse shape g′12

,in which ϕ1(t) = η is a constant that depends only on thetype of target gate. Thus, within the two-qubit subspace|00〉12, |01〉12, |10〉12, |11〉12, the final evolution operator

U2(T ) =

1 0 0 00 − cos ϑ2 sin ϑ

2 e−iϕ0 0

0 − sin ϑ2 eiϕ0 − cos ϑ2 0

0 0 0 1

, (14)

with the minimal time cost of T = T0/[2√

1 + cot2(ϑ/2)]

can be obtained by determining ϕ1(T ) = −π. At this point,a nontrivial two-qubit time-optimal geometric gate can be re-alized. Obviously, the gate time T is also faster than the cor-responding conventional geometric operation, the time beingT0 = 2π/g′

12[19, 21]. We next take the two-qubit geometric

gate with ϑ = π/2 and ϕ0 = π/2 as an typical example tofully evaluate its gate performance (for details of the simula-tion, see Appendix D). Realistically, we choose the couplingstrength of the two adjacent transmons as g

12= 2π× 8 MHz,

the anharmonicity of the second transmon as α2 = 2π × 180MHz, and the relaxation and dephasing rates of the transmonto be identical, as κ = κ1

− = κ1z = κ2

− = κ2z = 2π × 4

kHz. However, from the energy spectra of the two capaci-tively coupled transmons T1 and T2 as shown in Fig. 2(d), wefind that the effect of leakage into the noncomputational sub-space |02〉12, |20〉12 cannot be negligible completely [11].To avoid this type of leakage error as much as possible, it isnecessary to optimize the qubit parameters to obtain a param-eter range in which a high-fidelity two-qubit geometric gatecan be achieved. As shown in Fig. 5, we find an ellipticalregime, within which a two-qubit gate with a fidelity higherthan 99.80% can be realized. In particular, the numerical re-sults show that when ∆1 = 2π × 320 MHz, ν = 2π × 340MHz, and β ' 1.3, our two-qubit geometric gate fidelity canbe as high as 99.84%, which compares favorably with the bestperformance of the currently reported experiments.

V. DISCUSSION AND CONCLUSION

In summary, we proposed a practical implementation ofGQC in a simple experimental setup, which only utilizesan experimentally accessible two-body interaction and avoidsthe introduction of additional auxiliary energy levels beyondthe qubit states and additional auxiliary coupling elements.Meanwhile, our scheme is robust against the main gate errorsources and is less affected by the decoherence effect. As therequired interaction in Eq. (13) is the same as that for thesingle-excitation subspace of exchange-coupled spin systems,our scheme can be readily extended to these systems, e.g.,quantum dots, cavity QED, trapped ions, etc. Therefore, ourimplementation uses only existing experimental technologiesto remedy the main drawbacks of GQC, leading to an ultra-high gate fidelity that compares favorably with the best per-formance of the current reported experiments and thus makesit a promising strategy on the route towards robust and scal-able solid-state QC.

ACKNOWLEDGMENTS

We thank B.-J. Liu for helpful discussions. This work wassupported by the Key-Area Research and Development Pro-gram of GuangDong Province (Grant No. 2018B030326001),the National Natural Science Foundation of China (GrantNo. 11874156), the National Key R&D Program of China(Grant No. 2016YFA0301803), Science and Technology Pro-gram of Guangzhou (Grant No. 2019050001), and the In-novation Project of Graduate School of SCNU (Grant No.2019LKXM006).

Appendix A: Derivation of the Global geometric phase γg

Let us first consider the dressed state |ψ+(t)〉 to exam-ine its geometric nature of the evolution process. For ournoncyclic evolution path C1, the total relative phase betweenthe initial state |ψ+(0)〉 and the final state U(τ)|ψ+(0)〉 (i.e.,eiγ |ψ+(τ)〉) can be defined as γt=arg〈ψ+(0)|U(τ)|ψ+(0)〉.Thus, following Refs. [45–48], the geometric Pancharatnamphase in path C1 can be determined as

γp(C1) =γt − γd=arg〈ψ+(0)|U(τ)|ψ+(0)〉

+

∫ τ

0

〈ψ+(t)|H(t)|ψ+(t)〉dt, (A1)

where γd is the corresponding dynamical phase. In addi-tion, under a gauge transformation |ψ+(t)〉 → |ψ′+(t)〉 =

6

eiς(t)|ψ+(t)〉, we have

γ′p(C1) =arg〈ψ′+(0)|U(τ)|ψ′+(0)〉

+

∫ τ

0

〈ψ+(t)|H(t)|ψ+(t)〉dt

=arg〈ψ+(0)|U(τ)|ψ+(0)〉+ ς(τ)− ς(0)

+

∫ τ

0

〈ψ+(t)| (H(t)− ς(t)) |ψ+(t)〉dt

=γp(C1); (A2)

i.e., the Pancharatnam phase γp(C1) is gauge invariant. Fol-lowing the geodesic rule [45, 48], for the noncyclic geometricevolution, it is assumed that there is a geodesic line that con-nects the initial [χ(0), ξ(0)] and final [χ(τ), ξ(τ)] points ofthe actual evolution path, so that the actual evolution path C1

plus the geodesic lineC2 can form a closed pathC. Accordingto the similar formula above, we can solve for the accumulatedPancharatnam phase γp(C2) under the geodesic line, which isalso gauge invariant. Thus, the overall Pancharatnam phase is

γp(C) = γp(C1)+γp(C2)=1

2

∫C

sinχdχdξ, (A3)

which just represents half of the solid angle enclosed by theevolution path and the geodesic line. As dξ = 0 on the addi-tional geodesic line C2, the result of γp(C) will reduce to

γp(C) = −1

2

∫ τ

0

ξ(t) [1− cosχ(t)] dt = γg

which is the global geometric phase in Eq. (6) of the maintext; thus it also has the geometric nature. A similar discussionis applicable for the state |ψ−(t)〉.

Appendix B: Comparison of gate robustness

The pursuit of high-fidelity and strongly robust quantumgates in the previous geometric schemes is usually circum-scribed by complex interactions among multiple levels orqubits and the longer time required than for the dynamicalcounterpart. Therefore, it is desirable to prove that our geo-metric gate has a stronger robustness than the dynamical one,which can be realized only by changing φ(t) as a constant,i.e., φ(t) = φ0, to ensure the geometric phase γg = 0. Inparticular, to ensure the fairness of our gate-robustness com-parison, here we define the pulse shape of Ω(t) to be the sameas that of geometric gates. The resulting dynamical evolutionoperator can be obtained as

Ud(λ, θd, φ0)

= cosλ

2− i sin

λ

2

(− cos θd2 sin θd

2 e−iφ0

sin θd2 e

iφ0 cos θd2

), (B1)

with λ =∫ τ

0

√Ω2(t) + ∆2(t)dt and θd =

2 tan−1[Ω(t)/∆(t)]. In this way, arbitrary dynamicalX- and Y-axis rotation operations can be all realized byUd(π, 2π, 0)Ud(π, θx,−π/2) and Ud(π, 2π, 0)Ud(π, θy, 0).Unlike the implementation of our geometric gate, in the

dynamical case, one needs to restrict the phase variable φ(t)to be time independent in order to realize a target gate: thusthere is no additional degree of freedom to combine withTOC.

Appendix C: DRAG correction

In the realistic physical implementation, due to the weakanharmonicity of transmons, when our target microwave fieldis applied to the two lowest levels of transmon, it will alsosimultaneously induce the sequential transitions among thehigher excited states, resulting in the leakage error. Thus,to achieve independent manipulation of qubit states, we ap-ply the recent theoretical exploration of DRAG [43, 44] tosuppress this type of leakage error. Here, we consider the in-fluence of the third energy level, which is the main leakagesource of our qubit states. To this end, the Hamiltonian de-scribing a single-qubit system can be written as

H1(t) =1

2B(t) · S− α1|2〉〈2|, (C1)

where α1 is the intrinsic anharmonicity of the target transmon,B(t) = B0(t) + Bd(t) is the vector of the total microwavefield including the original and additional DRAG-correctingmicrowave fields, i.e.,

B0(t) = (Bx, By, Bz)

= (Ω(t) cos(φ0 + φ1(t)),Ω(t) sin(φ0 + φ1(t)),−∆),

Bd(t) = (Bd;x, Bd;y, Bd;z)

= − 1

2α1(−By +BzBx, Bx +BzBy, 0),

respectively, and the operator vector S is given by

Sx =∑m=0,1

√m+ 1(|m+ 1〉〈m|+ |m〉〈m+ 1|),

Sy =∑m=0,1

√m+ 1(i|m+ 1〉〈m| − i|m〉〈m+ 1|),

Sz =∑

m=0,1,2

(1− 2m)|m〉〈m|.

Meanwhile, through numerical simulation, we find that, forall the implemented single-qubit geometric gates, their infi-delities caused by leakage to the third energy level are all lessthan 0.01%, which is almost negligible, thus confirming thatit is feasible for the DRAG correction in our simulation.

Appendix D: Master-equation simulation

In practical physical implementations, the performance ofour implemented geometric gate is inevitably limited by thedecoherence effect of the target qubit system. In addition,to quantitatively evaluate the validity of the final effectiveHamiltonian, all of our simulations hereafter are based on theoriginal interaction Hamiltonian without any approximation.Therefore, here we consider the effects of decoherence and

7

the high-order oscillating terms. The quantum dynamics canbe simulated by the Lindblad master equation,

ρn

= −i[Hd(t), ρn] +

n∑i=1

κi−2

L (|0〉i〈1|+√

2|1〉i〈2|)

+κiz2

L (|1〉i〈1|+ 2|2〉i〈2|), (D1)

where ρn

is the density matrix of the quantum system un-der consideration, L (A) = 2Aρ

nA† − A†Aρ

n− ρ

nA†A

is the Lindblad operator for operator A, and κi−, κiz arethe relaxation and dephasing rates of the ith transmon, re-spectively. For the cases of a single qubit and two coupledqubits, the forms of the driving Hamiltonians are expressed asHd(t) = H1(t) with n = 1 and Hd(t) = H12(t) with n = 2,respectively.

To fully evaluate our implemented geometric gates, for the

general initial state of the single qubit, |ψ1〉 = cos θ1|0〉 +sin θ1|1〉, in which |ψfk=x,y

〉 = U(τk)|ψ1〉 is the ideal finalstate, we can define the single-qubit gate fidelity as FGk =1

2π

∫ 2π

0〈ψfk |ρ1|ψfk〉dθ1, where the integration is done nu-

merically for 1001 input states, with θ1 being uniformly dis-tributed within [0, 2π], and ρ1 is the numerically simulateddensity matrix of the qubit system. Furthermore, in the sameway, in the two-qubit case, for a general initial state of thetwo qubits according to |ψ2〉 = (cosϑ1|0〉 + sinϑ1|1〉) ⊗(cosϑ2|0〉 + sinϑ2|1〉), in which |ψfU2

〉 = U2(T )|ψ2〉 is theideal final state, we can also define the two-qubit gate fidelityas

FGU2=

1

4π2

∫ 2π

0

∫ 2π

0

〈ψfU2|ρ2|ψfU2

〉dϑ1dϑ2, (D2)

with the integration done numerically for 10001 input states,with ϑ1 and ϑ2 uniformly distributed over [0, 2π].

[1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press, Cam-bridge, 2000).

[2] J. I. Cirac and P. Zoller, Quantum computations with coldtrapped ions, Phys. Rev. Lett. 74, 4091 (1995).

[3] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J.Kimble, Measurement of Conditional Phase Shifts for QuantumLogic, Phys. Rev. Lett. 75, 4710 (1995).

[4] G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch,Quantum Logic Gates in Optical Lattices, Phys. Rev. Lett. 82,1060 (1999).

[5] D. Jaksch, H.-J. Briegel, J. I. Cirac, C.W. Gardiner, and P.Zoller, Entanglement of atoms via cold controlled collisions,Phys. Rev. Lett. 82, 1975 (1999).

[6] Y. Makhlin, G. Schon, and A. Shnirman, Josephson-junctionqubits with controlled couplings, Nature (London) 398, 305(1999).

[7] Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Coherent controlof macroscopic quantum states in a single-Cooper-pair box, Na-ture (London) 398, 786 (1999).

[8] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van derWal, and S. Lloyd, Josephson persistent-current qubit, Science285, 1036 (1999).

[9] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey,T. C. White, J. Mutus, A. G. Fowler, B. Campbell et al., Super-conducting quantum circuits at the surface code threshold forfault tolerance, Nature (London) 508, 500 (2014).

[10] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V.Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K. Aryaet al., A blueprint for demonstrating quantum supremacy withsuperconducting qubits, Science 360, 195 (2018).

[11] M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroat-modjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. daSilva et al., Demonstration of universal parametric entanglinggates on a multi-qubit lattice, Sci. Adv. 4, eaao3603 (2018).

[12] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R.Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buellet al., Quantum supremacy using a programmable supercon-ducting processor, Nature (London) 574, 505 (2019).

[13] P. Zanardi and M. Rasetti, Holonomic quantum computation,

Phys. Lett. A 264, 94 (1999).[14] J. Pachos, P. Zanardi, and M. Rasetti, Non-Abelian Berry con-

nections for quantum computation, Phys. Rev. A 61, 010305(1999).

[15] L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric manipulationof trapped ions for quantum computation, Science 292, 1695(2001).

[16] W. Xiang-Bin and M. Keiji, Nonadiabatic conditional geomet-ric phase shift with NMR, Phys. Rev. Lett. 87, 097901 (2001).

[17] S.-L. Zhu and Z. D. Wang, Implementation of universal quan-tum gates based on nonadiabatic geometric phases, Phys. Rev.Lett. 89, 097902 (2002).

[18] S. L. Zhu and Z. D. Wang, Unconventional geometric quantumcomputation, Phys. Rev. Lett. 91, 187902 (2003).

[19] P. Z. Zhao, X. D. Cui, G. F. Xu, E. Sjoqvist, and D. M. Tong,Rydberg-atom-based scheme of nonadiabatic geometric quan-tum computation, Phys. Rev. A 96, 052316 (2017).

[20] T. Chen and Z.-Y. Xue, Nonadiabatic geometric quantum com-putation with parametrically coupled transmons, Phys. Rev.Appl. 10, 054051 (2018)

[21] E. Sjoqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M.Johansson, and K. Singh, Non-adiabatic holonomic quantumcomputation, New J. Phys. 14, 103035 (2012).

[22] G. F. Xu, J. Zhang, D. M. Tong, E. Sjoqvist, and L. C. Kwek,Nonadiabatic holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. 109, 170501 (2012).

[23] B.-J. Liu, X.-K. Song, Z.-Y. Xue, X. Wang, and M.-H.Yung,Plug-and-Play Approach to Nonadiabatic Geometric Quantumgates, Phys. Rev. Lett. 123, 100501 (2019).

[24] G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral,Detection of geometric phases in superconducting nanocircuits,Nature (London) 407, 355 (2000).

[25] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J.Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband,and D. J. Wineland, Experimental demonstration of a robust,high-fidelity geometric two ion-qubit phase gate, Nature (Lon-don) 422, 412 (2003).

[26] J. Du, P. Zou, and Z. D. Wang, Experimental implementation ofhigh-fidelity unconventional geometric quantum gates using anNMR interferometer, Phys. Rev. A 74, 020302(R) (2006).

8

[27] A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal,S. Berger, A. Wallraff, and S. Filipp, Experimental realizationof non-Abelian non-adiabatic geometric gates, Nature (Lon-don) 496, 482 (2013).

[28] G. Feng, G. Xu, and G. Long, Experimental realization of nona-diabatic holonomic quantum computation, Phys. Rev. Lett. 110,190501 (2013).

[29] C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang,and L.-M. Duan, Experimental realization of universal geomet-ric quantum gates with solid-state spins, Nature (London) 514,72 (2014).

[30] Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y. P.Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, Single-Loop Realiza-tion of Arbitrary Nonadiabatic Holonomic Single-Qubit Quan-tum Gates in a Superconducting Circuit, Phys. Rev. Lett. 121,110501 (2018).

[31] T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z.Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, Ex-perimental realization of nonadiabatic shortcut to non-Abeliangeometric gates, Phys. Rev. Lett. 122, 080501 (2019).

[32] Z. Zhu, T. Chen, X. Yang, J. Bian, Z.-Y. Xue, and X.Peng, Single-loop and composite-loop realization of nonadia-batic holonomic quantum gates in a decoherence-free subspace,Phys. Rev. Appl. 12, 024024 (2019).

[33] Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y.Ma, H. Wang, Y. Song, Z.-Y. Xue, and L. Sun, Experimen-tal Implementation of Universal Nonadiabatic Geometric Quan-tum Gates in a Superconducting Circuit, Phys. Rev. Lett. 124,230503 (2020).

[34] X. Wang, M. Allegra, K. Jacobs, S. Lloyd, C. Lupo, and M.Mohseni, Quantum brachistochrone curves as geodesics: Ob-taining accurate minimum-time protocols for the control ofquantum systems, Phys. Rev. Lett. 114, 170501 (2015).

[35] J. Geng, Y. Wu, X. Wang, K. Xu, F. Shi, Y. Xie, X. Rong, and J.Du, Experimental time-optimal universal control of spin qubitsin solids, Phys. Rev. Lett. 117, 170501 (2016).

[36] X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. P.Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, Perfect quantum statetransfer in a superconducting qubit chain with parametricallytunable couplings, Phys. Rev. Appl. 10, 054009 (2018).

[37] W. Cai, J. Han, F. Mei, Y. Xu, Y. Ma, X. Li, H. Wang, Y. P.Song, Z.-Y. Xue, Z.-q. Yin, S. Jia, and L. Sun, Observation oftopological magnon insulator states in a superconducting cir-cuit, Phys. Rev. Lett. 123, 080501 (2019).

[38] J. Chu, D. Li, X. Yang, S. Song, Z. Han, Z. Yang, Y. Dong, W.Zheng, Z. Wang, X. Yu, D. Lan, X. Tan, and Y. Yu, Realizationof Superadiabatic Two-Qubit Gates Using Parametric Modula-tion in Superconducting Circuits, Phys. Rev. Appl. 13, 064012(2020).

[39] H. R. Lewis and W. B. Riesenfeld, An exact quantum theory ofthe time-dependent harmonic oscillator and of a charged par-ticle in a time-dependent electromagnetic field, J. Math. Phys.10, 1458 (1969).

[40] X. Chen, E. Torrontegui, and J. G. Muga, Lewis-Riesenfeld in-variants and transitionless quantum driving, Phys. Rev. A 83,062116 (2011).

[41] A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, Optimallyrobust shortcuts to population inversion in two-level quantumsystems, New J. Phys. 14, 093040 (2012).

[42] A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Time-optimalunitary operations, Phys. Rev. A 75, 062116 (2011).

[43] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.Wilhelm,Simple Pulses for Elimination of Leakage in Weakly NonlinearQubits, Phys. Rev. Lett. 103, 110501 (2009).

[44] T. Wang, Z. Zhang, L. Xiang, Z. Jia, P. Duan, W. Cai, Z. Gong,Z. Zong, M. Wu, J. Wu, L. Sun, Y. Yin, and G. Guo, The exper-imental realization of high-fidelity “shortcut-to-adiabaticity”quantum gates in a superconducting Xmon qubit, New J. Phys.20, 065003 (2018).

[45] J. Samuel and R. Bhandari, General Setting for Berry’s Phase,Phys. Rev. Lett. 60, 2339 (1988).

[46] S. L. Zhu, Z. D. Wang, and Y. D. Zhang, Nonadiabatic non-cyclic geometric phase of a spin- 1

2particle subject to an arbi-

trary magnetic field, Phys. Rev. B 61, 1142 (2000).[47] A. Friedenauer and E. Sjoqvist, Noncyclic geometric quantum

computation, Phys. Rev. A 67, 024303 (2003).[48] F. De Zela, The Pancharatnam-Berry Phase: Theoretical and

Experimental Aspects, in Theoretical Concepts of QuantumMechanics, edited by M. R. Pahlavani (InTech, London, 2012).