High-fidelity geometric quantum gates with short paths on superconducting circuits
HHigh-fidelity geometric quantum gates with short paths on superconducting circuits
Sai Li, ∗ Jing Xue, ∗ Tao Chen, and Zheng-Yuan Xue
1, 2, † Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,and School of Physicsand Telecommunication Engineering, South China Normal University, Guangzhou 510006, China Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China (Dated: February 9, 2021)Geometric phases are robust against certain types of local noises, and thus provide a promising way towardshigh-fidelity quantum gates. However, comparing with the dynamical ones, previous implementations of nona-diabatic geometric quantum gates usually require longer evolution time, due to the needed longer evolutionpath. Here, we propose a scheme to realize nonadiabatic geometric quantum gates with short paths based onsimple pulse control techniques, instead of deliberated pulse control in previous investigations, which can thusfurther suppress the influence from the environment induced noises. Specifically, we illustrate the idea on asuperconducting quantum circuit, which is one of the most promising platforms for realizing practical quantumcomputer. As the current scheme shortens the geometric evolution path, we can obtain ultra-high gate fidelity,especially for the two-qubit gate case, as verified by our numerical simulation. Therefore, our protocol suggestsa promising way towards high-fidelity and roust quantum computation on a solid-state quantum system.
Keywords: Nonadiabatic geometric phases, short path, quantum gates, superconducting circuits
I. INTRODUCTION
Quantum computation is based on a universal set of quan-tum gates [1], including arbitrary single-qubit gates and a non-trivial two-qubit gate. Recently, superconducting circuits sys-tem has shown its unique merits of high designability andscalability [2], which represents one of the promising plat-forms for realizing quantum computer. First, a superconduct-ing transmon device [3] is easily addressed as a two-level sys-tem, i.e., the ground and first-excited states {| (cid:105) , | (cid:105)} servingas qubit states, which can be operated by a driving microwavefield. Second, due to the circuit nature of superconductingqubits, they can be directly coupled by capacitance or induc-tance [4–6], for nontrivial two-qubit gate operations.On the other hand, in quantum computation, the realizedquantum gates are preferred to be robust against the intrinsicerrors as well as manipulation imperfections of the quantumsystems. To achieve this, realization of quantum gates usingthe well-known geometric phases [7–9] is one of the promis-ing strategies. This is because geometric phases are only re-lied on the global property of the evolution path in a quantumsystem and not sensitive to the evolution details, thus providesa promising way towards robust quantum gates against somelocal control errors [10–15].Due to the superiority of geometric phases, geometric quan-tum computation (GQC) [16] and holonomic quantum com-putation (HQC) [17, 18] have been proposed based on theadiabatic evolution. In adiabatic GQC and HQC require thatthe evolution process in inducing the geometric phases to beslowly enough so that the nonadiabatic transition among evo-lution states are greatly suppressed. Therefore, this adiabaticcondition results in long evolution time for desired geomet-ric manipulations, where qubit states will be considerably in-fluenced by environment noises, and thus difficult to realize ∗ These authors contributed equally to this work. † [email protected] high-fidelity geometric gates. To remove this main obstacle,nonadiabatic GQC [19, 20] and HQC [21, 22] were then pro-posed with fast evolutions. These kinds of geometric gatesshare both merits of geometric robustness and rapid evolution,and thus has attracted much attention. Theoretically, nonadia-batic GQC [23–26] and HQC [27–42] have been extensive ex-plored. Meanwhile, nonadiabatic GQC [15, 43–46] and HQC[47–58] have also also been experimentally demonstrated indifferent quantum systems.Comparing with nonadiabatic HQC realized with multi-level quantum systems, nonadiabatic GQC (NGQC) can onlyinvolve two-level systems, and thus is more easily realizedand controlled. However, in previous NGQC schemes, e.g.,orange-slice-shaped loops [23–25], to null dynamical phasesfrom the total phases, they usually require unnecessary longgate-time due to the fact that the evolution states need tosatisfy the cyclic evolution and parallel transport conditions.Recently, new conventional/unconventional NGQC schemeswith short evolution paths have been proposed [59–62], withdeliberated and correlated time-dependent parameter controlof the govern Hamiltonian, i.e., these schemes need complexpulse control [63].Here, to shorten unnecessary long evolution time of previ-ous NGQC without complex pulse control, we propose an ap-proach to realize universal nonadiabatic geometric quantumgates with short path based on simple pulse control, whichis experimentally preferred, where short evolution path corre-sponds to short evolution time under the same driving pulsecondition. Comparing with previous NGQC schemes, ourscheme extends the experimental feasible geometric quantumgates to the case with shorter gate-time, and thus leads tohigh-fidelity quantum gates as it can reduce the influence fromenvironment-induced decoherence.Moreover, we illustrate our approach by the realizationof arbitrary single-qubit and non-trivial two-qubit geometricgates on superconducting quantum circuits system, which isone of the promising platforms for realizing practical quan-tum computer. Finally, as our scheme is based on simple setup a r X i v : . [ qu a n t - ph ] F e b (b) (a) FIG. 1. Illustration of the implementation of single-qubit geometricquantum gates. (a) The two lowest energy levels of a superconduct-ing transmon qubit can be geometrically manipulated by a drivenmicrowave field. (b) Geometric illustration of the half-orange-slice-shaped evolution path on a Bloch sphere to induce the geometricphase − γ/ on | µ + (cid:105) . and exchange-only interactions, it is also suitable for manyother quantum systems that are potential candidates for phys-ical implementation of quantum computation. Therefore, ourscheme suggests a promising way towards high-fidelity androust quantum computation on solid-state quantum systems. II. SHORT PATH GEOMETRIC GATES
In this section, we present our general approach to realizenonadiabatic geometric quantum gates with short path basedon simple pulse control. We consider a general setup with aqubit is driven by a classical microwave field, as shown in Fig.1(a). Assuming ¯ h = 1 hereafter, in the interaction picture andwithin the computational subspace {| (cid:105) , | (cid:105)} , the interactionHamiltonian of this system is H ( t ) = 12 (cid:18) ∆ Ω e − iη Ω e iη − ∆ (cid:19) (1)where Ω and η are the amplitude and phase of the drivingmicrowave field, ∆ is a frequency difference between the mi-crowave field and the qubit. Here, different with previous in-vestigations, we choose an evolution path as shown in Fig.1(b) to induce our target geometric phases. During this cyclicevolution, which is divided into four parts to acquire a puregeometric phase at final time T , parameters of the Hamilto-nian is chosen as follows, (cid:90) T Ω dt = π − θ, ∆ = 0 , η = φ + π , t ∈ [0 , T ] , Ω = 0 , (cid:90) T T ∆ dt = γ, η = φ + γ − π , t ∈ [ T , T ] , (cid:90) T T Ω dt = π , ∆ = 0 , η = φ + γ − π , t ∈ [ T , T ] , (cid:90) TT Ω dt = θ, ∆ = 0 , η = φ + π , t ∈ [ T , T ] , (2)where parameters θ , φ and γ in each part are easily controlledby external microwave field, and usually the detuning ∆ is chosen as a fixed constant for easily experimental control pur-pose. Then, at the end of the evolution, the evolution operatorcan be obtained as U ( T )= U ( T, T ) U ( T , T ) U ( T , T ) U ( T , γ − i sin γ (cid:18) cos θ sin θe − iφ sin θe iφ − cos θ (cid:19) = e − i γ n · σ , (3)which represents rotation operations around axis n =(sin θ cos φ, sin θ sin φ, cos θ ) with an angle γ , where σ =( σ x , σ y , σ z ) are the Pauli matrixes. Due to phase γ/ ex-actly corresponding to half of the solid angle formed by theevolution path, the evolution operator along this path is nat-urally induced in a geometric way. Obviously, this evolutionpath in our approach is half-orange-slice-shaped loops, whichis shorter than previous nonadiabatic GQC with orange-slice-shaped loops. Meanwhile, during the whole evolution pro-cess, the Hamiltonian parameters are only required to meetthe conditions as prescribed in Eq. (2), which can be in arbi-trary shape and thus can be chosen as experimental-friendlyas possible. That is, our scheme is based on simple and easypulse control over the external microwave field, instead of de-liberately complex time modulation in previous schemes.To clearly show the geometric nature of the evolution oper-ator, we give further demonstration of the evolution operator U ( T ) as a geometric gate. Here, the two-dimensional orthog-onal eigenstates | µ + (cid:105) = cos θ | (cid:105) + sin θ e iφ | (cid:105) , | µ − (cid:105) = sin θ e − iφ | (cid:105) − cos θ | (cid:105) (4)of n · σ are selected as our evolution states in dressed repre-sentation, after the cyclical evolution, the evolution operatorcan be expressed in dressed basis {| µ + (cid:105) , | µ − (cid:105)} as U ( T ) = e − i γ | µ + (cid:105)(cid:104) µ + | + e i γ | µ − (cid:105)(cid:104) µ − | . (5)It is clear that the orthogonal eigenstates | µ + (cid:105) and | µ − (cid:105) strictly meet cyclic evolution condition as | µ m ( T ) (cid:105) = U ( T ) | µ m (cid:105) = | µ m (cid:105) (6)with ( m = + , − ) . Meanwhile, the parallel transport conditionis also satisfied during the cyclic evolution process as (cid:104) µ m | U † ( t ) H ( t ) U ( t ) | µ m (cid:105) = 0 , (7)which also means that there is no dynamical phases accumu-lated on the orthogonal eigenstates | µ m (cid:105) . Therefore, after go-ing through a half-orange-slice-shaped loop, as shown in Fig.1(b), at the end of the evolution, only pure geometric phasesare obtained on the orthogonal eigenstates | µ m (cid:105) without anydynamic phases. In other words, the total phase is equal to thegeometric phase. Thus, arbitrary geometric manipulation canbe achieved. t/T / max / max / t/T |0|1|2F R X ( /2) t/T -0.500.51 / max / max / t/T |0|1|2F R Z ( /4) (b)(a)(c) (d) FIG. 2. Performance of nonadiabatic single-qubit geometric gates.(a) and (c) The shape of parameters Ω , ∆ and φ for the R x ( π/ and R z ( π/ gates, respectively. (b) and (d) The qubit-state populationsand the state-fidelity dynamics of the R x ( π/ and R z ( π/ gateoperations, respectively. III. IMPLEMENTATION ON SUPERCONDUCTINGCIRCUITS
In this section, we present the implementation of ourshort-path NGQC (SNGQC) on superconducting quantum cir-cuits system. We first illustrate the single-qubit implemen-tation with numerically demonstration of the gate perfor-mance. By faithful numerical simulations, we also compareour approach against environment-induce decoherence withprevious NGQC under the same maximum driving amplitude.Moreover, we compare our approach against different errorswith previous NGQC under the same maximum driving am-plitude. Then, we proceed to the realization of nontrivialnonadiabatic geometric two-qubit gates based on two capaci-tively coupled transmon qubits, with numerical verifications.
A. Single-qubit gates
Now, we proceed to the implementation of the nonadiabaticgeometric single-qubit gates on superconducting circuits. Ina superconducting transmon device [3], the ground and first-excited states {| (cid:105) , | (cid:105)} serve as a qubit. A detuned classicalmicrowave field driving, as in Fig. 1(a), leads the interactingHamiltonian as in Eq. (1). Choosing the Hamiltonian param-eter according to Eq. (2), one can obtain the geometric quan-tum gate as in Eq. (3), which is an arbitrary rotation alongthe direction set by the parameters θ , φ and γ . Thus, arbitrarygeometric single-qubit gates are achieved.Then, we numerically demonstrate the performance of ourgate performance with faithful numerical simulation. Atfirst, the influence of environment-induced decoherence in thepractical superconducting circuits systems is a non-negligiblefactor to evaluate gate performance, which is key error inquantum systems, thus, decoherence must be considered for /2 (kHz)0.9970.99750.9980.99850.9990.99951 G a t e F i d e lit y F GR Z ( /4) F GR X ( /2) F GR PZ ( /4) F GR PX ( /2) FIG. 3. Gate fidelities with respect to the different uniform decoher-ence rate κ for the R x ( π/ and R z ( π/ gates of our approach andthe R Px ( π/ and R Pz ( π/ gates of previous NGQC under the samemaximum amplitude. faithful simulation. Meanwhile, as a superconducting trans-mon device only has weak anharmonicity, when microwavefield is added to drive the transmon qubit with states {| (cid:105) , | (cid:105)} ,it also drives states {| (cid:105) , | (cid:105)} with | (cid:105) being second-excitedstate, this can cause leakage error from the qubit basis tosecond-excited state | (cid:105) . Therefore, the DRAG correction[64, 65] must be introduced to suppress the leakage error be-yond the qubit basis. Overall, considering both the decoher-ence and the leakage term, we introduce the Lindblad masterequation as ˙ ρ = i [ ρ , H ( t ) + H L ( t )] + [ κ L (Λ ) + κ L (Λ )] , (8)with leakage term as H L ( t ) = ( − α − ∆2 ) | (cid:105)(cid:104) | + [ Ω √ e iφ | (cid:105)(cid:104) | + H . c . ] , (9)where all the unwanted imperfections are considered by us-ing Hamiltonian H ( t ) + H L ( t ) to faithfully evaluate the gateperformance, ρ represents the density matrix for the consid-ered system and L ( O ) = O ρ O † − O † O ρ / − ρ O † O / is the Lindblad operator of O with Λ = | (cid:105)(cid:104) | + √ | (cid:105)(cid:104) | and Λ = | (cid:105)(cid:104) | + 2 | (cid:105)(cid:104) | , and κ and κ are the decay anddephasing rates of the transmon qubit, respectively.Here, to achieve faithful simulation, we set all parametersof the transmon qubit being easily accessible with current ex-perimental technologies [2], including decay and dephasingrates as κ = κ = 2 π × kHz, the weak anharmonicity of thetransmon as α = 2 π × MHz, the maximum amplitude as Ω max = 2 π × MHz, and the frequency difference betweenmicrowave field and qubit as ∆ = 2 π × MHz. Meanwhile,due to the weak anharmonicity α of the transmon qubit, weneed to use DRAG correction to suppress the leakage error inorder to realize ultra-high gate fidelity, thus, we set the sim-ple form of the driving amplitude as Ω( t ) = Ω max sin ( πt/τ ) ,where τ is the duration in each part.Then, we choose two geometric single-qubit quantum gates R x ( π/ and R z ( π/ as typical examples, with gate pa-rameters γ = π/ , θ = π/ , and φ = 0 for R x ( π/ gate and γ = π/ , θ = 0 , and φ = 0 for R z ( π/ gate. G a t e F i d e lit y SNGQCNGQC A NGQC B G a t e F i d e lit y (b)(d)(c) (a) SNGQCNGQC A NGQC B SNGQCNGQC A NGQC B SNGQCNGQC A NGQC B -0.1 -0.05 0.10 0.05-0.1 -0.05 0.10 0.05 FIG. 4. Gate robustness comparison. (a) and (c) respectively com-paring the gate robustness for the S gate and H gate under the Rabierror (cid:15) , (b) and (d) respectively comparing the gate robustness forthe S gate and H gate under the detuning error δ , obtaining from ourSNGQC, NGQC in paths A (NGQC A ) and B (NGQC B ) with deco-herence rates κ = κ = 2 π × kHz under the same maximumamplitude of the driving field. The cyclic evolution time T is about 63 ns for the R x ( π/ gate and 56 ns for the R z ( π/ gate. The shape of param-eters Ω , ∆ and φ ( t ) for the R x ( π/ and R z ( π/ gatesare shown in Figs. 2(a) and 2(c), respectively. Assumingthe initial states of quantum system are | ψ (0) (cid:105) = | (cid:105) and | ψ (0) = ( | (cid:105) + | (cid:105) ) / √ for the R x ( π/ and R z ( π/ gates,respectively. These geometric gates can be evaluated by us-ing the state fidelity defined by F = (cid:104) ψ ( T ) | ρ | ψ ( T ) (cid:105) with | ψ ( T ) (cid:105) = ( | (cid:105) − i | (cid:105) ) / √ and | ψ ( T ) (cid:105) = ( | (cid:105) + e iπ/ | (cid:105) ) / √ being the corresponding ideal final states of the R x ( π/ and R z ( π/ gates, respectively. The state fidelities are as highas F R x ( π/ = 99 . and F R z ( π/ = 99 . , as shownin Figs. 2(b) and 2(d), respectively. In addition, for the gen-eral initial state | ψ (0) (cid:105) = cos ϑ | (cid:105) + sin ϑ | (cid:105) , the R x ( π/ and R z ( π/ gates should result in the ideal final states | ψ ( T ) (cid:105) = (cos ϑ − i sin ϑ ) / √ | (cid:105) + (sin ϑ − i cos ϑ ) / √ | (cid:105) and | ψ ( T ) (cid:105) = cos ϑ | (cid:105) + e iπ/ sin ϑ | (cid:105) , respectively. Tofully evaluate the gate performance, we define gate fidelity as F G = ( π ) (cid:82) π (cid:104) ψ ( T ) | ρ | ψ ( T ) (cid:105) dϑ with the integration nu-merically performed for 1001 input states with ϑ being uni-formly distributed over [0 , π ] . We find that the gate fideli-ties of the R x ( π/ and R z ( π/ gates can reach as high as F GR x ( π/ = 99 . and F GR z ( π/ = 99 . .Furthermore, to clearly show the merits of our approachin decreasing the influence of environment-induced decoher-ence compared with previous nonadiabatic GQC, we depictthe trend of gate fidelities under uniform decoherence rate κ/ π ∈ [0 , kHz for the R x ( π/ and R z ( π/ gates of ourapproach and the R Px ( π/ and R Pz ( π/ gates of previousNGQC under the same maximum amplitude Ω max = 2 π × MHz, as shown in Fig. 3, which directly demonstrate the ad-vantage of our approach over previous ones.Moreover, we compare the gate robustness of our SNGQCscheme for the S gate with gate parameters γ = π/ , θ = 0 ,and φ = 0 and H gate with gate parameters γ = π , θ = π/ ,and φ = 0 with previous NGQC in both paths A and B (see - E n e r gy L e v e l A Q B Q )a( )b()c( FIG. 5. Illustration of the realization of the geometric two-qubitgates. (a) Energy level of coupled transmon qubits Q A and Q B .(b) State dynamics and the gate fidelity of a nontrivial geometriccontrol-phase gate with γ (cid:48) = π/ . (c) Gate fidelities with respectto the different uniform decoherence rate κ for the U ( π/ gate ofour approach and previous NGQC under the same maximum Rabiamplitude. Appendix A for details) at different error conditions, includingthe Rabi error case of H (cid:15) ( t ) = [∆ σ z +(1+ (cid:15) )(Ω e − iη | (cid:105)(cid:104) | + H.c. )] with different Rabi errors (cid:15) ∈ [ − . , . and the detun-ing error case of H δ ( t ) = H ( t ) + δ Ω max | (cid:105)(cid:104) | with differentdetuning errors δ ∈ [ − . , . . Here, we simulate above pro-cess with the decoherence rates κ = κ = 2 π × kHz underthe same maximum amplitude Ω max = 2 π × MHz, andthe results are shown in Fig. 4. Due to the different evolutionpaths between SNGQC and NGQC respectively dominated bydifferent Hamiltonian, the Hamiltonian in NGQC case is notcommuted with detuning error, however, the Hamiltonian inSNGQC has σ z component, these can lead to the differentasymmetry and symmetry fidelity behavior respect to zero de-tuning point with regard to NGQC and SNGQC case in Figs.4(b) and (d). While our SNGQC scheme does not performswell under the detuning error δ than previous NGQC in bothpath A and B, as shown in Figs. 4(b) and (d), our SNGQCscheme does perform better under the Rabi error (cid:15) , as shownin Figs. 4(a) and (c). To sum up, our SNGQC scheme canshare the merits of decreasing the influence of environment-induced decoherence and the gate robustness against the Rabierrors than previous NGQC, and thus is promising for quan-tum systems where the X error is dominated. B. Nontrivial two-qubit gates
We further proceed to introduce the realization of nontrivialnonadiabatic geometric two-qubit gates with short path basedon two capacitively coupled transmon qubits [4–6], labeledas Q A and Q B with qubit frequency ω A,B and anharmonic-ity α A,B . Usually, the frequency difference ζ = ω B − ω A and coupling strength g between these two transmon qubits Q A and Q B are fixed and not adjustable. To achieve tun-able coupling and desired interaction between them [4–6], anac driving can be added on the transmon qubit Q B , whichresults in periodically modulating the frequency of Q B as ω B ( t ) = ω B + (cid:15) sin( νt + ϕ ) . Then, in the interaction pic-ture, the Hamiltonian of coupled system can be expressed as H C ( t )= g [ | (cid:105) AB (cid:104) | e iζt + √ | (cid:105) AB (cid:104) | e i ( ζ + α A ) t + √ | (cid:105) AB (cid:104) | e i ( ζ − α B ) t ] e − iβ cos( νt + ϕ ) + H.c. , (10)where β = (cid:15)/ν and | jk (cid:105) = | j (cid:105) ⊗ | k (cid:105) . Here, as shown in Fig.5(a), only the interaction in the subspace {| (cid:105) AB , | (cid:105) AB } isconsidered by choosing the driving frequency ν = ζ − α B +∆ (cid:48) with g (cid:28) { ν, ζ − ν, ζ + α A − ν } , and then using Jacobi-Angeridentity exp[ iβ cos( νt + ϕ )] = (cid:80) ∞ n = −∞ i n J n ( β )exp[ in ( νt + ϕ )] with J n ( β ) being the Bessel function of thefirst kind, and neglecting the high-order oscillating terms, theobtained effective Hamiltonian can be reduced to H = 12 (cid:18) ∆ (cid:48) g (cid:48) e iη (cid:48) g (cid:48) e − iη (cid:48) − ∆ (cid:48) (cid:19) , (11)in the two-qubit subspace {| (cid:105) AB , | (cid:105) AB } , where g (cid:48) =2 √ gJ ( β ) is effective coupling strength between transmonqubits Q A and Q B , ∆ (cid:48) is the energy difference between states | (cid:105) AB and | (cid:105) AB , and η (cid:48) = ϕ + π/ .Then, the Hamiltonian H can be directly applied to acquirea pure geometric phase e − iγ (cid:48) / on two-qubit state of | (cid:105) AB by a cyclic evolution beyond the computation basis like theway of constructing geometric single-qubit rotation opera-tions R z ( γ ) around axis σ z . Thus, within the two-qubit com-putation subspace {| (cid:105) AB , | (cid:105) AB , | (cid:105) AB , | (cid:105) AB } , the fi-nal nontrivial geometric two-qubit control-phase gates can beacquired as U ( γ (cid:48) ) = e − i γ (cid:48) . (12)Here, we use Hamiltonian H C ( t ) in Eq. (10) consider-ing all the unwanted imperfections to faithfully evaluate thenontrivial two-qubit geometric control-phase gates, and wealso apply the Lindblad master equation with γ (cid:48) = π/ as a typical example. Then, to achieve faithful simulation,we also set all parameters of the transmon qubits being eas-ily accessible with current experimental technologies [2], in-cluding the frequency difference ζ = 2 π × MHz, an-harmonicity of qubits α A = 2 π × MHz and α B =2 π × MHz, g = 2 π × MHz, the driving frequency ν = ζ − α B + ∆ (cid:48) = 2 π × MHz with ∆ (cid:48) = 0 and ν = ζ − α B + ∆ (cid:48) = 2 π × MHz with ∆ (cid:48) = 2 π × MHz, effective coupling strength of g (cid:48) max ≈ π × MHzwith β = 1 . , and the decoherence rate of transmons be-ing the same as the single-qubit case [2]. Then, the cyclicevolution time T for the two-qubit gate is about ns. Thestate dynamics of subspace {| (cid:105) AB , | (cid:105) AB } are verified in Fig. 5(b). To faithfully evaluate the gate performanceof two-qubit gates, for the general initial state | ψ (0) (cid:105) =(cos ϑ | (cid:105) A + sin ϑ | (cid:105) A ) ⊗ (cos ϑ | (cid:105) B + sin ϑ | (cid:105) B ) with | ψ ( T ) (cid:105) = U ( π/ | ψ (0) (cid:105) being the ideal final state, we candefine the two-qubit gate fidelity as F G = 14 π (cid:90) π (cid:90) π (cid:104) ψ ( T ) | ρ | ψ ( T ) (cid:105) dϑ dϑ , (13)with the integration numerically done for 10001 input stateswith ϑ and ϑ uniformly distributed over [0 , π ] . As shownin Fig. 5(b), we can obtain the gate fidelity F G = 99 . ,where the infidelity is caused by decoherence about . andleakage errors about . .In addition, we also simulate the trend of two-qubit gate fi-delities under uniform decoherence rate κ/ π ∈ [0 , kHz forthe U ( π/ gate of both our and previous NGQC approaches,under the same maximum amplitude g (cid:48) max = 2 π × MHz,as shown in Fig. 5(c), which also demonstrate the advantageof our approach over previous ones. Notably, under the fixedparameters anharmonicity of qubits α A = 2 π × MHz, α B = 2 π × MHz, g = 2 π × MHz, and the samemaximum amplitude, we can only optimize the frequency dif-ference ζ = 2 π × MHz to suppress the leakage errors to . for previous NGQC. IV. CONCLUSION
In summary, we propose an approach to realize universalnonadiabatic geometric quantum gates with short path basedon simple pulse control to shorten unnecessary long evolu-tion time of previous MGQC without complex pulse control.In addition, our scheme can perform better under the Rabierrors than NGQC with orange-slice loops. Our approachextends experimental feasible geometric quantum gates withshorter evolution process to futher reduce the influence ofenvironment-induced decoherence compared with previousNGQC. Meanwhile, our approach is suitable for many quan-tum physical systems, e.g., superconducting circuits systems.We further demonstrate our approach by the realization of ar-bitrary single-qubit geometric gates and non-trivial two-qubitgeometric gates on superconducting circuits systems.
APPENDIXAppendix A: Previous NGQC with orange-slice loops
In this appendix, we present the details in implementingprevious NGQC with orange-slice-shaped loops in path A andPath B. In path A, the orange-slice-shaped evolution path isdivided into three parts with resonant driving by microwavedrive with the amplitude Ω and phase η , which satisfy (cid:90) T Ω dt = θ, η = φ − π , t ∈ [0 , T ] , (cid:90) T T Ω dt = π, η = φ − γ + π , t ∈ [ T , T ] , (cid:90) TT Ω dt = π − θ, η = φ − π , t ∈ [ T , T ] , (A1)then, the geometric evolution operator can be obtained as U A ( T ) = e − iγ n · σ . And in path B, the geometric evolutionis realized by setting η = φ − γ − π at ∈ [ T , T ] , while thecorresponding geometric evolution operator keeps the same form of U A ( T ) . However, these two geometric paths havedifferent gate robustness against different type of errors showin Fig. 4 in the main text. ACKNOWLEDGEMENTS
This work was supported by the Key-Area Research andDevelopment Program of GuangDong Province (Grant No.2018B030326001), the National Natural Science Foundationof China (Grant No. 11874156), the National Key R&DProgram of China (Grant No. 2016 YFA0301803), and theScience and Technology Program of Guangzhou (Grant No.2019050001). [1] M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W.Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne,
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