Noncyclic Geometric Quantum Gates with Smooth Paths via Invariant-based Shortcuts
NNoncyclic and Nonadiabatic Geometric Quantum Gates with Smooth Paths
Li-Na Ji, Cheng-Yun Ding, 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 Guangdong-Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics,South China Normal University, Guangzhou 510006, China (Dated: February 2, 2021)Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quan-tum gates, which is the necessity of fault-tolerant quantum computation. However, it is limited by cyclic andmutative evolution path, which usually requires longer gate-time and abrupt pulse control, and thus weaken thegate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadia-batic evolutions, which effectively shorten the gate duration, and universal quantum gates can be realized in onestep without path mutation. Our numerical simulation shows that, comparing with the conventional dynamicalgates, the constructed geometric gates have stronger resistant not only to noise but also to decoherence effect.In addition, our scheme can also be implemented on a superconducting circuit platform, with the fidelities ofsingle-qubit and two-qubit gates higher than 99.97 % and 99.84 % , respectively. Therefore, our scheme providesa promising way to realize high-fidelity fault-tolerant quantum gates for scalable quantum computation. I. INTRODUCTION
Quantum computation can handle some hard problems ef-ficiently [1, 2] that classical computation can not, due to itsintrinsic parallel computation in a quantum way. Its basic ne-cessity is to implement a set of universal quantum gates basedon quantum mechanic principle [2]. However, the coherenceof quantum systems cannot be maintained perfectly, due tothe inevitable interaction with their surrounding environment,leaving limited effective operation time and intrinsic imper-fect quantum gates for quantum tasks. Therefore, to realizelarge-scale fault-tolerant quantum computation, it is challeng-ing and significant to complete quantum gates with fidelityas high as possible and with robustness as strong as possible,under the limited coherent times.In 1984, geometric phase under adiabatic and cyclic evolu-tion conditions was discovered by Berry [3], which only de-pends on the global feature of the evolution path and is insen-sitive to the detail of the accompanied noises. So, quantumcomputation with quantum gates that are constructed by geo-metric phases, the so-called geometric quantum computation[4, 5], has the distinct merit of being resilient to certain noises.Subsequently, elementary geometric quantum gates based onadiabatic Abelian [3] and non-Abelian [6] geometric phaseshave been demonstrated [7–10]. However, this slow adiabaticevolution process may lead to serious gate infidelity, due tothe prolonged exposure of the target quantum system into theenvironment.In 1987, Aharonov and Anandan [11] proposed the geomet-ric phase without adiabatic constraint, then the nonadiabaticgeometric quantum computation (NGQC) schemes based onAbelian [12–14] and non-Abelian [15, 16] geometric phaseshave been proposed, which accelerate the adiabatic process,and thus have soon attracted much attention and been imple-mented experimentally in various systems, such as trapped ∗ [email protected] ions [17–19], superconducting quantum circuits [20–23], nu-clear magnetic resonance [24–27] and nitrogen-vacancy cen-tres [28–32]. Currently, the orange-slice-shaped geometricpath [33] is widely used in NGQC [34–38] for single-shot re-alization of arbitrary quantum gates, which undergoes a multi-segment process with different parameters. However, compar-ing with the dynamical quantum gates from Rabi oscillation,this implementation has the following drawbacks. First, dueto the cyclic condition, the needed gate-time is the same forevery possible gate, even a very small geometric rotation op-eration, which is at least twice of the dynamical ones. Mean-while, there needs abrupt change of the parameters that deter-mine the evolution path, which increase the gate infidelity andthe complexity of experimental control.To avoid these drawbacks, the noncyclic geometric phase[39] could help. It not only shortens the evolution path, i.e.,shortens the gate-time, but also brings flexibility to the evolu-tion path design in constructing quantum gates [40–44]. Here,we propose a scheme for noncyclic NGQC with smooth geo-metric path and detail its implementation on superconductingquantum circuits. In our construction, we target to find shorternoncyclic path for arbitrary rotation gates, where decoherenceinduced gate infidelity is greatly suppressed. Meanwhile, wesucceed in avoiding the abrupt pulse control in constructingthe geometric path, which can decrease the error caused byparameter mutation and the demands of experimental real-ization. In addition, we also intend to conceal the accom-panied dynamical phase, instead of using it to construct un-conventional geometric quantum gates [44]. Due to thesemerits, the numerical simulations show that our scheme canimprove the gate performance remarkably compared with thecyclic NGQC and dynamical schemes. Finally, we implementour scheme on a superconducting circuit consists of trans-mon qubits, where the gate fidelities of single-qubit gates andtwo-qubit gate can be as high as 99.98 % and 99.84 % , respec-tively. Therefore, our scheme provides a promising way tofault-tolerant quantum computation. a r X i v : . [ qu a n t - ph ] F e b II. NONCYCLIC GEOMETRIC QUANTUM GATESA. The geometric phase
Consider a two-level system with lower and upper energylevels denote by {| (cid:105) = (1 , † , | (cid:105) = (0 , † } , with its en-ergy gap being ω , the transition between the two is inducedby a microwave field with coupling strength Ω( t ) , frequency ω d ( t ) and phase φ ( t ) . Assume (cid:126) = 1 hereafter, the interactionHamiltonian is H ( t ) = 12 (cid:18) − ∆( t ) Ω( t ) e − iφ ( t ) Ω( t ) e iφ ( t ) ∆( t ) (cid:19) , (1)where ∆( t ) = ω − ω d ( t ) is detuning. Take a pair of auxiliarydressed states | ψ ( t ) (cid:105) = cos χ ( t )2 | (cid:105) + sin χ ( t )2 e iξ ( t ) | (cid:105) , | ψ ( t ) (cid:105) = sin χ ( t )2 e − iξ ( t ) | (cid:105) − cos χ ( t )2 | (cid:105) (2)as evolution states, to induced a target geometric phase, whichare eigenstates of Lewis-Riesenfeld invariant of [45–47] I ( t ) = µ (cid:18) cos χ ( t ) sin χ ( t ) e − iξ ( t ) sin χ ( t ) e iξ ( t ) − cos χ ( t ) (cid:19) , (3)with µ being an arbitrary constant. By solving dynamic equa-tion i∂I ( t ) /∂t − [ H ( t ) , I ( t )] = 0 , the parameter relationshipsbetween { Ω( t ) , ∆( t ) , φ ( t ) } describing H ( t ) and { χ ( t ) , ξ ( t ) } describing | ψ , ( t ) (cid:105) are ˙ χ ( t ) = Ω( t ) sin[ φ ( t ) − ξ ( t )] , ˙ ξ ( t ) = − ∆( t ) − Ω( t ) cot χ ( t ) cos[ φ ( t ) − ξ ( t )] . (4)As parameters χ ( t ) and ξ ( t ) in Eq. (2) can be regarded as thepolar and azimuth angles on a Bloch sphere with χ ( t ) ∈ [0 , π ] and ξ ( t ) ∈ [0 , π ) ± nπ with n = 0 , , , ... , thus the pathevolution process of the evolution states | ψ , ( t ) (cid:105) over timecan be visualized on a Bloch sphere, as shown in Fig. 1.Furthermore, by setting the states | ψ ( t ) (cid:105) = e − iγ ( t ) | Ψ ( t ) (cid:105) and | ψ ( t ) (cid:105) = e iγ ( t ) | Ψ ( t ) (cid:105) with γ (0) = 0 , where | Ψ , ( t ) (cid:105) satisfy Schr¨odinger equation i∂ | Ψ , ( t ) (cid:105) /∂t = H ( t ) | Ψ , ( t ) (cid:105) .Then, at a final evolution time τ , we can get the overall phaseas γ ( τ ) = 12 (cid:90) τ ∆( t ) + ˙ ξ sin χ cos χ dt − (cid:90) τ (1 − cos χ ) ˙ ξdt. (5)The corresponding evolution operator at time τ is U ( τ ) = | Ψ ( τ ) (cid:105)(cid:104) Ψ (0) | + | Ψ ( τ ) (cid:105)(cid:104) Ψ (0) | = e iγ ( τ ) | ψ ( τ ) (cid:105)(cid:104) ψ (0) | + e − iγ ( τ ) | ψ ( τ ) (cid:105)(cid:104) ψ (0) | = (cid:18) u u − u ∗ u ∗ (cid:19) , (6)where u =(cos Γ cos χ − i sin Γ cos χ + e − i ξ − ,u =( − cos Γ sin χ − i sin Γ sin χ + e − i ξ +2 , (a) (b) Path 1Path 2
FIG. 1. (a) Schematic diagrams of noncyclic smooth geometric evo-lution path (Path 1) and cyclic geometric evolution path (Path 2) on aBloch sphere. χ and ξ represent the polar angle and azimuth angle ofevolution states, respectively. The two blue dots represent initial andfinal positions of the evolution state | ψ ( t ) (cid:105) . (b) Top view of diagram(a), Path 2 has mutation points obviously, while Path 1 is smooth. with ξ ± = ξ ( τ ) ± ξ (0) , χ ± = χ ( τ ) ± χ (0) , Γ = γ ( τ )+ ξ − / .For a general noncyclic evolution path, the dynamical part ofthe overall phase can be calculated as γ d ( τ ) = − (cid:90) τ (cid:104) ψ ( t ) |H ( t ) | ψ ( t ) (cid:105) dt = 12 (cid:90) τ ∆( t ) + ˙ ξ sin χ cos χ dt. (7)and the geometric phase of noncyclic path C with its geodesic C connecting the initial and final points of the actual evolu-tion path is γ (cid:48) g = γ C g + γ C g = − (cid:73) C + C (1 − cos χ ) ˙ ξdt, (8)which can be converted into a surface integral, see AppendixA for details. Therefore, the geometric property of γ (cid:48) g can beexplained by the solid angle bounded by C and C .Geometric phase has an intrinsic positive contribution to theresistance of local noise, so for achieving NGQC based a puregeometric phase, we will completely eliminate the dynamicalphase in Eq. (7) to be γ d = 0 , different from Ref. [44] whichinvolves the dynamical phase to construct the unconventionalgeometric phase. When we target to eliminate the dynamicalphase, one method is to let the dynamical phase always equalsto zero, i.e., ∆( t ) + ˙ ξ ( t ) sin χ ( t ) = 0 in Eq. (7). The othermethod is to null the dynamical phase at the final time, in thiscase, the detuning ∆( t ) can be time-independently, i.e., ∆( t ) ≡ ∆ = − τ (cid:90) τ ˙ ξ ( t ) sin χ ( t ) dt, (9)which means the frequency ω d of the driving field or the qubitfrequency can be fixed, remove the need of deliberate controlin the former case. Thus, this method simplifies the experi-mental control. In Fig. 2(a), we show the time-dependence of ∆( t ) when eliminating the dynamical phase using these twomethods, under a simple pulse shape of Ω( t ) = Ω m sin( πt/τ ) in which Ω m is pulse amplitude and τ is gate duration. sin = − sin / dt = − FIG. 2. The curve graph of the Hamiltonian parameters (a) ∆( t ) and(b) φ ( t ) over normalized time t/τ , with τ being the gate-operationtime, for the case of Hadamard-like gate. (a) Two methods to elimi-nate dynamical phase. Dotted line means γ d ( t ) = 0 , t ∈ [0 , τ ] , solidline means γ d ( τ ) = 0 and ∆ is a constant. B. Smooth geometric gates
After eliminating the dynamical phase, universal geometricquantum gates can be realized by assigning Γ , χ ± and ξ ± inEq. (6), but the different assignments of the shape of χ ( t ) and ξ ( t ) correspond to different evolution paths for a same gate.In the following, we present a new scheme of NGQC using anoncyclic geometric path evolving along latitude line, whichis smooth without the mutation of control parameters. Basedon smooth geometric path scheme we designed below, a set ofarbitrary and equivalent single-qubit geometric X-, Y- and Z-axis rotation operations, denoted as R x ( θ x ) -like, R y ( θ y ) -likeand R z ( θ z ) gates can all be realized, where θ x,y,z are rotationangles.Geometric property in Eq. (8) exists if ˙ ξ ( t ) (cid:54) = 0 and χ (cid:54) = 0 ,besides, χ (cid:54) = π/ when cancel the dynamical phase in Eq.(7). Based on the above restricted conditions, we consider anoncyclic and smooth path along latitude line, as Path 1 shownin Fig. 1. The path parameter we designed is ˙ χ = 0 , and thecorresponding evolution operator elements in Eq. (6) can besimplified to u =(cos Γ + i sin Γ cos χ ) e − i ξ − ,u = i sin Γ sin χe − i ξ +2 , (10)where γ ( τ ) = − (cid:82) τ (1 − cos χ ) ˙ ξdt/ ξ − (cos χ − / dueto γ d = 0 , and Γ = ξ − cos χ/ . R x ( θ x ) -like, R y ( θ y ) -like and R z ( θ z ) can be achieved by setting Γ = π/ , ξ (0) = π/ , χ = θ x / π/ , ξ (0) = π, χ = θ y / π, ξ − = θ z + 2 π, (11)respectively. Therefore, geometric operations of any rotationscan be achieved. Note that, R x,y ( θ x,y ) -like gates include a R z ( ξ (cid:48)− ), with ξ (cid:48)− = ξ − − π , operation in front of itself, whichcan be cancelled out by inverse operation R z ( − ξ (cid:48)− ) . Take a setof universal single-qubit gate, Hadamard-like gate, Phase gateand π/ gate, as typical examples. Hadamard-like gate can berealized by setting Γ = π/ , ξ (0) = 0 and χ = π/ includinga R z ( √ π ) operation in front of itself. Phase gate and π/ gate can be realized by R z ( π/ and R z ( π/ , respectively. FIG. 3. Gate fidelities of (a) Hadamard (-like) and (b) π/ gates asfunction of decoherence rate κ for noncyclic smooth geometric path(NSGP), cyclic orange-slice-shaped geometric path (COGP) and dy-namical path (DP). Next, we deduce the Hamiltonian parameters ∆ and φ ( t ) byEqs. (4) and (9). Set φ ( t ) = φ + φ ( t ) , where φ equals π for Hadamard-like gate and arbitrary value for Phase gate and π/ gate. For our path, ˙ χ = 0 , so we can set φ ( t ) = ξ ( t ) ± π .Therefore, the shapes of ∆ and φ ( t ) are ∆ = − τ (cid:90) τ Ω( t ) tan χdt,φ ( t ) = − ∆ t + (cid:90) t Ω( t ) cot χdt, (12)where the pulse shape Ω( t ) can be selected according to cer-tain actual experiment systems. We set Ω( t ) = Ω m sin( πt/τ ) which is easy achieved experimentally. As shown in Fig. 2,we plot the shapes of ∆ and φ ( t ) of Hadamard-like gate.In addition, the effect of decoherence is further considered,we next evaluate our gate performance by simulating Lindbladmaster equation [48] ˙ ρ = − i [ H ( t ) , ρ ] + κ A ( b − ) + κ A ( b z ) , (13)where ρ is the density operator, κ and κ are represented asthe decay and dephasing rates, the Lindblad operator A ( b ) =2 bρ b † − b † bρ − ρ b † b , b m − = (cid:80) + ∞ k =1 √ k | k − (cid:105) m (cid:104) k | and b mz = (cid:80) + ∞ k =1 k | k (cid:105) m (cid:104) k | are the standard lower operator andthe projector of k th level for m th qubit, respectively. Consideran ideal two-level system generalized by single qubit here,then k, m = 1 . To test the performance of quantum gates,we take a general initial state form ψ i = cos θ | (cid:105) + sin θ | (cid:105) ,under Hadamard-like and π/ gate operations, the final states ψ τ are [(cos θ + sin θ ) | (cid:105) + (cos θ − sin θ ) exp( i √ π ) | (cid:105) ] / √ and cos θ | (cid:105) +sin θ exp( iπ/ | (cid:105) , respectively. We define gatefidelity as F G = (1 / π ) (cid:82) π (cid:104) ψ τ | ρ | ψ τ (cid:105) dθ with the integra-tion numerically performed for 1001 initial states with θ beingevenly distributed over [0, 2 π ]. In the following, to prove thesuperiority of our scheme, we also conclude the cyclic orange-slice-shaped geometric path (see Appendix B) and the dynam-ical scheme (see Appendix C) as references. Fig. 3 shows thereduction of gate fidelity due to decoherence for three pathschemes, where κ = κ = κ . One can see that the line of oursmooth geometric path is very flat, when decoherence rate in-creases to − Ω m , the gate fidelity is still greater than 99.9 % .We also calculate the effective pulse area S = (cid:82) τ Ω( t ) dt/ . (a) (b) (c) (d)(e)(f)Smooth geometric pathDynamical pathOrange-slice-shaped geometric path FIG. 4. Gate fidelities as functions of qubit-frequency drift δ and thedeviation (cid:15) of driving amplitude for Hadamard (-like) (left column)and π/ (right column) gates for different schemes, under decoher-ence of κ = 4 × − Ω m . For our present scheme, S = √ π/ , π/ and √ π/ forHadamard-like gate, Phase gate and π/ gate, respectively;for dynamical path, they are π/ , π/ and π/ ; for cyclicorange-slice-shaped path, they are all equals π . Thus we canfind that our smooth geometric scheme has the shortest evolu-tion time, this is one of the important factors contributing toexcellent gate performance.Finally, we evaluate our gate robustness. The noise causedby qubit-frequency drift δ and the deviation (cid:15) of driving am-plitude and the decoherence caused by environment are themain culprits of gate infidelity. Therefore, in our simulation,the system affected by noise takes the Hamiltonian of H (cid:48) ( t ) = 12 (cid:18) − (∆ + δ Ω m ) (1 + (cid:15) )Ω( t ) e − iφ ( t ) (1 + (cid:15) )Ω( t ) e iφ ( t ) ∆ + δ Ω m (cid:19) . (14)For noise effect under the same decoherence condition κ = κ = 4 × − Ω m , Figs. 4(a) and 4(d) show the gate fidelitiesversus δ and (cid:15) for Hadamard-like and π/ gates. In the sameway, to demonstrate the superiority of our scheme in terms ofgate-robustness, we also show the results of gate-robustnessfor the cyclic orange-slice-shaped geometric path in Figs. 4(b)and 4(e) and the dynamical scheme in Figs. 4(c) and 4(f). Bycomparison, our scheme are insensitive to noise errors δ and (cid:15) for both Hadamard-like and π/ gate. Therefore, our schemealso shows stronger gate robustness. Note that, from the man-ner based on cyclic orange-slice-shaped geometric path alongthe longitude line, a open smooth path along the longitudeline can also be used to realize NGQC as seen in Appendix D, however, the gate-performance shown is not good comparedwith our designed geometric path using latitude line. This isthe meaning of our path design, effectively avoiding paths thatare seriously affected by noise. In the above comparison, theperformance of Phase gate is very similar to that of the π/ gate, thus not present here. III. PHYSICAL IMPLEMENTATIONA. Universal single-qubit geometric gates
In this section, we will demonstrate the feasibility of ourscheme on a superconducting quantum circuit, consists of ca-pacitively coupled transmon qubits. The two lowest energylevels of the single transmon qubit T , labelled by {| (cid:105) , | (cid:105)} ,is used as our computational subspace. However, when thedriving field excites the computational subspace, it will in-evitably cause the simultaneous coupling of higher energystates and result in the qubit-information leakage to | (cid:105) orhigher energy levels. Therefore, “derivative removal via adia-batic gate” (DRAG) technology [49–51] is used by us to sup-press this type of leakage. In this way, the Hamiltonian of atransmon which is excited by driving field with frequency ω d and adjustable phase φ ( t ) can be expressed as H ( t ) = + ∞ (cid:88) k =1 { [ kω − k ( k − α ] | k (cid:105) (cid:104) k | +[ 12 Ω D ( t ) √ k | k − (cid:105) (cid:104) k | e i ( ω d t − φ ( t )) + H . c . ] } , (15)where ω and α are the qubit-frequency and anharmonicityof transmon qubit T , respectively; and Ω D ( t ) = Ω( t ) − [ i ˙Ω( t )+Ω( t ) ˙ φ ( t ) + ∆Ω( t )] / (2 α ) is the corrected pulse by applyingDRAG technology with the original pulse Ω( t ) , where detun-ing parameter ∆ = ω − ω d .Next, based on the above superconducting quantum circuit,we further simulate the performance of Hadamard-like gateand π/ gate. We here consider the qubit-information leakageto state | (cid:105) which is regarded as the main leakage-error source,i.e., the case of k = 1 , in the master equation in Eq. (13)with the Hamiltonian H ( t ) . Following the current state-of-art experiment [52], we set the qubit parameters as κ = κ =2 π × kHz and α = 2 π × MHz. By searching Ω m tofind the corresponding highest gate fidelity, resulting in thegate fidelities 99.98 % and 99.97 % for Hadamard-like gate and π/ gate the optimal Ω m = 2 π ×
44 MHz and π ×
30 MHz,respectively, as shown in Fig. 5(b). Meanwhile, suppose thequbit is initially in the states ψ i = | (cid:105) and ( | (cid:105) + | (cid:105) ) / √ forHadamard-like gate and π/ gate, the ideal final states can be ψ τ = [ | (cid:105) +exp( i √ π ) | (cid:105) ] / √ and [ | (cid:105) +exp( iπ/ | (cid:105) ] / √ ,respectively. We evaluate these gates by state populations andthe state fidelities defined by F S = (cid:104) ψ τ | ρ | ψ τ (cid:105) in which ρ isthe final simulation result of density operator. State populationand state-fidelity dynamics are given for Hadamard-like gatein Fig. 5(c) and π/ gate in Fig. 5(d), the state fidelities canalso all reach to about 99.96 % , where the detuning parameters − ∆ / π ≈ . MHz and . MHz for Hadamard-like and π/ FIG. 5. (a) Gate fidelities as function of Ω m for Hadamard-like and π/ gates, optimal value of Ω m can be selected according to thehighest gate fidelity. (b) Dynamics of the gate fidelity for Hadamard-like and π/ gates using the optimal Ω m . The qubit-state populationand the state-fidelity dynamics are shown under the operations of (c)Hadamard-like gate with initial state | (cid:105) and (d) π/ gate with initialstate ( | (cid:105) + | (cid:105) ) / √ . gates, respectively. Similarly, the results of Phase gate aresimilar with π/ gate, and are not shown here. B. Nontrivial two-qubit geometric gate
We are going to construct the nontrivial two-qubit geomet-ric gate using two capacitively coupled transmon qubits T and T . The Hamiltonian of two coupled qubits with couplingstrength g is H ( t ) = (cid:88) m =1 , ∞ (cid:88) k =1 [ kω m − k ( k − α m ] | k (cid:105) m (cid:104) k | + g ( b − b † − + b † − b − ) , (16)where ω m and α m are the qubit frequency and anharmonic-ity of transmon T m , respectively. However, due to the fixedqubit-frequency difference and coupling strength between twocapacitively coupled transmon, Hamiltonian parameters haveno degree of freedom to be optimized, leading low gate fi-delity. To achieve better gate performance through the para-metrically tunable coupling [53–55], we add an ac drivingon T , experimentally induced by biasing the qubit by anac magnetic flux, which results in periodically modulatingT transition frequency in the form of ω ( t ) = ω + ˙ F ( t ) ,where F ( t ) = β sin[ νt + ϕ ( t )] , in which ν and ϕ ( t ) rep-resent frequency and phase of longitudinal field, respec-tively. Replace ω with ω ( t ) based on H ( t ) , and denotethe new Hamiltonian as H (cid:48) ( t ) . Then, move H (cid:48) ( t ) to the in-teraction picture, use Jacobi-Anger identity exp( iβ cos θ ) = (cid:80) n i n J n ( β )exp( inθ ) in which J n ( β ) is n th Bessel functionwith n being integer. Take first order of Bessel function, T T
FIG. 6. The performance of √ iSWAP like gate. (a) Dynamics of thegate fidelity. (b) Qubit-state population and state-fidelity dynamics,with the initial state ( | (cid:105) + | (cid:105) ) / √ . and truncate the Hamiltonian H (cid:48) ( t ) into the single- and two-excitation subspaces, then the Hamiltonian can be written as H = g J ( β ) e − i [ νt + ϕ ( t )] {| (cid:105)(cid:104) | e i ∆ t + √ | (cid:105)(cid:104) | e i (∆ − α ) t + √ | (cid:105)(cid:104) | e i (∆ + α ) t } + H . c ., (17)where | ml (cid:105) = | m (cid:105) ⊗ | l (cid:105) , and ∆ = ω − ω is the qubit-frequency difference between two capacitively coupled trans-mons T and T . Set ∆ L = ν − ∆ ( | ∆ L | (cid:28) | ν | , | ∆ | ), so asto produce an off-resonance coupling interaction with a slightdetuning ∆ L between the states | (cid:105) and | (cid:105) , while makingthe interactions between the computational basis | (cid:105) and thenon-computational subspace {| (cid:105) , | (cid:105)} in large-detuning, sothat the state | (cid:105) has rarely leakage to the non-computationalsubspace. Consider the single-excitation subspace {| (cid:105) , | (cid:105)} of H , and rotate the picture of Hamiltonian with operator U ∆ L = exp[ − i ∆ L ( | (cid:105)(cid:104) | − | (cid:105)(cid:104) | ) t/ , we can obtain aneffective two-level Hamiltonian H (cid:48) = 12 (cid:18) − ∆ L g (cid:48) e − iϕ ( t ) g (cid:48) e iϕ ( t ) ∆ L (cid:19) , (18)similar as Eq. (1), where g (cid:48) =2 g J ( β ) is effective couplingstrength which can achieve tunable by the optional value β .Next, we will construct the nontrivial two-qubit geometricgate √ iSWAP like in the subspace P = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} .In the single-excitation subspace {| (cid:105) , | (cid:105)} , similar to thesingle-qubit case, by setting Γ = π/ , ξ (0) = − π/ and χ = π/ in Eq. (10), we can also obtain R z ( ξ (cid:48)− ) (cid:18) ii (cid:19) / √ with ξ (cid:48)− = ( √ − π . Meanwhile, in the subspace {| (cid:105) , | (cid:105)} , wecan get an identity operator under the premise that no leakageof state | (cid:105) . Therefore, a two-qubit √ iSWAP like geometricgate can be achieved.Finally, we will test the performance of this two-qubit gate.To further simulate the decoherence effects of our consideredtwo coupled transmons, we also use the master equation of ˙ ρ = − i [ H (cid:48) ( t ) , ρ ] + κ A ( b − ) + κ A ( b z )+ κ (cid:48) A ( b − ) + κ (cid:48) A ( b z ) , (19)where κ (cid:48) and κ (cid:48) are represented as the decay and dephasingrates of the second transmon, respectively. For a general initialstate | ψ i (cid:105) = (cos θ | (cid:105) +sin θ | (cid:105) ) ⊗ (cos θ | (cid:105) +sin θ | (cid:105) ) ,the ideal final state is | ψ T (cid:105) = √ iSWAP like | ψ i (cid:105) , we de-fine F G = (1 / π ) (cid:82) π (cid:82) π (cid:104) ψ T | ρ | ψ T (cid:105) dθ dθ as the for-mula of two-qubit gate fidelity with the integration numeri-cally performed for 10001 initial states with θ and θ evenlydistributed over [0 , π ] . Realistically, we set the couplingstrength as g = 2 π × MHz, α = 2 π × MHz, de-cay and dephasing rates as κ = κ = κ (cid:48) = κ (cid:48) = 2 π × kHz. In addition, to reduce the influence from the high-order oscillating terms, optimizing parameter β for high gate-fidelity is necessary. We choose driving parameter β = 1 . and qubit-frequency difference ∆ = 2 π × MHz thatare easy achieved experimentally. Therefore, we can get ∆ L ≈ − π × . MHz. In this case, the gate fidelity canreach to 99.84 % as shown in Fig. 6(a). In Fig. 6(b), we showthe qubit-state population and state-fidelity dynamics with theinitial state ( | (cid:105) + | (cid:105) ) / √ , the oscillation of state | (cid:105) isvery slight, that means the leakages from the state | (cid:105) tothe non-computational subspace {| (cid:105) , | (cid:105)} are suppressedeffectively. At the final time, the state fidelity can reach to99.78 % . IV. CONCLUSION
In summary, we proposed a scheme for robust and high-fidelity geometric quantum gates, using the noncyclic smoothgeometric paths. Our path-design scheme has the shorter evo-lution time and experimental friendly, i.e., do not needs pa-rameter mutation and time-dependent modulation of the de-tuning. Therefore, the gate performance in our scheme canbe better than conventional dynamical gates, which thus canbe a promising alternative towards scalable and fault-tolerantquantum computation.
ACKNOWLEDGMENTS
This work is supported by the Development Pro-gram of GuangDong Province (No. 2018B030326001),the National Natural Science Foundation of China (No.11874156), the National Key R & D Program of China (No.2016YFA0301803), and Science and Technology Program ofGuangzhou (No. 2019050001).
Appendix A: Calculation of the geometric phase
Following Ref. [56], we take | ψ ( t ) (cid:105) as an example, thetotal relative phase from the initial state | ψ (0) (cid:105) to final state U ( τ ) | ψ (0) (cid:105) can be defined as γ t ( τ ) =arg (cid:104) ψ (0) | U ( τ ) | ψ (0) (cid:105) =arg (cid:104) ψ (0) | e iγ ( τ ) | ψ ( τ ) (cid:105) = γ ( τ ) + arg (cid:104) ψ (0) | ψ ( τ ) (cid:105) , (A1)removing the dynamical phase in Eq. (7) from the total rela-tive phase, the remaining item called Pancharatnam geometric (a) (b)(c) (d) FIG. 7. (a) Smooth geometric path along the longitude line. (b)View in z-y plane of diagram (a). Gate fidelity as functions of qubitfrequency drift δ and the deviation (cid:15) of driving amplitude for (c)Hadamard-like gate and (d) π/ gate under the decoherence effect. phase, that is γ g ( τ ) = γ t ( τ ) − γ d ( τ )= γ ( τ ) − γ d ( τ ) + arg (cid:104) ψ (0) | ψ ( τ ) (cid:105) (A2) = i (cid:90) τ (cid:104) ψ ( t ) | ∂∂t | ψ ( t ) (cid:105) dt + arg (cid:104) ψ (0) | ψ ( τ ) (cid:105) , which is gauge invariant. Specially, for cyclic path, | ψ (0) (cid:105) = | ψ ( τ ) (cid:105) , i.e., arg (cid:104) ψ (0) | ψ ( τ ) (cid:105) = 0 in Eq. (A2), so the phaseaccumulated through noncyclic path is a generalization exten-sion of cyclic case. In addition, when meet to open evolutionpath, auxiliary geodesic closing the initial and final states isintroduced to explain the geometric property [39, 56]. Then,the geometric phase of noncyclic path C with it’s geodesic C is calculated to be Eq. (8). Appendix B: Cyclic path
In previous works of NGQC, cyclic evolution condition isoften necessary when construct a set of universal geometricquantum gates. Take resonant system (∆ = 0) for example.To make dynamical phase always zero, the path evolves alongthe longitude line so that ˙ ξ = 0 , but at the same time, the ge-ometric phase is also null. In this case, two longitude linescorresponding to different azimuth angle ξ and ξ are neces-sary in path so that enclose a solid angle, and geometric phasecan produce on mutation point (for example, the south pole).Thus, the orange-slice-shaped geometric evolution path hasbeen designed: start at point ( χ , ξ ), go along the longitudeline to north pole ( , ξ ), then change ξ to ξ at the north pole,go along the longitude line to south pole ( π, ξ ), change ξ to ξ at the south pole, then return to the start point. The timeand corresponding path go through three segments, as shownin Fig. 1 (Path 2). The evolution operator is U c ( γ g , χ , ξ )= (cid:18) cos γ g + i sin γ g cos χ i sin γ g sin χ e − iξ i sin γ g sin χ e iξ cos γ g − i sin γ g cos χ (cid:19) , (B1)where γ g = ξ − ξ is the total geometric phase producedwhen the path mutates at the south pole, and dynamical phaseequals zero all the path. Eq. (B1) is one of the design methodsof Eq. (6), i.e., Γ = γ g , and universal single-qubit gates canbe realized [36, 37]. The parameters of Hamiltonian satisfy t ∈ [0 , τ ) : φ = ξ − π , (cid:82) τ Ω( t ) dt = χ ,t ∈ [ τ , τ ] : φ = ξ + π , (cid:82) τ τ Ω( t ) dt = π,t ∈ ( τ , τ ] : φ = ξ − π , (cid:82) ττ Ω( t ) dt = π − χ . (B2)However, the qubit parameters need to be mutated, which mayincrease control complexity and thus cause control error in theexperimental implementation. Appendix C: Dynamical gates
Previously, the pursuit of better performance of geometricquantum gates is usually limited by the complex multiple lev-els/qubits interactions and the longer gate duration requiredcompared to dynamical counterparts. Therefore, we take dy-namical gates as typical contrast to highlight higher fidelityand stronger robustness our geometric gates own. General dy-namical evolution operator constructed by Hamiltonian in Eq.(1) with ∆ = 0 and φ = φ d is a constant can be written as U d ( θ d , φ d ) = e − i (cid:82) τ H ( t ) dt = (cid:18) cos( θ d / − i sin( θ d / e − iφ d − i sin( θ d / e iφ d cos( θ d / (cid:19) , (C1)where parameters θ d = (cid:82) τ Ω( t ) dt , and φ d is a constant thatensures the geometric phase is zero. In this way, the arbitrary dynamical X-, Y- and Z-axis rotation operations can all berealized by R dx ( θ x ) = U d ( θ x , , R dy ( θ y ) = U d ( θ y , π/ and R dz ( θ z ) = U d ( π/ , π ) U d ( θ z , − π/ U d ( π/ , , respectively.In addition, a set of typical universal single-qubit gates, i.e.,Hadamard gate, Phase gate and π/ gate can all be realized by U d ( π, π ) U d ( π/ , π/ , R dz ( π/ and R dz ( π/ , respectively. Appendix D: Noncyclic geometric path along longitude