High-contrast ZZ interaction using superconducting qubits with opposite-sign anharmonicity
Peng Zhao, Peng Xu, Dong Lan, Ji Chu, Xinsheng Tan, Haifeng Yu, Yang Yu
aa r X i v : . [ qu a n t - ph ] A ug High-contrast ZZ interaction using superconducting qubits with opposite-sign anharmonicity
Peng Zhao, ∗ Peng Xu,
1, 2, 3
Dong Lan, Ji Chu, Xinsheng Tan, † Haifeng Yu, and Yang Yu National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 230039, China Institute of Quantum Information and Technology, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China State Key Laboratory of Quantum Optics and Devices, Shanxi University, Taiyuan, 030006, China (Dated: August 21, 2020)For building a scalable quantum processor with superconducting qubits, ZZ interaction is of great concernbecause its residual has a crucial impact to two-qubit gate fidelity. Two-qubit gates with fidelity meeting thecriterion of fault-tolerant quantum computationhave been demonstrated using ZZ interaction. However, as theperformance of quantum processors improves, the residual static-ZZ can become a performance-limiting factorfor quantum gate operation and quantum error correction. Here, we introduce a superconducting architectureusing qubits with opposite-sign anharmonicity, a transmon qubit and a C-shunt flux qubit, to address this issue.We theoretically demonstrate that by coupling the two types of qubits, the high-contrast ZZ interaction can berealized. Thus, we can control the interaction with a high on/off ratio to implement two-qubit CZ gates, orsuppress it during two-qubit gate operation using XY interaction (e.g., an iSWAP gate). The proposed architec-ture can also be scaled up to multi-qubit cases. In a fixed coupled system, ZZ crosstalk related to neighboringspectator qubits could also be heavily suppressed.
Engineering a physical system for fault-tolerant quantumcomputing demands quantum gates with error rates belowthe fault-tolerant threshold, which has been demonstratedin small-sized superconducting quantum processors [1]. Atpresent, high-performance quantum processors with dozensof superconducting qubits have become available [2], but re-alizing fault-tolerant quantum computing is still out of reach,mainly because of the heavy overhead needed for error-correction with state-of-the-art gate performance. Therefore,further improving gate performance is essential for realiz-ing fault-tolerant quantum computing with supercondcutingqubits.With today’s superconducting quantum processors, apartfrom increasing qubit coherence times, speeding up gatescan also fundamentally improve gate performance. However,there is a speed-fidelity trade-off imposed by parasitic interac-tions. Since the current two-qubit gates typically have lowergate speeds and worse fidelity than single-qubit gates [3], thisissue is particularly relevant to two-qubit gates. For imple-menting a fast two-qubit gates with strong two-qubit coupling,one of the major parasitic interactions is ZZ coupling, whichis mostly caused by the coupling between higher energy lev-els of qubits [4, 5]. Thus, for qubits with weak anharmonicity,such as transmon qubits [6] and C-shunt flux qubits (in sin-gle well regions) [7–9], the non-zero parasitic ZZ couplingexists inherently due to the intrinsic energy level diagramsof qubits. This ZZ interaction has been shown to act as adouble-edged sword for quantum computing: it can be usedto implement high-speed and high-fidelity controlled-Z (CZ)gates [1, 10–12], yet it can also degrade performance of two-qubit gates through XY interaction [2, 8, 11–17]. Moreover,in fixed coupled multi-qubit systems, such as the one shownin Fig.1(a), gate operations in the two qubits enclosed by therectangle involve six neighboring spectator qubits, and the ZZcoupling related to these qubits cannot be fully turned off bytuning qubits out of resonance [1]. The residual is typicallymanifested as crosstalk, which results in addressing errors and phase errors during gate operations and error correction[18–24]. Furthermore, these errors are correlated multi-qubiterrors, which are particularly harmful for realizing a fault-tolerant scheme [25]. Given the fidelity and performance limi-tations related to parasitic ZZ interaction, it is highly desirableto achieve high-contrast control over this parasitic coupling.To address this challenge, in this work, we introduce asuperconducting architecture using qubits with opposite-signanharmonicity. We theoretically demonstrate our protocolwith coupled transmon and C-shunt flux qubits, which havenegative and positive anharmonicity, respectively. We showthat high-contrast ZZ interaction can be achieved by engineer-ing the system parameters. By utilizing ZZ interaction with ahigh on/off ratio, we can implement the CZ gate with a speedhigher than that of the traditional setup using only one type ofqubit (e.g., full transmon systems). Parasitic ZZ coupling canalso be deliberately suppressed during two-qubit gate opera-tions using XY interaction (e.g., an iSWAP gate), while leav-ing the XY interaction completely intact. The proposed archi-tecture can be scaled up to multi-qubit cases, and in fixed cou-pled systems, ZZ crosstalk related to spectator qubits couldalso be heavily suppressed.To start, let us consider a superconducting architecture(hereinafter, the AB-type) where two qubits with opposite-sign anharmonicities are coupled together. The architecturecan be treated as a module that can be easily scaled up tomulti-qubit case. In Fig. 1(a), we show a case of a nearest-neighbor-coupled qubit lattice, where circles with A and B aretwo-type qubits with opposite-sign anharmonicities arrangedin an -A-B-A-B- pattern. As shown in Fig. 1(b), both qubitscan be modeled as a three-level (i.e., | i , | i , | i ) anharmonicoscillator for which the Hamiltonian is given as ( ~ = 1 ) H l = ω l q † l q l + α l q † l q l ( q † l q l − , (1)where the subscript l = a, b labels an anharmonic oscillatorwith anharmonicity α l and frequency ω l , and q l ( q † l ) is the as-sociated annihilation (creation) operator truncated to the low- (a) B BA AA B B BA AA B a a < b w a w b a > A B B A (c) (d) (b)
FIG. 1: (a) Layout of a two-dimensional nearest-neighbor lattice,where circles at the vertices denote qubits, and gray lines indicatecouplers between adjacent qubits. The lattice consists of two-typequbits arranged in an -A-B-A-B- pattern in each row and column.The circles with A and B are qubits with opposite-sign anharmonic-ities, and each one can be treated as a three-level anharmonic oscil-lators (b). Typically, transmon qubits and C-shunt flux qubits canbe modeled as anharmonic oscillators with negative and positive an-harmonicity, respectively. Qubits can be coupled to each other (c)directly via a capacitor or (d) indirectly using a resonator. est three-level. Usually, qubits can be coupled directly viaa capacitor or indirectly through a bus resonator, as shownin Fig. 1(c) and 1(d). For clarity and without loss of gen-erality, unless explicitly mentioned, we focus on the direct-coupled case in the following discussion, and the dynamicsof two-coupled qubits can be described by the Hamiltonian H = H a + H b + H I , where H I = g ( q † a q b + H.c. ) describesthe inter-qubit coupling with strength g .Before describing our main idea of engineering high-contrast ZZ interaction in our architecture, we need to firstexamine the origin of parasitic ZZ interaction in the tradi-tional setup (hereinafter, the AA-type), where two transmonqubits are coupled directly. By ignoring higher energy levels,we can model the transmon qubit as an anharmonic oscilla-tor with negative anharmonicity [6], thus the system Hamilto-nian can be expressed as H with α a,b < . Fig. 2(a) showsnumerically calculated energy levels of the system for an-harmonicities α a,b = − α ( α/ π = 250 MHz , which is apositive number throughout this work) [26]. One can findthat there are four avoided crossings, one corresponds to theXY interaction in the one-excitation manifold (i.e., interac-tion | i ↔ | i ), and the other three (from left to right) as-sociate with interactions among the two-excitation manifold {| i , | i , | i} (i.e., interactions | i ↔ | i , | i ↔ | i ,and | i ↔ | i ). The interactions between qubit states | i and non-qubit states ( | i , | i ) change the energy of state | f i , where | e ij i denotes the eigenstate of the Hamiltonian H that has the maximum overlap with the bare state | ij i , and thecorresponding eigenenergy is E e ij , thus leading to ZZ couplingwith strength ζ = ( E f − E f ) − ( E f − E f ) = J (tan θ b − tan θ a , (2) ⟩a) E / π ⟩ G H z ) ⟩b) |20⟩|11⟩|02⟩−500 −250 0 250 500 Δ/2π ⟩MHz)
Δ/2π ⟩MHz)
150 250 Δ5011.111.211.Δ11.4
FIG. 2: Numerical calculation of the energy levels of the coupledqubit system as a function of the qubit detuning ∆ = ω a − ω b for qubit frequency ω b / (2 π ) = 5 . , anharmonicity α/ π =250 MHz [8, 26], and coupling strength g/ π = 15 MHz . (a)Qubits with same-sign anharmonicity α a ( b ) = − α ; (b) Qubitswith opposite-sign anharmonicity α a ( b ) = ∓ α . The inset high-lights the triple degeneracy point in the two-excitation manifold {| i , | i , | i} . where tan θ a,b = 2 J/ (∆ ± α a,b ) , ∆ = ω a − ω b denotesqubit detuning, and J = √ g is the coupling strength of | i ↔ | i ( | i ) . When J ≪ | ∆ ± α a,b | , Eq. (2) can beapproximated by ζ = J / (∆ + α b ) − J / (∆ − α a ) [1].With the expression in Eq. (2), there are two terms con-tributing to ZZ interaction, and each is independently asso-ciated with the interaction | i ↔ | i ( | i ) . Replacing oneof the two transmon qubits with a qubit for which the valueof anharmonicity is comparable but the sign is positive candestructively interfere the two terms in Eq. (2), thus heavilysuppressing ZZ coupling. A promising qubit to implementingsuch a AB-type setup is the C-shunt flux qubit in single wellregime [7–9], where the qubit can be modeled as an anhar-monic oscillator that has positive anharmonicity with mag-nitude comparable to that of the transmon qubit [8, 26]. InFig. 2(b), we show numerically calculated energy levels forthis AB-type setup with α b / π = 250 MHz [8] and keepall other parameters the same as in Fig. 2(a). Compared withthe AA-type setup, the avoided crossing associated with inter-action | i ↔ | i is completely intact, but the interactionamong two-excitation manifold forms an avoided crossingwith triplets, as shown in the inset of Fig. 2(b). At the tripledegeneracy point, the eigenstates are ( | i + | i−√ | i ) / , ( | i − | i ) / √ , ( | i + | i + √ | i ) / , with the corre-sponding energies of E − √ J , E , and E + √ J [31].Fig. 3(a) shows the numerical result of ZZ coupling strengthas a function of qubit detuning ∆ in the AB-type setup. Theresult for the AA-type setup is also shown for comparison.In the AB-type setup, ZZ coupling is completely removedaway from the triple degeneracy point at ∆ = α b , whilefor regions close to the degeneracy point, coupling is pre-served, and the strength at the degeneracy point is larger thanthat of the AA-type setup ( g vs √ g ) [31]. In Fig. 3(b),we have also shown ZZ coupling strength as a function ofthe anharmonicity asymmetry δ α = | α b | − | α a | for typi-cal coupling strength g/ π = 15 MHz and qubit detuning ∆ / π = −
150 MHz . One can find that the ZZ couplingstrength is suppressed below . for the anharmonicityasymmetry around − ∼
600 MHz . In addition, since theanharmonicities of both types of qubits (the transmon qubitand the C-shunt flux qubit) depend primarily on geometriccircuit parameters, the typical anharmonicity asymmetry δ α around − ∼
20 MHz could be achieved with current qubitfabrication techniques [26]. In this case, ZZ coupling strengthcould be further suppressed below
60 KHz , as shown in theinset of Fig. 3(b), whereas for the traditional setup, the typicalstrength of the residual ZZ coupling is about , as shownin Fig. 3(a).For a more comprehensive analysis of ZZ coupling inthe AB-type setup, we explore the full parameter range inFig. 3(c) with varying qubit detuning ∆ and ahharmonicityasymmetry δ α . We identify three regions in parameter spacewith prominent characteristic. The two lighter regions indi-cate that ZZ coupling becomes strong when qubit detuningapproaches qubit anharmonicity, i.e., ∆ = ∓ α a,b , and the in-tersection region corresponds to the triple degeneracy point.The darker region shows where ZZ coupling is heavily sup-pressed and is zero for δ α = 0 . In Fig. 3(d), we show theresult for an indirect-coupled case, where qubits are coupledvia a resonator [32]. In this case, the strength of the effec-tive inter-qubit coupling depends on qubit detuning, thus thezero ZZ coupling point depends not only on the anharmonicityasymmetry, but also on the qubit detuning.Having shown high contrast ZZ interaction in the AB-typesetup, we now turn to study the implementation of two-qubitgates with a diabatic scheme in this setup [4, 11]. Here, we fo-cus on the direct-coupled system with always-on interactionsdescribed by the Hamiltonian H with α a < and α b > , butthe method is generalizable to other coupled systems [32]. Forillustration purposes and easy reference, we use the same pa-rameters as those in Fig. 2(b). In this case, during the gateoperations, the frequency of qubit b remains at its parkingpoint, while the frequency of qubit a changes from its park-ing point to the interaction point and then back, according toa time-dependent function [35, 38], as shown in Fig. 4(a) or4(d) where the full width at half maximum is defined as holdtime. We note that at the parking (idling) point where theinter-qubit coupling is effectively turned off, the logical ba-sis state | ij i is defined as the eigenstates of the system biasedat this point [31, 38], which is adiabatically connected to thebare state | ij i . Expressed in the logical basis, the target gateoperations can be expressed as U ( θ, φ ) = e − i ( | ih | + | ih | ) θ e − i | ih | φ , (3)where θ denotes the swap angle associated with the bareexchange interaction | i ↔ | i , and φ represents theconditional phase resulting from ZZ coupling. To quantify −500−250 0 250 500 Δ/2π ΔMHz) | ζ |/ π Δ M H z ) Δa)
AB − typeAA − type δ α /2π ΔMHz) | ζ |/ π Δ M H z ) Δb)
NumericalPerturbative −250 0 250
Δ/2π ΔMHz) δ α / π Δ M H z ) Δc) −300 0 300
Δ/2π ΔMHz)
Δd) −4 −2 −4 −2 | ζ |/ π Δ M H z ) −20−10 0 10 200.00.030.06 FIG. 3: Numerical results for ZZ coupling strength | ζ | . (a) | ζ | ver-sus qubit detuning ∆ for anharmonicity asymmetry δ α = | α b | −| α a | = 0 , where the dashed line shows result for the AA-type.(b) | ζ | versus δ α for ∆ / π = −
150 MHz , where the dashed lineshows perturbational result. The inset shows that | ζ | could be sup-pressed below
60 KHz with the typical anharmonicity asymmetry( − ∼
20 MHz ). (c) | ζ | versus ∆ and δ α . Horizontal (vertical) cutthrough (c) denotes the result plotted in a (b). (d) ZZ coupling in asystem comprising two qubits that are coupled via a resonator [32]. the intrinsic performance of the implemented gate operation,we use the metric of state-average gate fidelity F ( θ, φ ) =[Tr( U † U ) + | Tr( U ( θ, φ ) † U ) | ] / [39], where U is the ac-tual evolution operator (ignoring the decoherence process) upto single qubit phase gates.We first consider the implementation of the CZ gate U (0 , π ) , and the main idea is as follows. By tuning the fre-quency of qubit a from its parking point ω P = 6 . to theinteraction point ω I = ω b + α according to the time-dependentfunction shown in Fig. 4(a), the CZ gate can be realized af-ter a full Rabi oscillation between | i and [ | i + | i ] / √ .As mentioned before, the Rabi rate is larger than in the AA-type setup ( g vs √ g ), thus allowing a higher gate speed.We note that an additional small overshoot to the interactionfrequency ω I is critical to optimize the leakage to non-qubitstates [11, 35]. By taking the optimal overshoot and initial-izing the system in states | i and | i , Fig.4(b) shows theleakage ε leak = 1 − P ¯11 ( P ¯ ij denotes the population in thelogical state | ij i at the end of the gate operations) and swaperror ǫ swap = 1 − P ¯01 as a function of the hold time. In presentsystem with fixed inter-qubit coupling, it is nearly impossibleto have an optimal hold time to simultaneously minimize theswap error and leakage, as shown in Fig.4(b). Thus, here,we choose to minimize the leakage, and find that with a holdtime of . , a CZ gate with fidelity above . canbe achieved, and both the leakage and swap error can be sup-pressed to below − . However, as shown in Fig. (c), whenthe system is considered with typical anharmonicity asymme-try δ α , gate fidelity worsens. In order to identify the perfor-mance limiting factors, we extract phase error δ θ , δ φ with re- ω / ( π )( G H z ) (a) Hold time ω a ω b ω / ( π )( G H z ) (d) Hold time ω a ω b −4 −2 ε l e a k , s w a p (b) ε leak ε swap −4 −2 ε l e a k , s w a p (e) ε leak ε swap −20 −10 0 10 20 δ α /2π (MHz) −8 −6 −4 −2 E rr o r & P h a s e / π (c) ε leak θ ||δ ϕ | −20 −10 0 10 20 δ α /2π (MHz) −7 −5 −3 (f) ε leak θ ||δ ϕ | FIG. 4: Numerical results for CZ gate and iSWAP gate implementa-tion in our architecture. The system parameters used are same as inFig. 2(b). (a),(d) Typical pulses with small overshoots for realizingCZ gates and iSWAP gates, where the full width at half maximumis defined as hold time. (b),(e) Leakage ε laek and swap error ε swap versus hold time for system with anharmonicity asymmetry δ α = 0 and optimal overshoots [35]. (c),(f) Gate error − F versus typicalanharmonicity asymmetry. The phase error δ θ , δ φ with respect to theideal phase parameters ( , π for CZ gate, and π/ , for iSWAP gate)and leakage error ε leak are also presented for identifying the majorsource of error. spect to the ideal phase parameters θ = 0 , φ = π for CZgate and the leakage, and find that in the current case, gateerror primarily result from conditional phase error δ φ . Thiscan be explained by the fact that in systems with typical an-harmonicity asymmetry, the resonance condition for having afull Rabi oscillation between | i and non-qubit state breaksdown. Off-resonance Rabi oscillation is thus presented, caus-ing conditional phase error [35].An iSWAP gate U ( π/ , can be realized by tuning thetwo qubits into resonance according to the control pulseshown in Fig. 4(d). Given an optimal overshoot with respectto the interaction point ω a = ω b , an iSWAP gate with fidelityexceeding . can be realized with a hold time of . [40], and leakage ε leak and swap error ε swap = P ¯01 can besuppressed to below − , as shown in Fig. 4(e). As shown inFig. 4(f), we also study the effect of anharmonicity asymmetryon gate fidelity, and find that gate fidelity in excess of . can be achieved for system with typical anharmonicity asym-metry. By extracting phase error δ θ , δ φ and leakage for theiSWAP gate, we find that leakage error becomes the majorsource of error. Finally, we note that apart from leakage andswap error, phase error δ φ resulted from parasitic ZZ coupling limits the performance of iSWAP gates in the traditional AA-type setup [2, 11, 12]. The high-fidelity iSWAP gate and thelow conditional phase error δ φ demonstrated above indicatethat parasitic ZZ coupling is indeed heavily suppressed in theAB-type setup.In summary, we have studied parasitic ZZ coupling in asuperconducting architecture [41–43] where two qubits withopposite-sign anharmonicities are coupled together and foundthat high-contrast control over parasitic ZZ coupling can berealized. We further show that CZ gates with higher gatespeed and iSWAP gates with dramatically lower conditionalphase error can be realized with diabatic schemes in the pro-posed architecture. Moreover, as shown in Fig. 4(b) and 4(c),XY gates with arbitrary swap angles [44], leakage error be-low − , and negligible phase error is achievable , as is arbi-trary control phase gate with swap error below − . Sincethese errors are caused by off-resonant Rabi oscillation re-lated to the associated parasitic interaction (i.e., | i ↔ | i for CZ gates, and | i ↔ | i ( | i ) for iSWAP gates), evenlower error rates should be possible by increasing the valueof anharmonicity [43] or using the synchronization procedure[11]. Implementing these continuous set of gates nativelycould be useful for near-term applications of quantum pro-cessors [12, 44]. As one may expect, the high-contrast con-trol over ZZ coupling could also improve the performance ofparametric activated gates [15–17] and cross-resonance gates[8, 14]. In multi-qubit systems and with fixed coupled cases,the crosstalk resulted from ZZ coupling could be heavily sup-pressed, thus, gate operations can be implemented simultane-ously with low crosstalk. For tunable coupled cases [45–48],XY gates with arbitrary swap angles can be implemented na-tively with negligible conditional phase error [2, 11, 12].We would like to thank Songy for helpful suggestion on themanuscript. This work was partly supported by the NKRDPof China (Grant No. 2016YFA0301802), NSFC (Grant No.61521001, and No. 11890704), and the Key R & D Programof Guangdong Province (Grant No.2018B030326001). P. Xacknowledges the supported by Scientific Research Founda-tion of Nanjing University of Posts and Telecommunications(NY218097), NSFC (Grant No. 11847050), and the Youngfund of Jiangsu Natural Science Foundation of China (GrantNo. BK20180750). Haifeng Yu acknowledges support fromthe BJNSF (Grant No.Z190012). ∗ Electronic address: [email protected] † Electronic address: [email protected][1] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E.Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell,Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P.O’Malley, P. 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