Double-Transmon Coupler: Fast Two-Qubit Gate with No Residual Coupling for Highly Detuned Superconducting Qubits

Although two-qubit entangling gates are necessary for universal quantum computing, they are notoriously difficult to implement with high fidelity. Recently, tunable couplers have become a key component for realizing high-fidelity two-qubit gates in superconducting quantum computers. However, it is still difficult to achieve tunable coupling free of unwanted residual coupling for highly detuned qubits, which are desirable for mitigating qubit-frequency crowding or errors due to crosstalk between qubits. We thus propose a design for this kind of tunable coupler, which we call a double-transmon coupler, because this is composed of two transmon qubits coupled through a common loop with an additional Josephson junction. Controlling the magnetic flux in the loop, we can achieve not only fast high-fidelity two-qubit gates, but also no residual coupling during idle time, where computational qubits are highly detuned fixed-frequency transmons. The proposed coupler is expected to offer an alternative approach to higher-performance superconducting quantum computers.

Tunable couplers have recently become a key component for high-fidelity two-qubit gates in superconducting quantum computers [21-24, 27-29, 32, 41-48]. Tunable couplers allow us not only to implement fast two-qubit gates, but also to turn off an energy-exchange interaction called transverse or XY coupling [49]. Major tunable couplers, including those used for demonstration of quantum supremacy (advantage) [21,24], are based on the cancellation between a direct coupling via a capacitor and an indirect coupling via a frequency-tunable transmon qubit [50], which we refer to as a single-transmon coupler. This coupler is regarded as a capacitor or transmon version of a previously proposed inductor-based coupler for flux qubits [51,52]. (Other kinds of inductorbased tunable couplers have also been proposed [53][54][55][56].) In the single-transmon coupler, there exists an unwanted correlated energy shift due to residual coupling called longitudinal or ZZ coupling [49]. Residual ZZ coupling has recently become a central issue in the field of superconducting quantum computers [23, 27-32, 41-46, 57-63]. Some research groups have found special conditions under which the ZZ coupling in the single-transmon coupler vanishes [28,42]. However, the vanishing points exist only in a region of small detunings between two computational qubits [28] (see Appendix E). In other words, in the single-transmon coupler, there is inevitable ZZ coupling for highly detuned qubits [23,43]. Thus, zero ZZ coupling in the single-transmon coupler results in qubitfrequency crowding or crosstalk between qubits [64].
In this paper, we theoretically propose a new kind of tunable coupler, which we call a double-transmon coupler. Our coupler consists of two fixed-frequency transmons coupled through a common loop with an additional Josephson junction. We can control the coupling between the two coupler transmons by controlling the magnetic flux in the loop, and consequently tune the coupling strength between computational qubits. A remarkable feature of this coupler is that the ZZ coupling vanishes even for highly detuned computational qubits, unlike the single-transmon coupler. Our numerical simulations indicate that this coupler allows us to achieve not only high two-qubit gate fidelities of over 99.99% with a short gate time of 24 ns, but also no residual ZZ coupling during idle time for highly detuned fixed-frequency transmons with detuning of 0.7 GHz. Thus, the double-transmon coupler is expected to be promising for improving the performance of superconducting quantum computers. Figure 1 shows a diagram of the proposed coupler. This consists of two fixed-frequency transmons [Transmons 3 and 4 in Fig. 1 computational qubits [Transmons 1 and 2 in Fig. 1] are capacitively coupled to the coupler, as shown in Fig. 1.

II. DESIGN AND MECHANISM
The mechanism of this coupler is qualitatively explained from a classical point of view under rough approximations as follows [65]. The Lagrangian describing the total system is given by where ϕ i , φ i = φ 0 ϕ i , and ω Ji = φ 0 I ci are, respectively, the phase difference, flux variable, and Josephson energy for the ith Josephson junction with critical current of I ci [66], the dots denote time derivatives, and we have included unwanted parasitic capacitances.
Neglecting the parasitic capacitances and using the constraint that φ 5 = φ 4 − φ 3 − Φ ex (Φ ex is the external flux in the loop) [67], we can approximate K and V as where Θ ex = Φ ex /φ 0 . Note that the two qubits are coupled only through the coupling between the two coupler transmons given by the last term in Eq. (4). This coupling can approximately be turned off by tuning the external flux Φ ex , as follows.
Here we focus on the potential for the coupler V c . In the transmon regime where Josephson energies are much larger than charging energies [68], low-energy states concentrate around a potential minimum, and hence the following second-order approximation is valid: where ϕ

III. ZZ COUPLING
In order to accurately evaluate the properties of the double-transmon coupler, here we numerically investigate it using a fully quantum-mechanical model with finite parasitic capacitances. In this work, the qubits are assumed to be detuned. The qubit states are then welldefined by the energy eigenstates of the total Hamiltonian. However, there can be an unwanted correlated en-ergy shift due to residual ZZ coupling. The ZZ coupling strength ζ ZZ is defined as where ω ij = E ij / is the frequency corresponding to the energy, E ij , of the two-qubit state |ij . We also set the origin of energy as ω 00 = 0. When ζ ZZ = 0, the two qubits are completely independent. By numerically diagonalizing the quantum-mechanical Hamiltonian derived from the Lagrangian given by Eqs.
(1) and (2) (see Appendix A), we evaluate ζ ZZ for two situations: larger and smaller detunings between the qubits than the anharmonicities (Kerr coefficients) of the qubits, which are, respectively, called "out of the straddling regime" and "in the straddling regime." The results in the two situations are, respectively, shown in Figs. 2(a) and 2(b), where the parameters are set to experimentally feasible values [43]. From these results, it turns out that the double-transmon coupler can have the vanishing points of the ZZ coupling in both the regimes. This is a / 2 kHz / 2 (GHz) remarkable feature of the double-transmon coupler, because for the conventional single-transmon coupler, the ZZ-coupling vanishing points exist only in the straddling regime [28] (see Appendix E). It is also interesting that the coupler-transmon frequency required for the zero ZZ coupling is lower bounded out of the straddling regime, but upper bounded in the stradding regime [69].

IV. TWO-QUBIT GATE
We evaluate two-qubit gate performance by numerical simulations with the parameter values in Fig. 2(a) and ω 4 /(2π) = 8.5 GHz [indicated by the horizontal dashed line in Fig. 2(a)] [70]. The Θ ex dependence of ζ ZZ for these parameter values is shown in Fig. 3(a). As shown in the inset, the ZZ coupling vanishes at Θ ex 0.61π and 0.63π. We thus define the qubit states by the energy eigenstates at Θ ex = 0.61π. In other words, we set Θ ex = 0.61π during idle time, as indicated in Fig. 3(a).
In Fig. 3(a), it is also notable that |ζ ZZ |/(2π) becomes as large as 40 MHz at Θ ex = π. This property can be used for a fast two-qubit gate called the controlledphase (CPHASE) gate including the controlled-Z (CZ) gate [23,28,43], where ζ ZZ is adiabatically increased and then decreased by controlling the external flux Φ ex . The flux pulse shape in the present simulations is shown in Fig. 3(b), which is designed according to a technique for reducing nonadiabatic errors [71] (see Appendix C). The simulation results are shown in Figs. 3(c) and 3(d).
Figure 3(c) shows that the rotation angle, θ CPHASE , of the CPHASE gate increases linearly as the gate time T g increases. The CZ gate corresponding to θ CPHASE = π can be achieved when T g 24 ns, as indicated by the horizontal dashed line in Fig. 3(c). The average fidelity of the CPHASE gate is shown in Fig. 3 suggesting that the CZ-gate fidelity, indicated by the vertical dashed line in Fig. 3(d), will surpass 99.99%. Thus, the double-transmon coupler allows us to simultaneously achieve fast high-fidelity two-qubit gates and no residual coupling during idle time for highly detuned qubits [72]. The infidelity is mainly due to leakage errors caused by nonadiabatic transitions from |01 and |11 to higher levels outside the qubit subspace (see Appendix C). For instance, when the gate time is 24 ns, 20% and 73% of the average infidelity are due to the leakage errors from |01 and |11 , respectively.

V. FLUX NOISE
Although the qubits are fixed-frequency transmons, the qubit frequencies vary a little depending on the flux in the coupler (see Appendix A). Here we examine the influence of the flux noise on the qubit coherence.
The coherence time T 2 for Qubit 1 in terms of the flux noise is formulated as [68] T where the coefficient A Φ is typically 10 −5 Φ 0 [66, 68] (T 2 for Qubit 2 is given similarly). Using the parameter values in Fig. 3  suggest the robustness of the proposed scheme against flux noise.
During the two-qubit gate, Θ ex changes from 0.61π to π. In this range of Θ ex , the minimum values of T 2 estimated as above are 30 µs and 5µs, respectively, for Qubits 1 and 2. Even in the worst-case scenario where the coherence time is assumed to be 5µs, the infidelity of the CPHASE gate with the gate time of 24 ns may increase to about 0.5%, which is still small. This rough estimation suggests that the flux noise may not degrade the gate performance very much.

VI. CONCLUSIONS
We have theoretically proposed a new kind of tunable coupler for superconducting quantum computers. We call this a double-transmon coupler, because this consists of two fixed-frequency transmons coupled through a common loop with an additional Josephson junction. We have numerically found that by tuning the external flux in the loop, residual ZZ coupling vanishes even for highly detuned computational qubits, in contrast to the conventional single-transmon coupler. Numerical simulations have also shown that the proposed coupler enables twoqubit gates with high fidelity of over 99.99% and a short gate time of 24 ns. The next step is experimental realization of this proposal, where relaxation and decoherence in transmons will degrade the performance. However, from its short gate time (24 ns) and recently reported long coherence times of transmons (over 300 µs) [73,74], the proposed coupler is expected to achieve high two-qubit gate fidelity. Another important issue in experiments is the unwanted deviation of critical currents of Josephson junctions from design values. The precision of the critical currents is known to be about 2%, though this can be reduced by laser annealing [75]. The effects of the critical-current deviation on the coupler performance are left as an important issue for future work.

Appendix A: Quantum-mechanical model
Using the constraint that φ 5 = φ 4 − φ 3 − Φ ex , the kinetic energy term in Eq. (1) can be expressed as and M is a capacitor matrix. The canonical conjugate variables for the flux variables, namely, charge variables Q, and the Hamiltonian are obtained as Introducing the Cooper-pair number variables as n = Q/(2e) (e is the elementary charge), H is rewritten as where W = e 2 2 M −1 and ω C34 = e 2 2C 34 have been introduced as frequency parameters. The variables are quantized by the commutation relation [φ i ,n j ] = iδ ij as follows.n i is represented by −i ∂ ∂ϕi and the eigenfunction ofn i is proportional to e iniϕi . In the basis of these eigenfunctions, we have the following matrix representation of operators: where we have truncated the number of Cooper pairs at ±N . Since the total system is composed of four subsystems (transmons), each operator in Eq. (A4) is represented by a tensor product of four operators, such asn 1 ⊗Î 2 ⊗Î 3 ⊗Î 4 , whereÎ i is the identity operator for the ith subsystem ofφ i andn i . From the addition theorem, cos(ϕ 4 − ϕ 3 − Θ ex ) can be expressed as In the matrix representation,Î i is given by the (2N + 1) × (2N + 1) unit matrix and the tensor product ⊗ is replaced by the Kronecker product of matrices. Thus, we obtain a (2N + 1) 4 × (2N + 1) 4 matrix representation of the Hamiltonian in Eq. (A4). In this work, we choose N = 10 for sufficient convergence of energies.
Numerically diagonalizing the Hamiltonian matrix withΘ ex = 0, we can obtain the energies, E ij,kl , of the state |ij |kl , where |ij and |kl denote the qubit state (Qubits 1 and 2) and the coupler state (Transmons 3 and  4), respectively. These energies lead to the ZZ-coupling strength ζ ZZ in Figs. 2 and 3(a). For example, the energies in the case of Fig. 3  The average gate fidelity is a standard metric for evaluating the performance of quantum gates. This is defined by averaging gate fidelities over uniformly distributed initial states. The average fidelities in Fig. 3(d) are obtained using the formula in Ref. 76, which is an extension of the formula in Ref. 77 to cases where there exist leakage errors and the norm of the qubit-subspace vector is not preserved. In the case of two-qubit gates, the formula for the average fidelityF is given bȳ where U id is a 4 × 4 unitary matrix corresponding to the ideal gate operation and U is a 4 × 4 matrix defined as follows. Suppose that we simulate the gate operation on four initial states each of which is one of the four two-qubit basis vectors denoted by |ψ ij (i, j = 0, 1). Using the resultant vectors |ψ ij , U is defined as U 2i+j,2i +j = ψ ij |ψ i j . Note that U is not a unitary matrix in general because of leakage errors. In the case of the CPHASE gate, we define U id as U id = diag(e iθ0 , e iθ1 , e iθ2 , e iθ3 ), where diag(· · · ) represents a diagonal matrix and e iθ k = U k,k /|U k,k |. By eliminating the overall phase factor and single-qubit phase rotations from U id , we define the rotation angle of the CPHASE gate as θ CPHASE = θ 3 − θ 1 − θ 2 + θ 0 .
Appendix C: Flux pulse shape As shown in Fig. 4, higher energy levels E 00,10 and E 10,10 approach the qubit levels E 01,00 and E 11,00 , respectively, around Θ ex = π. Thus, the infidelity of the CPHASE gate is mainly due to the leakage errors from |01 |00 to |00 |10 and from |11 |00 to |10 |10 . To reduce these leakage errors, we design the flux pulse shape based on the technique proposed in Ref. 71. Here we explain how we designed the pulse shape shown in Fig. 3(b).
The technique is based on the two-level system, |g and |e , with a constant coupling rate g and a timedependent detuning ∆(t), where the energy gap between the two energy eigenstates of this sysmte is given by ω gap = ∆ 2 + 4g 2 . We first focus on the two levels of |01 |00 and |00 |10 . In this case, we have ω gap = (E 00,10 − E 01,00 )/ , and 2g is given by the minimum of ω gap . The energy gap is shown in Fig. 5 together with other energy gaps. From this, we obtain g and the Θ ex dependences of ω gap and ∆.
However, we found that this pulse shape leads to relatively high leakage error probabilities from |11 |00 , though the energy gap between |11 |00 and |10 |10 is close to that between |01 |00 and |00 |10 , as shown in Fig. 5. The leakage errors may be due to the smaller energy gap between |11 |00 and |02 |00 around Θ ex = 0.61π. Also, slower change of Θ ex around Θ ex = π may be more desirable for increasing the rotation angle, because the ZZ-coupling strength becomes maximum there. Inspired by these, we modified the energy gap used for which is shown by the bold solid curve (in blue) in Fig. 5 [Θ (g) ex is Θ ex satsfying (E 00,10 − E 01,00 )/ = 2g]. Thus, we obtain the pulse shape shown in Fig. 3(b).

Appendix D: Parameter values in numerical studies
The parameter values used for the present numerical studies are set as follows. Note that the present system can be regarded as a network of capacitively coupled four transmons, except for the interaction between Transmons 3 and 4 through the additional Josephson junction given by the last term in Eq. (4). The transmon network is quantized by the standard method using bosonic operators [79]. Its Hamiltonian is given bŷ whereâ i andâ † i are the annihilation and creation operators, respectively, for the ith transmon. In this work, we set the transmon frequencies ω i and capacitances as design values, as given in Fig. 2. The parasitic capacitances not shown in Fig. 1 are ideally set to zero, but this is impossible in actual experiments. Among the parasitic capacitances, C 34 is set to a relatively large value, because Transmons 3 and 4 are directly coupled via the Josephson junction with critical current of I c5 . On the other hand, the other parasitic capacitances, which comprise nonadjacent transmons, are set to small values, which may be feasible by placing the nonadjacent transmons as far from each other as possible. The other parameters are determined by their definitions, together with ω J5 = (ω J3 + ω J4 )/8 (a quarter of the mean value of ω J3 and ω J4 ). Table I   The conventional single-transmon coupler is shown in Fig. 6(a), in which the dc SQUID for the frequencytunable transmon in the coupler is replaced by a single Josephson junction for simplicity. Figure 6(b) shows the ZZ-coupling strength, ζ ZZ , of the coupler with typical parameter values [43]. Note that ZZ-coupling vanishing points [the white region in Fig. 6(b)] exist only in the straddling regime [the dashed yellow box in Fig. 6(b)]. Thus, the single-transmon coupler cannot realize zero ZZ coupling for highly detuned qubits. This is an essential contrast to the proposed double-transmon coupler.