Josephson qubits with gate and measurement fidelities surpassing the threshold of fault-tolerant quantum computing are attractive candidates for manufacturing scalable quantum computers. Microwave electronics, as a traditional way for qubit control and readout, have succeeded in obtaining gate fidelities beyond 99.9% [1] and realizing quantum supremacy [2]. However, the bottleneck of interconnection becomes significant when the number of qubits increases beyond a thousand due to quantitative restrictions on the input and output ports of the quantum processor and cryogenic transmission lines. It is desirable to introduce single flux quantum (SFQ) digital logic circuits [3] for control and readout [4]to overcome this bottleneck. Digital coherent XY control based on SFQ pulses to transmon qubits was proposed [5], and the fidelities of digital single-qubit gates were measured to be about 95% [6]. Methods of optimization of SFQ pulse sequences for single- [7–9] and two-qubit gates, such as cross-resonance and controlled phase (CZ) gates [10–13], have also been studied.

In order to control qubits with flexibility, SFQ-based devices for Z control have become a frontier that requires further research. McDermott et al. [4] proposed an SFQ-based coprocessor that operates at 3 K for controlling and measuring a quantum processor. This coprocessor requires SFQ-based flux digital-to-analog converters (flux DACs) [14] for Z control, which is an inspiring idea for creating a scalable superconducting quantum processor. Recently, Mohammad et al. [10] proposed an SFQ-based digital controller called DigiQ for superconducting qubits. In DigiQ, the Z control of qubits is performed with bias currents generated by an array of SFQ/DCs. These SFQ/DCs in DigiQ are placed at the 4 K plate of the dilution refrigerator, and bias currents need to be transmitted in superconducting microstrip flex lines to the 10-mK plate where the quantum processor works, which is similar to the approach used in [4].

To further advance the integration of qubits, it is crucial to integrate superconducting SFQ logic circuits for the control and measurement of qubits with quantum processors using 3D integration technologies in future developments [15,16]. Consequently, there is a need to research and design SFQ-based devices for on-chip Z control and ensure that the devices are as simple as possible for scalable quantum processors. These lean devices should be capable of converting SFQ pulse signals into flux signals. The circuits for Z control in DigiQ may be slightly more complex than flux DACs. Therefore, it is intuitive to consider flux DACs as the base for developing SFQ-based devices for Z control. However, the employment of a single flux DAC, as defined in [14], presents challenges in providing flux bias and completing a high-precision Z-control gate simultaneously because of the following reasons: 1) the resetting of a flux DAC at the end of a Z control operation will eliminate not only the flux performing the gate but also the bias flux setting the idle frequency of the controlled qubit and 2) the resetting is achieved by applying Φ₀/2 (half of a flux quantum) to the two-junction reset superconducting quantum interference device (SQUID) loop, which may necessitate another flux DAC, an SFQ/DC, or a coprocessor pin for an external current source. This ultimately increases the physical footprint and complexity of the coprocessor or exacerbates the interconnection bottleneck situation.

Here, we introduce a novel SFQ-based device for Z control, which stems from the improvement of flux DACs. The primary function of the qubit energy tuner (QET) is to adjust the energy levels of qubits. By supplying flux from a QET, the energy levels or frequency of a flux-tunable transmon qubit can be precisely set to specific values. Concurrently, QET enables the execution of gate operations that require flux bias, such as a Z gate or an iSWAP gate. Following a gate operation, the QET can seamlessly return the qubit frequency to its idle state.

The circuit structure of QETs is first presented in Section 2. Then, the key features of a specific QET are analyzed, and a formula is derived to calculate its inductor loop current responsible for providing flux. Section 3 describes the ideal Z control method, which employs square-wave-like currents for flux-tunable transmon. Subsequently, in Section 4, simulations are performed to demonstrate the variation in the inductor loop current of a QET providing flux due to SFQ pulses, as well as the performance of a QET in executing Z and iSWAP gates. Finally, in Section 5, the work is concluded, and the challenges and opportunities associated with QETs in the future are discussed.

QET comprises an inductor loop and several flux bias units, which can be either positive or negative, as illustrated in Figure 1. The inductor loop is weakly coupled to the SQUID of a flux-tunable transmon, supplying flux for tuning its energy levels. Each flux bias unit includes a Josephson junction shunted with an inductor coupled to the inductor loop. The Josephson junction can be made of an intrinsic Josephson junction in parallel with a resistor to be an overdamped Josephson junction. The node connected to the Josephson junction and the inductor is treated as an input port of QET for the SFQ pulse signal. After receiving an SFQ pulse, a positive flux bias unit increases the external flux through the SQUID by a specific amount, whereas a negative flux bias unit increases it in the opposite direction or decreases it by the same specific amount. This is realized by making the direction of dotted terminals of positive flux bias units the same as that of the corresponding inductor in the inductor loop but making the direction of dotted terminals of negative flux bias units opposite to that of the corresponding inductor in the inductor loop.

A QET should have at least a pair of flux bias units, one positive and the other negative. In order to tune the energy levels of a transmon more precisely, QET can be designed to have two or more pairs of flux bias units, among which some pairs are used for coarse tuning and others are used for fine-tuning. Inductors in different flux bias units can be coupled. The inspiration for the QET is from the design of the flux DAC proposed in [14] and [17]. Therefore, its circuits have similar but simpler structures compared with those of flux DACs.

The QET shown in Figure 2 is an example of the following analysis. It has a pair of flux bias units for coarse tuning and another pair for fine-tuning. The parameters of elements in the example are listed in Table 2. The symbol for QET in Figure 2 is drawn as in Figure 3. This kind of QET with two pairs of flux bias units is chosen to be analyzed because it combines accuracy, simplicity, and speed better than other cases with only one or over two pairs of flux bias units. On one hand, the QET with only a pair of flux bias units has only one precision, which causes a low speed of high-precision tuning or a low precision of high-speed tuning. On the other hand, the QET with three or more pairs of flux bias units has more ports and circuit elements, which means more complicated control, reduced reliability, and a larger footprint.

According to Kirchhoff’s voltage law, the electric potentials of nodes A, B, C, D, and E in Figure 2 are V A t ′ = L 1 d i 1 t ′ d t ′ + M 1 d i p t ′ d t ′ + M 12 d i 2 t ′ d t ′ , (1) V B t ′ = L 2 d i 2 t ′ d t ′ − M 2 d i p t ′ d t ′ + M 12 d i 1 t ′ d t ′ , (2) V C t ′ = L 3 d i 3 t ′ d t ′ + M 3 d i p t ′ d t ′ + M 34 d i 4 t ′ d t ′ , (3) V D t ′ = L 4 d i 4 t ′ d t ′ − M 4 d i p t ′ d t ′ + M 34 d i 3 t ′ d t ′ , (4) V E t ′ = L Σ d i p t ′ d t ′ + M 1 d i 1 t ′ d t ′ − M 2 d i 2 t ′ d t ′ + M 3 d i 3 t ′ d t ′ − M 4 d i 4 t ′ d t ′ + M d i q t ′ d t ′ , (5) where L Σ = L n0 + L n 1 + L n 2 + L n 3 + L n 4 + L n 5 (6) is the total inductance of the inductor loop obtained by summing self-inductances of all parts of the inductor loop. L ₁, L ₂, L ₃, and L ₄ are self-inductances of inductors in flux bias units. M ₁, M ₂, M ₃, and M ₄ are mutual inductances of flux bias units and the inductor loop, as shown in Figure 2. M is the mutual inductance between the inductor loop and the SQUID of flux-tunable transmon. i ₁ (t′), i ₂ (t′), i ₃ (t′), and i ₄ (t′) are currents of inductors in flux bias units at the moment t′. i _p (t′) and i _q (t′) are the currents of the inductor loop and the SQUID at the moment t′.

Under the zero initial condition, integrating both sides of Eqs 1–5 with 0 as the lower bound and time t as the upper bound yields ∫ 0 t V A t ′ d t ′ = L 1 i 1 t + M 1 i p t + M 12 i 2 t , (7) ∫ 0 t V B t ′ d t ′ = L 2 i 2 t − M 2 i p t + M 12 i 1 t , (8) ∫ 0 t V C t ′ d t ′ = L 3 i 3 t + M 3 i p t + M 34 i 4 t , (9) ∫ 0 t V D t ′ d t ′ = L 4 i 4 t − M 4 i p t + M 34 i 3 t , (10) ∫ 0 t V E t ′ d t ′ = L Σ i p t + M 1 i 1 t − M 2 i 2 t + M 3 i 3 t − M 4 i 4 t + M i q t . (11) The mutual inductance M is designed to be much smaller than the total inductance of the inductor loop L _Σ and other mutual inductances like M ₁ for weak coupling to the SQUID of the qubit. Additionally, the ring current of the SQUID i _q(t) should be less than the critical current of its Josephson junctions, which is about tens of nA for Al/AlOx/Al junctions and smaller than the current in the inductance loop i _p(t) (about several or tens of mA) by two or more orders of magnitude. Therefore, the influence of the SQUID on the inductance loop, Mi _q, can be ignored in Eq. 11, and the electric potential of node E is rewritten as ∫ 0 t V E t ′ d t ′ = L Σ i p t + M 1 i 1 t − M 2 i 2 t + M 3 i 3 t − M 4 i 4 t . (12)

Then, let Φ A t = ∫ 0 t V A t ′ d t ′ , (13) Φ B t = ∫ 0 t V B t ′ d t ′ , (14) Φ C t = ∫ 0 t V C t ′ d t ′ , (15) Φ D t = ∫ 0 t V D t ′ d t ′ , (16) Φ E t = ∫ 0 t V E t ′ d t ′ , (17) and we get Φ t = L i t , (18) where Φ t = Φ A t Φ B t Φ C t Φ D t Φ E t , (19) i t = i 1 t i 2 t i 3 t i 4 t i p t , (20) and L = L 1 M 12 0 0 M 1 M 12 L 2 0 0 − M 2 0 0 L 3 M 34 M 3 0 0 M 34 L 4 − M 4 M 1 − M 2 M 3 − M 4 L Σ . (21)

Then, to get i (t), we have i t = L − 1 Φ t , (22) where L − 1 = 1 F A . 1 F is the common factor of the elements in the inverse matrix of L . F is F = L 2 M 34 2 L 1 L Σ − M 1 2 + 2 L 1 M 3 M 34 M 4 + L 3 − L 4 L 1 L Σ − M 1 2 + L 1 M 4 2 + L 1 L 4 M 3 2 + − L 1 M 2 2 − L Σ M 12 2 − 2 M 1 M 12 M 2 M 34 2 − 2 M 12 2 M 3 M 34 M 4 + L 3 L 1 M 2 2 + L Σ M 12 2 + 2 M 1 M 12 M 2 L 4 − M 12 2 M 4 2 − L 4 M 12 2 M 3 2 . (23)

The elements a _ij (i, j = 1, 2, 3, 4, 5) of A are a 11 = L 2 M 34 2 L Σ + 2 M 3 M 4 M 34 + − L 4 L Σ + M 4 2 L 3 + L 4 M 3 2 + M 2 2 L 3 L 4 − M 34 2 , a 12 = M 12 − M 34 2 L Σ − 2 M 3 M 4 M 34 + L 4 L Σ − M 4 2 L 3 − L 4 M 3 2 + M 1 M 2 L 3 L 4 − M 34 2 , a 13 = − L 4 M 3 + M 34 M 4 L 2 M 1 + M 12 M 2 , a 14 = L 2 M 1 + M 12 M 2 L 3 M 4 + M 3 M 34 , a 15 = L 3 L 4 − M 34 2 L 2 M 1 + M 12 M 2 , a 21 = M 12 − M 34 2 L Σ − 2 M 3 M 4 M 34 + L 4 L Σ − M 4 2 L 3 − L 4 M 3 2 + M 1 M 2 L 3 L 4 − M 34 2 , a 22 = L 1 M 34 2 L Σ + 2 M 3 M 4 M 34 + − L 4 L Σ + M 4 2 L 3 + L 4 M 3 2 + M 1 2 L 3 L 4 − M 34 2 , a 23 = L 4 M 3 + M 34 M 4 L 1 M 2 + M 1 M 12 , a 24 = − L 1 M 2 + M 1 M 12 L 3 M 4 + M 3 M 34 , a 25 = − L 3 L 4 − M 34 2 L 1 M 2 + M 1 M 12 , a 31 = − L 4 M 3 + M 34 M 4 L 2 M 1 + M 12 M 2 , a 32 = L 4 M 3 + M 34 M 4 L 1 M 2 + M 1 M 12 , a 33 = L 4 L Σ M 12 2 + 2 M 1 M 2 M 12 + − L 2 L Σ + M 2 2 L 1 + L 2 M 1 2 + M 4 2 L 1 L 2 − M 12 2 , a 34 = M 34 − L Σ M 12 2 − 2 M 1 M 2 M 12 + L 2 L Σ − M 2 2 L 1 − L 2 M 1 2 + M 3 M 4 L 1 L 2 − M 12 2 , a 35 = L 1 L 2 − M 12 2 L 4 M 3 + M 34 M 4 , a 41 = L 2 M 1 + M 12 M 2 L 3 M 4 + M 3 M 34 , a 42 = − L 1 M 2 + M 1 M 12 L 3 M 4 + M 3 M 34 , a 43 = M 34 − L Σ M 12 2 − 2 M 1 M 2 M 12 + L 2 L Σ − M 2 2 L 1 − L 2 M 1 2 + M 3 M 4 L 1 L 2 − M 12 2 , a 44 = L 3 L Σ M 12 2 + 2 M 1 M 2 M 12 + − L 2 L Σ + M 2 2 L 1 + L 2 M 1 2 + M 3 2 L 1 L 2 − M 12 2 , a 45 = − L 1 L 2 − M 12 2 L 3 M 4 + M 3 M 34 , a 51 = L 3 L 4 − M 34 2 L 2 M 1 + M 12 M 2 , a 52 = − L 3 L 4 − M 34 2 L 1 M 2 + M 1 M 12 , a 53 = L 1 L 2 − M 12 2 L 4 M 3 + M 34 M 4 , a 54 = − L 1 L 2 − M 12 2 L 3 M 4 + M 3 M 34 , a 55 = − L 3 L 4 − M 34 2 L 1 L 2 − M 12 2 . Therefore, we have i p t = 1 F a 51 Φ A t + a 52 Φ B t + a 53 Φ C t + a 54 Φ D t + a 55 Φ E t , (26) that is, i p t = 1 F Φ A t L 3 L 4 − M 34 2 L 2 M 1 + M 12 M 2 − Φ B t L 3 L 4 − M 34 2 L 1 M 2 + M 1 M 12 + Φ C t L 1 L 2 − M 12 2 L 4 M 3 + M 34 M 4 − Φ D t L 1 L 2 − M 12 2 L 3 M 4 + M 3 M 34 − Φ E t L 3 L 4 − M 34 2 L 1 L 2 − M 12 2 . (27) Because node E is connected to the ground, Φ _E should always be zero. In order to make the flux bias units of coarse tuning be able to increase or decrease the external flux through the SQUID by the same amount, the following requirements should be met: L 1 = L 2 = L c , (28a) M 1 = M 2 = M c . (28b)

Similarly, for the flux bias units of fine-tuning, we have L 3 = L 4 = L f , (29a) M 3 = M 4 = M f . (29b) Therefore, i _p(t) becomes i p t = 1 F Φ A − Φ B L f 2 − M 34 2 L c + M 12 M c + Φ C − Φ D L c 2 − M 12 2 L f + M 34 M f , (30) where Φ _A, Φ _B, Φ _C, and Φ _D are the integral of the voltage at nodes A, B, C, and D over time t, respectively. Because the input signal to these nodes is SFQ, Φ _A, Φ _B, Φ _C, and Φ _D are multiples of flux quantum Φ₀. F becomes F = L c L f − L Σ L f + M f 2 L c + L f M c 2 + L f + M 34 L Σ L f − L Σ M 34 − 2 M f 2 M 12 2 + 2 L f 2 M c 2 − 2 M 34 2 M c 2 M 12 + L c L Σ L c − 2 M c 2 M 34 2 + 2 L c M f M f M 34 + L c L f M f 2 + L f 2 M c 2 . (31) Hence, the relationship between the current of the inductor loop i _p(t) and the external flux through the SQUID Φ _e is Φ e = M i p t . (32)

Denoting Φ A − Φ B = n c Φ 0 , (33) Φ C − Φ D = n f Φ 0 , (34) M F L f 2 − M 34 2 L c + M 12 M c = r c , (35) M F L c 2 − M 12 2 L f + M 34 M f = r f , (36) 1 F L f 2 − M 34 2 L c + M 12 M c Φ 0 = Δ i pc , (37) 1 F L c 2 − M 12 2 L f + M 34 M f Φ 0 = Δ i pf , (38) M Δ i pc = Φ ec , (39) M Δ i pf = Φ ef (40) yields i p = n c Δ i pc + n f Δ i pf , (41) Φ e = n c Φ ec + n f Φ ef , (42) Φ ec = r c Φ 0 , (43) Φ ef = r f Φ 0 . (44) Eqs 42–44 mean that the flux provided by QET can be divided into two parts, n _c Φ _ec and n _f Φ _ef, which are, respectively, created by coarse tuning and fine-tuning. Φ _ec can be regarded as the flux unit of coarse tuning, and Φ _ef can be regarded as the flux unit of fine-tuning. If n _c (or n _f) SFQ pulses are inputted to port A (or C) of the QET, then the external flux through the SQUID will increase by n _c times of Φ _ec (or n _f times of Φ _ef). Subsequently, if this external flux needs to be eliminated, n _c (or n _f) SFQ pulses should be inputted to port B (or D). Usually, for fine-tuning, r _f is smaller than r _c. If L c = L 1 = L n 1 = L 2 = L n 2 , (45) L f = L 3 = L n 3 = L 4 = L n 4 , (46) then the ratio of the flux unit of coarse tuning to the flux unit of fine-tuning can be defined as r cf = Φ ec Φ ef . (47) With Eq. 33 ∼Eq. 47, there is r cf = r c r f = Δ i pc Δ i pf = K c 1 − K 34 K f 1 − K 12 , (48) where the coupling coefficients are K c = M c L 1 L n 1 = M c L 2 L n 2 = M c L c , (49) K f = M f L 1 L n 1 = M f L 2 L n 2 = M f L f , (50) K 12 = M 12 L 1 L 2 = M 12 L c , (51) K 34 = M 34 L 3 L 4 = M 34 L f . (52)

The parameter r _cf indicates the ratio of the flux precision of coarse tuning to that of fine-tuning. It should have an appropriate value greater than 1, like 10, to distinguish the two precisions. The parameter r _f is the ratio of the smallest variation of the flux Φ _e, Φ _ef, to flux quantum Φ₀; it determines the flux precision of fine-tuning. The parameter r _c is the ratio of Φ _ec to flux quantum Φ₀ and can be set by r _c = r _cf ⋅ r _f.

The parameters r _cf, r _f, and r _c are the main concerns and should be first determined to design a QET. Then, with constraints including Eqs 35, 36, 45, 46, 48 ∼Eq. 52, all parameter values of circuit elements should be tried and iterated to meet the requirements from the higher-level design; for example, the footprint of the QET on the chip is matched with the footprint of the qubit.

A QET has almost zero static power dissipation in theory because no current will flow through its resistors when Josephson junctions are not switched. Assuming that a QET is used for a gate operation per 20 ns and about 20 SFQ pulses are required for each gate operation, the average switching frequency of Josephson junctions in the QET, f _s, is about 1 GHz. With the critical current I _C = 160 μA and a flux quantum Φ ₀ ≈ 2.06783385 × 10⁻¹⁵Wb, the dynamic power dissipation P _D of a QET can be estimated to be 0.33 nW according to the following formula [18]: P D = f s Φ 0 I C . (53)

In order to simplify the analysis and understand the main factors affecting the results of Z control, the square-wave-like waveforms of currents producing external flux Φ _e are considered as the ideal cases for a flux-tunable transmon. This section reviewed and discussed how the ideal Z control is performed.

The Hamiltonian of a flux-tunable transmon [19] is H ̂ = 4 E C n ̂ − n g 2 − E JS φ e cos ϕ ̂ , (54) where E JS φ e = E J Σ | cos φ e | 1 + d 2 ⁡ tan 2 φ e (55) is the effective Josephson energy of the SQUID of the transmon with a total Josephson coupling energy of two junctions E J Σ = E J 1 + E J 2 , (56) an asymmetry coefficient d = E J 2 − E J 1 E J Σ , (57) and a reduced external flux φ e = π Φ e t Φ 0 . (58) E _C is the charging energy of the transmon. n _g is the effective offset charge. n ̂ and ϕ ̂ are, respectively, the number operator and the phase operator of Cooper pairs. For convenience, the hats of all operators, including Hamiltonian, are left out in the following derivation.

The solution for the k ^th eigen energy of Eq. 54 with the first-order approximation of perturbation theory [19] is E k = k 8 E C E JS φ e − E C 12 6 k 2 + 6 k + 3 − E JS φ e . (59) Usually, the external flux Φ _e(t) at the moment t for Z control is provided by a conductor line beside the SQUID with its current, i _z(t), and a mutual inductance between the line and the SQUID, M. Because the current in the SQUID is much smaller than i _z(t), its influence on i _z(t) can be ignored. There is Φ e t = M i z t . (60) Therefore, E _JS (φ _e) can be written as E JS i z t = E J Σ cos π M i z t Φ 0 1 + d 2 ⁡ tan 2 π M i z t Φ 0 . (61) By changing i _z(t), E _JS (i _z(t)) can be set to a target value. Then, the energy level E _k, especially E ₀ and E ₁, can be tuned so that the qubit frequency is set to the corresponding target value. When E _JS (i _z(t)) is set, the condition E _JS (i _z(t))/E _C ≫ 1 should be guaranteed to make sure that the qubit is a transmon. According to Eq. 59, we have E 0 = − E C 4 − E JS φ e , (62) E 1 = 8 E C E JS φ e − E C − E C 4 − E JS φ e , (63) E 2 = 2 8 E C E JS φ e − 3 E C − E C 4 − E JS φ e . (64) Therefore, the differences of energy levels are E 10 = E 1 − E 0 = 8 E C E JS φ e − E C , (65) E 21 = E 2 − E 1 = 8 E C E JS φ e − 2 E C , (66) and the anharmonicity of the qubit is α = E 21 − E 10 = − E C . (67) Without losing generality, i _z(t) has a square-wave-like waveform and is set as i z t = i w , t s ⩽ t ⩽ t e , i i , 0 ⩽ t < t s o r t > t e , (68) where i _w and i _i are the currents for setting working frequency ω _qw and idle frequency ω _qi of a transmon, respectively. ω _qw is the qubit frequency used for Z control. ω _qi is the qubit frequency when it is idle and is determined to be the frequency of the rotating frame [20]. t _s and t _e are the moments when a gate operation starts and ends, respectively. Then, the qubit frequency becomes ω q t = ω qw , t s ⩽ t ⩽ t e , ω qi , 0 ⩽ t < t s o r t > t e , (69) where ω qw = 8 E C E JS i w − E C / ℏ , (70) ω qi = 8 E C E JS i i − E C / ℏ . (71) Here, we denote Δ ω q = ω qw − ω qi . (72) With Eqs 70–72, we have Δ ω q = 8 E C E JS i w − E JS i i . (73)

For an idle qubit, its Hamiltonian is H 0 = ℏ ω qi a † ⁡ a + α 2 a † a † a a . (74) Actually, the time-dependent Hamiltonian of the qubit is H = ℏ Δ ω t a † a + ω qi a † a + α 2 a † a † a a , (75) where a ^† and a are the creation and annihilation operators, respectively, and Δω(t) is defined by Δ ω t = ω q t − ω qi . (76) ω _q(t) is the actual frequency of the qubit. We denote H dz = ℏ Δ ω t a † a (77) as the drive Hamiltonian for Z control. Therefore, there is H = H 0 + H dz . (78) In the rotating frame, the drive Hamiltonian for Z control becomes H ̃ = H dz , (79) and the corresponding evolution operator in the rotating frame is U ̃ dz = T ⁡ exp − i ∫ t s t e H ̃ ℏ d t , (80) where T is the chronological operator. With Eqs 77–80, we have U ̃ dz = T ⁡ exp − i a † a ∫ t s t e Δ ω t d t . (81) In the ideal situation in which i _w and i _i are constants, when the qubit is working (t _s⩽t⩽t _e), there is ω _q(t) = ω _qw, so Δω(t) becomes the constant Δω _q: Δ ω t = Δ ω q = ω qw − ω qi , (82) and the Hamiltonian H becomes H = H w = ℏ ω qw a † a + α 2 a † a † a a . (83) We define φ = − ∫ t s t e Δ ω t d t (84) as the phase shift realized by Z control, and define t z = t e − t s (85) as the gate operation time for Z control. With Eqs 73, 82, 84, 85, we have φ = − Δ ω q t z = 8 E C E JS i i − E JS i w t z . (86) According to Eq. 81, the corresponding evolution operator for the qubit becomes U ̃ dz = 1 0 0 e i φ = 1 0 0 exp i 8 E C E JS i i − E JS i w t z . (87) To realize on-chip Z control by SFQ, instead of choosing the Z control line, i _i and i _w can be produced by the inductor loop current i _p(t) of a QET, which means making i _z(t) = i _p(t).