Weakly Flux-Tunable Superconducting Qubit

Flux-tunable qubits are a useful resource for superconducting quantum processors. They can be used to perform cphase gates, facilitate fast reset protocols, avoid qubit-frequency collisions in large processors, and enable certain fast readout schemes. However, flux-tunable qubits suffer from a trade-off between their tunability range and sensitivity to flux noise. Optimizing this trade-off is particularly relevant for enabling fast, high-fidelity, all-microwave cross-resonance gates in large, high-coherence processors. This is mainly because cross-resonance gates set stringent conditions on the frequency landscape of neighboring qubits, which are difficult to satisfy with nontunable transmons due to their relatively large fabrication imprecision. To solve this problem, we realize a coherent, flux-tunable, transmonlike qubit, which exhibits a frequency tunability range as small as $43\mathrm{MHz}$, and whose frequency, anharmonicity and tunability range are set by a few experimentally achievable design parameters. Such a weakly tunable qubit may be used to avoid frequency collisions in a large lattice while exhibiting minimal susceptibility to flux noise.

Figure 1

Transmon circuits. (a) Fixed-frequency transmon consisting of a single Josephson junction with energy $E_J$ shunted by a capacitor $C$. (b) Widely tunable transmon consisting of a symmetric dc SQUID with identical junctions $E_{J1}=E_{J2}=E_J/2$ shunted by a capacitor $C$. The qubit frequency can be tuned using an external magnetic flux ${\mathrm{\Phi }}_x$ threading the SQUID loop. (c) Medium-range tunable transmon consisting of an asymmetric dc SQUID with dissimilar junctions $E_{J2}\gg E_{J1}$, where $E_{J2}\cong E_J$, shunted by a capacitor $C$.

Figure 2

Circuit diagram of the WTQ. It consists of three Josephson junctions with inductances $L_{J_1}$, $L_{J_2}$, $L_{J_3}$ and self-capacitances $C_{J_1}$, $C_{J_2}$, $C_{J_3}$, respectively. The critical currents of the junctions are related to their inductances by the relation $I_{c,i}={\mathrm{\Phi }}_0/2\pi L_{J_i}$, for $i=1,2,3$. The SQUID loop formed by the junctions $J_2$ and $J_3$ is biased by the external dc flux ${\mathrm{\Phi }}_x$. The sensitivity of the SQUID loop to the flux noise is reduced by making the areas of the junctions $J_2$ and $J_3$ unequal, i.e., $I_{c,3}={\rho }_J{I}_{c,2}$ for some ${\rho }_J>1$. Capacitors $C_1$ and $C_2$ shunting the junctions create the two main modes of the circuit: the qubit mode formed by $J_1$ shunted with $C_1$ and the high-frequency mode formed by $C_2$ shunting the SQUID loop. The electrostatic interaction of the qubit mode with the SQUID through the capacitance $C_c$ provides the tunability of the WTQ.

Figure 3

Detailed WTQ circuit diagram. The flux bias in the SQUID loop is generated by the dc current source $I_B$ with impedance $Z(\omega )$. $L_c$ is the inductance of the coil that produces the magnetic field threading the SQUID loop. Two partial inductances [31] $L_1$ and $L_2$ are introduced to model the linear inductance of the SQUID loop, which are inductively coupled to the coil with mutual inductances $M_1$ and $M_2$, respectively. Such a finely detailed construction of the WTQ circuit avoids inconsistenties in the calculation of the decoherence rates.

Figure 4

Multiport Belevitch transformer with a turns-ratio matrix $\mathbf{T}$ is introduced to reduce the number of capacitors by one. The capacitance values are $C_A=C_1+C_{J_1}+C_c{C}_2/C_2+C_c$ and $C_B=C_2+C_c$. The turns-ratios are $t_{11}=1$, $t_{12}=C_c/C_2+C_c$, $t_{21}=0$, and $t_{22}=1$. Note that $C_{J_1}$ is added with $C_1$ and contributes to the value of the capacitance $C_A$. In blue are the branches corresponding to the spanning tree. The remaining branches are the chord branches that define the fundamental loop matrix in Eq. (3) [29]. The multiport Belevitch transformer is eliminated according to Ref. [34] to arrive at the effective fundamental loop matrix in Eq. (3).

Figure 5

A WTQ example. (a) Qubit frequency and (b) anharmonicity of the WTQ as a function of the normalized flux threading the SQUID loop. In this example, the WTQ has a maximum frequency and anharmonicity of about $5\mathrm{GHz}$ and $300\mathrm{MHz}$, respectively, and exhibits a frequency tunability $\delta =50\mathrm{MHz}$. Blue curves are obtained by the diagonalization of the Hamiltonian in Eq. (60) in the charge basis. Red dashed curves are calculated using the analytical formulas in Eqs. (64) and (73). (c) The high-frequency mode of the WTQ $f_{10}$ versus normalized flux. The WTQ circuit parameters employed in this example are $I_{c1}=I_{c2}=26\mathrm{nA}$, $I_{c3}={\rho }_J{I}_{c2}$, ${\rho }_J=3.5$, $C_1^{\prime }=50\mathrm{fF}$, $C_2^{\prime }=20\mathrm{fF}$, $C_c=20\mathrm{fF}$, where $C_{J_1}=C_{J_2}=1\mathrm{fF}$ and $C_{J_3}={\rho }_J{C}_{J_2}$. In plots (d)–(f), we evaluate the dependence of the WTQ coherence times, i.e., $T_1$, $T_{\varphi }$, and $T_2$ on the applied flux. In this calculation, we assume $T=0.02\mathrm{K}$ and $Z(\omega )=R=0.1\mathrm{ohm}$ and that the maximum values for $T_1$ and $T_2=1.5T_1$, i.e., $100\mu \mathrm{s}$ and $150\mu \mathrm{s}$, respectively, are set by loss or noise sources that are independent of flux or the flux biasing circuit. The $T_2$ curves in (f) are calculated using the relation $1/T_2=1/2T_1+1/T_{\varphi }$, which also sets the maximum for $T_{\varphi }$ in (e). The analytical calculation of $T_1$ and $T_{\varphi }$ uses the matrix element results of Eqs. (96)–(103). The exact calculation employs Eqs. (74) and (75) with a diagonalization of the Hamiltonian in Eq. (60).

Figure 6

Dependence of WTQ dephasing time $T_{\varphi }$ on the normalized flux threading the SQUID loop for varying parameter $R$ (a) and $T$ (b) of the flux biasing circuit, where $Z(\omega )=R$ is assumed. In (a),(b) the values of $T$ and $R$ are set to $0.02\mathrm{K}$ and $0.1\mathrm{ohm}$, respectively. The WTQ circuit parameters employed in this calculation are the same as Fig. 5. The curves in both plots are obtained by the diagonalization of the Hamiltonian in Eq. (60) in the charge basis along with Eqs. (74) and (75).

Figure 7

(a) Photo of one of the two seven-qubit chips measured in this work. The chip consists of six WTQs and one single-JJ transmon (Q4). The WTQs are realized using two gap-capacitance geometries shown in (b),(c). (b) P-shape WTQ (Q2, Q3, Q7), in which the capacitance pads are parallel. (c) U-shape WTQ (Q1, Q5, Q6), in which one capacitance pad is curved. The fabrication of the WTQ chips follows the same planar device fabrication process described in Ref. [16]. (d) Equivalent WTQ circuit. $C_1^{\prime }$ and $C_2^{\prime }$ represent the total capacitance shunting the JJs (including the self capacitance of the JJs). The small junction of the SQUID is comparable to the junction of the single-JJ transmon $E_{J2}\cong E_{J1}\cong E_J$, whereas the other JJ of the SQUID is slightly larger, giving ${\rho }_J=2-5$. Rounded corners used in the electrodes are an effort to minimize $E$-field concentration and are not specific to the WTQ design.

Figure 8

Representative qubit spectroscopy measurements of chip $A$ plotted versus normalized external flux. The data shows $f_{01}$, and $f_{02}/2$ curves. (a) WTQ2 (P shape). (b) WTQ3 (P shape). (c) WTQ5 (U shape). (d) WTQ6 (U shape). WTQ2 and WTQ5, having ${\rho }_J=3.5$, exhibit a small tunability of $50\mathrm{MHz}$ and $43\mathrm{MHz}$, respectively. WTQ3 and WTQ6, having ${\rho }_J=2.8$, exhibit a slightly larger tunability of $89$ and $86\mathrm{MHz}$. Dashed black curves are solutions to the Hamiltonian in Eq. (60).

Figure 9

Coherence measurements for chip $A$ taken for Q4 (a), WTQ2 (b), WTQ3 (c), and WTQ5 (d). Top to bottom rows show the qubit frequency $f_q$ (blue circles), $T_1$ (black stars), $T_{2E}$, $T_{2R}$ (red circles and magenta squares, respectively), and $T_{\phi }$ (blue circles) as a function of the applied coil bias current $I_B$. The solid black curves in the first row plots represent theory fits for the qubit frequency based on the diagonalization of the Hamiltonian. The solid blue curves drawn in the second row plots, represent fits based on Eq. (74). The solid red curves shown in the third row plots correspond to the calculated $T_{2E}$ based on the $T_1$ and $T_{\phi }$ fits (see main text). The dashed red, solid blue, and solid black curves shown in the fourth row plots represent the calculated $T_{\varphi ,D}$, $T_{\phi ,F}$, and $T_{\phi }$ fits, respectively (see main text and Table 5 for details). The observed maximum in $T_{\phi }$ of WTQs seen in fourth row plots around $-0.25\mathrm{mA}$ can be attributed to the increase of $T_{\phi ,F}$ near the net zero flux bias around $-0.5\mathrm{mA}$ (that cancels the nonzero stray magnetic field in the setup), and the decrease of $T_{\varphi ,D}$ due to the increase in the applied current magnitude responsible for heating.

Figure 10

Coherence measurements for chip $B$ taken for WTQ2 (a), WTQ3 (b), and WTQ5 (c). Top to bottom rows show the qubit frequency $f_q$ (blue circles), $T_1$ (black stars), $T_{2E}$, $T_{2R}$ (red circles and magenta squares, respectively), and $T_{\phi }$ (blue circles) as a function of the applied coil bias current $I_B$. The solid black curves in the first row plots represent theory fits for the qubit frequency based on the diagonalization of the Hamiltonian. The solid blue curves drawn in the second row plots, represent fits based on Eq. (74). The solid red curves shown in the third row plots correspond to the calculated $T_{2E}$ based on the $T_1$ and $T_{\phi }$ fits (see main text). The dashed red, solid blue, and solid black curves shown in the fourth row plots represent the calculated $T_{\varphi ,D}$, $T_{\phi ,F}$, and $T_{\phi }$ fits, respectively (see main text and Table 6 for details).

Figure 11

(a) WTQ flux sensitivities of qubits on chip $A$ and of example device, found by numerical differentiation of Eq. (64), using device parameters listed in Table 2 and Fig. 5. (b) Flux-noise-limited dephasing times $T_{\varphi }$, assuming $1/f$ flux-noise amplitude $A_{\mathrm{\Phi }}^{1/2}\sim 2\mu {\mathrm{\Phi }}_0$ at $1\mathrm{Hz}$.