Coupler-Assisted Controlled-Phase Gate with Enhanced Adiabaticity

High-fidelity two-qubit entangling gates are essential building blocks for fault-tolerant quantum computers. Over the past decade, tremendous efforts have been made to develop scalable high-fidelity two-qubit gates with superconducting quantum circuits. Recently, an easy-to-scale controlled-phase gate scheme that utilizes the tunable-coupling architecture with fixed-frequency qubits [Phys. Rev. Lett. 125, 240502 (2020); Phys. Rev. Lett. 125, 240503 (2020)] has been demonstrated with high fidelity and attracted broad interest. However, in-depth understanding of the underlying mechanism is still missing, preventing us from fully exploiting its potential. Here we present a comprehensive theoretical study, explaining the origin of the high-contrast $ZZ$ interaction. Based on improved understanding, we develop a general yet convenient method for shaping an adiabatic pulse in a multilevel system, and identify how to optimize the gate performance from design. Given state-of-the-art coherence properties, we expect the scheme to potentially achieve a two-qubit gate error rate near ${10}^{-5}$, which would drastically speed up the progress towards fault-tolerant quantum computation.

Figure 1

(a) Conceptual sketch of the tunable-coupling architecture, where the direct coupling strength between two fixed-frequency qubits $Q_1$ and $Q_2$ ($g_{12}$) is considered much weaker than the coupling between the qubits and the coupler $C$ ($g_{1c}$, $g_{2c}$). (b) Circuit diagram for implementing the architecture described in (a) with superconducting quantum circuits. Single-junction transmon qubits and a split-transmon coupler are all capacitively coupled to each other. Both the qubit-coupler coupling capacitances $C_{ic}(i=1,2)$ and the qubit-qubit coupling capacitance $C_{12}$ are much smaller than the shunt capacitances of the transmons $C_i(i=1,2,c)$.

Figure 2

(a) Energy-level spectra of the circuit Hamiltonian in Eq. (1) as a function of the coupler frequency ${\omega }_c/2\pi$. Solid (dashed) lines correspond to energies (frequencies) of adiabatic (diabatic) states. Points $A$, $B$, and $C$ indicate level crossings between different pairs of diabatic states. The horizontal range is divided into the near-resonance regime and the dispersive regime. Circuit parameters are: ${\omega }_{1,2}/2\pi =6.0,5.4$ GHz, ${\alpha }_{1,2,c}/2\pi =-250,-250,-300$ MHz, ${\rho }_{1c,2c,12}=0.018,0.018,0.0015$. Inset: enlargement of the interested region with avoided crossings. The dotted black line denotes the sum frequency of $|100⟩$ and $|001⟩$, so the difference between the dotted black line and the solid red line ($|101⟩$)—the shaded region—indicates the $ZZ$ interaction strength $\zeta$. (b) Absolute value of $\zeta /2\pi$ versus ${\omega }_c/2\pi$ with (solid) and without (dashed) direct qubit-qubit coupling.

Figure 3

(a) Absolute value of the $ZZ$ interaction strength $|\zeta |$ versus qubit-coupler detuning ${\mathrm{\Delta }}_{1c}$ and direct qubit-qubit coupling strength $g_{12}$, given qubit-qubit detuning ${\mathrm{\Delta }}_{12}/2\pi =600$ MHz (left panel) and 150 MHz (right panel). The coupling strength between the qubits and the coupler are $g_{1c,2c}/2\pi =120,100$ MHz and the anharmonicities are ${\alpha }_{1,2,c}/2\pi =-250,-250,-300$ MHz. The dashed black lines indicate zero effective qubit-qubit coupling, i.e., $\tilde{g}=0$. (b) The $ZZ$ interaction strength $\zeta$ (numerically simulated) versus the effective qubit-qubit coupling strength $\tilde{g}$ for different qubit-qubit detunings ${\mathrm{\Delta }}_{12}$. The direct qubit-qubit coupling strength $g_{12}/2\pi =8$ MHz. According to Eq. (4), $\zeta$ is approximately a quadratic function of $\tilde{g}$. The parabolas open upward in the small-detuning regime (${\mathrm{\Delta }}_{12}<|{\alpha }_q|$) or downward in the large-detuning regime (${\mathrm{\Delta }}_{12}<|{\alpha }_q|$). These parabolas share a common point (black dot) at $\tilde{g}={\alpha }_q\nu$ and $\zeta =(8{\alpha }_c+4{\alpha }_q){\nu }^2$, where $\nu =g_{1c}{g}_{2c}/(2{\mathrm{\Delta }}_{1c}{\mathrm{\Delta }}_{2c})$.

Figure 4

Simulated nonadiabatic error of cz gates. Unless specified otherwise, the circuit parameters in the plots are: ${\omega }_{1,2}/2\pi =6.0$, 5.4 GHz, ${\alpha }_{1,2,c}/2\pi =-250,-250,-300$ MHz, ${\rho }_{1c,2c,12}=0.018,0.018,0.0015$. (a) Gate error versus gate time for different pulse shapes, including the Fourier basis and AWP with different component numbers. The coupler idles at ${\omega }_c/2\pi =7.87$ GHz for minimizing $|\zeta |$. For each gate time, we numerically search for the minimum error rate using the Nelder-Mead method. (b) Gate error of the 30-ns cz gate generated from one-component AWP versus deviations in frequencies (left), anharmonicities (center), and coupling coefficients (right) with respect to the configuration in (a). (c) Gate error of the 30-ns cz gate with distorted waveform. As for distortion, we consider a reflected pulse added onto the original waveform with a delay time $T_d=10$ ns and a varying reflection coefficient $r$. Inset: an example pulse waveform before (dashed curve) and after (solid curve) distortion. We include a few configurations targeting different qubit-coupler coupling strengths: $g_{1c,2c}/2\pi \approx g_{\mathrm{QC}}=70$ MHz (${\rho }_{1c,2c,12}=0.012,0.012,0.0006$; solid light green line), 105 MHz (${\rho }_{1c,2c,12}=0.018,0.018,0.0015$; solid blue and orange lines), and 175 MHz (${\rho }_{1c,2c,12}=0.03,0.03,0.0036$; solid pink line). The resulting residual $ZZ$ interaction ${\zeta }^{\mathrm{idle}}$ at ${\omega }_c/2\pi \approx 8$ GHz is also listed for each case. For comparison, we also plot the results from the traditional coupler-free design (dashed lines). We assume that ${\omega }_{1,2}/2\pi =6.0,8.0$ GHz when idling (${\omega }_2$ is tunable) and the qubit-qubit coupling strength $g_{\mathrm{QQ}}/2\pi =30$ MHz (${\rho }_{12}=0.005$). Considering the finite bandwidth of control electronics, all pulses are filtered by a Gaussian low-pass filter with a 300-MHz cutoff frequency.

Figure 5

(a) Optimized cz gate error using one-component AWP ($T_g=30$ ns) versus the qubit-qubit detuning ${\mathrm{\Delta }}_{12}$ and coupler anharmonicity ${\alpha }_c$. The circuit parameters are ${\omega }_1/2\pi =6.0$ GHz, ${\alpha }_{1,2}/2\pi =-250$ MHz, and ${\rho }_{1c,2c,12}=0.018,0.018,0.0015$. The two dashed lines indicate the conditions ${\mathrm{\Delta }}_{12}+{\alpha }_c=0$ and ${\mathrm{\Delta }}_{12}+{\alpha }_1=0$, dividing the plot into four different regions. (b)–(e) Level diagrams of the four typical cases marked in (a). The circuit parameters are ${\mathrm{\Delta }}_{12}/2\pi =100$ MHz, ${\alpha }_c/2\pi =200$ MHz for the case in (b); ${\mathrm{\Delta }}_{12}/2\pi =100$ MHz, ${\alpha }_c/2\pi =-800$ MHz for the case in (c); ${\mathrm{\Delta }}_{12}/2\pi =800$ MHz, ${\alpha }_c/2\pi =-500$ MHz for the case in (d); ${\mathrm{\Delta }}_{12}/2\pi =800$ MHz, ${\alpha }_c/2\pi =-1200$ MHz for the case in (e). Points $A$ and $B$ indicate the same level crossings as in Fig. 2. Point $D$ indicates the crossing between $|101{⟩}^0$ and $|020{⟩}^0$. (f),(g) Magnitude of the $ZZ$ interaction $|\zeta |$ (left panel) and the maximum instantaneous $D$-factor $D^{\ast }({\omega }_c)$ (right panel) versus the coupler frequency ${\omega }_c$ and anharmonicity ${\alpha }_c$ in the small-detuning regime (${\mathrm{\Delta }}_{12}/2\pi =100$ MHz) and in the large-detuning regime (${\mathrm{\Delta }}_{12}/2\pi =800$ MHz). The two dashed lines indicate the conditions of crossing $D$ and $A$. The arrows indicate the coupler anharmonicities used in (b)–(e).

Figure 6

(a) Layout of the 2D qubit array in a square lattice. Red and blue squares representing qubits in different frequency bands are alternately arranged to keep large detuning between neighboring qubits. Gray circles denote tunable couplers. Arrows indicate stray couplings. (b) Error rates of $\mathrm{CZ}\otimes I$ (left) and $I\otimes \mathrm{CZ}$ (right) versus the stray coupling strength. Different colors correspond to different types of stray coupling. The parameters for the five-mode circuit are: ${\omega }_{1,2,3}/2\pi =6.0,5.4,6.1$ GHz, ${\alpha }_{q,c}/2\pi =-250,-300$ MHz, and ${\rho }_{\mathrm{QC,\; QQ}}=0.018,0.0015$. Gate time is 30 ns. The AWP method is used in pulse finding. (c) Spectra of energy levels in the single-excitation manifold (left panel). Enlargement of the dashed area is shown for the cases of zero (top right) and finite (bottom right) stray couplings between $Q_3$ and $C_1$.

Figure 7

(a) Categorization of major decoherence processes according to various properties. The coupling axis refers to the direction along which the noise coupled to the corresponding mode (isolated). Noise frequency indicates the relevant spectral components that contribute to the corresponding decoherence processes. Most intermanifold transitions are transitions within the computational subspace ($\mathcal{S}\to \mathcal{S}$) with the exception of $|101⟩\to |010⟩$. Most intramanifold transitions are leakage transitions ($\mathcal{S}\to \mathcal{L}$) with the exception of $|100⟩\leftrightarrow |001⟩$. (b) Calculated instantaneous transition rates versus the coupler frequency for various types of transitions. The coupler is assumed to be a split transmon with a maximum frequency of 8.2 GHz. Other parameters are the same as those used in Fig. 4. Both qubits have $T_1=20\mu \mathrm{s}$. The coupler $T_1$ is assumed to be shorter, $10\mu \mathrm{s}$. For longitudinal noise, we assume flux noise with a $1/f$-type spectrum, $S_{\mathrm{\Phi }}(\omega )=A_{\mathrm{\Phi }}/(|\omega |/2\pi )$, with $A_{\mathrm{\Phi }}=(10\mu {\mathrm{\Phi }}_0{)}^2$. Vertical lines are references to level crossings $A$ and $B$, the same as those introduced in Fig. 2. (c) Calculated instantaneous pure-dephasing rates versus the coupler frequency. We assume a total quasistatic flux fluctuation of ${\sigma }_{\mathrm{\Phi }}=60\mu {\mathrm{\Phi }}_0$ that corresponds to the same $1/f$ noise used above. (d) Error rate per gate (EPG) in the RB metric versus gate time for different decohering processes. Noise-induced errors—transitional (solid red line) and phase (solid blue line)—are calculated from the model described in Eqs. (12)–(13) using the same noise assumed above. Also shown is the case assuming $T_1=1$ ms (dashed red line) and ${\sigma }_{\mathrm{\Phi }}=6\mu {\mathrm{\Phi }}_0$ (dashed blue line). Coherent (nonadiabatic) errors are from the numerical simulation.

Figure 8

(a) Sketch of four coupler-assisted control schemes. The solid lines represent the absolute $ZZ$ interaction strength. Circuit parameters: ${\omega }_1/2\pi =6.0$ GHz, ${\omega }_2/2\pi =5.4$ GHz for the large-detuning cases (red and blue), 5.9 GHz for the small-detuning cases (orange and green), ${\rho }_{\mathrm{QC,\; QQ}}=0.03,0.004$ for the CAQ cases (red and orange), ${\rho }_{\mathrm{QC,\; QQ}}=0.04,-0.004$ for the CBQ cases (blue and green), and ${\alpha }_{1,2,c}/2\pi =-250,-250,-300$ MHz. The valleys in the small detuning cases correspond to two roots of $\zeta =0$, as discussed in Sec. 2c. (b) Plot of $D({\omega }_c)$ for the four cases. The black dots indicate where the coupler is idly biased.

Figure 9

Level structures of a tunable coupling system using the two-level approximation (a) and full model (b). The lowest three manifolds are shown with gray (${\mathcal{M}}_0$), cyan (${\mathcal{M}}_1$), and yellow (${\mathcal{M}}_2$) backgrounds. Solid and dashed arrows represent strong qubit-coupler coupling ($\propto g_{ic},i=1,2$) and weak qubit-qubit coupling ($\propto g_{12}$), respectively. The frequency shift of a computational state is illustrated as the rectangle between the corresponding adiabatic state (solid horizontal) and diabatic state (dashed horizontal).

Figure 10

Absolute $ZZ$ interaction strength versus ${\omega }_c$ and $g_{12}$, calculated using numerical diagonalization (a),(b) and Eq. (4) (c),(d). The detuning between the qubits (${\mathrm{\Delta }}_{12}/2\pi$) is set at 600 MHz for the large-detuning case (a),(c) and 150 MHz for the small-detuning case (b),(d). The other parameters are: ${\alpha }_{1,2,c}/2\pi =-250,-250,-300$ MHz and $g_{1c,2c}/2\pi =120,100$ MHz.

Figure 11

(a) Sketch of noise-induced transitions. The black and gray wavy lines represent transitions within the computational subspace ($\mathcal{S}\to \mathcal{S}$) and leakage ($\mathcal{S}\to \mathcal{L}$). (b) Phase covariance sensitivities to flux noise on the coupler at different frequencies, in a 40 ns unipolar (solid) and net-zero (dashed) cz gate. The circuit parameters are the same as those used in Fig. 7. The AWP method is used in generating both the unipolar and net-zero pulse, with the coupler idly biased at 7.87 GHz (minimum residual $ZZ$) and 8.2 GHz (the sweet spot), respectively.