Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime

For a variety of superconducting qubits, tunable interactions are achieved through mutual inductive coupling to a coupler circuit containing a nonlinear Josephson element. In this paper, we derive the general interaction mediated by such a circuit under the Born-Oppenheimer approximation. This interaction naturally decomposes into a classical part, with origin in the classical circuit equations, and a quantum part, associated with the coupler's zero-point energy. Our result is nonperturbative in the qubit-coupler coupling strengths and in the coupler nonlinearity. This can lead to significant departures from previous, linear theories for the interqubit coupling, including nonstoquastic and many-body interactions. Our analysis provides explicit and efficiently computable series for any term in the interaction Hamiltonian and can be applied to any superconducting qubit type. We conclude with a numerical investigation of our theory using a case study of two coupled flux qubits, and in particular study the regime of validity of the Born-Oppenheimer approximation.

Figure 1

A generic circuit for inductive coupling of two or more superconducting circuits (left column). Each smaller circuit $\{q_i,L_i\}$ represents a single qubit. The strength and type of coupling can be tuned via an external magnetic flux ${\mathrm{\Phi }}_{cx}$ applied through the main coupler loop (right-hand side). The coupler's junction may alternatively be a dc SQUID forming an effective Josephson junction with tunable $I_c^{\left(c\right)}$ via a separate flux bias.

Figure 2

(a) Coupler ground-state energy as a function of external flux bias ${\phi }_x$. Solid lines: exact ground-state energy of ${\widehat{H}}_c$ [Eq. (6)] computed by diagonalizing in the first 50 harmonic oscillator basis states. The coupler parameters correspond to ${\beta }_c=0.95$ and ${\zeta }_c=0.1$ (dark blue), 0.05 (magenta), 0.01 (light orange), respectively. Dashed, black line: classical component of the coupler ground-state energy, computed using the scalar function $U_{\min }\left({\phi }_x\right)={\beta }_c{cos}_{{\beta }_c}\left({\phi }_x\right)$. (b) Coupler zero-point energy as a function of external flux bias ${\phi }_x$. Solid lines: difference between the exact ground-state energy $E_g/E_{{\tilde{L}}_c}$ (computed numerically as above) and the classical energy contribution $U_{\min }\left({\phi }_x\right)$. Overlayed dashed lines: linearized approximation to the coupler zero-point energy, computed using Eq. (28) and truncating the series at $\left|\nu \right|\le {\nu }_{\max }=100$. Inset are the same curves, restricted to the bias range ${\phi }_x\in [0,0.03]\times 2\pi$.

Figure 3

Standard flux qubits with interaction mediated by an inductive coupler.

Figure 4

The response of various approximations to increases in specific circuit parameters. ${}^{*}$: for $k$ identical qubits, the mutual inductance is physically bounded as $M_j\lesssim \frac{1}{k}\sqrt{L_j{L}_c}$, so ${\alpha }_j=M_j/L_j\le \frac{1}{k}{\left(E_{{\tilde{L}}_c}/E_{L_j}\right)}^{-1/2}$. Physically, increasing $\left(E_{{\tilde{L}}_c}/E_{L_j}\right)$ (by decreasing the coupler length scale) should correspond to a proportional decrease in $M_j$.

Figure 5

All Born-Oppenheimer theories accurately predict the low-energy spectra in the “reference” regime. We consider a single coupler circuit interacting with two identical flux qubits for varying qubit nonlinearity ${\beta }_j$. (All circuits are at zero bias, ${\phi }_{cx}={\phi }_{jx}=0$.) Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. (See the Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 6

Excluding a small discrepancy near the maximal coupling bias ${\phi }_{cx}=0$, all Born-Oppenheimer theories predict the same qubit dynamics in the reference regime. Shown are coupler-induced qubit coefficients for ${\widehat{H}}_{\mathrm{int}}=E_g\left({\widehat{\phi }}_x\right)$ at the reference parameters (Fig. 5, with ${\beta }_j=1.05$). The solid dark blue, dashed magenta, and dotted black curves correspond to the predictions of the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories, respectively. Plots (a), (b), and (c) correspond to the $xx,xI$, and $zI$ terms, respectively. All calculations were carried out in the “parity” basis (see the Appendix, Sec. pp1-s1 for more details).

Figure 7

Discrepancy between the different Born-Oppenheimer theories near the maximal coupling bias, ${\phi }_{cx}=0$. Solid curves: $xx$ coupling predicted by the nonlinear analytic (NA) theory, for coupler impedance ${\zeta }_c=0.05$ (dark blue), ${\zeta }_c=0.03$ (magenta), and ${\zeta }_c=0.01$ (light orange). Overlayed dotted curves correspond to the $xx$ coupling predicted by the linear numerical (LN) theory at the same coupler parameters. The top curves in plot (a) correspond to the “reference” coupler parameters described in the text (${\beta }_c=0.75,{\alpha }_j=0.05,{\zeta }_c=0.05$). The curves in plots (b) and (c) correspond to the weak coupling (${\alpha }_j\to 0.01$) and low nonlinearity ${\beta }_c\to 0.5$ limits. In all calculations, the qubit parameters were fixed at ${\beta }_j=1.05,{\zeta }_j=0.05,{\phi }_{jx}=0$. Since the “parity” basis was used to define the Hamiltonian coefficients, the $g_{xx}$ interaction is strictly stoquastic (i.e., it is a $zz$ coupling in the computational, “persistent current” basis). All calculations were carried out as done for Fig. 6 (see the Appendix, Sec. pp1-s1 for more details).

Figure 8

Increasing coupler impedance decreases the accuracy of the analytic (NA and LA) theories, while leaving the numerical theory unchanged. We consider the low-energy spectrum of two coupled flux qubits, but double the coupler impedance relative to the reference regime (Fig. 5). Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. (See the Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 9

Increasing coupler impedance ${\zeta }_c$ increases discrepancy between the analytic and numerical theories (relative to the reference regime, Fig. 6). For $xx$ and $zI$ terms [plots (a) and (c)], a discrepancy between analytic (NA and LA, solid dark blue and dashed magenta) and numerical (NL, dotted black) theories exists near maximum coupling ${\phi }_{cx}=0$. The theories match closely for the local $xI$ term [plot (b)]. Calculations were carried out for qubit parameters ${\zeta }_j={\alpha }_j=0.05,{\beta }_j=1.05,{\phi }_{jx}=0$ and coupler parameters ${\beta }_c=0.75,{\zeta }_c=0.1$ (twice the impedance of the reference regime). All calculations were carried out in the parity basis (see the Appendix, Sec. pp1-s1 for more details).

Figure 10

Born-Oppenheimer theories fail to predict the low-energy spectrum for high coupler nonlinearity (near ${\phi }_{cx}=0$). We consider a single coupler circuit interacting with two identical flux qubits for varying coupler bias ${\phi }_{cx}\ll 1$. Circuit parameters are identical to the reference regime (Fig. 5), except qubit nonlinearity is fixed at ${\beta }_c=1.05$ and coupler nonlinearity ${\beta }_c$ is increased from 0.75 to 0.95. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. The NA theory agrees well with exact diagonalization for ${\phi }_{cx}\gtrsim 0.01\times 2\pi$. The large oscillations observed in the LA spectrum are due to the divergences in the analytic expressions for the first and second derivatives of $E_g$ as ${\beta }_c\to 1$ [Eqs. (49) and (50)]. Figure 16 shows the same calculation for a larger range of bias values ${\phi }_{cx}\in [0,0.2]\times 2\pi$. (See the Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 11

Increasing coupler nonlinearity ${\beta }_c$ increases discrepancy between the analytic and numerical theories (relative to the reference regime, Fig. 6). Plots (a), (b), and (c) correspond to the $xx,xI$, and $zI$ terms, respectively, with coupler nonlinearity increased from ${\beta }_c=0.75$ to $0.95$ relative to the reference regime. The solid dark blue, dashed magenta, and dotted black curves correspond to the predictions of the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories, respectively. For ${\phi }_{cx}\lesssim 0.01\times 2\pi$ none of the theories are expected to be accurate (Fig. 10). The LA and LN theories agree for ${\phi }_{cx}\gtrsim 0.01\times 2\pi$, indicating that the harmonic approximation to the zero-point energy converges (Fig. 15). Thus, the NL theory (making only the harmonic approximation) is expected to be accurate for ${\phi }_{cx}\gtrsim 0.01\times 2\pi$. The discrepancy between the NA and LN theories for ${\phi }_{cx}\approx 0.01\times 2\pi$ indicates that higher-order terms neglected by the LN theory are significant. The divergence of the LA calculation is due to the divergences in the analytic expressions for the first and second derivatives of $E_g$ as ${\beta }_c\to 1$ [Eqs. (49) and (50)]. All calculations were carried out in the parity basis. To account for higher coupler nonlinearity, the sums used in the NA calculated [Eq. (42)] were truncated at $\left|\nu \right|\le 200$ (see the Appendix, Sec. pp1-s1 for more details).

Figure 12

Coupler-induced qubit coefficients for ${\widehat{H}}_{\mathrm{int}}=E_g\left({\widehat{\phi }}_x\right)$ at strong coupling ${\alpha }_j$. Shown are coupler-induced qubit coefficients for ${\widehat{H}}_{\mathrm{int}}=E_g\left({\widehat{\phi }}_x\right)$ in the strong coupling limit (Fig. 5, with ${\beta }_j=1.05$ and ${\alpha }_j$ increased from 0.05 to 0.1). The solid dark blue, dashed magenta, and dotted black curves correspond to the predictions of the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories, respectively. Plots (a), (b), and (c) correspond to the $xx,xI$, and $zI$ terms, respectively. All calculations were carried out in the parity basis (see the Appendix, Sec. pp1-s1 for more details).

Figure 13

Coupler-mediated 3-local interactions are small for typical parameter regimes. Comparison of $k$-qubit $x$-type couplings for three interacting qubits (in the parity basis): the value of $g_{\overline{\eta }}$ was computed for $\overline{\eta }=(x,I,I)$ (dark blue), $(x,x,I)$ (magenta), and $(x,x,x)$ (light orange) using the nonlinear, analytic theory (Sec. 3). (Inset is a semilogarithmic plot of $|g_{\overline{\eta }}|/E_{{\tilde{L}}_c}$.) The qubit and coupler parameters were ${\beta }_j=1.05,{\zeta }_j=0.05$, and ${\phi }_{jx}=0$ and ${\beta }_c=0.5$ and ${\zeta }_c=0.05$, respectively. All calculations were carried out in the parity basis (see the Appendix, Sec. pp1-s1 for more details).

Figure 14

The nonlinear theory predicts small but non-negligible nonstoquastic couplings. Main figure: comparison of two-qubit couplings depending on coupling type. (Inset is the same plot for the reduced bias range ${\phi }_{cx}\in [0,0.04]\times 2\pi$, focused on only the $xz$ and $zz$ couplings.) The value of $g_{\overline{\eta }}$ was computed for $\overline{\eta }=(x,x),(x,z),$ and $(z,z)$. The physical and numerical parameters used in this calculation were identical to those in Fig. 13, except that we assume a coupler ${\beta }_c=0.95$. Note that the interaction Hamiltonian of the nonlinear, analytic (NA) theory closely predicts the two-qubit spectrum only for ${\phi }_{cx}\gtrsim 0.01\times 2\pi$ (cf. Fig. 10). All calculations were carried out in the parity basis, so that the nonstoquastic interactions correspond to $(x,z)$ and $(z,z)$. To account for higher coupler nonlinearity, the sums used in the NA calculated [Eq. (42)] were truncated at $\left|\nu \right|\le 200$ (see the Appendix, Sec. pp1-s1 for more details).

Figure 15

The harmonic approximation to the coupler zero-point energy converges for small but nonzero biases (${\phi }_{cx}\gtrsim 0.01\times 2\pi$). Comparison of coupler zero-point energies for different coupler nonlinearities ${\beta }_c$ and fixed impedance ${\zeta }_c=0.05$. Solid curves correspond to the numerically exact zero-point energy. This is computed as the difference between the numerically exact ground-state energy $E_g\left({\phi }_x\right)/E_{{\tilde{L}}_c}$ and the classical potential minimum $U_{\min }\left({\phi }_x\right)={\beta }_c{cos}_{{\beta }_c}\left({\phi }_x\right)$. Dashed curves correspond to the harmonic approximation to the zero-point energy ${\zeta }_c\sqrt{1-{\beta }_{c}cos\left({\phi }_c^{(*)}\right)}$, discussed in Sec. 2d. From the top, each pair of solid and dashed curves corresponds to coupler nonlinearities ${\beta }_c=0.4$ (very light green), 0.75 (light orange), 0.85 (magenta), and 0.95 (dark blue), respectively. The exact calculation was carried out using 50 harmonic oscillator basis states, as discussed in the Appendix, Sec. pp1-s1.

Figure 16

Even for high coupler nonlinearity, all theories predict the correct low-energy spectrum at sufficiently large coupler bias. Circuit parameters are identical to the reference regime (Fig. 5), except qubit nonlinearity is fixed at ${\beta }_c=1.05$ and coupler nonlinearity ${\beta }_c$ is increased from 0.75 to 0.95. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. The NA theory agrees well with exact diagonalization for ${\phi }_{cx}\gtrsim 0.01\times 2\pi$. The large oscillations observed in the LA spectrum are due to the divergences in the analytic expressions for the first and second derivatives of $E_g$ as ${\beta }_c\to 1$ [Eqs. (49) and (50)]. Figure 10 shows the same calculation for bias values focused near ${\phi }_{cx}=0$. (See Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 17

Born-Oppenheimer theories fail to predict the low-energy spectrum for strong coupling and at maximum bias. We consider a single coupler circuit interacting with two identical flux qubits for varying qubit nonlinearity ${\beta }_j$. Circuit parameters are identical to the reference regime (Fig. 5), except the coupling strength ${\alpha }_j=M_j/L$ is increased from 0.05 to 0.1. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. Figure 18 considers the same parameter regime, but for varying coupler bias ${\phi }_{cx}$. (See Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 18

Away from maximum bias ${\phi }_{cx}=0$, the NA theory accurately predicts the low-energy spectrum even for strong coupling, while the linear theories fail. (Although not shown, for sufficiently large bias, the LA and LN theories do converge to the exact spectrum.) We consider a single coupler circuit interacting with two identical flux qubits for varying coupler bias ${\phi }_{cx}$. Circuit parameters are identical to the reference regime (Fig. 5), except the coupling strength ${\alpha }_j=M_j/L$ is increased from 0.05 to 0.1 and the qubit nonlinearity is fixed at ${\beta }_j=1.05$. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. Figure 17 considers the same parameter regime, but fixed at maximum coupling ${\phi }_{cx}=0$ and for varying qubit nonlinearity ${\beta }_j$. (See Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 19

Born-Oppenheimer theories break down in the limit of small coupler impedance. We consider a single coupler circuit interacting with two identical flux qubits for varying qubit nonlinearity ${\beta }_j$. Circuit parameters are identical to the reference regime (Fig. 5), except the coupler impedance ${\zeta }_c=\frac{2\pi e}{{\mathrm{\Phi }}_0}\sqrt{{\tilde{L}}_c/C}$ is decreased from 0.05 to 0.02. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. (See Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 20

Born-Oppenheimer theories break down in the limit of large coupler nonlinearity and at maximum bias ${\phi }_{cx}=0$. A single coupler circuit interacting with two identical flux qubits for varying qubit nonlinearity ${\beta }_j$. Circuit parameters are identical to the reference regime (Fig. 5), except the coupler nonlinearity ${\beta }_c$ is increased from 0.75 to 0.95. Solid curves represent exact numerical diagonalization of the full Hamiltonian [Eq. (4)]. The black dashed, dark blue crossed, and light green dotted curves correspond to the nonlinear analytic (NA), linear analytic (LA), and linear numerical (LN) theories of the Born-Oppenheimer approximation, respectively. (See Appendix, Sec. pp1-s1 for a detailed description of each calculation.)

Figure 21

Ratio of the zero-point energy $U_{\mathrm{ZPE}}$ to the integral in the Born-Oppenheimer diagonal correction compared to the linearized Hamiltonian approximation $4{\zeta }_c^2{(1-{\beta }_c)}^2$. These calculations we carried out at flux bias ${\phi }_{cx}=0$ (which minimizes the ratio). Solid curves (starting from the top) correspond to the numerically exact ratio at coupler impedances ${\zeta }_c=0.1$ (dark blue), 0.05 (magenta), and 0.01 (light orange), respectively. The Hamiltonian ${\widehat{H}}_c$ [Eq. (6)] was diagonalized in the harmonic oscillator basis truncated at 70 basis states, and the vector ${\partial }_{{\phi }_x}\left|{\psi }_g\right.〉$ was then computed using Eq. (A35). The value of $U_{\mathrm{ZPE}}$ was computed by subtracting the classical energy contribution ${\beta }_c{cos}_{{\beta }_c}\left(0\right)={\beta }_c$ from the ground-state energy $E_g/E_{{\tilde{L}}_c}$. Overlayed dashed curves correspond to the linear approximation $4{\zeta }_c^2{(1-{\beta }_c)}^2$ [Eq. (A38)].

Figure 22

A more complicated coupler implementation involving two distinct junctions.