Variational tight-binding method for simulating large superconducting circuits

We generalize solid-state tight-binding techniques for the spectral analysis of large superconducting circuits. We find that tight-binding states can be better suited for approximating the low-energy excitations than charge basis states, as illustrated for the interesting example of the current-mirror circuit. The use of tight binding can dramatically lower the Hilbert-space dimension required for convergence to the true spectrum, and allows for the accurate simulation of larger circuits that are out of reach of charge basis diagonalization.

Figure 1

Comparison between tight binding as applied to solids and superconducting circuits. On the left is an example two-dimensional lattice with a two-atom basis, signified by the gray and blue atoms. On the far left are example atomic wave functions. On the right is the potential of the flux qubit, which has two inequivalent minima in each unit cell at the chosen value of flux. We only color the potential below a cutoff value to draw the eye to the potential minima locations. Near the minima the potential is approximately harmonic, therefore the local wave functions take the form of harmonic oscillator states. Example wave functions are shown on the right.

Figure 2

Schematic of ansatz construction schemes. (a) Improper, where local wave functions are defined according to the curvature of the $m=0$ minimum and are reused to form the local wave functions of other inequivalent minima. (b) Proper, where local wave functions for every minimum are defined according to the local curvature. (c) Improper with anharmonicity correction, where harmonic length(s) of the ansatz ground-state wave function of the $m=0$ minimum are optimized to account for anharmonicity corrections to the potential. The resulting wave functions are then also used for $m\ne 0$ minima as in (a). The dashed lines show the unoptimized local ground-state wave function defined for the $m=0$ minimum (the change in the harmonic length due to anharmonicity corrections has been exaggerated). (d) Proper with anharmonicity correction of only the $m=0$ minimum. Wave functions for the $m=0$ minimum are defined according to the local curvature and anharmonicity correction scheme, while wave functions for $m\ne 0$ minima are defined only according to the local curvature.

Figure 3

Spectrum of the flux qubit as a function of (a) flux and (b) offset charge $n_{g1}$, calculated using charge basis diagonalization (solid) and improper tight binding (dashed). At the magnification level of the two figures, the spectra almost exactly overlap. Below each spectrum is the absolute error of the tight-binding calculation relative to the exact spectrum for each of the four lowest-energy eigenstates. Sub-MHz level agreement is achieved in all cases considered here with ${\mathrm{\Sigma }}_{\text{max}}=5$, and sub-kHz level absolute error is possible for both parameter sets by increasing ${\mathrm{\Sigma }}_{\text{max}}$. For (a) flux modulation, we choose parameters $E_J/h=1$ GHz, $E_J/E_{C_J}=60,E_{C_g}/E_{C_J}=50,\alpha =0.8$, and $n_{gi}=0$ [41]. For (b) tuning $n_{g1}$, we use parameters $E_J/h=1$ GHz, $E_J/E_{C_J}=5,E_{C_g}/E_{C_J}=50,\alpha =0.8$ and have set $n_{g2}=0$, ${\phi }_{\text{ext}}=0.5$.

Figure 4

Comparison of convergence to the exact low-energy flux qubit spectrum between improper tight binding (blue circle, solid line) and approximate diagonalization in the charge basis (green circle, dashed line) as a function of $n_H$. The colored circles represent the average relative deviation ${\eta }_{\text{avg}}$, while the colored lines are merely a guide to the eye. The colored shaded regions encompass the range between ${\eta }_{\text{min}}$ and ${\eta }_{\text{max}}$. The gray shaded region represents the $n_H$ values for which tight binding yields an advantage over charge basis diagonalization, comparing ${\eta }_{\text{avg}}$ for a given $n_H$. Improper tight binding allows for an accurate estimate of the low-energy eigenspectrum already at ${\mathrm{\Sigma }}_{\text{max}}=1$, yielding ${\eta }_{\text{avg}}<7\times {10}^{-3}$ and maximum absolute error of less than 25 MHz. We choose the parameters of Fig. 3, as well as ${\phi }_{\text{ext}}=0.47,n_{g1}=0.2,n_{g2}=0.3$. We can perform the same calculation for the parameters of Fig. 3 and obtain similar results, with tight binding outperforming charge basis diagonalization for small $n_H$. The inset shows a schematic of the flux-qubit circuit.

Figure 5

Performance of the tight-binding method as applied to the $N_B=3$ current-mirror circuit. Similarly to the case of the flux qubit, we plot ${\eta }_{\text{avg}}$ for improper tight binding (blue circle, solid line), improper tight binding with anharmonicity correction (red triangle, solid line), and approximate diagonalization in the charge basis (green circle, dashed line) as a function of $n_H$. Improper tight binding with anharmonicity correction outperforms charge basis diagonalization across approximately four orders of magnitude in $n_H$, as indicated by the shaded region. The sharp cliff in ${\eta }_{\text{min}}$ for improper tight binding with anharmonicity correction at $n_H\approx {10}^5$ is due to the inclusion of new basis states that contribute to the ground state, yielding ${\eta }_{\text{min}}\approx {10}^{-4}$. The inset shows a schematic of the $N_B=3$ current-mirror circuit. We choose ${\phi }_{\text{ext}}=0$ and $n_{gi}=0$, with circuit parameters given in the main text.

Figure 6

Comparison of computed ground-state energies for the $N_B=5$ current-mirror circuit. Individual curves correspond to results obtained with tight-binding schemes improper (blue solid circle, solid line), improper with anharmonicity correction (red triangle, solid line), proper (orange open circle, solid line), and proper with anharmonicity correction of the global minimum (pink star, solid line), as well as approximate diagonalization in the charge basis (green solid circle, dashed line). Tight-binding techniques consistently yield lower and hence more accurate eigenenergies as compared to charge basis diagonalization, with a difference of 163 MHz between the best results obtained with tight binding and approximate diagonalization in the charge basis (see the inset). We used the same current-mirror circuit parameters here as in Fig. 5.