Fast binomial-code holonomic quantum computation with ultrastrong light-matter coupling

We propose a protocol for bosonic binomial-code nonadiabatic holonomic quantum computation in a system composed of an artificial atom ultrastrongly coupled to a cavity resonator. In our protocol, the binomial codes, formed by superpositions of Fock states, can greatly save physical resources to correct errors in quantum computation. We apply to the system strong driving fields designed by shortcuts-to-adiabatic methods. This reduces the gate time to tens of nanoseconds. Noise induced by control imperfections can be suppressed by a systematic-error-sensitivity nullification method. As a result, this protocol can rapidly $(\sim 35\mathrm{ns})$ generate fault-tolerant and high-fidelity ($\gtrsim 98\%$ with experimentally realistic parameters) quantum gates.

Figure 1

(a) Schematic illustration of an atom-cavity combined system. (b) Level diagram of the bare three-level atom. The upper two levels ($|e\rangle ,|g\rangle$) of the atom are ultrastrongly coupled to the cavity mode with strength $g$. The lower two levels ($|g\rangle ,|\mu \rangle$) are off-resonantly driven by a composite pulse $\mathrm{\Omega }={\sum }_k{\mathrm{\Omega }}_{k}cos({\omega }_{k}t+{\varphi }_k)$.

Figure 2

(a) Illustration of the effective transitions according to Eq. (6). (b) Dynamical and geometric phases acquired by the evolution along $|{\psi }_{-}\left(t\right)\rangle$ with parameters given in Eq. (19) and ${\mathrm{\Theta }}_s=\pi /2$.

Figure 3

For $({\mathrm{\Theta }}_s,\theta ,\varphi )=(\pi /2,\pi /4,0)$, (i.e., the Hadamard gate): (a) finite-duration drives ${\mathrm{\Omega }}_0=(\mathrm{\Xi }/\sqrt{2}{c}_0^2)sin(\theta /2)$, ${\mathrm{\Omega }}_2=(\mathrm{\Xi }/c_2^2)cos(\theta /2)$, and ${\mathrm{\Omega }}_4=(\mathrm{\Xi }/\sqrt{2}{c}_4^2)sin(\theta /2)$ when $T=35\mathrm{ns}$. (b) Average fidelity $\overline{F}$ of the Hadamard vs the error coefficient ${\delta }_i$. The intermediate state is $|{\zeta }_2\rangle$ and the gate time is $T=150\mathrm{ns}$. Other parameters are $g=0.8{\omega }_c$ and ${\omega }_q={\omega }_c=2\pi \times 6.25\mathrm{GHz}$ [104].

Figure 4

(a) Average infidelities $(1-\overline{F})$ of arbitrary single-qubit gates when $\varphi =0$ and $T=150\mathrm{ns}$. Each circle denotes a single-qubit gate, e.g., $({\mathrm{\Theta }}_s,\theta )=(\pi /2,\pi /4)$ corresponds to the Hadamard gate. In each circle, the left (right) side denotes the infidelity in the absence (presence) of pulse imperfections. When considering pulse imperfections, the error coefficient is assumed to be ${\delta }_i=0.1$. (b) Average fidelities $\overline{F}$ of the Hadamard gate vs simulation counts with an additive white Gaussian noise (AWGN). Each cross denotes the average gate fidelity when the signal-to-noise ratio is $r=15$. Other parameters are the same as those in Fig. 3.

Figure 5

(a) Fidelity $F_{\mathrm{out}}$ of the output state of the Hadamard gate vs $\kappa$ and $\gamma$. The decay and dephasing rates in the red-dashed box have been achieved in experiments of superconducting circuits [53]. Other parameters are the same for Fig. 3. (b) Average fidelities of arbitrary two-qubit gates in the presence of pulse imperfections with an error coefficient ${\delta }_i=0.1$. We assume ${\mathrm{\Theta }}_s=\pi /2$ and $\varphi =\pi$ for simplicity. Each circle denotes a two-qubit gate, e.g., $({\theta }_0,{\theta }_1)=(0,\pi /2)$ corresponds to the controlled-not gate. The intermediate state is $|{\zeta }_0^{\prime }\rangle$; parameters are $T=750\mathrm{ns}$, $g_a=g_b=1.3{\omega }_c$, and ${\omega }_b=0.9{\omega }_a$, ${\omega }_a={\omega }_q=2\pi \times 6.25\mathrm{GHz}$.

Figure 6

Histograms of Fock-state populations calculated with the total Hamiltonian $H_{\mathrm{tot}}\left(t\right)$ (green) and the effective Hamiltonian $H_{\mathrm{eff}}\left(t\right)$ (yellow). The red-solid broken lines in each panel denote the Fock-state population calculated by the master equation in Eq. (23) when $(\kappa ,{\kappa }^{\varphi },{\gamma }_{g,\left(\mu \right)},{\gamma }_{g,\left(\mu \right)}^{\varphi })\simeq 2\pi \times (0.33,0.3,8,8)\mathrm{kHz}$ [55]. Insets show the Wigner function $W\left(\alpha \right)$ of the generated states. (a) The superposition state $|{\psi }_{\mathrm{in}}\rangle =\left(\right|\tilde{0}\rangle +\sqrt{2}|\tilde{1}\rangle )/\sqrt{3}$ with parameters ${\tilde{\beta }}_f=arccos\left(\sqrt{1/6}\right)$, ${\epsilon }_2=2/\sqrt{5}$, and $g=0.7{\omega }_c$. (b) The cat state $|{\mathcal{C}}_e^{\eta }\rangle$ with amplitude $\eta =g/{\omega }_c=\sqrt{2}$. The evolution time for each panel is $T=35\mathrm{ns}$.

Figure 7

Probability amplitudes of $|g\rangle |0\rangle$, $|g\rangle |2\rangle$, and $|g\rangle |4\rangle$ in dressed states (a) $|{\zeta }_0\rangle$ and (b) $|{\zeta }_2\rangle$ of $H_R$ as a function of the coupling strength $g$ when ${\omega }_q={\omega }_c$. (c) Energy spectrum of the Hamiltonian $H_R$ when ${\omega }_q={\omega }_c$. The red circle in panel (c) denotes an avoided level crossing.