Single-Period Floquet Control of Bosonic Codes with Quantum Lattice Gates

Abstract

Bosonic codes constitute a promising route to fault-tolerant quantum computing. {Existing Floquet protocols enable analytical construction of bosonic codes but typically rely on slow adiabatic ramps with thousands of driving periods.} In this work, we circumvent this bottleneck by introducing an analytical and deterministic Floquet method that directly synthesizes arbitrary unitaries within a single period. The phase-space unitary ensembles generated by our approach reproduce the Haar-random statistics, enabling practical pseudorandom unitaries in continuous-variable systems. We prepare various prototypical bosonic codes from vacuum and implement single-qubit logical gates with high fidelities using quantum lattice gates. By harnessing the full intrinsic nonlinearity of Josephson junctions, quantum lattice gates decompose quantum circuits into primitive operations for efficient continuous-variable quantum computing.

Figures (7)

• Figure 1: Single-period Floquet control with quantum lattice gates (QLGs). (a) Wigner function of the target state $|\psi_{\rm tar}\rangle$. (b) Target Hamiltonian $\hat{H}_{\rm tar}= i\lambda\log(\hat{U}_{\rm tar})$ in the Fock basis, where the target unitary $\hat{U}_{\rm tar}$ is constructed from Eq. (\ref{['eq:unitary learning']}). (c) Amplitude $A(k,\tau)$ and phase $\phi(k,\tau)$ of the driving potential over one Floquet period $T$. (d) State-preparation fidelity $\mathcal{F}=|\bra{\psi_{\rm tar}} \hat{U}_{\rm tar} |\psi_0\rangle|^2$ as a function of Trotter depths $N_k$ and $N_t$. We use $N_t = 2^6$ and $N_k=40$ in (a-c). Other parameters: $N_p=15$, $k_f=40$, and $\beta_0=1$.
• Figure 2: Haar-state preparation. (a) Average fidelity $\mathcal{F}_{\rm QLG}\equiv|\langle \psi_{\rm QLG} | \psi_{\rm Haar} \rangle|^2$ of Haar-state preparation versus Trotter depth $N_t$ and Hilbert subspace dimension $d$, where each marker is averaged over $10^2$ randomly sampled Haar unitaries. The blue dashed line indicates the empirical scaling $N_t \sim \mathcal{O}(2^{d/4})$ required to reach high-fidelity ($\mathcal{F}_{\rm QLG}>0.95$). (b) Fidelity $\mathcal{F}_{\rm QLG}$ versus $N_t$ for different dimension $d$ ($10^3$ samples per point). (c) Probability density distribution $\mathbb{P}(\mathcal{F})$ of the stochastic fidelity $\mathcal{F}=|\langle \psi_{\rm QLG} | \psi_{\rm ref} \rangle|^2$ over $2 \times 10^3$ from QLG realizations (markers) compared with the analytical Haar statistics $\mathbb{P}_{\rm Haar}(\mathcal{F})$ given by Eq. ( \ref{['eq:Haar']} ) (solid curves) for different dimensions. (d) Mean fidelity $\mathbb{E}[\mathcal{F}]$ versus $N_t$, with the theoretical prediction $1/d$ shown as dashed lines. Other parameters are the same as in Fig. \ref{['fig:figure1']}.
• Figure 3: Bosonic code preparation and operation. (a)-(b): Infidelity of prepared code states versus (a) Trotter depth $N_t$ and (b) control bound $\delta / \beta_0$ for the binomial (triangle), cat (circle), and GKP (square) codes. (c) Violin plots for the binomial (Bio), Cat and GKP codes by randomly sampling over $10^2$ logical gates. The width of each violin (colored region) for a given infidelity logarithm value $\log_{10}(1 - \mathcal{F}_{\mathrm{gate}})$ is proportional to the probability density of the infidelity. The white line marks the median and the black box represents the interquartile range. (d) Infidelity of elementary single-qubit gates $\{ {\tt H}, {\tt S}, {\tt T} \}$ realized via quantum optimal control for each code. The error bars indicate standard deviation over $10^2$ data points. Parameters: $\delta = 1$ for (a), $N_t = 2^6$ for (b), and $N_t = 2^8$, $\delta = 2$ for (c,d). Other parameters follow Fig. \ref{['fig:figure1']}.
• Figure 4: Two-dimensional Haar-random states. The distribution of $10^4$ two-dimensional Haar-random states ($d=2$) is illustrated on the Bloch sphere in panel (a), and the corresponding probability distribution is shown in panel (b), in comparison with the theoretical prediction given in Eq. \ref{['Aeq:Haar']}.
• Figure 5: Bosonic code-state preparation via optimal pulse engineering (OPE). (a-c) Snapshots of the Wigner functions during state preparation from vacuum to target bosonic code states using the optimal control technique for (a) binomial, (b) cat, and (c) GKP codes, respectively. (d) Piecewise trajectories of the optimized control parameter $\beta(\tau)$, normalized by $\beta_0$, for the three bosonic codes. All final infidelities approach $10^{-5}$. Parameters: $N_k=50$, $\sigma=0.35$, $N_t=64$, $d=32$, $\delta =1$, and $\alpha=2.3447$.