Engineering Precise and Robust Effective Hamiltonians

Abstract

Engineering effective Hamiltonians is essential for advancing quantum technologies including quantum simulation, sensing, and computing. This paper presents a general framework for effective Hamiltonian engineering, enabling robust, precise, and efficient quantum control strategies. To achieve efficiency, we focus on creating target zeroth-order effective Hamiltonians while minimizing higher-order contributions and enhancing robustness against systematic errors. The control design identifies the minimal subspace of the toggling-frame Hamiltonian and the full set of achievable, zeroth-order, effective Hamiltonians. The framework also enables robust state transfer, characterization of achievable density matrices, and extension to stochastic parameter fluctuations via a cumulant expansion. Examples are included to illustrate the process flow and resultant precision and robustness.

Figures (18)

• Figure 1: Illustration of a very simplified control system. The input signals $\left\{{a_k(t)}\right\}$ are designed by the user. The control system outputs a continuous control field $\vec{B}(t)$ acting on the qubit system in the lab frame. The design of $\left\{{a_k(t)}\right\}$ should account for the model of the system so that it captures the distortion of the hardware along with additive noise. In this case, the control parameters $a_k(t)$ are transformed via convolution with a kernel reflecting the bandwidth of the low- and band-pass filters. More complete mappings are explored in Sec. \ref{['nlsec']}.
• Figure 2: Schematic for finding achievable scaling factors. The set of achievable zeroth-order average Hamiltonians $\mathscr{O'}(\mathbf{g}_\text{pri}, H_\text{pert})$ (light blue area) is a convex set in $\mathscr C({\mathbf{g}_\text{pri}},H_\text{pert})$ (spanned by $h_1$ and $h_2$). The convex set is approximated by the polygon (dark blue area) formed by $\left\{{\vec{v}_i=U_i^\dagger H_\text{pert}U_i}\right\}$. The achievable scaling factors for $H_\text{target}$ can then be found via linear programming.
• Figure 3: (a) Control parameters of a Hadamard gate that is robust to variations in Rabi field strength and detuning. (b) Contour and 3D landscapes of fidelity for the robust sequence. (c) Same as (b), but for a non-robust Hadamard gate of the same $T_\text{seq}$.
• Figure 4: (a) Control sequence for Hadamard gate that is robust to $W$ and $\delta$ variations. The blue and orange curves represent the relative amplitudes of $B_x(t)$ and $B_y(t)$ before and after distortion. (b) Contour and 3D landscape of fidelity for the robust sequence. (c) Same as (b) but for a non-robust sequence of the same $T_\text{seq}$.
• Figure 5: A simple resonant circuit with $R_s=50 ~\Omega$, $R=0.01~\Omega$, $C_m=11$ fF, $C_t=1.479$ pF, and $L=170$ pH. The resonance frequency is 10 GHz.

Theorems & Definitions (7)

• Theorem C.1
• proof
• Theorem D.1
• proof
• Theorem D.2
• Theorem J.1
• proof