Arbitrary state preparation in quantum harmonic oscillators using neural networks

TL;DR

This work tackles scalable arbitrary state preparation in a quantum harmonic oscillator (HO) by coupling the HO to an auxiliary qubit and using a neural network to predict a sequence of rectangular pulses that yield the target HO state. The method relies on a Jaynes–Cummings–type Hamiltonian $\hat{H}$ and a Carr–Purcell sequence of $N$ pulses, with the neural network $f_{\boldsymbol{\eta}}: \mathbb{R}^d \to \mathbb{R}^{4N+1}$ taking $d=n^2-1$ $SU(n)$ expectation values and outputting pulse parameters $(\zeta,\xi,\phi,\varphi)$ for each pulse plus the total time $T$. Targets are sampled from the Haar measure on $SU(n)$, and training minimizes average infidelity with a regularization term to discourage excitation of higher HO levels. Key results show average fidelities of $F\approx0.999$ for qubits ($n=2$) and $F\approx0.97$ for qutrits ($n=3$), with seven pulses delivering the highest qubit fidelity and post-processing eliminating problematic fidelity-drop regions. The work highlights limitations for higher-dimensional qudits and the impact of using rectangular pulses, while suggesting architectural enhancements and extensions to Gaussian pulses, as well as incorporating noise, to broaden practical applicability. $\hat{H}$ and pulse parameters $\boldsymbol{\Theta}$ (and total time $T$) form the core mathematical framework, enabling a scalable, data-driven approach to on-demand state preparation in quantum harmonic oscillators.

Abstract

Preparing quantum states is a fundamental task in various quantum algorithms. In particular, state preparation in quantum harmonic oscillators (HOs) is crucial for the creation of qudits and the implementation of high-dimensional algorithms. In this work, we develop a methodology for preparing quantum states in HOs. The HO is coupled to an auxiliary qubit to ensure that any state can be prepared in the oscillator [J. Math. Phys. 59, 072101]. By applying a sequence of square pulses to both the qubit and the HO, we drive the system from an initial state to a target state. To determine the required pulses, we use a neural network that predicts the pulse parameters needed for state preparation. Specifically, we present results for preparing qubit and qutrit states in the HO, achieving average fidelities of 99.9% and 97.0%, respectively.

Figures (15)

• Figure 1: Scheme of the proposed methodology. The target state passes through a layer that computes the expectation values of the elements of the $SU(n)$ basis. These values are the input to a neural network whose output consists of the parameters of the sequence of $N$ pulses applied to the HO ($\vb*{\xi}, \vb*{\varphi}$) and to the ancillary qubit ($\vb*{\zeta}, \vb*{\phi}$), and the total preparation time $T$.
• Figure 2: Average preparation infidelity as a function of the training set size for a qubit and a qutrit. The fidelity is evaluated using a separate validation set of $100$ states, also sampled from the Haar measure. The shaded area represents the range between the maximum and minimum infidelity values, while the point indicates the average infidelity value.
• Figure 3: Trajectory of the state in the HO when attempting to prepare the target state $\ket{1}$ with a sequence of five pulses. Additionally, the pulse profiles and the evolution of the purity in function of the time are presented. The trajectory is colored according to the purity to make it clearer when the qubit state lies inside the sphere.
• Figure 4: Preparation infidelity using the proposed neural network for 100 different states and with three models with different parameters for the same number of pulses. The red line corresponds to preparing qubit states in the HO, and the blue line corresponds to preparing qutrit states in the HO. Shaded regions correspond to 75% confidence intervals using bootstrapping virginia_bootstrap
• Figure 5: Base-10 logarithm of the infidelity on the Bloch sphere, using the model with a pulse sequence of $N=7$. Panel (a) shows the front view of the sphere, while panel (b) presents the rear view. A total of 62,500 states were sampled uniformly on the Bloch sphere using the Haar measure.