Hamiltonian-based data loading with shallow quantum circuits

Data loading with shallow quantum circuits is a highly desirable ingredient for efficient execution of many quantum algorithms before large-scale quantum error corrections with full fault tolerance become readily available. The need for efficient data loading is especially urgent for the study of quantum machine learning. In this work, we propose a protocol that only uses a parameterized shallow quantum circuit without ancilla qubits for loading data into the amplitude of a quantum state with high fidelity. We term this data-loading method Hamiltonian-based data loading (HDL), which comprises two stages. First, the HDL algorithm identifies a Hamiltonian $\widehat{H}$ whose unique ground state $|\psi \rangle$ represents the normalized data $\vec{x}$ in the form of amplitude encoding. Next, the target state is reconstructed with a parameterized quantum circuit by utilizing methods like variational quantum eigensolver to minimize energy. In this work, we provide three convincing examples to demonstrate the effectiveness of HDL for loading an $N$-dimensional classical data with minimal quantum resources ($O\left[\mathrm{poly}\left({log}_{2}N\right)\right]$-depth quantum circuit without ancilla qubits). Our approach is particularly useful for quantum hardware without large-scale error corrections, and shall benefit the development of quantum machine learning algorithms.

Figure 1

The schematic of the HDL protocol. (a) At first, the input data $x$ is provided to construct a Hamiltonian $H={\sum }_i{\omega }_i{L}_i$ with a set of observables $L_i$ and parameters ${\omega }_i$ by using the correlation-matrix method illustrated in Sec. 2a. (b) Then the quantum processing unit (QPU) will be invoked to estimate the energy of $H$. With these estimations, a classical central processing unit (CPU) will evaluate the loss function and provide updates to the gate parameters. These hybrid quantum-classical loops will continue until the loss function gets below the tolerance $\varepsilon$. (c) Once the optimization stops, ideally, the learned parameterized quantum circuit $U\left({\theta }^{*}\right)$ will output a state that closely resembles the desired quantum state $|\psi \rangle$ with high fidelity.

Figure 2

The ansatz in the variational quantum circuit, where $n$ represents the number of qubits.

Figure 3

Comparison of the number of quantum gates for data loading in the HDL and Ref. [21].

Figure 4

Experimental results of quantum state representations of ten different grayscale handwritten digits. (a) An example of a grayscale picture of handwritten digits with $16\times 16$ pixels. (b) The grayscale pictures represent ten different input handwritten digits. (c) The grayscale pictures represent the output with a 16-layer quantum circuit, with fidelity of each of at least $95\%$. (d) The grayscale pictures represent the output with a 24-layer quantum circuit, with fidelity of each of at least $99\%$.

Figure 5

The schematic of loading multiple data via the HDL protocol. Four images with 128 pixels will be loaded into a seven-qubit quantum circuit. Then the same HDL protocol is executed on a seven-qubit parameterized circuit to prepare the state $|{\psi }_{\mathrm{qram}}\rangle ={\sum }_i|i\rangle |{\psi }_T^i\rangle$, where $|{\psi }_T^i\rangle$ corresponds to the $i\mathrm{th}$ data point. The qubits in the top circuit in the blue square act as the address qubits, and the lower part of the circuit in the orange square as each encoded data point (conditioned on the state of the address qubits).