MEMO-QCD: Quantum Density Estimation through Memetic Optimisation for Quantum Circuit Design

TL;DR

A memetic algorithm is proposed to find the architecture and parameters of a variational quantum circuit that implements the quantum feature map, along with a variational learning strategy to prepare the training state.

Abstract

This paper presents a strategy for efficient quantum circuit design for density estimation. The strategy is based on a quantum-inspired algorithm for density estimation and a circuit optimisation routine based on memetic algorithms. The model maps a training dataset to a quantum state represented by a density matrix through a quantum feature map. This training state encodes the probability distribution of the dataset in a quantum state, such that the density of a new sample can be estimated by projecting its corresponding quantum state onto the training state. We propose the application of a memetic algorithm to find the architecture and parameters of a variational quantum circuit that implements the quantum feature map, along with a variational learning strategy to prepare the training state. Demonstrations of the proposed strategy show an accurate approximation of the Gaussian kernel density estimation method through shallow quantum circuits illustrating the feasibility of the algorithm for near-term quantum hardware.

Figures (7)

• Figure 3: Example of the decoding of a bit string (chromosome) of 8 gates in a quantum circuit (individual). The bit string is divided into sections of 5 bits. Each section encodes a gate. The first gate in the chromosome is applied to the first qubit. The second gate is applied to the second qubit, and so on. When the last qubit is reached, one starts from the first qubit again, until all gates in the chromosome are applied. The specific rules for building a gate from a section of 5 bits are given in the main text and in \ref{['tab:bitstrings_left']} and \ref{['tab:bitstrings_right']}.
• Figure 4: Flowchart of the evolutionary optimisation algorithm used to find the best variational quantum circuit that approximates the kernel, and consequently the QFM. The distinction between a genetic and a memetic algorithm lies in the inclusion of local improvers in the latter. In our approach, the local improvement entails optimising the decoded angles of each circuit prior to evaluating their fitness function. This enables the memetic algorithm to explore both the space of variational circuit architectures and parameter space. Boxes with dashed lines denote operations performed when memetic optimisation is selected, utilising \ref{['eq:single_feature_cost_fn']}. Otherwise, genetic optimisation is conducted without local improvement, employing \ref{['eq:genetic_fitness']}.
• Figure 5: Mean squared error (MSE) of the Gaussian kernel approximation, as a function of the number of qubits $n_x$ that encode a single feature. The genetic algorithm (blue circles) optimises the loss by exploring the variational ansatz space with coarse-grained variational parameters (MSE calculated using \ref{['eq:genetic_fitness']}). Optimisation of the variational parameters of HEA (purple squares) is done through gradient-descent (MSE calculated using \ref{['eq:hea_loss']}). The memetic algorithm (green triangles) uses gradient-descent to fine-tune the variational parameters of ansätze explored by the genetic algorithm (MSE calculated using \ref{['eq:single_feature_cost_fn']}).
• Figure 6: Gate count of variational quantum circuits architectures found by genetic (blue circles) and memetic (green triangles) algorithms compared to the fixed HEA architecture (purple squares).
• Figure 7: Density maps for a 2D two-moon dataset composed of $1000$ points (shown as small semi-transparent white dots) as a function of the number of qubits of the QFM and the layers of the HEA training state circuit. In this case, two features and one auxiliary qubit are considered, resulting in a total of $2n_x+1$ qubits for the training circuit, where $n_x$ represents the number of QFM qubits. The continuous Kullback-Leibler Divergence (KLD), displayed in the white boxes of each figure, serves as a metric for comparing the density estimates of each model. This metric measures the distance between the estimated probability density and the actual probability density of the training set.