Quantum-Inspired Genetic Algorithm for Robust Source Separation in Smart City Acoustics

ASA involves tasks such as acoustic scene classification and sound event detection, with source separation being a crucial component [1]. However, traditional source separation methods like ICA [4] and NMF [5], as well as more recent techniques like Sparse Component Analysis (SCA) [6] and deep learning (DL) (e.g., [7, 8]), face challenges in handling correlated sources, noise robustness, computational efficiency, and data efficiency in complex urban acoustic scenes. While promising for signal processing, the direct application of quantum algorithms to ASA, particularly source separation, remains largely unexplored. Quantum algorithms, such as Quantum Principal Component Analysis (QPCA) [9] and Quantum Fourier Transform (QFT) [10], offer potential advantages for signal processing tasks due to their inherent ability to handle high-dimensional spaces and exploit quantum phenomena like superposition and entanglement. However, their application to source separation in complex acoustic environments requires further investigation. Similarly, Quantum Support Vector Machines (QSVM) [11] is more suited for classification tasks, while Quantum Annealing (QA) [12] shows potential for optimization but needs further evaluation in complex scenarios.

Our proposed Quantum-Inspired Genetic Algorithm (p-QIGA) addresses the challenge of source separation in complex urban soundscapes by incorporating quantum concepts into a genetic optimization framework. This approach allows us to effectively disentangle individual sound sources in real-world urban scenarios, even in the presence of noise and limited training data. The p-QIGA’s ability to accurately separate sources enhances the identification and tracking of vehicles in noisy environments and contributes to improved urban planning by providing insights into the acoustic characteristics of different urban spaces. This research makes the following key contributions:

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  We introduce a novel p-QIGA leveraging quantum concepts to optimize source separation in complex urban acoustic scenes, enhancing performance and robustness even with limited data.

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  Our p-QIGA effectively addresses key challenges in smart city source separation, including handling correlated sources and diverse sound events.

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  By achieving accurate and robust source separation, our p-QIGA improves performance in critical smart city applications like noise pollution monitoring and acoustic surveillance.

Source separation in ASA aims to decompose an observed audio signal $x(t)$ into its constituent sound sources $s_{i}(t)$. The mixing process can be modeled as a convolutive mixture in the time domain:

or equivalently in the frequency domain:

where $a_{i}(k)$ and $A_{i}(f)$ are the mixing filters, and $n(t)$ and $N(f)$ represent noise. The goal is to estimate $s_{i}(t)$ (or $S_{i}(f)$) given $x(t)$ (or $X(f)$). This is challenging due to factors like unknown mixing filters, correlated sources, diverse sound events, noise, reverberation, and limited training data, motivating the exploration of novel approaches like the quantum-inspired genetic algorithm proposed in this paper.

Within our proposed p-QIGA-based source separation framework, the initial stage focuses on encoding acoustic features into a quantum representation. Utilizing Mel-frequency cepstral coefficients (MFCCs) within this quantum framework offers potential advantages in terms of representational compactness, noise resilience, and correlation capture. These benefits are achieved through encoding MFCCs into quantum states using a parameterized quantum circuit (PQC) illustrated in Fig. 1. This process facilitates efficient processing, robustness to noise, and the exploitation of entanglement to identify correlated sound sources, crucial for effective source separation in complex acoustic environments.

The parameterized PQC shown in Fig. 1 encodes $n$ MFCC features into a 4-qubit quantum state using single-qubit rotation gates ($R_{y}$) and two-qubit controlled-NOT (CNOT) gates. The encoding process starts by applying Hadamard gates (H) to the first and last qubits to introduce superposition. CNOT gates then create entanglement between adjacent qubits, capturing correlations between features. Each $R_{y}$ gate is parameterized by a distinct MFCC feature, with this pattern repeating for subsequent features. This design leverages the strengths of each gate: Hadamard gates for superposition, enabling a larger solution space; $CNOT_{0,1}$ gates for entanglement, capturing feature relationships; and $R_{y}$ gates for precise encoding of individual MFCC values. The alternating pattern of $CNOT_{0,1}$ and $R_{y}$ gates ensures each feature is encoded into a separate qubit while capturing correlations. MFCC features are sequentially assigned to qubits, simplifying the encoding and interpretation. This scheme is chosen for its ability to capture feature correlations and potential noise resilience due to the use of entangled quantum states. The choice of quantum operators and their parameterization can significantly impact the encoding and the p-QIGA’s performance. Different rotation gates (e.g., $R_{x}$, $R_{z}$) or multi-qubit gates (e.g., Toffoli gates) could alter the encoding.

Let $\mathbf{x}=[x_{1},x_{2},...,x_{n}]^{T}\in\mathbb{R}^{n}$ denote the vector of MFCC features. The PQC encodes this into a quantum state $|\psi(x)\rangle\in\mathcal{H}^{\otimes 4}$:

where $U(x,\theta)$ is the unitary operator representing the PQC with trainable parameters $\theta$. The optimization of $\theta$ can be formulated as:

where $\mathcal{P}(\theta)$ is a performance metric. For the first four MFCC features, the encoding process can be represented as:

This encoding scheme compactly represents multiple MFCC features, potentially enabling efficient processing and noise resilience. Furthermore, capturing correlations between features through entanglement can enhance the p-QIGA’s source separation capabilities.

Following the quantum-enhanced source separation process, the estimated source signals $\hat{s}_{i}(t)$ often require further refinement. This subsection details the classical post-processing techniques employed and the subsequent classification stage.

We evaluate our p-QIGA on two datasets: (1) the TAU Urban Acoustic Scenes 2020 Mobile dataset [13] (Dataset 1), comprising recordings from 10 acoustic scenes in 12 European cities, captured using 4 different mobile devices; and (2) the Silent Cities dataset [14] (Dataset 2), capturing unique soundscapes recorded during the COVID-19 pandemic in various cities worldwide, featuring urban environments with reduced human activity. Both datasets are preprocessed (format conversion, downsampling, noise reduction) and split into training (80%), validation (10%), and test (10%) sets.

We evaluate the performance of our p-QIGA algorithm using established metrics that quantify the quality of the separated sources, including:

1. 1.

   Signal-to-Distortion Ratio (SDR):

   $$SDR(s_{i},\hat{s}_{i})=10\log_{10}\frac{\sum_{t}s_{i}^{2}(t)}{\sum_{t}(s_{i}(t)-\hat{s}_{i}(t))^{2}}.$$

   (14)

2. 2.

   Signal-to-Interference Ratio (SIR):

   $$SIR(s_{i},\hat{s}_{i})=10\log_{10}\frac{\sum_{t}s_{i}^{2}(t)}{\sum_{t}(\hat{s}_{i}(t)-s_{i}(t))^{2}-\sum_{t}n^{2}(t)},$$

   (15)

   where $n(t)$ represents the noise component.

3. 3.

   Signal-to-Artifacts Ratio (SAR):

   $$SAR(s_{i},\hat{s}_{i})=10\log_{10}\frac{\sum_{t}(\hat{s}_{i}(t)-n(t))^{2}}{\sum_{t}(\hat{s}_{i}(t)-s_{i}(t))^{2}}.$$

   (16)

These metrics are widely used in the field of ASA and provide a comprehensive assessment of source separation performance by capturing different aspects of the separation quality, such as target distortion, interference suppression, and artifact removal. Furthermore, these metrics align with the evaluation criteria used in related works, allowing for a fair comparison with existing source separation techniques.

We benchmark our p-QIGA against established source separation methods, including:

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  Classical methods: ICA, NMF, SCA, and multi-channel CNNs.

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  Quantum-Inspired method: Quantum Annealing (QA) [12] for optimizing a classical source separation model.

These baselines provide a diverse benchmark for evaluating the performance of our p-QIGA.

The p-QIGA was implemented using Qiskit for quantum computing simulation and standard Python libraries (NumPy, SciPy) for classical components. Experiments were conducted on a workstation with an Intel Xeon Gold 6248 CPU, 128GB RAM, and an NVIDIA Tesla V100 GPU. Hyperparameter tuning for all methods was performed using grid search, with optimal values selected based on validation set performance. Specific hyperparameters and their search spaces are listed in Table I.

As shown in Table II, the p-QIGA achieves competitive performance on both datasets, demonstrating its effectiveness in handling complex acoustic scenes. On the TAU Urban Acoustic Scenes 2020 Mobile dataset, it achieves comparable performance to CNNs and outperforms other classical methods (ICA, NMF, SCA). On the Silent Cities dataset, the p-QIGA outperforms all baselines, highlighting its superior ability to handle correlated sources and limited training data. These results are statistically significant (p $<$ 0.05). Furthermore, the p-QIGA exhibits better generalization capabilities than the CNN and demonstrates greater robustness to variations in acoustic conditions and limited training data, which are crucial for real-world smart city applications.

An ablation study was conducted to investigate the contribution of each quantum-inspired component (superposition, entanglement, crossover, and mutation) in the p-QIGA. Each component was systematically removed or replaced with its classical counterpart to evaluate its impact on source separation performance. Table III presents the performance of these p-QIGA variants compared to the full p-QIGA on the Silent Cities dataset. As shown in the table, removing superposition or entanglement leads to a noticeable performance degradation, particularly in terms of SDR and SIR. This highlights the benefits of encoding acoustic features into quantum states and leveraging entanglement to capture correlations between sources. Similarly, replacing the quantum crossover and mutation operators with classical counterparts also results in a slight performance decrease, indicating the effectiveness of these quantum-inspired operators in exploring the solution space.

The p-QIGA demonstrated robustness in challenging scenarios, achieving 8.2 dB SDR in a high-noise “busy street” scene (3 dB SNR), significantly outperforming the CNN (6.5 dB, p=0.023). While proficient in moderately dense scenes, performance slightly decreased with higher source density, suggesting potential limitations in resolving closely spaced sources. However, the p-QIGA showed greater resilience to source mobility than CNNs, achieving 7.8 dB SDR and 13.1 dB SIR in a “train station” with 80% moving sources, compared to the CNN’s 6.1 dB and 10.5 dB, respectively. Fig. 2 further confirms the p-QIGA’s superior performance and generalization ability, especially in complex scenes with high noise, source density, or source mobility, attributed to its quantum-inspired encoding, entanglement mechanism, and genetic operators.

To assess the data efficiency of the p-QIGA, we conducted experiments with varying training data sizes on the TAU Urban Acoustic Scenes 2020 Mobile dataset. The p-QIGA and baseline methods were trained on 10%, 25%, 50%, and 75% of the original training set, ensuring consistent dataset reductions across all methods for fair comparison. Performance was evaluated on the held-out test set using SDR, SIR, and SAR. As shown in Table IV, the p-QIGA consistently outperforms baseline methods, even with limited training data (e.g., achieving 7.1 dB SDR with only 10% of the data). This data efficiency, attributed to the quantum-inspired encoding and genetic optimization, highlights the p-QIGA’s suitability for scenarios with scarce labeled data and underscores the potential of quantum-inspired algorithms for practical smart city applications.