Introduction
Discussions of quantum machine learning (QML) are often sidetracked by hardware considerations, like the limits and possibilities of quantum computers, and their timelines. While these are undoubtedly important, especially for near-term applications, this type of discussion can overshadow a more fundamental question: what are the algorithmic and logical advantages of QML over classical machine learning?
This analysis examines the representational differences between classical neural networks and parameterized quantum circuits (PQCs), with a focus on quantum entanglement and how it enables novel forms of feature correlation not easily replicated by classical correlation structures. Rather than focusing on hardware feasibility, this post aims to explore the mathematical foundations that distinguish quantum and classical learning paradigms.
Classical vs Quantum Information Processing
The primary difference between quantum and classical computing with regard to machine learning paradigms lies not in their physical substrates, but the mathematical frameworks they employ for the representation and manipulation of information.
Classical neural networks operate within the bounds of linear algebra over real vector spaces. They encode information as real-valued feature vectors in ℝⁿ,with parameters consisting of weights and biases that are optimized during training. At a high level we can describe the standard computational process as a series of affine transformations followed by elementwise nonlinear activations. The network's expressiveness emerges from the arrangement of these operations across multiple consecutive layers. That is, the representational basis relies on classical correlations between features learned through parameter optimization.
Quantum circuits, on the other hand, use a fundamentally different computational paradigm. They manipulate quantum states in exponentially large Hilbert spaces, where information is encoded using quantum superposition and entanglement rather than classical probability distributions. Parameters take the form of rotation angles in parameterized unitary operators, and the core computations consists of unitary evolution via sequences of quantum gates. The computational space grows exponentially with the number of qubits (in theory, this creates an exponential memory advantage over classical networks, although the information is inaccessible until measurement). While classical networks introduce non-linearity through explicit activation functions, the approach in QML is more nuanced. The core quantum evolution, governed by sequences of unitary gates, is fundamentally linear. Non-linearity is introduced through other parts of the process: first, in the (often non-linear) encoding of classical data into quantum states; and second, through the act of measurement, which collapses the state vector in a non-linear way. The most important difference is that the representational basis relies on quantum correlations from entanglement, which are mathematically distinct from classical feature dependencies.
This difference in computational substrate leads to qualitatively different representational capacities. Quantum systems can encode exponentially more information in superposition (operating on all possible computational paths simultaneously) until measurement collapses the states and we can extract classical outputs. While classically describing the state of an n-qubit system requires tracking 2ⁿ complex amplitudes, a quantum computer inherently operates on this entire complex vector space, which provides immense computational leverage. Still, it is not equivalent to exponential information storage as a maximum of only n classical bits of information can be retrieved from measuring n qubits.
The Role of Entanglement in Expanding Representational Capacity
To understand how entanglement enhances the expressivity of machine learning systems, consider the example of modeling joint dependencies between input features, for example, predicting protein folding stability based on amino acid sequences, where the stability depends not just on individual residues but on complex multi-way interactions between distant positions that collectively determine the three-dimensional structure.
In classical systems, such dependencies must be learned through parameter optimization, a process that is difficult to scale as the complexity of the relationship increases. With classical neural networks, interactions between features xᵢ and xⱼ are mediated through learned weights; a deep neural network can learn to approximate these interactions implicitly, but it must build them up layer-by-layer from simpler linear combinations and non-linear activations, which can require a massive number of parameters for complex problems.
Quantum circuits approach this issue differently. Consider two qubits that start as independent quantum bits, each carrying its own separate information. After applying an entangling gate (such as CNOT), the two qubits become inseparably linked—you can no longer describe them as independent pieces.
CNOT(|+⟩ ⊗ |0⟩) = (|00⟩ + |11⟩)/√2
This identity makes the claim concrete: we begin with two independent qubits in the product state |+⟩ ⊗ |0⟩, but after the CNOT we land in (|00⟩ + |11⟩)/√2, a Bell state that cannot be factored into a tensor product of single-qubit states. Any attempt to describe one qubit on its own now loses information, since the reduced state of each qubit is maximally mixed, yet measurements of the two are perfectly correlated (if the first is 0, the second is 0; if the first is 1, the second is 1). That inseparability is exactly what we mean by the two qubits no longer being independent pieces.
This entangled state creates correlations that are intrinsically non-local: when you measure one qubit, you instantly know something definitive about the other qubit, no matter how far apart they are. For machine learning, this means that entangling gates can create joint feature representations where the "features" (qubits) become inherently interdependent. That is to say, their states are tied together in ways that classical bits simply cannot replicate.
Information-Theoretic Perspective
From an information-theoretic standpoint, entanglement enables the encoding of correlations that cannot be reduced to classical shared randomness. While classical systems can simulate quantum correlations given sufficient computational resources, quantum simulation requires exponential overhead which quickly becomes intractable.
Consider a quantum state with n qubits. The classical description of this state requires 2ⁿ complex amplitudes, but a quantum system can prepare, manipulate, and measure such states using in polynomial time (in terms of gate count). This exponential compression of classical information into quantum states is the source of quantum computational advantage in many algorithms.
Suppose we want to model a Boolean function f(x₁, x₂, x₃) that depends on complex three-way interactions between input bits. Classically, we might need to explicitly enumerate interaction terms or learn them through deep network training.
Quantumly, we can encode the inputs into qubit states, apply a sequence of entangling gates that create correlations between all three qubits simultaneously, and then measure the resulting state. The entangling gates effectively perform a parallel exploration of all possible correlation structures, with the measurement statistics revealing the learned function. No classical hidden variable model can reproduce these correlations without exponential resources.
Entanglement as Network Depth
In classical deep learning, network depth enables hierarchical feature composition. Lower layers learn simple patterns (edges, textures), while higher layers combine these into complex representations (objects, scenes). Each additional layer expands the space of representable functions.
In quantum circuits, depth corresponds to the number of entangling gates applied. Each additional entangling gate increases the complexity of the joint feature representation, allowing the circuit to capture more intricate correlations. The depth of a quantum circuit can be thought of as a measure of its representational capacity, analogous to the depth of a classical neural network.
The training process is analogous at a high level, typically relying on classical optimizers to adjust gate parameters based on a calculated cost function. However, this analogy has limits; the optimization of quantum circuits presents unique and formidable challenges, such as the 'barren plateau' problem, where gradients can vanish exponentially in deep circuits, making effective training significantly harder than for most classical networks.
Expressivity of Quantum Models
The expressivity of quantum circuits depends critically on the entangling structure of the quantum circuit. Circuits without entanglement produce separable states that can be efficiently simulated classically. Circuits with sufficient entangling depth can map classical data into a quantum feature space in ways that are difficult to replicate. The resulting probability distributions can be used to classify data, and for certain problems, these quantum-derived distributions cannot be efficiently simulated or reproduced by any known classical algorithm.
Recent work has shown that certain quantum methods can achieve better results than can classical algorithms for specific learning tasks, particularly those related to physical systems (Huang et al., 2021). These results provide grounding for the hypothesis that quantum machine learning offers genuine computational advantages, not merely alternative implementations of classical algorithms.
Implications for Algorithm Design
Understanding the mathematical differences between classical and quantum machine learning suggests several principles for designing quantum algorithms:
- Strategically Add Entanglement: Not all machine learning tasks benefit from quantum entanglement. Problems with inherently classical correlation structures may see little advantage from quantum approaches. However, tasks involving complex joint dependencies, high-dimensional feature interactions, or exponentially large state spaces may be natural candidates for quantum enhancement.
- Design for Measurement: Unlike classical neural networks that can access all internal states, quantum algorithms must carefully design measurement schemes to extract useful information from quantum states. The measurement basis and timing significantly impact the algorithm's performance.
- Optimize Circuit Depth vs Width Trade-offs: Quantum circuits face unique constraints from decoherence and gate errors. Algorithm design must therefore balance the expressivity gained from deep, highly entangled circuits against the practical limitations of near-term hardware. Finding effective strategies to manage this trade-off is a central challenge in the field and is the primary focus of my paper, "Bridging Quantum and Classical Computing in Drug Design"
- Hybrid Approaches: Consider hybrid quantum-classical models that combine the strengths of both quantum and classical paradigms.
Conclusion
The primary innovation of quantum machine learning is not in the underlying hardware, but in the expansion of mathematical paradigms available for processing information. By leveraging the power of quantum entanglement, QML algorithms can encode and calculate correlations that are intractable classically.
This post shows how QML should be interpreted as a computational paradigm, rather than an alternative implementation of classical algorithms on a different physical substrate. The exponential scaling of quantum Hilbert spaces, the non-local correlations enabled by entanglement, and the rich interference patterns of quantum amplitudes create novel computational opportunities.
While significant engineering challenges remain for practical quantum computing, the theoretical foundations of quantum machine learning provide compelling evidence for its long-term potential. As quantum hardware continues to mature, understanding these structural advantages will be crucial for identifying the problem domains where quantum approaches offer the greatest promise.
The path forward requires continued theoretical development alongside experimental validation, but the mathematical foundations already suggest that quantum machine learning may unlock new computational capabilities.