In Search Of The Essence Of Quantum Machine Learning

How parameterized quantum circuits turn exponential Hilbert spaces into learning models

Classic machine learning already provides universal approximators and massive models. So why resort to quantum circuits? This is the search for the real challenge, namely to design problem-specific circuits that translate theoretical capacities into practical learning.

by Frank Zickert

September 2, 2025

What could Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. offer that would justify the effort involved in establishing the new field of Quantum Machine Learning?Quantum Machine Learning is the field of research that combines principles from Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. with traditional Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. to solve complex problems more efficiently than classical approaches. The bar is set high. Over the past decade, classical Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. has achieved remarkable results. TransfomersTransfomer now capture far-reaching dependencies in texts with more than billion parameters. Their results—texts, source code, images, even videos—are often indistinguishable from human work.

And that's only one part of it. Classic Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. encompasses much more than Generative AIGenerative Artificial Intelligence. Neural NetworksAn artificial neural network is a computational model of interconnected nodes inspired by biological neurons, used to approximate functions and recognize patterns. are universal function approximators: with sufficient capacity, they can represent any continuous function. Kernel methodsKernel is... map data to high-dimensional Hilbert SpacesHilbert Space where linear separation is possible. OptimizationOptimization is... algorithms such as Stochastic Gradient DescentStochastic Gradient Descent and AdamAdam can be reliably scaled to millions or billions of parameters. Together, these advances form an arsenal of techniques that dominate tasks in the fields of image processing, language, and structured data alike.

Against this backdrop, we must critically examine the core approach of Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. with traditional Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. to solve complex problems more efficiently than classical approaches. , which focuses on a Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. as a machine learning modelModel is.... The Quantum CircuitQuantum Circuit is trained variationallyA Variational Quantum Algorithm is a hybrid quantum–classical algorithm in which a Quantum CircuitQuantum Circuit is paramterized by a classical routine. This means, it usually computes the values for rotation angles used inside this Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. during a classical pre-processing step. Additionally, the measurement results are interpreted during a classical post-processing. and its parameters are updated through a classical OptimizationOptimization is... loop. This design turns the Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. itself into a function approximator that competes directly with Neural NetworksAn artificial neural network is a computational model of interconnected nodes inspired by biological neurons, used to approximate functions and recognize patterns., KernelsKernel is..., and other classical architectures.

For such an approach to be viable, a Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. must offer some representation capacity or computational efficiency that goes beyond what these established ModelsModel is... already achieve. Without this, they run the risk of becoming expensive imitations of methods that already work, and the rationale for Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. with traditional Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. to solve complex problems more efficiently than classical approaches. disappears.

So what does a Quantum AlgorithmA quantum algorithm is a step-by-step computational procedure designed to run on a quantum computer, exploiting quantum phenomena such as superposition, entanglement, and interference to solve certain problems more efficiently than classical algorithms. offer that a classical algorithm does not? The first real clue comes from comparing how classical and quantum ModelsModel is... expand data. In a classical ModelModel is..., an input VectorVector is assigned to a Feature SpaceFeature Space. In this space, the number of Basis StatesBasis State grows polynomially (usually linearly) with the dimension .

A quantum ModelModel is..., on the other hand, embeds an input into the state space of qubits. This state is a SuperpositionSuperposition is... of all computational Basis StatesBasis State. Instead of a polynomially growing set of features, the Quantum CircuitQuantum Circuit opens up access to an exponentially large basis. Each Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. doubles the number of accessible states, creating a representation landscape that no classical ModelModel is... can efficiently match.

This shift puts the problem in a new context. The promising potential of Parameterized Quantum CircuitsA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. lies not in marginal improvements, but in the fact that they open up a Feature SpaceFeature Space whose size grows exponentially with the number of QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.. This exponential expansion is the structural reason for assuming that a quantum ModelModel is... could offer an advantage for Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. .

However, access to a Hilbert SpaceHilbert Space of dimension is meaningless if the ModelModel is... does not use it. A quantum KernelKernel is... must do more than embed inputs in a huge space. It must use this space in such a way that a separation is achieved that cannot be achieved with any efficient classical construction.

So even though we know that capacity is exponential, when we use naive EncodingsEncoding we fall back on forms that also classical KernelsKernel is... can approximate, too. Any possible Quantum AdvantageQuantum Advantage vanishes. The unresolved problem therefore lies in finding out how data EmbeddingsEmbedding and algorithms can be designed that actually make use of the exponential structure. Until then, the exponentially growing Hilbert SpaceHilbert Space remains a theoretical promise rather than a practical advantage.

Fortunately, Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. with traditional Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. to solve complex problems more efficiently than classical approaches. is an active area of research. And there is a number of publications that show how access to an exponentially large Hilbert SpaceHilbert Space.

Yet, unfortunately, that research is not precisely beginner-friendly. Because the articles sound something like this:

The findings in the literature are rigorous, but not always practical. They prove that certain Quantum KernelsQuantum Kernel are difficult to approximate classically. Yet, they do not explain how to develop working models for real data sets. There are two main challenges here.

Challenge 1: Choosing useful encodings

It is easy to write a function with the corresponding kernel . The difficult part is designing so that the kernel does two things at once:
• captures the structure of the actual learning problem and
• remains provably hard to approximate classically, while still being trainable on real quantum hardware.

Most scientific papers provide examplesSchuld, M., 2019, Physical Review Letters, Vol. 122, pp. 040504 and hardness proofs,Havlicek, V., 2019, Nature, Vol. 567, pp. 209-212 but they stop short of giving recipes that align naturally with typical datasets.

Challenge 2: Making circuits trainable

Parameterized Quantum CircuitsA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. with very high expressivity often behave like random unitariesAn Unitary operator is a reversible quantum transformation.McClean, J.R., 2018, Nature Communications, Vol. 9, pp. 4812 In this setting, GradientsGradient disappear exponentially fast with the number of Quantum BitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states., a problem known as Barren PlateausBarren Plateau. This makesOptimizationOptimization is... practically impossible without additional structure or careful initialization.

Therefore, it is not enough to simply place a model in the vast Hilbert SpaceHilbert Space. But the Quantum CircuitQuantum Circuit architecture must be chosen in such a way that learning remains possible.

The key mechanism for accessing the exponential Hilbert SpaceHilbert Space is InterferenceInterference is.... When a Quantum CircuitQuantum Circuit is running, the system evolvesUnitary evolution is the reversible, deterministic time evolution of a quantum system governed by a unitary operator. into a SuperpositionSuperposition is... of many computational paths. Each path contributes a complex amplitudeAmplitude and the probability of an outcome depends on the square of the sum of these AmplitudesAmplitude.

By tuning the Quantum GateQuantum Gate parameters, you change how the paths reinforce or cancel each other out. This selective InterferenceInterference is... allows the Quantum CircuitQuantum Circuit to highlight certain patterns in the input and suppress others. The effect is a Decision BoundaryDecision Boundary that can be highly nonlinear and structured. In classical Models,Model is... capturing the same pattern would often require deep Neural NetworksAn artificial neural network is a computational model of interconnected nodes inspired by biological neurons, used to approximate functions and recognize patterns. or a large number of parameters.

This explanation is quite mathematical. It's more than time to make the principle tangible and look at it in practical terms.

The following listing depicts the core of the circuit implementation in QiskitQiskit.

1. We define a quantum circuit qc with two registers.
   • qr defines a quantum register with a single Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states..
   • cr defines a classical register with one classical bit to store the measurement outcome.
2. We add three Quantum GatesQuantum Gate and a MeasurementMeasurement to our circuit:
   • The first Hadamard operator puts the Quantum BitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. into a SuperpositionSuperposition is... state in which the AmplitudesAmplitude of both Basis StatesBasis State are equal. In other words, if we were to measure this Quantum StateQuantum State is..., we would measure and with the same probability.
   • The rz-function applies a rotation around the -axis by the angle that we take as an external parameter theta. This means, it is not yet fixed at a certain value but is specified when we use the quantum circuit. The rotation does not affect the measurement probability of either Basis StateBasis State. But it changes the Quantum PhaseQuantum Phase of is a basis state..
   • The second Hadamard operator moves the Quantum State VectorQuantum State Vector away from a SuperpositionSuperposition is... state in which both Basis StatesBasis State have an AmplitudeAmplitude of the same size. Depending on the Quantum PhaseQuantum Phase applied, it moves either in the direction of is a basis state. or in the direction of is a basis state..
   • The final measurement measures the qubit in the Computational BasisComputational Basis and puts the result into the classical register cr.
3. This function returns the quantum circuit qc and the parameter theta.

If you look at the mathematical explanation of our circuit again, you will see that we do not feed ("theta") directly into the - operator, but rather . This a function ("phi") of .

This function serves as a feature mapping that translates the external variable into the actual gate angle used in the circuit. It implements a simple linear transformation:

where controls how many oscillations of the InterferenceInterference is... pattern occur over the interval . introduces a phase shift along the -axis.

This mapping ensures that the measurement probabilities and exhibit the characteristic sinusoidal patterns associated with the InterferenceInterference is... of individual Quantum BitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states., with adjustable frequency and phase shift. These parameters and are optional and are therefore sometimes omitted for the sake of conciseness.

The corresponding function in Python is straightforward:

In the next step, we use our double_slit circuit.

1. We start with a Classical PreprocessingClassical Preprocessing, we create the actual circuit to be executed.
   • We create the circuit and obtain its instance and an instance of the parameter theta.
   • We spcify x=pi and map it into a circuit parameter through the feature map phi. This step turns classical data into a phase angle. With w=3.0, we specify to pack three oscillations into the range between and . Further, we specify the starting offset to be b=0.1.
   • Next the symbolic placeholder theta is replaced by the actual number val. This makes the circuit concrete: when executed, it rotates exactly by this phase. Without this binding, the circuit is just a template.
2. In the next block, we run the quantum circuit. In our case, a localQuantum SimulationQuantum Simulation will do.
   • We specify an Aer simulator,
   • run the circuit times,
   • obtain the counts dictionary,
   • and extract the probabilities of measuring the qubit as either or .
3. Finally, we compare the measured results with the theoretical values.

The following listing depicts the measurement output.

As we can see, our empiric results are close to the calculated theoretical values.

Before you try all the different values for manually, let's write a brief convenience function to do that for us. When we run our circuit in a loop for x in np.linspace(0, 2*np.pi, 50), we get points between and .

The following figure shows the plotted empirical results along the calculated theoretial values.

The output of the double-slit circuit is not a linear threshold but a smooth, periodic curve. The measurement probability oscillates (here times) as a cosine squared of the input phase. That periodicity is the direct result of InterferenceInterference is... between the two computational paths.

To mimic this behavior in a classical ModelModel is..., you would need to generate the same sinusoidal dependence on the input. That is why, when comparing to a feedforward Neural NetworkAn artificial neural network is a computational model of interconnected nodes inspired by biological neurons, used to approximate functions and recognize patterns., the relevant question becomes: how well can it approximate functions like ? A ReLU network can only generate piecewise linear functions, so approximating a smooth periodic curve like requires hidden units to reach accuracy . The quantum circuit does it with depth and one parameter.

The circuit in QiskitQiskit is minimal. But it impressively shows how InterferenceInterference is... patterns encode input-dependent Decision BoundariesDecision Boundary. It is the most direct method to see how Parameterized Quantum CircuitsA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. use SuperpositionSuperposition is... and InterferenceInterference is... to convert Quantum PhaseQuantum Phase information into a nonlinear classification signal.

Parameterized Quantum CircuitsA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. do not strive for general universality. They target specific structures that are costly for classical ModelsModel is..., such as periodic decision boundaries. Therefore, Quantum CircuitsQuantum Circuit are precise tools that are valuable not everywhere, but precisely where classical methods are oversized in scope and depth.

The big challenge in Quantum Machine LearningQuantum Machine Learning is the field of research that combines principles from Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics. with traditional Machine LearningMachine Learning is an approach on solving problems by deriving the rules from data instead of explicitly programming. to solve complex problems more efficiently than classical approaches. lies not in pure expressiveness, but in modeling. A Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. has exponential capacity, but this capacity is useless if it is not shaped by the structure of the learning task. Just as with classical KernelKernel is... methods, the effectiveness of a quantum kernel or Parameterized Quantum CircuitA parameterized quantum circuit (PQC) is a Quantum CircuitQuantum Circuit whose Quantum GatesQuantum Gate depend on adjustable realReal Number parameters. These parameters are optimized by a classical algorithm to minimize a Cost Function,Cost Function making parameterized quantum circuits the central building block of variational quantum algorithms. They serve as an interface between quantum hardwareQuantum Computer and OptimizationOptimization is... tasks, connecting abstract algorithm design with practical implementation. depends on the problem domain. A useful model must exploit quantum correlations in a way that is tailored to the specific structure of the task.

Progress is achieved through architectures and training methods tailored to real-world problems, not by striving for larger Hilbert SpacesHilbert Space. The advantage of Quantum CircuitsQuantum Circuit will only be realized if they are developed to solve specific learning tasks and not to demonstrate Quantum MechanicsQuantum mechanics is the branch of physics that describes the behavior of matter and energy at atomic and subatomic scales..