Parameterized Quantum Circuit

The tunable gate toward practical Quantum Machine Learning

Parameterized quantum circuits let us embed classical data into quantum states, tune them with adjustable gates, and read out predictions through measurement. The real question is whether these simple building blocks can be scaled into architectures that deliver genuine quantum advantage.

by Frank Zickert
September 10, 2025
Parameterized Quantum Circuit

A parameterized quantum circuit is a Quantum Circuit with Quantum Gate that depend on adjustable Real Number parameters. The values of these parameters are controlled externally and enable the parameterized quantum circuit to generate a Cerezo, M., 2021, Nature Reviews Physics, Vol. 3, pp. 625-644.

Importantly, a parameterized quantum circuit only makes sense as a computational Model is... if the states it generates cannot be efficiently captured by classical means. Otherwise, its role is Schuld, M., 2020, Physical Review A, Vol. 101, pp. 032308.

Just as antennas range from a simple car whip to large parabolic arrays, parameterized quantum circuits span from underparameterized to overparameterized designs. A minimal Quantum Circuit resembles the car antenna. This is easy to operate but too limited to capture genuinely quantum signals. At the other extreme, a highly complex Quantum Circuit mirrors the massive array. That is capable in principle but difficult to calibrate, highly sensitive to noise, and expensive to operate. The central challenge is to find Ansatz that, like a well-engineered antenna, balance simplicity with precision. By using only a few parameters we aim to reliably tune into regions of nonclassical behavior. Only then can parameterized quantum circuits provide a practical route to Quantum Advantage on Noisy Intermediate-Scale Quantum refers to the current generation of quantum devices that have enough qubits to run non-trivial algorithms but are still small and error-prone, limiting their reliability and scalability.

    A parameterized quantum circuit consists of two main parts:
  • Encoding
  • Measurement

The encoding maps the Real Number input to a Quantum State is... via the An Unitary operator is a reversible quantum transformation. is a An Unitary operator is a reversible quantum transformation. (therefore ) that Encoding the data value according to a Quantum Feature Map is... ("phi"). Different choices of correspond to different ways of embedding data into a Quantum State is..., such as Angle Encoding, Amplitude Encoding, or more elaborate Quantum Feature Map is...). So means apply the unitary defined by encoding rule to input . The is not a variable like , but a tag to remind you how the data is being encoded..

Suppose your parameterized quantum circuit uses a single parameter . In the problem domain, could be something meaningful, such as the intensity of a signal, the value of a feature in a dataset, or a physical measurement. However, for the parameterized quantum circuit itself, is just an external Real Number. To integrate it into the Quantum Circuit, must be translated into the language of Quantum State is.... Schuld, M., 2019, Physical Review Letters, Vol. 122, pp. 040504.

is a An Unitary operator is a reversible quantum transformation. (therefore ) that Encoding the data value according to a Quantum Feature Map is... ("phi"). Different choices of correspond to different ways of embedding data into a Quantum State is..., such as Angle Encoding, Amplitude Encoding, or more elaborate Quantum Feature Map is...). So means apply the unitary defined by encoding rule to input . The is not a variable like , but a tag to remind you how the data is being encoded. is simply a collection of quantum gates that rotate and/or entangle qubits so that information about becomes embedded in the quantum state. Once this encoding is done, the circuit can act on the state with its trainable parameters.

Let's look at a concrete code example that shows how this encoding and subsequent processing are implemented in practice.

? depicts the encoding of a parameter in a Quantum Circuit in Qiskit.

pqc.py
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from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
 
qc = QuantumCircuit(1)
x = Parameter("x")
qc.ry(x, 0)
Listing 1 The simplest case of a parameterized quantum circuit
    In this code listing, we
  1. the required functions from Qiskit,
  2. a Quantum Circuit with a single A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.,
  3. , and
  4. the value of x into a quantum state by using x as the rotation angle of the ry Quantum Gate (The Y-Rotation Operator () turns the Quantum State Vector of a single A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. around the -axis. ).
As you can see, no specific value has been assigned to the parameter x. Instead, this can be done dynamically before we execute the Quantum Circuit.

The measurement is the next essential part of a parameterized quantum circuit.

We extract information about the parameter by measuring an Observable , such as a Pauli Operator (, , or ). The Quantum Circuit output is then the Expectation Value of with respect to the encoded Quantum State is...:

    If this equation doesn't scare you off, then I don't know what will. But actually, this is just the mathematical definition of a simple concept (read from the right to the left).
  1. First, we apply the encoding is a An Unitary operator is a reversible quantum transformation. (therefore ) that Encoding the data value according to a Quantum Feature Map is... ("phi"). Different choices of correspond to different ways of embedding data into a Quantum State is..., such as Angle Encoding, Amplitude Encoding, or more elaborate Quantum Feature Map is...). So means apply the unitary defined by encoding rule to input . The is not a variable like , but a tag to remind you how the data is being encoded. onto the computational basis start state is a basis state..
  2. Then measure the resulting quantum state (using the Observable ).
  3. is the Adjoint (Conjugate Transpose) of the Encoding. It is not an instruction you actually run after Measurement. Instead, it appears as Nielsen, M.A., 2010, Cambridge university press, . So, it is an artifact of Linear Algebra that essentially tells us to compute the Expectation Value.

In Qiskit the Measurement procedure simplifies considerably. A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. are always measured in the Computational Basis, so you don't have to specify a Measurement yourself. You only need to interpret the results accordingly. A single execution of the circuit yields one outcome, sampled according to the Quantum State is... probability distribution. To estimate theExpectation Value, the Quantum Circuit is executed many times, and the measurement outcomes are averaged.

Let's have a look at ?.

parameterized-quantum-circuit.py
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from math import asin, sqrt
from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
from qiskit_aer import Aer
 
qc = QuantumCircuit(1)
x = Parameter("x")
qc.ry(x, 0)
qc.measure_all()
 
probability = 0.7
 
simulator = Aer.get_backend("aer_simulator")
counts = (
simulator.run(qc.assign_parameters({x: 2 * asin(sqrt(probability))}), shots=1_000)
.result()
.get_counts()
)
 
print(f"Empirical P(1) = {counts.get("1", 0) / 1_000}, Theoretical: {probability}")
Listing 2 A complete parameterized quantum circuit
    In this code listing, we extend the parameterized quantum circuit with a measurement and check whether the observed results match the desired probability.
  1. We from Qiskit and Python's math library.
  2. Similar to the previous example, the , by an angle determined by the .
  3. After this rotation, we call measure_all(), which in the computational () basis.
  4. Next, we , here , for observing the outcome is a basis state.. To , we use the relation . Solving for gives , which we assign to the circuit parameter .
  5. The circuit is on Qiskit for shots. Each run produces a measurement of either or , and the .
  6. yields the empirical probability. This value is printed alongside the theoretical probability, demonstrating how the encoding and measurement steps reproduce the intended statistics.

The following ? denotes one possible output.

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Empirical P(1) = 0.715, Theoretical: 0.7
Listing 3 Output of running the full parameterized quantum circuit

As you can see in ?, the mathematical and the circuit representation correspond to each other. They are two descriptions of the same concept.

Figure 1 The equation and the circuit are two representations of the same concept

Parameterized quantum circuits offer a minimal but powerful template: classical data is Encoding into Quantum State is... further developed using tunable Quantum Gate, and extracted through Measurement The code listings show how this abstract recipe can be reduced to a few lines inQiskit with probabilities directly linked to Quantum Gate parameters and verified through repeated Measurement

The real challenge, however, is not to build such Quantum Circuit but to design them in such a way that they exploit the quantum structure instead of laboriously imitating classical models. Whether parameterized quantum circuits with a manageable number of parameters can consistently deliver non-classical behavior remains the central open research question.

For now, small examples like these make the mechanics transparent: data becomes a rotation, a A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states. becomes a probabilistic Model is..., andMeasurement convert Amplitude into statistics. From here, the task is to scale this simple picture to architectures that could one day demonstrate a true Quantum Advantage