Study proposes a hybrid quantum framework aimed at analyzing the impact of noise on Shor’s algorithm. The study approach is divided into five main stages: algorithmic framework construction, hybrid quantum-classical logic gate design, quantum circuit implementation, noise model simulation, and thorough performance assessment.

In order to mimic the factorization of a composite number (N = 15) using a selected base (a = 7), Shor’s algorithm is implemented using IBM’s Qiskit framework. A quantum modular exponentiation stage, an inverse Quantum Fourier Transform (QFT), and a classical pre-processing step comprise the procedure. The Euclidean method, a classical post-processing technique, is used to extract the factors from the periodic output.

Figure 2 illustrates the overall procedural flow.

The main novelty of this study is the development and implementation of a hybrid logic gate, which combines quantum control mechanisms with classical AND logic. The Gate and Quantum Circuit classes in Qiskit were used to create this gate, which uses several control qubits to carry out conditional actions. It is incorporated into the modular exponentiation subroutine to decrease noise-induced decoherence during entanglement and stabilize qubit interactions.

Study introduces a hybrid gate that combines classical AND logic with quantum control to enable conditional operations that improve multi-qubit interactions to noise. The gate functions as a controlled operation, similar to a multi-controlled Toffoli (AND) gate, in which the target qubit is only flipped if all control qubits are in the 1 state.

The quantum state is represented by $c_1,c_2,\cdots ,c_n,t$, where $t$ is the target qubit and $c_i$ are the control qubits. The hybrid gate ${\text{H}}_{\text{AND}}$ does the following:

The action of the hybrid gate ${\text{H}}_{\text{AND}}$ on the quantum state $c_1,c_2,\cdots ,c_n,t$ is defined as:

${\text{H}}_{\text{AND}}{c}_1,c_2,\cdots ,c_n,t=\{\begin{array}{ll}{c}_1,c_2,\cdots ,c_n,t\oplus 1\hfill & \text{if}{c}_1=c_2=\cdots =c_n=1\hfill \\ c_1,c_2,\cdots ,c_n,t\hfill & \text{otherwise}\hfill \end{array}$

The hybrid gate is equal to the conventional Toffoli gate in matrix form for the 3-qubit scenario (2 controls, 1 target):

The Toffoli gate (also known as the CCNOT gate) is represented by the following 8 × 8 unitary matrix:

$\text{Toffoli}=\left[\begin{array}{llllllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill \end{array}\right]$

A multi-controlled X gate is the generalization for n controls.

For one target (q2) and two controls (q0, q1): Figure 3 shows the hybrid (multi-controlled AND) gate’s circuit diagram, which has two controls and one target.

Table 3 illustrates the logical behavior of the hybrid gate. The flipping of the target qubit occurs only when both control qubits are in the $|1\rangle$ state. For all other input combinations, the target qubit remains unchanged.

This behavior is equivalent to traditional AND logic and is achieved using the QuantumCircuit and Gate modules from Qiskit.

The proposed hybrid gate is scalable, regardless of the main goal of this study, which is to improve modular exponentiation stability using a 2-control, 1-target configuration. It can be expanded to accommodate larger quantum systems with more intricate gate behavior. A conceptual design with one control and three data qubits is presented below to illustrate the generalizability of the gate ( Table 4 ).

The CNOT gate serves as a fundamental building block for implementing this logic in the quantum domain. It flips the target qubit only when its control qubit is in the state $|1\rangle$. Toffoli (CCNOT) gates extend this principle by enabling a three-qubit controlled AND operation, where the target qubit is flipped only if both control qubits are in the $|1\rangle$ state. Figure 4 shows a CNOT gate’s circuit diagram. The target qubit is conditionally flipped by this two-qubit gate when the control qubit is in the $|1\rangle$ condition. The hybrid AND gate architecture relies heavily on the CNOT gate to enable controlled quantum processes that are vital to the suggested framework.

Figure 5 illustrates a quantum circuit with multiple qubits and quantum gates. Each horizontal line represents a qubit, showing its state throughout the computation. The white circles serve as control points, commonly used in gates like the Controlled-NOT (CNOT) or controlled phase gates. The large circle signifies an operation on a target qubit, such as a NOT gate (X gate) or another single-qubit operation. Dashed lines indicate interactions or dependencies between qubits, such as those found in multi-qubit gates.

The dots connected to a target gate (like the large circle) suggest a controlled operation. For example, in a CNOT gate, the target qubit is flipped only if the control qubit is in the $|1\rangle$ state. A more general controlled operation, such as a controlled unitary CU, applies the unitary transformation U to the target qubit when the control qubit is in state $|1\rangle$. When multiple control qubits are connected to a single target, the circuit may represent a multi-controlled gate, such as a Toffoli gate (CCNOT), where the target qubit is flipped only if all control qubits are in the $|1\rangle$ state. If the target operation involves a phase shift, the circuit could represent a controlled phase gate, where a phase factor ei is applied based on the control qubit states. The specific purpose of the circuit depends on its context—it could form part of a quantum algorithm, such as Shor’s algorithm or Grover’s search, where each gate contributes to the overall quantum computation.

This circuit diagram shows the layout of the quantum circuit intended to use a hybrid classical-quantum logic gate to implement Shor’s algorithm. The evolution of the quantum state during the algorithm is represented by the horizontal lines, each of which represents a qubit and to which operations are applied from left to right.

The control qubits, represented by white circles, are commonly seen in gates such as the Controlled-NOT (CNOT) gate and controlled phase gates. Solid circles or X-symbols are used to represent target qubits, and a conditional operation, such as a NOT gate, is applied to them. Dashed lines or vertical connectors between multiple control and target qubits indicate multi-controlled gates, such as the Toffoli gate (CCNOT), which flips the target qubit only when all control qubits are in the state $|1\rangle$.

This visual framework is used to create the hybrid AND gate in the context of this investigation. It serves as a multi-controlled unitary gate and is incorporated into the modular exponentiation step of Shor’s algorithm. The objective of this gate is to enhance noise robustness in the quantum algorithm by integrating classical logical control.

Thus, the circuit diagram illustrates the construction of Shor’s algorithm and the integration of classical logic through custom gate design. Hadamard gates are used to generate superposition, followed by controlled modular exponentiation, the inverse Quantum Fourier Transform (QFT), and the final measurement.

In order to improve qubit stability and lessen sensitivity to noise in quantum computing systems, the suggested architecture incorporates a novel hybrid quantum logic gate into Shor’s algorithm. By addressing significant shortcomings in existing quantum computational models, this improvement aids in the creation of quantum algorithms that are more reliable and effective. $n_{\text{count}}+4$ qubits are initialized at the start of the quantum circuit. The phase estimation register uses the first $n_{\text{count}}$ qubits, and modular arithmetic operations employ the remaining four. The phase estimation qubits are subjected to Hadamard gates in order to produce superposition. Figure 6 illustrates how the inverse Quantum Fourier Transform (QFT), a crucial step in the phase estimation procedure, is implemented. After that, the customized hybrid gate and conventional quantum gates are used to carry out controlled modular exponentiation. The problem’s periodic structure is encoded into the quantum state in this step. The state is then converted into the frequency domain using the inverse Quantum Fourier Transform (QFT), which makes periodicity detection possible.

A depolarizing noise model is used to represent the circuit in order to replicate realistic quantum circumstances. This model mimics flaws in real quantum hardware by introducing random perturbations into the quantum states at the gate level. The noisy quantum circuit is executed using the Qiskit Aer simulator, which supports noise-aware simulation and circuit sampling. Statistical robustness is ensured by running each circuit several times (shots = 1024).

The primary testbed for this study was the depolarizing noise model, which provides an isotropic, hardware-agnostic model of Pauli-type errors and is commonly used as a baseline in quantum-noise benchmarking. Methodically comparing mitigation approaches is made easier by the depolarizing channel, which allows for a simple, flexible parameterization of noise strength. To assess device-dependent behavior and validate the generality of the present results, future research will investigate the different error signatures that can be generated by models representing device-specific processes (e.g., phase-damping for pure dephasing or amplitude-damping for energy relaxation).

Measurement counts, which indicate the probability distribution over observed quantum states, are compiled from the simulation outputs. Synthetic noise data is also produced at various noise levels to investigate their impact on algorithm performance. Two important performance indicators are assessed:

The fidelity and entropy of both observable (simulated) and genuine (theoretical) data are calculated. These are employed to examine how algorithmic accuracy gets compromised by noise.

Fidelity and entropy are plotted against varying noise levels to visualize how increasing noise affects the algorithm’s performance, offering a clear comparison between ideal and noisy conditions. To quantify these effects, residuals analysis are performed by comparing the observed metrics with the true values, and these deviations are also plotted to highlight the impact of noise. Finally, all generated data—including noise levels, true and observed metrics—is saved to a CSV file, ensuring it is preserved for further analysis and reporting.

Figure 7 shows how the suggested hybrid classical-quantum framework, intended to improve Shor’s algorithm in noisy environments, has an organized flow. With a focus on the function of hybrid logic and noise reduction techniques, the graphic describes every phase of the implementation, from initialization to the final performance analysis.

The process starts by setting up the quantum framework to examine Shor’s algorithm in a noisy environment. This involves preparing the computational setup, which includes defining quantum and classical registers and initializing qubits.

The initial computational step in Shor’s algorithm is Controlled Modular Exponentiation, a fundamental subroutine. This process performs modular exponentiation using quantum parallelism, playing a vital role in detecting periodicity in factorization problems. It is essential as it sets up the quantum state for subsequent transformations, optimizing the algorithm’s efficiency by harnessing quantum mechanics.

The Quantum Fourier Transform (QFT) is then applied to identify periodic patterns within the quantum state. As a fundamental quantum operation, QFT converts the computational basis into the Fourier basis, enabling efficient factorization. This transformation enhances the detection of periodicity, a crucial step in Shor’s algorithm for accurately determining prime factors.

A significant innovation of this framework is the incorporation of a hybrid logic gate, aimed at enhancing computational accuracy and assessing noise effects. This gate functions as a controlled AND operation, utilizing a Toffoli gate and implemented as a custom HybridGate in Qiskit. The integration is achieved through the addhybridgate() function and evaluated QASM to verify its accuracy and resilience to noise. This step is crucial as it introduces a hybrid quantum-classical approach, ensuring the system’s robustness even in noisy quantum environments.

To test Shor’s algorithm’s performance under several quantum noise models, such as depolarizing, amplitude damping, and dephasing noise, it is applied to synthetic data. This controlled simulation method makes it possible to conduct repeatable tests to evaluate the hybrid framework’s efficacy.

Once the algorithm is executed, the outcomes are evaluated through fidelity and entropy measurements. Fidelity serves as an indicator of how accurately the resulting quantum state aligns with the expected theoretical state, reflecting the precision of the computation. Meanwhile, entropy analysis assesses the extent of information loss caused by noise, providing insights into the effects of quantum state decoherence. These metrics are essential for determining the robustness and efficiency of quantum computations in the presence of noise.

At this stage, the fidelity of the quantum state is evaluated against a 0.9 threshold. If the fidelity surpasses 0.9, it signifies that the hybrid quantum framework effectively reduces noise while preserving computational accuracy. However, if the fidelity falls below this limit, further enhancements—such as advanced error correction methods or refinements in hybrid gate design—may be necessary to improve performance.

The process wraps up with the finalization of noise analysis in Shor’s algorithm. The findings offer crucial insights into the impact of noise on quantum computations and demonstrate how hybrid techniques can improve fault tolerance. This framework provides a structured approach to assessing noise resilience in quantum computing, contributing to the development of more robust and reliable implementations of Shor’s algorithm on noisy quantum systems.

The relationship between fidelity and noise levels in the depolarizing noise model is depicted in Figure 8 . From values around unity at low noise levels (0.025) to roughly 0.6 at higher noise intensity (0.200), fidelity drops dramatically. The model’s strong ability to capture fidelity degradation is demonstrated by the tight alignment of the predicted fidelity curve (shown as a smooth blue line) with the observed fidelity data points (cyan scatter). This close match demonstrates that the noise model is accurate and that, despite intrinsic quantum stochastic fluctuations, fidelity decays predictably with increasing noise. Building upon the fidelity analysis, we next examine the residuals to evaluate model accuracy.

Figure 9 displays residuals, which are the variations between observed and expected fidelity, to further evaluate the prediction accuracy of the model. This indicates a strong agreement between the model predictions and the observed data with no significant bias. In general, the results validate the robustness of the implemented noise-resilient Shor algorithm and the effectiveness of the noise model in capturing the impact of noise on fidelity.

The analysis of entropy about noise levels illustrated in Figure 10 reveals a strong linear relationship, as illustrated in the “Entropy vs. Noise Levels” graph. The entropy increases consistently with noise, starting near 0.0 at minimal noise levels and reaching approximately 0.35 at maximum noise levels. The true entropy curve, shown by the red line, aligns closely with the observed entropy values, represented as yellow dots, confirming the reliability and accuracy of the entropy model. This proportional increase in entropy with rising noise levels underscores the significant impact of noise on the system’s disorder.

Figure 11 “Entropy Residuals” graph further validates the model, with residuals scattered around zero and exhibiting no discernible bias or trend. The consistent variance in residuals indicates the robustness and stability of entropy predictions across various noise levels. Overall, the findings highlight the critical influence of noise on entropy and stress the importance of effective noise management to preserve the performance and reliability of quantum systems.

The proposed hybrid quantum framework successfully mimics the impact of noise on Shor’s algorithm, as shown by the fidelity and entropy evaluations. The correctness and resilience of the noise model and hybrid gate design are confirmed by the high agreement between the expected and observed data over a range of noise levels. These findings demonstrate how this approach may be used to increase the noise resilience of quantum calculations, opening the door for more dependable and scalable quantum algorithms in real-world noisy settings.