Noise accumulation remains a central limitation on the reliable execution of quantum algorithms. This study analyzes the sensitivity of Deutsch’s algorithm to global depolarizing noise using the density-matrix formalism. First, a general analytical expression is derived for the success probability of a quantum circuit subjected to repeated depolarizing channels. The expression relates algorithmic performance to the ideal success probability, the rank of the success projector, and the cumulative contraction factor across circuit layers. The framework is then applied to the two-qubit Deutsch circuit, with a global depolarizing channel introduced after each of its three effective unitary layers. This yields the closed-form success probability PD(λ)=(1+λ3)/2, where λ denotes the survival parameter of the channel. The analytical result is numerically verified in MATLAB for all four Boolean oracles over 404 parameter–oracle combinations. The maximum discrepancy between the analytical and numerical probabilities is 6.661×10-16, while the maximum trace error is 7.772×10-16, indicating agreement within double-precision floating-point accuracy. The results demonstrate how repeated global depolarization progressively reduces the ideal contribution to the output state and drives the classification probability toward the random-guessing limit. Although the model does not represent hardware-specific noise, it provides an exactly solvable benchmark for studying cumulative noise effects in elementary quantum circuits.
Noise accumulation remains a central limitation on the reliable execution of quantum algorithms. This study analyzes the sensitivity of Deutsch’s algorithm to global depolarizing noise using the density-matrix formalism. First, a general analytical expression is derived for the success probability of a quantum circuit subjected to repeated depolarizing channels. The expression relates algorithmic performance to the ideal success probability, the rank of the success projector, and the cumulative contraction factor across circuit layers. The framework is then applied to the two-qubit Deutsch circuit, with a global depolarizing channel introduced after each of its three effective unitary layers. This yields the closed-form success probability PD(λ)=(1+λ3)/2, where λ denotes the survival parameter of the channel. The analytical result is numerically verified in MATLAB for all four Boolean oracles over 404 parameter–oracle combinations. The maximum discrepancy between the analytical and numerical probabilities is 6.661×10-16, while the maximum trace error is 7.772×10-16, indicating agreement within double-precision floating-point accuracy. The results demonstrate how repeated global depolarization progressively reduces the ideal contribution to the output state and drives the classification probability toward the random-guessing limit. Although the model does not represent hardware-specific noise, it provides an exactly solvable benchmark for studying cumulative noise effects in elementary quantum circuits.
We analytically investigate the robustness of the Bernstein-Vazirani and Deutsch-Jozsa algorithms against single-qubit Pauli noise, of which bit-flip, phase-flip, and depolarizing noise are special cases, using the density-matrix formalism. For a general Pauli channel acting on each qubit after every gate, we derive exact closed-form expressions for the success probability of a single run as a function of the error probabilities and the number of qubits n, and validate them against Qiskit simulations. The noise enters only through the probability ε with which each output bit is flipped. Under bit-flip noise with error probability p>1/2 the most probable outcome of the Bernstein-Vazirani algorithm is the complement of the correct one, which classical post-processing corrects, whereas depolarizing and phase-flip noise never bias the output toward a wrong outcome. For the Deutsch-Jozsa algorithm, the success probability is the average over constant and balanced oracles, and the results are given for the single-variable balanced oracle, which is shown to be the least robust of all balanced oracles for ε<1/2. For a single run, the tolerable error probability of both algorithms decreases as 1/n. We then determine the minimum number of repetitions required for a given confidence and compare it with the number of queries of the classical algorithms. The threshold error probability below which the noisy Bernstein-Vazirani algorithm retains its query advantage increases with n, whereas that of the Deutsch-Jozsa algorithm with its standard readout decreases as 1/n. This contrast indicates that algorithms whose answer is encoded in many independently decodable output bits are considerably more robust to noise than algorithms whose verdict depends on a single global outcome.
This study examines how noise affects the reliability of quantum circuits through two complementary approaches: a controlled simulation using Qiskit Aer and an execution study on real IBM Quantum hardware. In simulation, five noise models (bit-flip, phase-flip, depolarizing, readout, and thermal-relaxation) were applied to a parameterized benchmark circuit across varying qubit counts, circuit depths, and noise strengths, producing 6,000 experimental observations. Success probability declined substantially as qubit count and circuit depth increased, with bit-flip, phase-flip, and depolarizing noise degrading performance far more severely (mean success probabilities of 0.597, 0.580, and 0.607) than readout noise (0.917) or thermal-relaxation noise (0.982). The same benchmark circuit was then executed on the ibm_marrakesh processor across 150 hardware runs (150,000 shots). Hardware performance remained consistently high (mean success probability 0.992), declining with qubit count (0.998 at two qubits to 0.986 at six) but showing little sensitivity to circuit depth. These results indicate that controlled noise mechanisms and real hardware degrade circuit performance through different dynamics, with implications for interpreting simulated noise models relative to physical device behavior.
Noise and errors are unavoidable in any realistic quantum process, including processes designed to reduce noise and errors in the first place. In particular, quantum thermodynamical protocols for cooling can be significantly affected, potentially altering both their performance and efficiency. Analytically characterizing the impact of such errors becomes increasingly challenging as the system size grows, particularly in deep quantum circuits where noise can accumulate in complex ways. To address this, we introduce a general framework for approximating the cumulative effect of gate-dependent noise using a global depolarizing channel. We specify the regime in which this approximation provides a reliable description of the noisy dynamics. Applying our framework to the thermodynamical two-sort algorithmic cooling (TSAC) protocol, we analytically derive its asymptotic cooling limit in the presence of noise. Using the cooling limit, the optimal cooling performance is achieved by a finite number of qubits--distinguished from the conventional noiseless TSAC protocol by an infinite number of qubits--and fundamental bounds on the achievable ground-state population are derived. This approach opens new avenues for exploring noisy quantum thermodynamical processes.
The performance of quantum algorithms on near-term devices is heavily constrained by hardware noise, yet systematic comparisons of algorithmic vulnerability across diverse noise types remain limited. In this study, we benchmark three representative algorithms: quantum teleportation, Grover’s search, and the Quantum Approximate Optimization Algorithm (QAOA), under depolarizing, amplitude damping, phase damping, and thermal relaxation channels. Using exact density-matrix simulations, we characterize algorithmic performance by evaluating state fidelity decay as a function of noise strength. To enable fair, structure-aware comparisons across circuits of varying depths, we introduce a novel metric, the per-gate decay rate. Our analysis reveals that teleportation exhibits the lowest per-gate vulnerability, whereas Grover’s algorithm demonstrates substantially larger, super-linear decay driven by its repeated oracle–diffuser architecture. QAOA displays intermediate and stable noise resilience across system sizes. Finally, we validate our simulation framework through hardware experiments on an IBM Heron processor, demonstrating strong agreement with theoretical predictions (Pearson’s correlation coefficient r = 0.981). This work establishes a quantitative, structure-aware framework for assessing circuit-level noise sensitivity and offers actionable insights for algorithm selection in the noisy intermediate-scale quantum era.
Evaluating the reliability of noisy quantum circuits is essential for implementing quantum algorithms on noisy quantum devices. However, current quantum hardware exhibits diverse noise mechanisms whose compounded effects make accurate and efficient reliability evaluation challenging. While state fidelity is the most faithful indicator of circuit reliability, it is experimentally and computationally prohibitive to obtain. Alternative metrics, although easier to compute, often fail to accurately reflect circuit reliability, lack universality across circuit types, or offer limited interpretability. To address these challenges, we propose a fine-grained, scalable, and interpretable framework for efficient and accurate reliability evaluation of noisy quantum circuits. Our approach performs a state-independent analysis to model how circuit reliability progressively degrades during execution. We introduce the Noise Proxy Circuit (NPC), which removes all logical operations while preserving the complete sequence of noise channels, thereby providing an abstraction of cumulative noise effects. Based on the NPC, we define Proxy Fidelity, a reliability metric that quantifies both qubit-level and circuit-level reliability. We further develop an analytical algorithm to estimate Proxy Fidelity under depolarizing, thermal relaxation, and readout error channels. The proposed framework achieves fidelity-level reliability estimation while remaining execution-free, scalable, and interpretable. Experimental results show that our method accurately estimates circuit fidelity, with an average absolute difference (AAD) ranging from 0.031 to 0.069 across diverse circuits and devices.
Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system’s noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. This work proposes a modified representation for a single-qubit depolarization channel. Our modified channel uses two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per channel execution. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model’s accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era.