Noise is an inherent part of quantum computation. Although there exists a fault-tolerant approach to quantum computing, this requires many computational resources. Hence, it is important to analyze how noise affects well-known algorithms. In this paper the effect of noise in Grover?s search algorithm is studied. The noise is modeled as both total depolarizing channel (TDCh), and local depolarizing channel (LDCh) in every qubit. An analysis of the order has been made analytically for the TDCh, and an approximation has been done for the LDCh.
Noise is an inherent part of quantum computation. Although there exists a fault-tolerant approach to quantum computing, this requires many computational resources. Hence, it is important to analyze how noise affects well-known algorithms. In this paper the effect of noise in Grover?s search algorithm is studied. The noise is modeled as both total depolarizing channel (TDCh), and local depolarizing channel (LDCh) in every qubit. An analysis of the order has been made analytically for the TDCh, and an approximation has been done for the LDCh.
In this paper, the effect of noise on Grover’s algorithm is analyzed, modeled as a total depolarizing channel (TDCh) and a local depolarizing channel (LDCh) in each qubit. The focus was not in error correction (e.g. by the fault-tolerant method), but to provide an insight to the kind of error, or degradation, that needs to be corrected. In the last years, analytical results regarding mainly the TDCh model have been obtained. In this paper, we extend these previous results to the local case, concluding that the degradation of Grover’s algorithm with the latter is worse than the former. It has been shown that for both cases with an N-dependent small enough error-width, smaller than [Formula: see text] for total error and [Formula: see text] for the local case, correction is not needed.
Abstract In quantum computing, Grover's technique is extremely vulnerable to several kinds of quantum noise in practice, which can seriously impair its functionality. This study examines how the algorithm's success probabilities is affected by several types of noise, such as rotational error, thermal noise, Two-State Purity (TSP) error, decay and decoherence, and depolarization variables. The findings demonstrate that the algorithm can accomplish flawless amplification of the goal state with negligible mistakes in non-target states when it is executed without noise. The most notable decrease among the noise kinds is caused by decay and decoherence errors and depolarization factors, which lower the likelihood of correctly detecting the target state from 1.0 to roughly 0.885 and 0.968, respectively. Thermal noise, TSP error, and rotational error, on the other hand, have less severe impacts and result in lesser deviations in both target and non-target states. These results demonstrate how crucial quantum error correction and noise reduction techniques are to guaranteeing Grover's algorithm's scalability and dependability in noisy quantum environments. To fully utilize the algorithm's promise in practical applications on both present-day and next quantum technology, these issues must be mitigated.
Abstract Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.
The quantum singular value transformation (QSVT) algorithm is a general framework to implement most of the known algorithms and provides a way forward for designing new algorithms. In the present work, the impact of noise on the QSVT algorithm is examined for bit flip, phase flip, bit-phase flip, and depolarizing noise models for a small number of qubits. The small number of noisy qubits approximates the currently available noisy quantum computers. For simulation results, the QSVT implementation of the Grover search and quantum phase estimation (QPE) algorithms is considered. These algorithms are among the basic quantum algorithms and form the building blocks of various applications of quantum algorithms. The results showed that the QSVT implementation of the Grover search and QPE algorithms has a consistently worse dependence upon noise than the original implementation for all four noise models. The probability of success of the Grover algorithm and phase measured by the QPE algorithm were found to exponentially depend upon the error probability in the noisy channels but only linearly dependent on the number of qubits.
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.
Abstract When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.