Reducing Depolarizing Noise in Grover’s Search Algorithm Using Quantum Switches

Abstract

Grover’s quantum search algorithm is one of the earliest discoveries in quantum computing. It is a fundamental algorithm that uses quantum superposition to provide a quadratic speedup for unstructured search tasks over the best possible classical method (linear search). However, this advantage is substantially affected due to the presence of quantum noise. Quantum noise arises due to the interaction between quantum systems and their environments. The resultant errors manifest between the execution of two quantum gates can be effectively modeled using quantum channels. In this study, we look at the effect of total partial depolarizing channel error, affecting the entire quantum register coherently with a certain probability. This modeling approach simulates continuous noise exposure in quantum computers and the stochastic nature of error introduction. In this scenario, implementing an error correction method is non-trivial and costly. Quantum theory allows for a scenario in which quantum channels, representing the flow of quantum information, are organized in a coherent superposition of alternative orders. This phenomenon, known as indefinite causal order, has sparked significant interest in recent years as researchers seek to understand its implications. A quantum switch is a device that operationalizes the concept of indefinite causal order. In a quantum switch, the advantages appear significantly due to the coherent superposition of alternative configurations of the quantum components, which are controlled by an additional control system. In this work, we propose a new approach to mitigate the impact of quantum noise on Grover’s search algorithm using quantum switches. For this, we assume the resultant noise at every iteration to be originating in discrete steps within the iteration such that it can be modeled as a composition of two depolarising channels. In particular, we propose two theoretical frameworks for incorporating these quantum switches in a noisy Grover’s algorithm and take the success probability of finding g, the desired element at any given iteration, as the sole quantifier of the switch’s action in diminishing the effect of noise in search space. In the first framework, we apply the superposition of channels’ orders using a quantum switch and make a measurement followed by post-selection at every iteration or application of Grover operator G. In the second framework, we delay the measurement until the very end. In other words, if we want to look at the quantum switch’s action at the k th step, we already have k − 1 th measurements followed by post-selections for the first framework. In contrast, we will only have a single measurement at the end of k th iteration in the second case. The measurements we make on the quantum switch intermittently between the iterations followed by post-selection destroy the quantum correlations between the target input system and the control qubit. Since these correlations are believed to capture the additional information in the combined system, the second framework is expected to retain more information than the first framework during the run of Grover’s Algorithm under the influence of noise. We observe in plots that the second framework gives a significant advantage in the success probability of Grover’s algorithm running on a search space of 24 elements or 4 qubits. Thus, we explore the quantum switch’s potential role as a tool for maintaining quantum computing’s promise against the challenges posed by quantum noise, marking a significant step forward in the field of quantum computing and information theory.