Quantum Search on Noisy Intermediate-Scale Quantum Devices Kun Zhang, 1 Kwangmin Yu, 2 and Vladimir Korepin 3 1 Department of Chemistry, Stony Brook university, Stony Brook, New York 11794, USA 2 Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, USA 3 C.N. Yang Institute for Theoretical Physics, Stony Brook university, Stony Brook, New York 11794-3840, USA (Dated: February 2, 2022) The Quantum search algorithm (also known as Grover’s algorithm) lays the foundation of many other quantum algorithms. It demonstrates an advantage (for unstructured search) over the classical algorithm. Although it is very simple, its implementation is limited on the noisy intermediate-scale quantum (NISQ) processors. The limitation is due to the circuit depth, rather than the number of qubits. For the first time, we present detailed and completed benchmarks of the five-qubit quantum search algorithm on different quantum processors, including IBMQ, IonQ and Honeywell quantum devices. Besides Grover’s algorithm, we also present the implementations of a recently proposed optimized quantum search algorithm [Phys. Rev. A 101, 032346 (2020)]. Noisy simulations are also applied to interpret the results. We report the highest success probability of the five-qubit search algorithm compared to previous works. Our study shows that different quantum processors, with different levels of errors, have different optimal ways to realize the quantum search algorithms. The original Grover’s algorithm might not be the best choice. To maximize the power of NISQ computers, designing error-aware algorithms is necessary. I. INTRODUCTION The ultimate goal of quantum computers is to imple- ment quantum algorithms that are superior to the classi- cal counterpart. During the last twenty years, quantum computers have vastly developed. Quantum processors with hundred qubits have been delicately designed and built. We have entered the noisy-intermediate-quantum (NISQ) era [1]. Quantum processors with hundred qubits have the ability to tackle problems far beyond the reach of classical computers. However, errors limit the num- ber of consecutive operations that can be applied. The number of consecutive operations is also called depth. To harness the power of NISQ processors, various quantum algorithms with shallow depths have been de- signed [2]. The practical power of the shallow depth al- gorithm is under extensive study now. Nevertheless, the promising of quantum computers largely relies on the application of functional quantum algorithms, such as Shor’s algorithm [3] and Grover’s algorithm [4]. Recently, more researches begin to benchmark the quantum com- puters based on those applications (application-oriented benchmarking) [5, 6]. Because of its simplicity, Grover’s algorithm is usu- ally the first quantum algorithm taught in the course of quantum computation. Grover’s algorithm solves the unstructured search problem [4, 7]. It finds the target item, also called the marked item, in an unstructured database. Classically, the unstructured search problem can be solved by querying each item in the database. The target item can be recognized by the black-box function (oracle). Therefore, the oracular complexity of the clas- sical search is O(N ), assuming that the number of items in the database is N . Grover’s algorithm has the oracu- lar complexity O( √ N ). The quadratic speedup is due to the superposition of quantum states. Grover’s algorithm is optimal in the number of queries to the oracle, because of the linearity of quantum me- chanics [8, 9]. The idea of Grover’s algorithm is not limited to the unstructured search problem. The am- plitude of wanted states can be amplified with a similar quadratic speedup [10, 11]. Therefore, it makes the gen- eralized quantum search algorithm applicable to a wide of problems, such as quantum machine learning [12]. During the last twenty years, many variants of Grover’s algorithm have been proposed [13–24]. Few of them have focused on the practical performance of quantum search algorithms on NISQ devices. It was Grover himself who firstly realized the trade-off between the number of ora- cles and the physical resource for real implementations (such as depths) [14]. And then from a more practical viewpoint, one can variational learning the physical re- source of quantum search algorithms by comparing all the possible implementations [21]. Besides, the divide- and-conquer strategy can also be applied to the quantum search algorithm, which prevents the error accumulations [21]. Note that the depth optimization of quantum search algorithms can also benefit their implementations in the post NISQ era. In other words, reducing the circuit depth also reduces the error-correction resources (and the run- ning time). The implementation of Grover’s algorithm (with the three-qubit search domain) was firstly reported in 2017 [25]. Since then, more researches have studied the perfor- mance of Grover’s algorithm on real quantum processors [26–30]. Most of the realizations are up to four qubits (N =2 4 ). In our study, we benchmark the state-of-art quantum processors via the quantum search algorithms. The sig- nificance of benchmark in our work is three-fold. First, the recent application-oriented benchmark reported in [5, 6] gives a general framework on how to benchmark quantum computers via various quantum algorithms. However, a more sophisticated design of the circuits can arXiv:2202.00122v1 [quant-ph] 31 Jan 2022