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Photonics for Quantum (PfQ)

Implementation of Grover's algorithm on the IBMQ: constraints and error assessment

Asma Eldridi, Amor Gueddana, Vasudevan Lakshminarayanan

Published Jul 9, 2021• 0 citations
Publisher

Abstract

We use the IBMQ machine to implement a quantum database circuit. The database is made of a single multi- column table with multiple superposed records using Toffoli, CNOT and single quantum qubit gates. We present an optimized quantum circuit which allows us to select an item from the table, according to a specific oracle. Our circuit is based only on a single auxiliary qubit to perform the Selective Inversion (SI) and the Inversion Around the Average (IAA), which constitute the Grover operator. We analyze the difference between the generated transpired circuits while executing the original circuit on different IBMQ machines, namely the Melbourne and Athens quantum machines. We discuss the physical constraints causing the difference between the simulation on the composer and the executions of the circuits on the real IBMQ devices.

DOI
10.1117/12.2600578

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We use the IBMQ machine to implement a quantum database circuit. The database is made of a single multi- column table with multiple superposed records using Toffoli, CNOT and single quantum qubit gates. We present an optimized quantum circuit which allows us to select an item from the table, according to a specific oracle. Our circuit is based only on a single auxiliary qubit to perform the Selective Inversion (SI) and the Inversion Around the Average (IAA), which constitute the Grover operator. We analyze the difference between the generated transpired circuits while executing the original circuit on different IBMQ machines, namely the Melbourne and Athens quantum machines. We discuss the physical constraints causing the difference between the simulation on the composer and the executions of the circuits on the real IBMQ devices.
We propose a quantum circuit for the quantum search problem. The quantum circuit is superior to Grover's algorithm in some realistic cases. The reasons for the superiority are in short as follows: In the quantum circuit proposed in this paper, all the operators except for the oracle can be written as direct products of single-qubit gates. Such separable operators can be executed much faster than multiparticle operators, such as controlled-NOT gates and Toffoli gates, in many realistic systems. The idea of this quantum circuit is inspired by the Hamiltonian used in the adiabatic quantum computer. In addition, the scaling of the number of oracle calls for this circuit is the same as that for Grover's algorithm, i.e., O(2n∕2)O({2}^{n∕2})O(2n∕2).
Unstructured searching, which is to find the marked element from a given unstructured data set, is a widely studied problem in computer science. It is well known that Grover algorithm provides a quadratic speedup to solve unstructured search problem compared with the classical algorithm. This algorithm has received a lot of attention due to the strong versatility. In this manuscript, we report experimental results of searching a unique target from 16 elements on five different quantum devices of IBM quantum Experience (IBMQ). We first implement the original Grover algorithm on these devices. However, the experiment probability of success of finding the correct target is almost the same as random choice. We then optimize the quantum circuit size of the search algorithm. The oracle operator and diffusion operator are two of the most costly operators in Grover algorithm. For the 16-element quantum search algorithm, both the oracle operator and diffusion operator consist of a triple controlled [Formula: see text] gate ([Formula: see text]) and some single-qubit gates. So we optimize the implementation of the [Formula: see text] gate according to the qubits layout of different quantum devices. On the ibmq_santiago, the experimental success rate of the 16-element quantum search algorithm is increased to [Formula: see text] by the optimization, which is better than all the published experiments implemented on IBMQ devices. For other IBMQ devices, the experimental success rate of 16-element quantum search also has been significantly improved. We then try to further reduce the size of the quantum circuit by modifying the Grover algorithm, with a tolerable loss of the theoretical success probability. On ibmq_quito, the experimental success rate is further improved from 25.23% to 27.56% after optimization. These experimental results show the importance of circuit optimization and algorithm optimization in the Noisy-Intermediate-Scale Quantum (NISQ) era.
Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.
In this paper, we present a straightforward implementation of a logical expression in quantum computing. We develop a quantum oracle for our logical expression and implement it using Grover's algorithm. The logical expression comprises two variables and two clauses involving the XOR and AND operators. For this expression, we require two qubits as variables, one qubit as the oracle, and one qubit as an ancilla. Additionally, we utilise CNOT and Toffoli gates to construct the quantum oracle. We then design its quantum circuit and execute it on a simulator. The implementation yields the quantum state of a two-qubit quantum system as the solution to our logical expression.
The discovery of polynomial time prime factorization, secure key distribution, and fast database search algorithm have recently made quantum computing the most rapidly expanding research field. For a quantum algorithm to be useful, it is essential that the algorithm can be implemented using quantum circuits. Nanotechnology, in particular quantum mechanics based devices, can be used to realize such an algorithm. In this paper, we show how quantum Boolean circuits can be used to implement the oracle circuit and the inversion-about-average function in Grover's search algorithm. We also show that a slight modification of the oracle circuit can be used to search multiple targets.
Abstract This laboratory work presents a practical implementation of Grover’s search algorithm using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is intended to provide hands-on experience with amplitude amplification and oracle-based quantum search. In Laboratory Work No. 7, participants construct Grover’s algorithm step by step. The implementation begins with defining a function that initializes the quantum registers and prepares the initial stage of the circuit. The output qubit associated with the oracle is set to the state ∣1⟩|1\rangle∣1⟩ to enable phase inversion through the phase kickback mechanism. Hadamard gates are then applied to create a uniform superposition over the search space. The oracle is designed to mark the target state by applying a phase shift, effectively flipping the sign of the amplitude corresponding to the solution. This is followed by the diffusion operator (inversion about the mean), which amplifies the probability amplitude of the marked state. The Grover iteration—composed of the oracle and diffusion operator—is repeated an optimal number of times to maximize the probability of measuring the correct result. The laboratory exercise emphasizes the principles of quantum amplitude amplification, circuit modularization, oracle construction, and measurement-based verification. By executing the algorithm on simulators and cloud-accessible quantum hardware, participants observe the quadratic speedup of Grover’s algorithm compared to classical linear search methods. This practical guide bridges theoretical understanding of quantum search with executable quantum circuits, reinforcing the operational aspects of quantum algorithm design within modern quantum computing