Algorithms for triangle finding, the smallest nontrivial instance of the $k$ -clique problem, have been proposed for quantum computers. Still, those algorithms assume the use of fixed access time quantum RAM. In this article, we present a practical gate-based approach to both the triangle-finding problem and its NP-hard $k$ -clique generalization. We examine both constant factors for near-term implementation on a noisy intermediate scale quantum computing (NISQ) device and the scaling of the problem to evaluate long-term use of quantum computers. We compare the time complexity and circuit practicality of the theoretical approach and actual implementation. We propose and apply two different strategies to the $k$ -clique problem, examining the circuit size of Qiskit implementations. We analyze our implementations by simulating triangle finding with various error models, observing the effect on damping the amplitude of the correct answer, and compare to execution on six real IBM quantum machines. Finally, we estimate the approximate quantum volume needed so that the smallest instance of our approach can be executable with minimal error on a real NISQ device.Algorithms for triangle finding, the smallest nontrivial instance of the$k$-clique problem, have been proposed for quantum computers. Still, those algorithms assume the use of fixed access time quantum RAM. In this article, we present a practical gate-based approach to both the triangle-finding problem and its NP-hard$k$-clique generalization. We examine both constant factors for near-term implementation on a noisy intermediate scale quantum computing (NISQ) device and the scaling of the problem to evaluate long-term use of quantum computers. We compare the time complexity and circuit practicality of the theoretical approach and actual implementation. We propose and apply two different strategies to the$k$-clique problem, examining the circuit size of Qiskit implementations. We analyze our implementations by simulating triangle finding with various error models, observing the effect on damping the amplitude of the correct answer, and compare to execution on six real IBM quantum machines. Finally, we estimate the approximate quantum volume needed so that the smallest instance of our approach can be executable with minimal error on a real NISQ device.

arXiv:2008.12525IEEE Transactions on Quantum EngineeringJan 1, 202016 citations

Intermediate-qudit assisted Improved quantum algorithm for string matching with an Advanced Decomposition of Fredkin gate

Amit Kumar Saha, Om Khanna

The circuit-level implementation of a quantum string-matching algorithm, which matches a search string (pattern) of length $M$ inside a longer text of length $N$ , has already been demonstrated in the literature to outperform its classical counterparts in terms of time complexity and space complexity. Higher-dimensional quantum computing is becoming more and more common as a result of its powerful storage and processing capabilities. In this article, we have shown an improved quantum circuit implementation for the string-matching problem with the help of higher-dimensional intermediate temporary qudits. It is also shown that with the help of intermediate qudits not only the complexity of depth can be reduced but also query complexity can be reduced for a quantum algorithm, for the first time to the best of our knowledge. Our algorithm has an improved query complexity of $O(\sqrt{N-M+1})$ with overall time complexity $O\left(\sqrt{N-M+1}\left((\log {(N-M+1)} \log N)+\log (M)\right)\right)$ as compared to the state-of-the-art work which has a query complexity of $O(\sqrt{N})$ with overall time complexity $O\left(\sqrt{N}\left((\log N)^{2}+\log (M)\right)\right)$ , while the ancilla count also reduces to $\frac{N}{2}$ from $\frac{N}{2}+M$ . The cost of state-of-the-art quantum circuit for string-matching problem is colossal due to a huge number of Fredkin gates and multi-controlled Toffoli gates. We have exhibited an improved gate cost and depth over the circuit by applying a proposed Fredkin gate decomposition with intermediate qutrits (3-dimensional qudits or ternary systems) and already existing logarithmic-depth decomposition of $n$ -qubit Toffoli or multi-controlled Toffoli gate (MCT) with intermediate ququarts (4-dimensional qudits or quaternary systems). We have also asserted that the quantum circuit cost is relevant instead of using higher dimensional qudits through error analysis.The circuit-level implementation of a quantum string-matching algorithm, which matches a search string (pattern) of length $M$ inside a longer text of length $N$, has already been demonstrated in the literature to outperform its classical counterparts in terms of time complexity and space complexity. Higher-dimensional quantum computing is becoming more and more common as a result of its powerful storage and processing capabilities. In this article, we have shown an improved quantum circuit implementation for the string-matching problem with the help of higher-dimensional intermediate temporary qudits. It is also shown that with the help of intermediate qudits not only the complexity of depth can be reduced but also query complexity can be reduced for a quantum algorithm, for the first time to the best of our knowledge. Our algorithm has an improved query complexity of $O(\sqrt{N-M+1})$ with overall time complexity $O\left(\sqrt{N-M+1}\left((\log {(N-M+1)} \log N)+\log (M)\right)\right)$ as compared to the state-of-the-art work which has a query complexity of $O(\sqrt{N})$ with overall time complexity $O\left(\sqrt{N}\left((\log N)^{2}+\log (M)\right)\right)$, while the ancilla count also reduces to $\frac{N}{2}$ from $\frac{N}{2}+M$. The cost of state-of-the-art quantum circuit for string-matching problem is colossal due to a huge number of Fredkin gates and multi-controlled Toffoli gates. We have exhibited an improved gate cost and depth over the circuit by applying a proposed Fredkin gate decomposition with intermediate qutrits (3-dimensional qudits or ternary systems) and already existing logarithmic-depth decomposition of $n$-qubit Toffoli or multi-controlled Toffoli gate (MCT) with intermediate ququarts (4-dimensional qudits or quaternary systems). We have also asserted that the quantum circuit cost is relevant instead of using higher dimensional qudits through error analysis.

arXiv:2304.03050Apr 6, 20231 citation

Circuit design for clique problem and its implementation on quantum computer

Arpita Sanyal, Amit Saha, Banani Saha +1 more

Abstract Finding cliques in a graph has a wide range of applications due to its pattern matching ability. The k ‐clique problem, a subset of the clique problem, determines whether or not an arbitrary network has a clique of size k . Modern‐day applications include a variation of the k ‐clique problem that lists all cliques of size k . However, the quantum implementation of such a variation of the k ‐clique problem has not been addressed yet. In this work, apart from the theoretical solution of such a k ‐clique problem, practical quantum‐gate‐based implementation has been addressed using Grover's algorithm. In a classical‐quantum hybrid architecture, this approach is extended to build the circuit for the maximum clique problem. Our technique is generalised since the program automatically builds the circuit for any given undirected and unweighted graph and any chosen k . For a small k with regard to a big graph, the proposed solution to addressing the k ‐clique issue has shown a reduction in qubit cost and circuit depth when compared to the state‐of‐the‐art approach. A framework is also presented for mapping the automated generated circuit for clique problems to quantum devices. Using IBM's Qiskit, an analysis of the experimental results is demonstrated.Abstract Finding cliques in a graph has a wide range of applications due to its pattern matching ability. The k ‐clique problem, a subset of the clique problem, determines whether or not an arbitrary network has a clique of size k . Modern‐day applications include a variation of the k ‐clique problem that lists all cliques of size k . However, the quantum implementation of such a variation of the k ‐clique problem has not been addressed yet. In this work, apart from the theoretical solution of such a k ‐clique problem, practical quantum‐gate‐based implementation has been addressed using Grover's algorithm. In a classical‐quantum hybrid architecture, this approach is extended to build the circuit for the maximum clique problem. Our technique is generalised since the program automatically builds the circuit for any given undirected and unweighted graph and any chosen k . For a small k with regard to a big graph, the proposed solution to addressing the k ‐clique issue has shown a reduction in qubit cost and circuit depth when compared to the state‐of‐the‐art approach. A framework is also presented for mapping the automated generated circuit for clique problems to quantum devices. Using IBM's Qiskit, an analysis of the experimental results is demonstrated.

arXiv:2004.10596IET Quantum CommunicationDec 9, 20218 citations

Finding Small and Large

k

-Clique Instances on a Quantum Computer

Algorithms for triangle finding, the smallest nontrivial instance of the

k

-clique problem, have been proposed for quantum computers. Still, those algorithms assume the use of fixed access time quantum RAM. In this article, we present a practical gate-based approach to both the triangle-finding problem and its NP-hard

k

-clique generalization. We examine both constant factors for near-term implementation on a noisy intermediate scale quantum computing (NISQ) device and the scaling of the problem to evaluate long-term use of quantum computers. We compare the time complexity and circuit practicality of the theoretical approach and actual implementation. We propose and apply two different strategies to the

k

-clique problem, examining the circuit size of Qiskit implementations. We analyze our implementations by simulating triangle finding with various error models, observing the effect on damping the amplitude of the correct answer, and compare to execution on six real IBM quantum machines. Finally, we estimate the approximate quantum volume needed so that the smallest instance of our approach can be executable with minimal error on a real NISQ device.

The circuit-level implementation of a quantum string-matching algorithm, which matches a search string (pattern) of length

M

inside a longer text of length

N

, has already been demonstrated in the literature to outperform its classical counterparts in terms of time complexity and space complexity. Higher-dimensional quantum computing is becoming more and more common as a result of its powerful storage and processing capabilities. In this article, we have shown an improved quantum circuit implementation for the string-matching problem with the help of higher-dimensional intermediate temporary qudits. It is also shown that with the help of intermediate qudits not only the complexity of depth can be reduced but also query complexity can be reduced for a quantum algorithm, for the first time to the best of our knowledge. Our algorithm has an improved query complexity of

O(\sqrt{N-M+1})

with overall time complexity

O\left(\sqrt{N-M+1}\left((\log {(N-M+1)} \log N)+\log (M)\right)\right)

as compared to the state-of-the-art work which has a query complexity of

O(\sqrt{N})

with overall time complexity

O\left(\sqrt{N}\left((\log N)^{2}+\log (M)\right)\right)

, while the ancilla count also reduces to

\frac{N}{2}

from

\frac{N}{2}+M

. The cost of state-of-the-art quantum circuit for string-matching problem is colossal due to a huge number of Fredkin gates and multi-controlled Toffoli gates. We have exhibited an improved gate cost and depth over the circuit by applying a proposed Fredkin gate decomposition with intermediate qutrits (3-dimensional qudits or ternary systems) and already existing logarithmic-depth decomposition of

n

-qubit Toffoli or multi-controlled Toffoli gate (MCT) with intermediate ququarts (4-dimensional qudits or quaternary systems). We have also asserted that the quantum circuit cost is relevant instead of using higher dimensional qudits through error analysis.