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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Comparison of Influence of Two Data-Encoding Methods for Grover Algorithm on Quantum Costs

Sidharth Dhawan, Marek A. Perkowski

Published May 1, 2011• 3 citations
Publisher

Abstract

It is important to be able to calculate realistic estimates of quantum costs for real oracles used in quantum algorithms. In this paper, we compare Perkowski's oracle data encoding method with Hogg's encoding method for Grover algorithm, to examine the decrease in Oracle gate cost, if any, for four common constraint satisfaction problems: Graph coloring, Satisfiability, Send-More-Money and Max Clique.

DOI
10.1109/ismvl.2011.29

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•Jun 18, 2021•0 citations
It is important to be able to calculate realistic estimates of quantum costs for real oracles used in quantum algorithms. In this paper, we compare Perkowski's oracle data encoding method with Hogg's encoding method for Grover algorithm, to examine the decrease in Oracle gate cost, if any, for four common constraint satisfaction problems: Graph coloring, Satisfiability, Send-More-Money and Max Clique.
Constraint satisfiability problems, crucial to several applications, are solved on a quantum computer using Grover's search algorithm, leading to a quadratic improvement over the classical case. The solutions are obtained with high probability for several cases and are illustrated for the cases involving two variables for both 3- and 4-bit numbers. Methods are defined for inequality comparisons, and these are combined according to the form of the satisfiability formula, to form the oracle for the algorithm. The circuit is constructed using IBM Qiskit and is verified on an IBM simulator. It is further executed on one of the Noisy Intermediate-Scale Quantum (NISQ) processors from IBM on the cloud. Noise levels in the processor at present are found to be too high for successful execution. Running the algorithm on the simulator with a custom noise model lets us identify the noise threshold for successful execution.
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over 105 relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over 103 . However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.
Many applications such as Graph coloring, Triangle finding, Boolean satisfiability, Traveling salesman problem can be solved by Grover's search algorithm, which is one of the remarkable quantum computing algorithm. To design a quantum circuit for a given quantum algorithm, which involves Grover's search, we need to define an oracle circuit specific to the given algorithm and the diffusion operator for amplification of the desired quantum state. In this paper, we propose a quantum circuit implementation for the oracle of the vertex coloring problem based on Grover's algorithmic approach. To the best of our knowledge, this is a first of its kind approach in regards to the quantum circuit synthesis of the vertex coloring oracle in binary quantum domain. We have performed the synthesis of the proposed oracle circuit for the six commonly available Physical Machine Description (PMD) gate libraries using the Fault Tolerant Quantum Logic Synthesis (FTQLS) tool. The synthesis results have been presented to understand the cost estimation of the oracle circuit for the various PMDs in terms of number of quantum operations and number of cycles.
The maximum clique of an undirected graph is the largest subgraph in which an edge exists between every vertex. The maximum clique problem presents itself in various fields and finding a tractable algorithm to solve the problem is important. This paper introduces an oracle to solve the maximum clique problem using Grover’s quantum algorithm. This approach potentially solves the maximum clique problem in O(∣V∣2∣V∣)O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)O(∣V∣2∣V∣​) time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.
Constraint satisfiability problems, crucial to several applications, are solved on a quantum computer using the Grover's search algorithm, leading to a quadratic improvement over the classical case. The solutions are obtained with high probability for several cases and are illustrated for the cases involving 2 variables for both 3-bit and 4-bit numbers. Methods are defined for inequality comparisons and these are combined according to the form of the satisfiability formula, to form the oracle for the algorithm. The circuit is constructed using IBM Qiskit and is verified on an IBM simulator. It is further executed on one of the Noisy Intermediate-Scale Quantum (NISQ) processors from IBM on the cloud. Noise levels in the processor at present are found to be too high for successful execution. Running the algorithm on the simulator with a custom noise model lets us identify the noise threshold for successful execution.