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arXiv:quant-ph/0301077

Stochastic Simulation of Grover's Algorithm

L. R. U. MANSSUR, Renato Portugal

Published Jan 16, 2003• 3 citations
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Abstract

We simulate Grover's algorithm in a classical computer by means of a stochastic method using the Hubbard-Stratonovich decomposition of n-qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. The equations are obtained automatically for any number of qubits by employing a computer algebra program. We present the numerical results of the simulation for a small search space.

DOI
10.48550/arxiv.quant-ph/0301077

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We simulate Grover's algorithm in a classical computer by means of a stochastic method using the Hubbard-Stratonovich decomposition of n-qubit gates into one-qubit gates integrated over auxiliary fields. The problem reduces to finding the fixed points of the associated system of Langevin differential equations. The equations are obtained automatically for any number of qubits by employing a computer algebra program. We present the numerical results of the simulation for a small search space.
Grover's search algorithm is one of the most significant quantum algorithms, which can obtain quadratic speedup of the extensive search problems. Since Grover's search algorithm cannot be implemented on a real quantum computer at present, its quantum simulation is regarded as an effective method to study the search performance. When simulating the Grover's algorithm, the storage space required is exponential, which makes it difficult to simulate the high-qubit Grover's algorithm. To this end, we deeply study the storage problem of probability amplitude, which is the core of the Grover simulation algorithm. We propose a novel memory-efficient method via amplitudes compression, and validate the effectiveness of the method by theoretical analysis and simulation experimentation. The results demonstrate that our compressed simulation search algorithm can help to save nearly 87.5% of the storage space than the uncompressed one. Thus under the same hardware conditions, our method can dramatically reduce the required computing nodes, and at the same time, it can simulate at least 3 qubits more than the uncompressed one. Particularly, our memory-efficient simulation method can also be used to simulate other quantum algorithms to effectively reduce the storage costs required in simulation.
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.
This paper develops and analyzes Grover’s quantum search algorithm in 2- and 3-qubit systems in terms of the number of gates, representation of state vectors, and state vector measurement probabilities. We designed, verified and simulated the quantum circuit using QCAD. We present a theoretical implementation of the search algorithm in different qubit systems. We may observe that our circuit design for 2-qubit systems is accurate and does not allow any error when a small error occurs in the design of a 3-qubit quantum system. it was considered to what extend given information about object of the research is used purposefully in the modeling process, as a result of mathematical analysis of Grover’s method of the quantum algorithm, i.e. it is important to provide the adequacy of the model. The results of the optimization performed with the quantum algorithm based on Grover’s method are presented.
We present a method for optimizing quantum circuit compilation by automating the allocation of auxiliary qubits for multi-qubit gate decompositions. This approach is implemented and evaluated within the high-level quantum programming platform Ket. Our results indicate that the decomposition of multi-qubit gates is more effectively handled by the compiler, which has access to all circuit parameters, rather than through a quantum programming API. To evaluate the approach, we compared our implementation against Qiskit, a widely used quantum programming platform, by analyzing two quantum algorithms. Using a 16-qubit QPU, we observed a reduction of 87% in the number of CNOT gates in Grover's algorithm for 9 qubits. For a state preparation algorithm with 7 qubits, the number of CNOT gates was reduced from 2.8×1072.8\times10^72.8×107 to 5.7×1035.7\times10^35.7×103, leveraging additional Ket optimizations for high-level quantum program constructions. Overall, a quadratic reduction in the number of CNOT gates in the final circuit was observed, with greater improvements achieved when more auxiliary qubits were available. These findings underscore the importance of automatic resource management, such as auxiliary qubit allocation, in optimizing quantum applications and improving their suitability for near-term quantum hardware.