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Reuse method for quantum circuit synthesis

Cyril Allouche, Marc Baboulin, Goubault de Brugière, Timothée, Benoît Valiron

Published Aug 20, 2017• 0 citations
OpenAlex

Abstract

The algebraic decomposition of a unitary operator is a key operation in the synthesis of quantum circuits. If most methods factorize the matrix into products, there exists a method that allows to reuse already existing optimized circuits to implement linear combinations of them. This paper presents an attempt to extend this method to a general framework of circuit synthesis. The method needs to find suitable groups for the implementation of new quantum circuits. We identify key points necessary for the construction of a comprehensive method and we test potential group candidates.

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The algebraic decomposition of a unitary operator is a key operation in the synthesis of quantum circuits. If most methods factorize the matrix into products, there exists a method that allows to reuse already existing optimized circuits to implement linear combinations of them. This paper presents an attempt to extend this method to a general framework of circuit synthesis. The method needs to find suitable groups for the implementation of new quantum circuits. We identify key points necessary for the construction of a comprehensive method and we test potential group candidates.
The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.
The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.
Until today, we have not found a general and effective algorithm for synthesis, especially to multi-variables quantum circuit.This is an important question needed to be resolved, because it can not only reduce the cost of manufacture quantum circuit, but also optimize many quantum algorithms.For a quantum circuit which contains N quantum bits, the latitude of the matrix is 𝟐 𝒏 to realize its function.If we directly design the circuits of the matrix, the workload is huge.Want to reduce the dimensions of the matrix, but also to ensure the unitary of matrix after decomposing, so it is need to use Kronecker product.
We apply numerical optimization and linear algebra algorithms for classical computers to the problem of automatically synthesizing algorithms for quantum computers. Using our framework, we apply several common techniques from these classical domains and numerically examine their suitability for and performance on this problem. Our methods are evaluated on single-qubit systems as well as on larger systems. While the first part of our proposed method outputs a single unitary matrix representing the composite effects of a quantum circuit or algorithm, we use existing tools—and assess the performance of these—to factor such a matrix into a product of elementary quantum gates. This enables our pipeline to be truly end-to-end: starting from desired input/output examples, our code ultimately results in a quantum circuit diagram. The code implementation is publicly available at https://github.com/DABH/qule .
Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for the translation of bigger unitary gates into elementary quantum operations, which is key to executing these algorithms on existing quantum computers. The decomposition can be used as an aggressive optimization method for the whole circuit, as well as to test part of an algorithm on a quantum accelerator. For the selection and implementation of the decomposition algorithm, perfect qubits are assumed. We base our decomposition technique on Quantum Shannon Decomposition, which generates O(344n) controlled-not gates for an n-qubit input gate. In addition, we implement optimizations to take advantage of the potential underlying structure in the input or intermediate matrices, as well as to minimize the execution time of the decomposition. Comparing our implementation to Qubiter and the UniversalQCompiler (UQC), we show that our implementation generates circuits that are much shorter than those of Qubiter and not much longer than the UQC. At the same time, it is also up to 10 times as fast as Qubiter and about 500 times as fast as the UQC.
Constructing appropriate unitary matrix operators for new quantum algorithms and finding the minimum cost gate sequences for the implementation of these unitary operators is of fundamental importance in the field of quantum information and quantum computation. Evolution of quantum circuits faces two major challenges: complex and huge search space and the high costs of simulating quantum circuits on classical computers. Here, we use the group leaders optimization algorithm to decompose a given unitary matrix into a proper-minimum cost quantum gate sequence. We test the method on the known decompositions of Toffoli gate, the amplification step of the Grover search algorithm, the quantum Fourier transform, and the sender part of the quantum teleportation. Using this procedure, we present the circuit designs for the simulation of the unitary propagators of the Hamiltonians for the hydrogen and the water molecules. The approach is general and can be applied to generate the sequence of quantum gates for larger molecular systems.