Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period $p$ . We conjecture that to create a simple periodic function of period $p$ , where $p$ is an $n$ -bit number, one needs at most $n$ Toffoli gates.Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period $p$. We conjecture that to create a simple periodic function of period $p$, where $p$ is an $n$-bit number, one needs at most $n$ Toffoli gates.

Quantum Information and ComputationJul 1, 20140 citations

An efficient algorithm to synthesize quantum circuits and optimization

Ömercan Susam, Mustafa Altun

Quantum computers, more specifically quantum circuits, take on the eyes with their computational promises such as reversibility. In this paper, we perform synthesis and optimization of quantum circuits. In the first part, we propose a fast synthesis algorithm that implements any given reversible Boolean function with quantum gates. Instead of an exhaustive search on every given function, our algorithm creates a library of essential functions and performs sorting. As an example, to implement 4 bit circuits we only use 120 essential functions out of all 20922789888000 functions. In the second part, we optimize our circuits by using new templates. The proposed templates mainly consist of Toffoli gates with negative and positive controlling lines. These templates also show us that optimum area solutions proposed in the literature are not actually optimum; they can be improved.Quantum computers, more specifically quantum circuits, take on the eyes with their computational promises such as reversibility. In this paper, we perform synthesis and optimization of quantum circuits. In the first part, we propose a fast synthesis algorithm that implements any given reversible Boolean function with quantum gates. Instead of an exhaustive search on every given function, our algorithm creates a library of essential functions and performs sorting. As an example, to implement 4 bit circuits we only use 120 essential functions out of all 20922789888000 functions. In the second part, we optimize our circuits by using new templates. The proposed templates mainly consist of Toffoli gates with negative and positive controlling lines. These templates also show us that optimum area solutions proposed in the literature are not actually optimum; they can be improved.

IEEE International Conference on Electronics, Circuits and Systems (ICECS)Dec 1, 20142 citations

A Key Component of the Shor's Algorithm and Its Possible Quantum Implementations

Jiří Tomčala

A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.

Dec 17, 20250 citations

Optimal Multi-Bit Toffoli Gate Synthesis

Young-Min Jun, In-Chan Choi

Multi-bit Toffoli gates form an essential quantum gate class for quantum algorithms. They should be efficiently decomposed into elementary single- or multi-qubit quantum gates, such as CNOT, T, and Hadarmard, for a scalable implementation of a quantum algorithm. We propose an engineering method for the practical synthesis of a multi-bit Toffoli gate. Two optimization models and their closed-form solutions are presented for optimal decomposition of the multi-bit Toffoli gate. These models are based on linearized multi-objective integer programming with parameters such as the number of target ancillae, ancillae states, and basis gates. The proposed method supports the systematic handling of quantum circuit constraints, including the total number of available qubits and maximum circuit depth, which depend on various quantum hardware specifications. Our approach exhibits promise in the noisy intermediate-scale quantum environment by providing a rapid and optimal method for synthesizing multi-bit Toffoli gates in diverse and unpredictable quantum hardware specifications.Multi-bit Toffoli gates form an essential quantum gate class for quantum algorithms. They should be efficiently decomposed into elementary single- or multi-qubit quantum gates, such as CNOT, T, and Hadarmard, for a scalable implementation of a quantum algorithm. We propose an engineering method for the practical synthesis of a multi-bit Toffoli gate. Two optimization models and their closed-form solutions are presented for optimal decomposition of the multi-bit Toffoli gate. These models are based on linearized multi-objective integer programming with parameters such as the number of target ancillae, ancillae states, and basis gates. The proposed method supports the systematic handling of quantum circuit constraints, including the total number of available qubits and maximum circuit depth, which depend on various quantum hardware specifications. Our approach exhibits promise in the noisy intermediate-scale quantum environment by providing a rapid and optimal method for synthesizing multi-bit Toffoli gates in diverse and unpredictable quantum hardware specifications.

IEEE AccessJan 1, 20233 citations

Synthesis of Quantum Arrays from Kronecker Functional Lattice Diagrams

Martin Lukáč, Dipal Shah, Marek A. Perkowski +1 more

Reversible logic is becoming more and more popular due to the fact that many novel technologies such as quantum computing, low power CMOS circuit design or quantum optical computing are becoming more and more realistic. In quantum computing, reversible computing is the main venue for the realization and design of classical functions and circuits. We present a new approach to synthesis of reversible circuits using Kronecker Functional Lattice Diagrams (KFLD). Unlike many of contemporary algorithms for synthesis of reversible functions that use n×n Toffoli gates, our method synthesizes functions using 3×3 Toffoli gates, Feynman gates and NOT gates. This reduces the quantum cost of the designed circuit but adds additional ancilla bits. The resulting circuits are always regular in a 4-neighbor model and all connections are predictable. Consequently resulting circuits can be directly mapped in to a quantum device such as quantum FPGA [14]. This is a significant advantage of our method, as it allows us to design optimum circuits for a given quantum technology.Reversible logic is becoming more and more popular due to the fact that many novel technologies such as quantum computing, low power CMOS circuit design or quantum optical computing are becoming more and more realistic. In quantum computing, reversible computing is the main venue for the realization and design of classical functions and circuits. We present a new approach to synthesis of reversible circuits using Kronecker Functional Lattice Diagrams (KFLD). Unlike many of contemporary algorithms for synthesis of reversible functions that use n×n Toffoli gates, our method synthesizes functions using 3×3 Toffoli gates, Feynman gates and NOT gates. This reduces the quantum cost of the designed circuit but adds additional ancilla bits. The resulting circuits are always regular in a 4-neighbor model and all connections are predictable. Consequently resulting circuits can be directly mapped in to a quantum device such as quantum FPGA [14]. This is a significant advantage of our method, as it allows us to design optimum circuits for a given quantum technology.

IEICE Transactions on Information and SystemsJan 1, 20143 citations

Reversible logic circuit synthesis

Vivek Shende, Aditya K. Prasad, Igor L. Markov +1 more

Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.

International Conference on Computer Aided DesignJun 25, 200368 citations

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