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Advances in systems analysis, software engineering, and high performance computing book series

Optimal Circuit Decomposition of Reversible Quantum Gates on IBM Quantum Computers

Hilal Ahmad Bhat, Farooq Ahmad Khanday, Khurshed Ahmad Shah

Published Mar 3, 2023• 1 citation
Publisher

Abstract

A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. In this chapter, quantum implementations of FRSG1, URG, JTF1 and R gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM have been discussed. FRSG1 and JTF1 gates are important in applications like Stochastic computing, fingerprint authentication system, and parity generation circuits. URG gate is better in terms of number of complex functions and can be utilized to design quantum comparator circuits. R gate plays an important role in inverting and duplicating a signal. In FRSG1, URG, JTF1 and R gates, the implementation count of single qubit gates decreases by 56%, 11%, 71%, and 62%, respectively and the count of two qubit gates reduces by 15%, 26%, 41%, and 5% respectively after optimization.

DOI
10.4018/978-1-6684-6697-1.ch008

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A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. In this chapter, quantum implementations of FRSG1, URG, JTF1 and R gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM have been discussed. FRSG1 and JTF1 gates are important in applications like Stochastic computing, fingerprint authentication system, and parity generation circuits. URG gate is better in terms of number of complex functions and can be utilized to design quantum comparator circuits. R gate plays an important role in inverting and duplicating a signal. In FRSG1, URG, JTF1 and R gates, the implementation count of single qubit gates decreases by 56%, 11%, 71%, and 62%, respectively and the count of two qubit gates reduces by 15%, 26%, 41%, and 5% respectively after optimization.
Currently, computing paradigm is ruled by the laws of classical physics and will continue till the dimensions of transistors reaches the size of atomic particles. Quantum computing (QC) is a new technology that employs laws of quantum mechanics to address issues such as irreversibility and power dissipation that are outside the scope of conventional computing models. QC offers a powerful stage for solving complex and compound problems like searching and number factoring. A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Unitary decomposition is a popular scheme for mapping quantum algorithms to any set of elementary quantum operations. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates, which is critical for running algorithms on prevailing quantum computers. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. This paper is the first to present quantum implementations of FRSG1 and JTF1 gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM. These gates are important in a variety of practical applications like Stochastic computing and parity generation circuits. In FRSG1 the implementation count of single qubit gates decreases by 56% and the count of two qubit gates by 15% after optimization, whereas in JTF1 gate the single qubit gate count reduces by 71% and the number of two qubit gates reduces by 41%.
IBM has made several quantum computers available to researchers around the world via cloud services. Two architectures with five qubits, one with 16, and one with 20 qubits are available to run experiments. The IBM architectures implement gates from the Clifford+T gate library. However, each architecture only implements a subset of the possible CNOT gates. In this paper, we show how Clifford+T circuits can efficiently be mapped into the two IBM quantum computers with 5 qubits. We further present an algorithm and a set of circuit identities that may be used to optimize the Clifford+T circuits in terms of gate count and number of levels. It is further shown that the optimized circuits can considerably reduce the gate count and number of levels and thus produce results with better fidelity.
Reversible logic computation and optimization of reversible circuits help synthesize applications characterized by ultra‐low power consumption, including quantum information processing, the field of nanotechnology, semiconductor technology, the field of optics, and very‐large‐scale integration (VLSI), among others. Reversible logic design is a growing research subject that evaluates and applies resources in new advancements. This article provides an optimized, unique, resource‐constrained universal quantum reversible gate for nanoscale applications that perform all fundamental logic operations. Optimization and gate relocation have enhanced a published and proposed design. Quantum reversible circuits are optimized for cost and depth via template matching, gate fusion (merging) and commutation, gate decomposition, and gate elimination. These methods reduce circuit complexity and size significantly. A step‐by‐step optimization of the quantum reversible logic design is first presented. The novel quantum reversible gates MRQ1, MRQ2, and M‐BUS are implemented and verified on the IBM Qiskit platform. The proposed reversible gates’ design complexity is assessed using 13 standard Boolean expressions. Proposed quantum reversible gates demonstrate an improvement in performance, covering a range of 12.5%–72.7% over prior designs in gate count, depth, and quantum cost. Gate count improved most, impacting other parameters and demonstrating the design’s resource efficiency.
We provide a method for compiling approximate multicontrolled single qubit gates into quantum circuits. Without ancilla qubits, the total number of elementary gates to decompose an n-qubit multicontrolled gate is proportional to 32n elementary operations. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multitarget and multicontrolled CNOT and SU(2) gates. We also provide an approximate decomposition with ancilla qubits with lower-circuit complexity. Computational experiments show the reduction of CNOT gates when multicontrolled U(2) gates are applied. As multicontrolled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications.
This paper proposes an optimized quantum block-ZXZZXZZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general nnn-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.
IBM's quantum computers implement gates from Clifford +T gate library. All single qubit gates are implemented, but only a subset of the possible CNOT are provided. It is well known that the functionally of the missing gates can be achieved by a sequence of gates. The sequence of gates is based on SWAP gates. Up to seven elementary gates are required to implement a SWAP gate. In this paper we show how the same effect can be achieved with fewer gates. To show the potential of the proposed transformations, an example is presented where a reduction of 44% in the gate count and a 26% reduction in the number of levels for IBM's QX5 computer is achieved. An algorithm that is considered state of the art, is used for the comparison.