Quantum circuit designs of integer division optimizing T-count and T-depth

Himanshu Thapliyal, Edgard Munoz-Coreas, T. S.S. Varun, Travis S. Humble

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


Quantum circuits for mathematical functions such as division are necessary to use quantum computers for scientific computing. Quantum circuits based on Clifford+T gates can easily be made fault-tolerant but the T gate is very costly to implement. The small number of qubits available in existing quantum computers adds another constraint on quantum circuits. As a result, reducing T-count and qubit cost have become important optimization goals. The design of quantum circuits for integer division has caught the attention of researchers and designs have been proposed in the literature. However, these designs suffer from excessive T gate and qubit costs. Many of these designs also produce significant garbage output resulting in additional qubit and T gate costs to eliminate these outputs. In this work, we propose two quantum integer division circuits. The first proposed quantum integer division circuit is based on the restoring division algorithm and the second proposed design implements the non-restoring division algorithm. Both proposed designs are optimized in terms of T-count, T-depth and qubits. Both proposed quantum circuit designs are based on (i) a quantum subtractor, (ii) a quantum adder-subtractor circuit, and (iii) a novel quantum conditional addition circuit. Our proposed restoring division circuit achieves average T-count savings from 79.03 to 91.69 percent compared to the existing works. Our proposed non-restoring division circuit achieves average T-count savings from 49.22 to 90.03 percent compared to the existing works. Further, both our proposed designs have linear T-depth. We also illustrate the application of the proposed quantum division circuits in quantum image processing with a case study of quantum bilinear interpolation.

Original languageEnglish
Article number8691552
Pages (from-to)1045-1056
Number of pages12
JournalIEEE Transactions on Emerging Topics in Computing
Issue number2
StatePublished - Apr 1 2021

Bibliographical note

Funding Information:
This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. Travis Humble acknowledges support from the Department of Energy, Office of Science Early Career Research Program.

Publisher Copyright:
© 2013 IEEE.


  • Quantum computing
  • clifford+T gates
  • integer division
  • non-restoring division
  • quantum arithmetic
  • quantum circuits
  • restoring division

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Information Systems
  • Human-Computer Interaction
  • Computer Science Applications


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