SQMS Center

Science Highlight

Enabling classical scientific computing on quantum hardware with exact fixed-point arithmetic

Scientific computing requires reliable and predictable arithmetic operations for reproducibility. Approaches to date for fixed-point arithmetic are either inexact or impractical to implement. 

The Science:

We proposed a novel approach for computing the exact product of superpositions of fixed-point numbers by employing ancillary qubits and the Quantum Fourier Transform. We demonstrated the effectiveness our approach by simulating a 2-by-2 system of ordinary differential equations (ODE). In simulations of a quantum computer, we used the trapezoidal rule for numerical integration to approximate the solution of the ODE, compared the error with the known analytic solution over a range of precisions, and observed exponential decay in the error of the solution with respect to the number of fractional qubits used for each numerical quantity. 

Figure (a) shows the fixed-point encoding scheme that we used in our work. Each quantum register consists of n qubits, with f qubits denoting the fractional part and n-f qubits denoting the integral part. We used a Two’s Complement representation to include negative numbers. Figure (b) shows the ordinary differential equation that we integrated numerically to demonstrate the effectiveness of this approach. Figure (c) shows the analytic solution of the system of ordinary differential equations. Figures (d) and (e) show the evolution over time of the individual components of the solution for a range of precisions or number of fractional qubits. Figure (f) shows the error in the L2 sense of the error in the solutions, for individual components as well as for the vector of components.

The Impact:

Exact arithmetic is essential for the reliable and reproducible use of quantum computers for scientific computing purposes. Reproducibility ensures that one will always obtain the same answer from a simulation or optimization. This work advances SQMS’s goals by providing a framework with which to adapt classical scientific computing problems to quantum computers. 

Summary:

We identified and addressed the shortcomings of classical fixed-point arithmetic on quantum hardware. We proposed the use of ancillary qubits to render the multiplication of superpositions of fixed-point numbers to be exact. Exactness in these operations is crucial for the use of quantum computers for scientific applications. We also presented strategies for implementing higher order operations such as division and exponentiation. We demonstrated the effectiveness of our proposed approach for multiplication by simulating these operations on a classical computer and by solving numerically a sample ordinary differential equation, as one would solve one on a classical computer. The approach presented in this work offers a framework or strategy with which to adapt existing problems in classical scientific computing to the quantum setting. 

Contact:

José E. Cruz Serrallés, Jose.CruzSerralles@nyulangone.org

Focus Area:

Quantum Algorithms, Simulations and Benchmarking

Institutions:

Fermi National Accelerator Laboratory, NASA Ames, NYU Langone Health

Citation:

J. E. C. Serrallés, O. Ogunkoya, D. M. Kürkçüog̃lu, N. Bornman, N. M. Tubman, S. Zorzetti, R. Lattanzi, “A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations,” 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), Montreal, QC, Canada, 2024, pp. 50-57, DOI: 10.1109/QCE60285.2024.00016

Funding Acknowledgement:

This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under Contract Number DE-AC02-07CH11359.