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Quantum CORDIC -- Arcsin on a Budget

Iain Burge, Michel Barbeau, Joaquin Garcia-Alfaro

TL;DR

This work introduces a quantum algorithm for computing the arcsine function to an arbitrary accuracy with CORDIC, a family of iterative algorithms that can approximate various trigonometric, hyperbolic, and elementary functions using only bit shifts and additions.

Abstract

This work introduces a quantum algorithm for computing the arcsine function to an arbitrary accuracy. We leverage a technique from embedded computing and field-programmable gate array (FPGA), called COordinate Rotation DIgital Computer (CORDIC). CORDIC is a family of iterative algorithms that, in a classical context, can approximate various trigonometric, hyperbolic, and elementary functions using only bit shifts and additions. Adapting CORDIC to the quantum context is non-trivial, as the algorithm traditionally uses several non-reversible operations. We detail a method for CORDIC which avoids such non-reversible operations. We propose multiple approaches to calculate the arcsine function reversibly with CORDIC. For n bits of precision, our method has space complexity of order n qubits, a layer count in the order of n times log n, and a CNOT count in the order of n squared. This primitive function is a required step for the Harrow-Hassidim-Lloyd (HHL) algorithm, is necessary for quantum digital-to-analog conversion, can simplify a quantum speed-up for Monte-Carlo methods, and has direct applications in the quantum estimation of Shapley values.

Quantum CORDIC -- Arcsin on a Budget

TL;DR

This work introduces a quantum algorithm for computing the arcsine function to an arbitrary accuracy with CORDIC, a family of iterative algorithms that can approximate various trigonometric, hyperbolic, and elementary functions using only bit shifts and additions.

Abstract

This work introduces a quantum algorithm for computing the arcsine function to an arbitrary accuracy. We leverage a technique from embedded computing and field-programmable gate array (FPGA), called COordinate Rotation DIgital Computer (CORDIC). CORDIC is a family of iterative algorithms that, in a classical context, can approximate various trigonometric, hyperbolic, and elementary functions using only bit shifts and additions. Adapting CORDIC to the quantum context is non-trivial, as the algorithm traditionally uses several non-reversible operations. We detail a method for CORDIC which avoids such non-reversible operations. We propose multiple approaches to calculate the arcsine function reversibly with CORDIC. For n bits of precision, our method has space complexity of order n qubits, a layer count in the order of n times log n, and a CNOT count in the order of n squared. This primitive function is a required step for the Harrow-Hassidim-Lloyd (HHL) algorithm, is necessary for quantum digital-to-analog conversion, can simplify a quantum speed-up for Monte-Carlo methods, and has direct applications in the quantum estimation of Shapley values.

Paper Structure

This paper contains 8 sections, 2 theorems, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Classical CORDIC for Arcsine
  5. Quantum CORDIC for Arcsine
  6. Binary to Amplitude Transformation
  7. Complexity and Simulations
  8. Conclusion

Key Result

Lemma 1

Suppose $i$ is even, at the beginning of the $i$th iteration of Algorithm 3's for loop, the state is, where $z$ is the initial value of in, and $r=-2^{-m}$.

Figures (5)

  • Figure 1: Unitary transformation $R$ which encodes the binary value of the aux register into the output register. Define $R_{y}(\omega)=(\cos\omega, -\sin\omega; \sin\omega, \cos\omega)$.
  • Figure 2: Unitary transformation $R'$ which transfers the binary encoding of CORDIC rotation direction stored d register into the amplitude of the out register. Define $R_{y}(\omega)=(\cos\omega, -\sin\omega; \sin\omega, \cos\omega)$, and $\mu_i=\tan^{-1}2^{-i}$.
  • Figure 3: Results of Classical simulations of Quantum compatible Algorithm for CORDIC Arcsine (see our GitHub repository at https://github.com/iain-burge/QuantumCORDIC for further details).
  • Figure :
  • Figure :

Theorems & Definitions (7)

  • Remark 6.1
  • Remark 6.2
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 2
  • proof