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A Novel Single-Layer Quantum Neural Network for Approximate SRBB-Based Unitary Synthesis

Giacomo Belli, Marco Mordacci, Michele Amoretti

Abstract

In this work, a novel quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and is subsequently redesigned with the number of CNOTs asymptotically reduced by an exponential contribution. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case of the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOT gates in the scalable variational quantum circuit, thus deriving a new implementable scaling scheme that requires only one layer of approximation. The single layer CNOT-reduced quantum neural network is implemented, and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.

A Novel Single-Layer Quantum Neural Network for Approximate SRBB-Based Unitary Synthesis

Abstract

In this work, a novel quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and is subsequently redesigned with the number of CNOTs asymptotically reduced by an exponential contribution. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case of the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOT gates in the scalable variational quantum circuit, thus deriving a new implementable scaling scheme that requires only one layer of approximation. The single layer CNOT-reduced quantum neural network is implemented, and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.

Paper Structure

This paper contains 44 sections, 8 theorems, 81 equations, 40 figures, 26 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Unitary approximation via SRBB: the algebraic structure
  3. Recursive construction of the matrix algebras
  4. Quantum circuit to approximate 2-qubit systems
  5. From the permutation group to ZYZ-decompositions
  6. Diagonal contributions
  7. Even contributions
  8. Odd contributions
  9. 2-qubit circuit to approximate SU(4)
  10. A scalable algorithm for optimizing CNOT-gates in the SRBB-based synthesis framework
  11. Simplifying CNOTs between diagonal contributions
  12. Gray-Code-based solution
  13. Solution for n=4
  14. Simplifying CNOTs between even/odd contributions
  15. Solution for n=4
  16. ...and 29 more sections

Key Result

Proposition 1

For $n=2$, $M_x^o\in SU(2^n)$ are block-diagonal matrices with $SU(2)$ blocks and admit a $ZYZ$-type decomposition.

Figures (40)

  • Figure 1: The single-layer VQC to approximate $SU(2^n)$ operators.
  • Figure 2: Complete set of elements $B_j^{(4)}$ for the Recursive Block Basis $\mathcal{B}^{(4)}$.
  • Figure 3: Complete set of elements $U_j^{(4)}$ for the Standard Recursive Block Basis $\mathcal{U}^{(4)}$.
  • Figure 4: Diagram to find the position of CNOTs and rotation gates.
  • Figure 5: Circuit representation of the diagonal basis $\mathcal{J}_Z$ for $n=2$.
  • ...and 35 more figures

Theorems & Definitions (16)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 6 more