Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A quantum neural network framework for scalable quantum circuit approximation of unitary matrices

Rohit Sarma Sarkar, Bibhas Adhikari

TL;DR

This work introduces a Lie-group–driven framework for parametrically representing and approximating unitary matrices acting on $n$ qubits. It defines a Recursive Standard Block Basis (SRBB) that generalizes the Pauli basis to $d=2^n$, enabling products of exponentials to express unitaries, and proves both exact representations for 2-level unitaries and algorithmic approaches for broader targets. A layered quantum neural-network view is adopted, where each layer corresponds to a set of exponentials, optimized to minimize Frobenius distance to a target unitary, with careful ordering to yield scalable circuit constructions. The authors derive explicit quantum circuits for transpositions, diagonal blocks, multi-controlled rotations, and block-diagonal structures, and provide gate-count bounds showing scalability with the number of qubits. Numerical experiments on Haar-random and standard unitaries demonstrate high-precision approximations, especially for sparse targets, and illustrate how layer depth improves accuracy, pointing to practical potential for scalable quantum compilation on NISQ devices.

Abstract

In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order $2^n$. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an $(n+1)$-qubit unitary can be constructed from the circuit of $n$-qubit system by adding a few CNOT gates and single-qubit gates.

A quantum neural network framework for scalable quantum circuit approximation of unitary matrices

TL;DR

This work introduces a Lie-group–driven framework for parametrically representing and approximating unitary matrices acting on qubits. It defines a Recursive Standard Block Basis (SRBB) that generalizes the Pauli basis to , enabling products of exponentials to express unitaries, and proves both exact representations for 2-level unitaries and algorithmic approaches for broader targets. A layered quantum neural-network view is adopted, where each layer corresponds to a set of exponentials, optimized to minimize Frobenius distance to a target unitary, with careful ordering to yield scalable circuit constructions. The authors derive explicit quantum circuits for transpositions, diagonal blocks, multi-controlled rotations, and block-diagonal structures, and provide gate-count bounds showing scalability with the number of qubits. Numerical experiments on Haar-random and standard unitaries demonstrate high-precision approximations, especially for sparse targets, and illustrate how layer depth improves accuracy, pointing to practical potential for scalable quantum compilation on NISQ devices.

Abstract

In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order . The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an -qubit unitary can be constructed from the circuit of -qubit system by adding a few CNOT gates and single-qubit gates.
Paper Structure (18 sections, 173 equations, 7 figures, 5 tables, 7 algorithms)

This paper contains 18 sections, 173 equations, 7 figures, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Recursive construction of Hermitian unitary basis
  3. Construction of Hermitian unitary basis for $d\times d$ complex matrices
  4. Hermitian unitary basis for $2^n\times 2^n$ complex matrices
  5. Lie group theoretic approach for parametric representation of unitary matrices
  6. Exact parametric representation of certain unitary matrices
  7. Algorithmic approximation of special unitary matrices
  8. Approximation of $n$-qubit unitaries
  9. Numerical simulations
  10. Quantum circuit representation of unitary matrices of order $2^n$
  11. Quantum circuits for product of transpositions
  12. Quantum circuit for diagonal unitaries
  13. Quantum circuit for multi-controlled rotation gates
  14. Quantum circuit for unitary block diagonal matrices
  15. Scalable quantum circuits for approximating special unitary matrices
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Errors in the Frobenius norm using Algorithm \ref{['algo1']} (a) and Algorithm \ref{['alg2']} (b), considering original and modified SRB basis elements for the decomposition of $2$-qubit unitary matrices sampled from Haar distribution.
  • Figure 2: The errors obtained from up to three iterations (layers) for approximating $3$-qubit Haar random unitaries. The error after $3$rd iteration lies between $10^{-4}$ to $10^{-6}.$
  • Figure 3: Errors in Frobenius norm for approximating random $4$-sparse and $6$-sparse $3$-qubit unitaries with two ordering of the SRBB, considering only one iteration of the algorithm.
  • Figure 4: Error for approximating unitaries of order $3$ and $5$ using Algorithm \ref{['algo1']} up to one iteration. The unitary matrices are sampled at random from Haar distribution and Nelder-Mead is employed for optimization.
  • Figure 5: Errors for approximating Haar random $8$-sparse $4$-qubit block-diagonal unitaries considering only one iteration of the Algorithm \ref{['alg2']}.
  • ...and 2 more figures