Scalable quantum circuits for $n$-qubit unitary matrices

TL;DR

This work tackles scalable representations of $n$-qubit unitaries by introducing the Standard Recursive Block Basis ($SRBB$) and a Lie-algebra–based parametric framework. It develops a recursive Hermitian unitary basis (RBB) and combines diagonal IZ-type elements with non-diagonal SRBB elements to form a complete basis for $n$ qubits, enabling a parametric decomposition of unitaries as products of exponentials of basis elements. An optimization-based, multi-layer approach is proposed to approximate a target unitary in ${\mathsf{SU}(2^n)}$ with controlled gate counts, supported by circuit constructions that realize SRBB-based components such as transpositions, diagonal rotations, and block-diagonal matrices, along with explicit scalability to larger $n$. Numerical experiments on $2$-, $3$-, and $4$-qubit targets show competitive accuracy, particularly for sparse unitaries, and demonstrate a practical path to scalable quantum circuit synthesis using SRBB-based parametrizations.

Abstract

This work presents an optimization-based scalable quantum neural network framework for approximating $n$-qubit unitaries through generic parametric representation of unitaries, which are obtained as product of exponential of basis elements of a new basis that we propose as an alternative to Pauli string basis. We call this basis as the Standard Recursive Block Basis, which is constructed using a recursive method, and its elements are permutation-similar to block Hermitian unitary matrices.

Figures (3)

• Figure 1: Approximation errors using Algorithm 1 for $2$-qubit unitary matrices sampled from random Haar distribution.
• Figure 2: The errors obtained from up to three iterations (layers) for $3$-qubit Haar random unitaries. The error after $3$rd iteration lies between $10^{-4}$ to $10^{-6}.$
• Figure 3: Errors for simulating random $8$-sparse $4$-qubit block-diagonal unitaries considering only one iteration of the Algorithm 1.