Linear-Size QAC0 Channels: Learning, Testing and Hardness

• URL: http://arxiv.org/abs/2510.00593v2
• Date: Sat, 08 Nov 2025 02:29:34 GMT
• Title: Linear-Size QAC0 Channels: Learning, Testing and Hardness
• Authors: Yangjing Dong, Fengning Ou, Penghui Yao,
• Abstract summary: Current noisy quantum hardware can only perform faithful quantum computation for a short amount of time.<n>We present the first algorithm for $mathbfQAC0$ channels using subexponential running time and queries.<n>We also establish exponential lower bounds on the query complexity of learning $mathbfQAC0$ channels under both the spectral norm distance of the Choi matrix and the diamond norm distance.
• Score: 13.101369903953804
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Shallow quantum circuits have attracted increasing attention in recent years, due to the fact that current noisy quantum hardware can only perform faithful quantum computation for a short amount of time. The constant-depth quantum circuits $\mathbf{QAC}^0$, a quantum counterpart of $\mathbf{AC}^0$ circuits, are the polynomial-size and constant-depth quantum circuits composed of only single-qubit unitaries and polynomial-size generalized Toffoli gates. The computational power of $\mathbf{QAC}^0$ has been extensively investigated in recent years. In this paper, we are concerned with $\mathbf{QLC}^0$ circuits, which are linear-size $\mathbf{QAC}^0$ circuits, a quantum counterpart of $\mathbf{LC}^0$. * We show that depth-$d$ $\mathbf{QAC}^0$ circuits working on $n$ input qubits and $a$ ancilla qubits have approximate degree at most $\tilde{O}((n+a)^{1-2^{-d}})$, improving the $\tilde{O}((n+a)^{1-3^{-d}})$ degree upper bound of previous works. Consequently, this directly implies that to compute the parity function, $\mathbf{QAC}^0$ circuits need at least $\tilde{O}(n^{1+2^{-d}})$ circuit size. * We present the first agnostic learning algorithm for $\mathbf{QLC}^0$ channels using subexponential running time and queries. Moreover, we also establish exponential lower bounds on the query complexity of learning $\mathbf{QAC}^0$ channels under both the spectral norm distance of the Choi matrix and the diamond norm distance. * We present a tolerant testing algorithm which determines whether an unknown quantum channel is a $\mathbf{QLC}^0$ channel. This tolerant testing algorithm is based on our agnostic learning algorithm. Our approach leverages low-degree approximations of $\mathbf{QAC}^0$ circuits and Pauli analysis as key technical tools. Collectively, these results advance our understanding of agnostic learning for shallow quantum circuits.

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