Multi-qubit controlled gate synthesis without T-count overhead in the small-error limit

Abstract

We study the Clifford+T synthesis of quantum multiplexers $U=\bigoplus_{i=1}^{2^n}U_i$: an $n$-qubit control register selects which single-qubit gate $U_i\in\mathrm{SU}(2)$ acts on the target qubit. For each block $U_i$, consider the minimum even T-count needed to approximate it within diamond distance $\varepsilon$ without ancillae, and let $m$ be the largest of these costs. For an arbitrary multiplexer $U$, we construct an approximation of $U$ within the same distance using at most $m+O(\sqrt{2^n(m+1)})$ $T$ gates. With one control qubit, the cost is $m+O(1)$ and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed $n$ and independent Haar-random blocks, the minimum T-count among these circuits lies between $(3-δ)\log_2(1/\varepsilon)$ and $(3+δ)\log_2(1/\varepsilon)$ for every fixed $δ>0$, except with probability $O(\varepsilon^{c_{n,δ}})$ as $\varepsilon\rightarrow0$. The optimal leading coefficient is therefore the same as for a single Haar-random $\mathrm{SU}(2)$ gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order $\varepsilon^2$. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random $\mathrm{SU}(4)$ gates within diamond distance $\varepsilon$, using at most $(9+δ)\log_2(1/\varepsilon)$ $T$ gates for every fixed $δ>0$, except with probability $O(\varepsilon^{c_δ})$ with $c_δ>0$.