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The Clifford$+T$ gate set is commonly used to perform universal quantumcomputation. In such setup the $T$ gate is typically much more expensive toimplement in a fault-tolerant way than Clifford gates. To improve thefeasibility of fault-tolerant quantum computing it is then crucial to minimizethe number of $T$ gates. Many algorithms, yielding effective results, have beendesigned to address this problem. It has been demonstrated that performing apre-processing step consisting of reducing the number of Hadamard gates in thecircuit can help to exploit the full potential of these algorithms and therebylead to a substantial $T$-count reduction. Moreover, minimizing the number ofHadamard gates also restrains the number of additional qubits and operationsresulting from the gadgetization of Hadamard gates, a procedure used by somecompilers to further reduce the number of $T$ gates. In this work we tackle theHadamard gate reduction problem, and propose an algorithm for synthesizing asequence of $\pi/4$ Pauli rotations with a minimal number of Hadamard gates.Based on this result, we present an algorithm which optimally minimizes thenumber of Hadamard gates lying between the first and the last $T$ gate of thecircuit.

Optimal Hadamard gate count for Clifford$+T$ synthesis of Pauli rotations sequences