The Clifford
+T gate set is commonly used to perform universal quantum computation. In such setup the
T gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of
T gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial
T-count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of
T gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of
π/4 Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last
T gate of the circuit.
The Clifford+ T gate set is commonly used to perform universal quantum computation. In such setup the T gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of T gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial T -count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of T gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of π /4 Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last T gate of the circuit.
Among the cost metrics characterizing a quantum circuit, the T -count stands out as one of the most crucial as its minimization is particularly important in various areas of quantum computation such as fault-tolerant quantum computing and quantum circuit simulation. In this work, we contribute to the T -count reduction problem by proposing efficient T -count optimizers with low execution times. In particular, we greatly improve the complexity of TODD, an algorithm currently providing the best T -count reduction on various quantum circuits. We also propose some modifications to the algorithm which are leading to a significantly lower number of T gates. In addition, we propose another algorithm which has an even lower complexity and that achieves a better or equal T -count than the state of the art on most quantum circuits evaluated. We also prove that the number of T gates in the circuit obtained after executing our algorithms on a Hadamard-free circuit composed of n qubits is upper bounded by n ( n + 1 ) / 2 + 1 , which improves on the worst-case T -count of existing optimization algorithms. From this we derive an upper bound of ( n + 1 ) ( n + 2 h ) / 2 + 1 for the number of T gates in a Clifford + T circuit where h is the number of internal Hadamard gates in the circuit, i.e. the number of Hadamard gates lying between the first and the last T gate of the circuit.