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Reconstructing the Hadamard Gate from Euler-Angle Parameters

P. K. Majumdar

Published Aug 19, 2026• 0 citations
Publisher

Abstract

This notebook implements the general single-qubit unitary matrix U(θ, φ, λ) and evaluates it for θ = π/2, φ = 0 and λ = π. It compares the generated array with the exact Hadamard matrix, uses matrix multiplication to verify their equivalence, and independently confirms that the resulting operator satisfies the unitarity condition U†U = I. The exercise demonstrates how familiar quantum gates can be recovered as particular cases of a general parameterised single-qubit transformation.

DOI
10.5281/zenodo.22013297

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This notebook implements the general single-qubit unitary matrix U(θ, φ, λ) and evaluates it for θ = π/2, φ = 0 and λ = π. It compares the generated array with the exact Hadamard matrix, uses matrix multiplication to verify their equivalence, and independently confirms that the resulting operator satisfies the unitarity condition U†U = I. The exercise demonstrates how familiar quantum gates can be recovered as particular cases of a general parameterised single-qubit transformation.
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Determining the optimal implementation of a quantum gate is critical for designing a quantum computer. We consider the crucial task of efficiently decomposing a general single-qubit quantum gate into a sequence of fault-tolerant quantum operations. For a given single-qubit circuit, we construct an optimal gate sequence consisting of fault-tolerant Hadamard (H) and π/8 rotations (T). Our scheme is based on a novel canonical form for single-qubit quantum circuits and the corresponding rules for exactly reducing a general single-qubit circuit to our canonical form. The result is optimal in the number of T gates. We demonstrate that a precomputed epsilon net of canonical circuits in combination with our scheme lowers the depth of approximation circuits by up to 3 orders of magnitude compared to previously reported results.
Abstract. Determining the optimal implementation of a quantum gate is critical for designing a quantum computer. We consider the crucial task of efficiently decomposing a general single-qubit quantum gate into a sequence of fault-tolerant quantum operations. For a given single-qubit circuit, we construct an optimal gate sequence consisting of fault-tolerant Hadamard (H) and π/8 rotations (T). Our scheme is based on a novel canonical form for single-qubit quantum circuits and the corresponding rules for exactly reducing a general single-qubit circuit to our canonical form. The result is optimal in the number of T gates. We demonstrate that a precomputed ɛ-net of canonical circuits in combination with our scheme lowers the depth and number of T gates of approximation circuits by up to three orders of magnitude compared to previously reported results.
Among the many proposals for the realization of a quantum computer, holonomic quantum computation is distinguished from the rest as it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the controlled-NOT gate and the SWAP gate, and the discrete Fourier transformation can be obtained with a single loop.