Logarithmic-Depth Quantum Circuits for Hamming Weight Projections

• URL: http://arxiv.org/abs/2404.07151v3
• Date: Thu, 24 Oct 2024 14:02:43 GMT
• Title: Logarithmic-Depth Quantum Circuits for Hamming Weight Projections
• Authors: Soorya Rethinasamy, Margarite L. LaBorde, Mark M. Wilde,
• Abstract summary: We propose several quantum algorithms that realize a coherent Hamming weight projective measurement on an input pure state. We analyze a depth-width trade-off for the corresponding quantum circuits, allowing for a depth reduction of the circuits at the cost of more control qubits.
• Score: 3.481985817302898
• License:
• Abstract: A pure state of fixed Hamming weight is a superposition of computational basis states such that each bitstring in the superposition has the same number of ones. Given a Hilbert space of the form $\mathcal{H} = (\mathbb{C}_2)^{\otimes n}$, or an $n$-qubit system, the identity operator can be decomposed as a sum of projectors onto subspaces of fixed Hamming weight. In this work, we propose several quantum algorithms that realize a coherent Hamming weight projective measurement on an input pure state, meaning that the post-measurement state of the algorithm is the projection of the input state onto the corresponding subspace of fixed Hamming weight. We analyze a depth-width trade-off for the corresponding quantum circuits, allowing for a depth reduction of the circuits at the cost of more control qubits. For an $n$-qubit input, the depth-optimal algorithm uses $O(n)$ control qubits and the corresponding circuit has depth $O(\log (n))$, assuming that we have the ability to perform qubit resets. Furthermore, the proposed algorithm construction uses only one- and two-qubit gates.

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