Logarithmic-Depth Quantum Circuits for Hamming Weight Projections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913655112597504 |
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| author | Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. |
| author_facet | Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07151 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Logarithmic-Depth Quantum Circuits for Hamming Weight Projections Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. Quantum Physics 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. |
| title | Logarithmic-Depth Quantum Circuits for Hamming Weight Projections |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2404.07151 |