Constant-depth circuits for Boolean functions and quantum memory devices using multi-qubit gates
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| Format: | Preprint |
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2023
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| _version_ | 1866912131284205568 |
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| author | Allcock, Jonathan Bao, Jinge Doriguello, Joao F. Luongo, Alessandro Santha, Miklos |
| author_facet | Allcock, Jonathan Bao, Jinge Doriguello, Joao F. Luongo, Alessandro Santha, Miklos |
| contents | We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates defined by $|x\rangle|b\rangle\mapsto |x\rangle|b\oplus f(x)\rangle$ for $x\in\{0,1\}^n$ and $b\in\{0,1\}$, where $f$ is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register $|x\rangle$, while the second is based on Boolean analysis and exploits different representations of $f$ such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices -- Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) -- of memory size $n$. The implementation based on one-hot encoding requires either $O(n\log^{(d)}{n}\log^{(d+1)}{n})$ ancillae and $O(n\log^{(d)}{n})$ Fan-Out gates or $O(n\log^{(d)}{n})$ ancillae and $16d-10$ Global Tunable gates, where $d$ is any positive integer and $\log^{(d)}{n} = \log\cdots \log{n}$ is the $d$-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires $8d-6$ Global Tunable gates at the expense of $O(n^{1/(1-2^{-d})})$ ancillae. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_08539 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constant-depth circuits for Boolean functions and quantum memory devices using multi-qubit gates Allcock, Jonathan Bao, Jinge Doriguello, Joao F. Luongo, Alessandro Santha, Miklos Quantum Physics Computational Complexity Emerging Technologies We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates defined by $|x\rangle|b\rangle\mapsto |x\rangle|b\oplus f(x)\rangle$ for $x\in\{0,1\}^n$ and $b\in\{0,1\}$, where $f$ is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register $|x\rangle$, while the second is based on Boolean analysis and exploits different representations of $f$ such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices -- Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) -- of memory size $n$. The implementation based on one-hot encoding requires either $O(n\log^{(d)}{n}\log^{(d+1)}{n})$ ancillae and $O(n\log^{(d)}{n})$ Fan-Out gates or $O(n\log^{(d)}{n})$ ancillae and $16d-10$ Global Tunable gates, where $d$ is any positive integer and $\log^{(d)}{n} = \log\cdots \log{n}$ is the $d$-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires $8d-6$ Global Tunable gates at the expense of $O(n^{1/(1-2^{-d})})$ ancillae. |
| title | Constant-depth circuits for Boolean functions and quantum memory devices using multi-qubit gates |
| topic | Quantum Physics Computational Complexity Emerging Technologies |
| url | https://arxiv.org/abs/2308.08539 |