On the Approximation of the Quantum Gates using Lattices
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arXiv
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| Format: | Preprint |
| Published: |
2015
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| _version_ | 1866917247151243264 |
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| author | Greene, A. Damelin, S. B. |
| author_facet | Greene, A. Damelin, S. B. |
| contents | A central question in Quantum Computing is how matrices in $SU(2)$ can be approximated by products over a small set of generators. A topology will be defined on $SU(2)$ so as to introduce the notion of a covering exponent which compares the length of products required to covering $SU(2)$ with $\varepsilon$ balls against the Haar measure of $\varepsilon$ balls. An efficient universal set over $PSU(2)$ will be constructed using the Pauli matrices, using the metric of the covering exponent. Then, the relationship between $SU(2)$ and $S^3$ will be manipulated to correlate angles between points on $S^3$ to give a conjecture on the maximum of angles between points on a lattice. It will be shown how this conjecture can be used to compute the covering exponent. Some extensions are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1506_05785 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On the Approximation of the Quantum Gates using Lattices Greene, A. Damelin, S. B. Quantum Algebra 81P68, 11K36, 68Q12, 28A78, 11E12, 11D09, 11H31 A central question in Quantum Computing is how matrices in $SU(2)$ can be approximated by products over a small set of generators. A topology will be defined on $SU(2)$ so as to introduce the notion of a covering exponent which compares the length of products required to covering $SU(2)$ with $\varepsilon$ balls against the Haar measure of $\varepsilon$ balls. An efficient universal set over $PSU(2)$ will be constructed using the Pauli matrices, using the metric of the covering exponent. Then, the relationship between $SU(2)$ and $S^3$ will be manipulated to correlate angles between points on $S^3$ to give a conjecture on the maximum of angles between points on a lattice. It will be shown how this conjecture can be used to compute the covering exponent. Some extensions are discussed. |
| title | On the Approximation of the Quantum Gates using Lattices |
| topic | Quantum Algebra 81P68, 11K36, 68Q12, 28A78, 11E12, 11D09, 11H31 |
| url | https://arxiv.org/abs/1506.05785 |