Short remarks on shallow unitary circuits
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912769123549184 |
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| author | Haah, Jeongwan |
| author_facet | Haah, Jeongwan |
| contents | (i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion using invertible subalgebra pumping. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short remarks on shallow unitary circuits Haah, Jeongwan Quantum Physics (i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion using invertible subalgebra pumping. |
| title | Short remarks on shallow unitary circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2504.14005 |