Operator Spreading in Random Unitary Circuits with Unitary-invariant Gate Distributions
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| Format: | Preprint |
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2025
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| _version_ | 1866915269657493504 |
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| author | Tan, Zhiyang Brouwer, Piet W. |
| author_facet | Tan, Zhiyang Brouwer, Piet W. |
| contents | Random unitary circuits have become a model system to investigate information scrambling in quantum systems. In the literature, mostly random circuits with Haar-distributed gate operations have been considered. In this work, we investigate operator spreading in random unitary circuits in which the elementary gate operations are drawn from general unitary-invariant ensembles, which include the well-studied Haar-distributed random unitary circuits as a special case. Similar to the Haar-distributed case, the long-time behavior of operator spreading with the more general unitary-invariant gate distribution is governed by drift-diffusion equations characterized by the butterfly velocity $v_{\rm B}$ and a diffusion constant $\mathcal{D}$. Differences with the Haar-random case are (i) that it takes a finite time $τ_{\rm b}$ until ensemble-averaged Pauli-string weights take a ``binary'' form, in which they depend only on whether Pauli operators inside the support of the Pauli strong are equal to the identity matrix, and (ii) that the operator spreading is characterized by a finite ``domain-wall width'' $n_{\rm DW}$ separating regions with a random-matrix-like Pauli-string distribution. To illustrate these findings, we perform explicit calculations for random unitary circuits distributed according to the Poisson kernel, which interpolates between the trivial and Haar-distributed circuits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operator Spreading in Random Unitary Circuits with Unitary-invariant Gate Distributions Tan, Zhiyang Brouwer, Piet W. Quantum Physics Random unitary circuits have become a model system to investigate information scrambling in quantum systems. In the literature, mostly random circuits with Haar-distributed gate operations have been considered. In this work, we investigate operator spreading in random unitary circuits in which the elementary gate operations are drawn from general unitary-invariant ensembles, which include the well-studied Haar-distributed random unitary circuits as a special case. Similar to the Haar-distributed case, the long-time behavior of operator spreading with the more general unitary-invariant gate distribution is governed by drift-diffusion equations characterized by the butterfly velocity $v_{\rm B}$ and a diffusion constant $\mathcal{D}$. Differences with the Haar-random case are (i) that it takes a finite time $τ_{\rm b}$ until ensemble-averaged Pauli-string weights take a ``binary'' form, in which they depend only on whether Pauli operators inside the support of the Pauli strong are equal to the identity matrix, and (ii) that the operator spreading is characterized by a finite ``domain-wall width'' $n_{\rm DW}$ separating regions with a random-matrix-like Pauli-string distribution. To illustrate these findings, we perform explicit calculations for random unitary circuits distributed according to the Poisson kernel, which interpolates between the trivial and Haar-distributed circuits. |
| title | Operator Spreading in Random Unitary Circuits with Unitary-invariant Gate Distributions |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2501.04091 |