Unitary k-designs from random number-conserving quantum circuits
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916699989606400 |
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| author | Hearth, Sumner N. Flynn, Michael O. Chandran, Anushya Laumann, Chris R. |
| author_facet | Hearth, Sumner N. Flynn, Michael O. Chandran, Anushya Laumann, Chris R. |
| contents | Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics. With a global $U(1)$ charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of $k$-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a $k_c$-design with $k_c = O(L^d)$ for a system in $d$ spatial dimensions with linear dimension $L$. Second, for $k < k_c$, we derive bounds on the depth $τ$ required for the circuit to converge to an approximate $k$-design. The depth is lower bounded by diffusion $k L^2 \ln(L) \lesssim τ$. In contrast, without number conservation $τ\sim \text{poly}(k) L$. The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks $k$ $U(1)$ symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound $τ\lesssim k L^{d+2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01035 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unitary k-designs from random number-conserving quantum circuits Hearth, Sumner N. Flynn, Michael O. Chandran, Anushya Laumann, Chris R. Statistical Mechanics Strongly Correlated Electrons High Energy Physics - Theory Chaotic Dynamics Quantum Physics Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics. With a global $U(1)$ charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of $k$-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a $k_c$-design with $k_c = O(L^d)$ for a system in $d$ spatial dimensions with linear dimension $L$. Second, for $k < k_c$, we derive bounds on the depth $τ$ required for the circuit to converge to an approximate $k$-design. The depth is lower bounded by diffusion $k L^2 \ln(L) \lesssim τ$. In contrast, without number conservation $τ\sim \text{poly}(k) L$. The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks $k$ $U(1)$ symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound $τ\lesssim k L^{d+2}$. |
| title | Unitary k-designs from random number-conserving quantum circuits |
| topic | Statistical Mechanics Strongly Correlated Electrons High Energy Physics - Theory Chaotic Dynamics Quantum Physics |
| url | https://arxiv.org/abs/2306.01035 |