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This research establishes constant lower bounds on the spectral gap for specific random walks within unitary groups, including Random Pauli Rotations and Brickwork circuits. The findings are significant because they hold uniformly across all system sizes $n$ and apply to representations relevant for constructing unitary $t$-designs using Clifford unitaries. These results advance our theoretical understanding of quantum circuit mixing times without relying on asymptotic approximations that fail at finite scales.
Random unitary circuits with constant spectral gap
arXiv:2607.20919v1 Announce Type: new Abstract: We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $\theta \in \mathbb R / 2\pi \mathbb Z$, both uniformly at random, and apply $e^{\mathrm i \theta P}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gaps are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs. We also prove analogous constant gap results for Clifford unitaries, which are indispensable for our result on Brickwork Random Unitary Circuit.
Abstract:We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $\theta \in \mathbb R / 2\pi \mathbb Z$, both uniformly at random, and apply $e^{\mathrm i \theta P}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gaps are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs. We also prove analogous constant gap results for Clifford unitaries, which are indispensable for our result on Brickwork Random Unitary Circuit.
24 pages. Julia and Mathematica code
Quantum Physics (quant-ph); Probability (math.PR)
arXiv:2607.20919 [quant-ph]
(or arXiv:2607.20919v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2607.20919 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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