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Quantum information processing in the presence of continuous symmetry is ofwide importance and exhibits many novel physical and mathematical phenomena.SU(d) is a continuous symmetry group of particular interest since it representsa fundamental type of non-Abelian symmetry and also plays a vital role inquantum computation. Here, we explicate the applications of SU(d)-symmetricrandom unitaries in three different contexts ranging from physics to quantumcomputing: information scrambling with non-Abelian conserved quantities,covariant quantum error correcting random codes, and geometric quantum machinelearning. First, we show that, in the presence of SU(d) symmetry, the localconserved quantities would exhibit residual values even at $t \rightarrow\infty$ which decays as $\Omega(1/n^{3/2})$ under local Pauli basis for qubitsand $\Omega(1/n^{(d+2)^2/2})$ under local symmetric basis for general quditswith respect to the system size, in contrast to $O(1/n)$ decay for U(1) caseand the exponential decay for no-symmetry case in the sense of out-of-timeordered correlator (OTOC). Second, we show that SU(d)-symmetric unitaries canbe used to construct asymptotically optimal (in the sense of saturating thefundamental limits on the code error that have been called the approximateEastin-Knill theorems) SU(d)-covariant codes that encodes any constant $k$logical qudits, extending [Kong \& Liu; PRXQ 3, 020314 (2022)]. Finally, wederive an overpartameterization threshold via the quantum neural tangent kernel(QNTK) required for exponential convergence guarantee of generic ansatz forgeometric quantum machine learning, which reveals that the number of parametersrequired scales only with the dimension of desired subspaces rather than thatof the entire Hilbert space. We expect that our work invites further researchon quantum information with continuous symmetries.

SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning