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

Understand

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena.

• SU(d) is a continuous group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation.
• Here, we explicate three particularly interesting applications of symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning.
• First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at $t \rightarrow \infty$ which decays as $\Omega(1/n^{3/2})$ under local Pauli basis for qubits and $\Omega(1/n^{(d+2)^2/2})$ under symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator.