On Unitary k-Design with SU(d) Symmetry

We propose, for the first time, an explicit unitary ensemble that can achieve unitary k-design with SU(d) symmetry by Schur-Weyl duality on SU(d) and Sn actions on qudits. From the perspective of investigating commutant subspaces, we establish relationship among well-known approaches like tensor product expander, kth fold channel and frame potential to study unitary design with or without an imposed symmetry. Based on which, we further explore the potential of applying Sn representation theory, especially the celebrated Okounkov-Vershik approach, to quantum physics, during which we define the kth order Convolutional Quantum Alternating group (CQA) with kth order CQA ensemble with at most 2k-local unitary and prove that they form SU(d) symmetric k-design in exact and approximate senses respectively. Our results indicate that, despite a "no-go theorem" between locality and universality in the presence of SU(d) symmetry, ensembles of local SU(d)-symmetric unitary would still form unitary k-designs under SU(d) symmetry.