A framework for semi-universality: Semi-universality of 3-qudit SU(d)-invariant gates

• URL: http://arxiv.org/abs/2407.21249v1
• Date: Tue, 30 Jul 2024 23:54:46 GMT
• Title: A framework for semi-universality: Semi-universality of 3-qudit SU(d)-invariant gates
• Authors: Austin Hulse, Hanqing Liu, Iman Marvian,
• Abstract summary: We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries. We also prove that these gates achieve full universality when supplemented with 3 ancilla qudits. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits.
• Score: 0.8192907805418583
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum circuits with symmetry-respecting gates have attracted broad interest in quantum information science. While recent work has developed a theory for circuits with Abelian symmetries, revealing important distinctions between Abelian and non-Abelian cases, a comprehensive framework for non-Abelian symmetries has been lacking. In this work, we develop novel techniques and a powerful framework that is particularly useful for understanding circuits with non-Abelian symmetries. Using this framework we settle an open question on quantum circuits with SU(d) symmetry. We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries, up to certain constraints on the relative phases between sectors with inequivalent representation of symmetry. Furthermore, we prove that these gates achieve full universality when supplemented with 3 ancilla qudits. Interestingly, we find that studying circuits with 3-qudit gates is also useful for a better understanding of circuits with 2-qudit gates. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits. Additionally, we investigate the statistical properties of circuits composed of random SU(d)-invariant gates. Our findings reveal that while circuits with 2-qudit gates do not form a 2-design for the Haar measure over SU(d)-invariant unitaries, circuits with 3-qudit gates generate a t-design, with t that is quadratic in the number of qudits.

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