PIRSA:20050048

Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates

Roth, I. (2020). Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates. Perimeter Institute. https://pirsa.org/20050048

Roth, Ingo. Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates. Perimeter Institute, May. 27, 2020, https://pirsa.org/20050048 

          @misc{ pirsa_PIRSA:20050048,
            doi = {10.48660/20050048},
            url = {https://pirsa.org/20050048},
            author = {Roth, Ingo},
            keywords = {Other},
            language = {en},
            title = {Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates},
            publisher = {Perimeter Institute},
            year = {2020},
            month = {may},
            note = {PIRSA:20050048 see, \url{https://pirsa.org}}
          }

Ingo Roth Freie Universität Berlin

DOI 10.48660/20050048
Collection
Talk Type Scientific Series
Subject

Abstract

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t^4 log^2(t) log(1/ε)) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε-approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators. Joint work with J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, and D. Gross.