Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates
Haferkamp J, Montealegre-Mora F, Heinrich M, Eisert J, Gross D, Roth (2023)
Publication Type: Journal article
Publication year: 2023
Journal
Book Volume: 397
Pages Range: 995-1041
Journal Issue: 3
DOI: 10.1007/s00220-022-04507-6
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(t4log 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.
Involved external institutions
Germany (DE)How to cite
APA:
Haferkamp, J., Montealegre-Mora, F., Heinrich, M., Eisert, J., Gross, D., & Roth, . (2023). Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates. Communications in Mathematical Physics, 397(3), 995-1041. https://doi.org/10.1007/s00220-022-04507-6
MLA:
Haferkamp, J., et al. "Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates." Communications in Mathematical Physics 397.3 (2023): 995-1041.
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