Haferkamp, J., Montealegre-Mora, F., Heinrich, M., Eisert, J., Gross, D. and Roth, I (2023). Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates. Commun. Math. Phys., 397 (3). S. 995 - 1042. NEW YORK: SPRINGER. ISSN 1432-0916
Full text not available from this repository.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/epsilon)) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an epsilon-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.
Item Type: | Journal Article | ||||||||||||||||||||||||||||
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URN: | urn:nbn:de:hbz:38-678285 | ||||||||||||||||||||||||||||
DOI: | 10.1007/s00220-022-04507-6 | ||||||||||||||||||||||||||||
Journal or Publication Title: | Commun. Math. Phys. | ||||||||||||||||||||||||||||
Volume: | 397 | ||||||||||||||||||||||||||||
Number: | 3 | ||||||||||||||||||||||||||||
Page Range: | S. 995 - 1042 | ||||||||||||||||||||||||||||
Date: | 2023 | ||||||||||||||||||||||||||||
Publisher: | SPRINGER | ||||||||||||||||||||||||||||
Place of Publication: | NEW YORK | ||||||||||||||||||||||||||||
ISSN: | 1432-0916 | ||||||||||||||||||||||||||||
Language: | English | ||||||||||||||||||||||||||||
Faculty: | Unspecified | ||||||||||||||||||||||||||||
Divisions: | Unspecified | ||||||||||||||||||||||||||||
Subjects: | no entry | ||||||||||||||||||||||||||||
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URI: | http://kups.ub.uni-koeln.de/id/eprint/67828 |
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