Universität zu Köln

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

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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
Creators:	Creators Email ORCID ORCID Put Code Haferkamp, J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Montealegre-Mora, F. UNSPECIFIED UNSPECIFIED UNSPECIFIED Heinrich, M. UNSPECIFIED UNSPECIFIED UNSPECIFIED Eisert, J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Gross, D. UNSPECIFIED UNSPECIFIED UNSPECIFIED Roth, I UNSPECIFIED UNSPECIFIED UNSPECIFIED
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
Uncontrolled Keywords:	Keywords Language RANDOM QUANTUM CIRCUITS; SPECTRAL GAP Multiple languages Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/67828