Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates
Jonas Haferkamp, Felipe Montealegre‐Mora, Markus Heinrich, Jens Eisert, D. Gross, Ingo Roth · Communications in Mathematical Physics · 2022
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 $$O(t^{4}\log ^{2}(t)\log (1/\varepsilon ))$$ O ( t 4 log 2 ( t ) log ( 1 / ε ) ) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an $$\varepsilon $$ 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.