Shallow Shadows: Expectation Estimation Using Low-Depth Random Clifford Circuits

Explore a 52-minute lecture on shallow shadows and expectation estimation using low-depth random Clifford circuits. Delve into Christian Bertoni's presentation from the Dahlem Center for Complex Quantum Systems at Freie Universitat Berlin, Germany. Learn about a new scheme that interpolates between random single-qubit Clifford measurements and random global Clifford measurements using a brickwork circuit of random two-qubit gates. Discover how this approach achieves sample efficiency comparable to global Clifford measurements at logarithmic depth while maintaining favorable characteristics of single-qubit Clifford measurements. Gain insights into the potential of logarithmically deep Clifford circuits for classical shadows in experimental implementations. Follow the lecture's progression through topics such as the HKP protocol, global Cliffords scheme, change in terminology, storage of classical shadow, measurement channel representation and inversion, protocol execution, and analytical bounds for locally scrambled shadow norm.

Intro
The HKP protocol: estimating averages
The HKP protocol: what is needed
The global Cliffords scheme
Shallow shadows
Change in terminology
Storage of classical shadow
The measurement channel: representation
The measurement channel: inversion
Running the protocol
Analytical bound for the locally scrambled shadow norm
Computing the shadow norm