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Operator Relaxation and the Optimal Depth of Classical Shadows
Matteo Ippoliti1, Yaodong Li1, Tibor Rakovszky1
1Department of Physics, Stanford University, Stanford, California 94305, USA.
Physical Review Letters
|June 24, 2023
Summary
Classical shadows efficiently learn quantum states using randomized measurements. Shallow shadows with optimized circuit depth exponentially improve sample complexity for learning Pauli operators, connecting quantum dynamics to information science.
Area of Science:
- Quantum Information Science
- Quantum Many-Body Dynamics
Background:
- Classical shadows are a sample-efficient method for characterizing quantum states using randomized measurements.
- Learning properties of quantum states is crucial for quantum information science.
Purpose of the Study:
- Investigate the sample complexity of learning Pauli operator expectation values using shallow shadows.
- Analyze the impact of local unitary circuit depth on learning efficiency.
Main Methods:
- Studied the shadow norm, which governs sample complexity, by analyzing operator evolution under twirling circuits.
- Examined operator spreading and relaxation dynamics for spatially contiguous Pauli operators.
- Derived bounds on shadow norm and quantitative results in 1D using mean-field approximation.
Main Results:
- Showed shadow norm depends on Heisenberg time evolution of operator weight distribution.
- Derived an upper bound for the shadow norm, yielding exponential sample complexity gains for shallow shadows (depth t~log(k)).
- Obtained quantitative 1D results, including a universal correction to optimal depth, matching numerical simulations.
Conclusions:
- Shallow shadows offer significant sample complexity advantages over the t=0 protocol.
- Connected quantum many-body dynamics (operator spreading/relaxation) to quantum information protocols.
- Paved the way for optimized protocols for learning quantum state properties.
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