Improved Classical Shadow Tomography Using Quantum Computation
Zahra Honjani, Mohsen Heidari
TL;DR
The paper addresses the high classical and quantum resource costs of shadow tomography by introducing a quantum-to-classical-to-quantum (QCQC) CST framework. In this approach, initial random measurements yield classical data, which is then used to prepare quantum shadow states that are directly measured by the target observables, eliminating expensive classical trace computations. The authors provide rigorous analyses for two measurement regimes: Clifford and Pauli, proving unbiasedness and variance bounds, and deriving concrete copy, time, and space complexities. For k-local or low-rank observables, the QCQC method achieves a quadratic speedup in running time and exponential reduction in space compared with standard CST, enabling more scalable shadow tomography on larger systems.
Abstract
Classical shadow tomography (CST) involves obtaining enough classical descriptions of an unknown state via quantum measurements to predict the outcome of a set of quantum observables. CST has numerous applications, particularly in algorithms that utilize quantum data for tasks such as learning, detection, and optimization. This paper introduces a new CST procedure that exponentially reduces the space complexity and quadratically improves the running time of CST with single-copy measurements. The approach utilizes a quantum-to-classical-to-quantum process to prepare quantum states that represent shadow snapshots, which can then be directly measured by the observables of interest. With that, calculating large matrix traces is avoided, resulting in improvements in running time and space complexity. The paper presents analyses of the proposed methods for CST, with Pauli measurements and Clifford circuits.