Shadows and Cutting and Quantum States

Classical shadow tomography is a quantum measurement technique that efficiently estimates properties of quantum systems by constructing compact classical representations, called ​“shadows,” from random measurements. These shadows allow the prediction of quantum observables with fewer measurements than traditional quantum state tomography. 

Circuit cutting is a technique for dividing large quantum circuits into smaller fragments that can be executed using fewer quantum resources. The fragments can be executed independently, and their outputs can be recombined through classical postprocessing. But circuit cutting comes at a cost: postprocessing overheads can grow exponentially with the number of cuts made to a circuit.

Researchers from Argonne National Laboratory and the quantum information company Infleqtion have now developed a divide-and-conquer method for constructing classical shadows while incorporating circuit cutting procedures in realistic scenarios where the number of measurements is limited. The method, which they call fragmented shadow tomography, combines efficient computation and storage of quantum states through classical shadows and the ability to be parallelized in independent quantum devices.

Working at the cutting edge

Briefly stated, the method combines shadow tomography with quantum circuit cutting, a technique that divides large circuits into smaller, manageable fragments. This method allows smaller circuits to be executed independently on the quantum devices while classical postprocessing recombines their results. 

The study provides a mathematical framework for this approach, analyzes its computational complexity, and demonstrates its advantages in estimating higher weight observables acting on many qubits. ​“This approach outperforms traditional shadow tomography when dealing with high-weight observables in many scenarios,” said Zain Saleem, an assistant computational scientist in Argonne’s Mathematics and Computer Science Division and a co-author of the study.

Putting it to the test

To evaluate their approach, the researchers examined two possible circuit structures: a cascade circuit and a clustered cascade circuit. Using the two circuit structures, the researchers ran experiments to numerically study the trade-off between the size of the observable and the level of fragmentation. As shown in Figure 1 above, with the clustered circuit structure, for a fixed number of samples, or ​“shots,” fragmentation significantly improves the estimated expectation when large observables are being estimated. 

Figure 2, below, shows the numerical scaling of error with the number of fragments in the cascade circuit structure. The results show that the absolute error decreases as the number of fragments increases. The lower the absolute error, the closer the estimated expectation is to the true expected value. 

But the researchers also issued a caveat: More fragments are not always better. 

“Both analytically and numerically, we see a trade-off between the number of fragments and the size of the observable,” Saleem said. ​“For small observables, excessive fragmentation can lead to an accumulation of error. On the other hand, our results suggest that fragmentation can lead to dramatically fewer errors when the observable is large.” 

What is the optimal number of fragments for a given quantum circuit and a quantum observable? That remains an open question that the researchers are interested in exploring with their quantum divide-and-conquer approach.

For the full paper, see D. T. S. Chen, Z. H. Saleem, M. A. Perlin, ​“Quantum Circuit Cutting for Classical Shadows,”ACM Transactions on Quantum Computing 5(2) 1–21, 2024