Shadow Tomography with 𝑁-Representability Conditions

One of the most sought-after quantum computing applications is molecular simulation. However, obtaining a full description of an 𝑁-qubit quantum system requires an exponential number of measurements and classical memory, translating into very long quantum circuits on quantum hardware. Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with direct applications in quantum computing. The output of this procedure is a classical shadow of an unknown quantum state that can be used to predict molecular properties. Here, we combine shadow tomography with additional physical constraints necessary for the measurements to be consistent with an 𝑁-electron quantum system. These constraints, known as 𝑁-representability conditions, significantly reduce the number of required measurements. We demonstrate the advantages of the algorithm by computing the ground- and excited-state properties of several molecular systems. The resulting tomography has the potential for significant applications in quantum many-body simulations on quantum devices.