Hybrid quantum-classical computers have the potential to simulate many-electron systems with non-exponential cost. In this lecture we consider the two-electron reduced density matrix (2-RDM) as the basic variable in both the state preparation and the tomography of hybrid quantum-classical simulation. First, we present the preparation of quantum states for superconductivity and exciton condensation as well as novel quantum states that simultaneously exhibit both superconductivity and exciton condensation. We verify the preparation of these states by measuring the 2-RDM and the particle-hole RDM whose O(N) large eigenvalues provide the condensate populations. The eigenvectors of the large eigenvalues reveal the quantum entanglement that enables the frictionless transport of electrons and energy. Second, we discuss the central role of the 2-RDM in both the variational quantum eigensolvers (VQE) as well as a new family of solvers, known as contracted quantum eigensolvers (CQE). In classical 2-RDM theory the 2-RDM can be computed from an approximate solution of a contraction (or projection) of the Schrödinger equation onto the space of two (or more) electrons, known as the contracted Schrödinger equation. The CQE is the hybrid-quantum-classical analog of purely classical methods for solving the anti-Hermitian part of the contracted Schrödinger equation for the energies and 2-RDMs of ground and excited states. We discuss advantages of the CQE as well as applications to molecular systems including the isomers of benzyne. Finally, we show how the <i>N</i>-representable 2-RDM can be uniquely parameterized in terms of qubit particles, the natural particles of the quantum computer, for many-electron quantum simulations without the additional expense of fermionic encoding of the wave function.

Two-electron reduced density matrix as the basic variable in the hybrid quantum-classical computing of many-electron systems

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