On the other hand, suppose the experimental data obeys not only the Pauli constraints but also the generalized Pauli constraints, then the smallest 3 eigenvalues of each measured 1-RDM will only describe the smaller generalized Pauli polytope, pictured as just the yellow region in Fig. Suppose the experimental data satisfies the Pauli constraints but not the generalized Pauli constraints, then the smallest 3 eigenvalues of each measured 1-RDM will describe the Pauli polytope which is shown as the combined yellow and blue regions in Fig. The boundaries or “flat” sides of the polytope are determined by the Pauli and generalized Pauli constraints. ![]() The eigenvalues of the 1-RDM, ordered from highest to lowest, form a special type of convex set with “flat” sides known as a polytope. The 1-RDM is diagonalized on a classical computer and the eigenvalues (natural occupation numbers) are inserted into the generalized Pauli constraints to check for satisfaction. We measure the matrix elements of the 1-RDM of each state |Ψ i(123)〉 generated on the quantum computer and check each 1-RDM to verify satisfaction of the generalized Pauli constraints for 3 fermions in 6 orbitals. The natural orbitals are the eigenfunctions of the 1-electron reduced density matrix (1-RDM) that is defined by integrating the many-electron density matrix over the coordinates of all electrons except one We first randomly prepare the quantum state of a 3-electron system and second measure the occupations of the natural orbitals. Rapid advances in quantum hardware and hybrid classical-quantum algorithms have led to multi-qubit experimental implementations 20, 21, 22. Algorithms which utilize these quantum states promise a large computational advantage over known classical algorithms for solving critically important problems such as integer factorization, eigenvalues estimation, and fermionic simulation 16, 17, 18, 19. Quantum computers differ from classical computers in that quantum states can be prepared on the quantum computer. In this work we use quantum states prepared on a quantum computer to provide experimental verification of the generalized Pauli constraints. These inequalities, which have become known as generalized Pauli constraints 5, 6, 7, 8, provide new insights into the electronic structure of many-electron atoms and molecules 7, 9, 10, 11, 12, the limitations of entanglement as a resource for quantum control 13, 14, as well as the fundamental distinctions between open and closed quantum systems 15. In 2006 Klyachko (and in 2008 with Altunbulak) presented a systematic mathematical procedure for generating these constraints for potentially arbitrary numbers of electrons and orbitals 3, 4. ![]() In their calculations Borland and Dennis discovered, however, that even in three-electron atoms and molecules there are additional constraints beyond the well-known exclusion principle. Formally, the Pauli exclusion principle implies that the spin-orbital occupations are rigorously bounded by zero and one. ![]() In 1926 Pauli had observed that no more than a single electron can occupy a given one-electron quantum state known as a spin orbital 2. For example, the electron shell structure of lithium is 1s 22s 1 (two electrons in ‘s’ sub-shell of the first shell, and one electron in ‘s’ sub-shell of the second shell the superscript indicates the number of electrons in the shell).While performing calculations with classical computers at IBM, Borland and Dennis discovered something unexpected and surprising about the electronic structure of atoms and molecules 1. A shell is given a number and a letter ( s,p,d,f,g,etc. ). Each shell contains sub-shells or energy sub-levels. The number of electrons allowed in a shell is 2n 2. These numbers are the ‘principle quantum numbers’.Īn increase in n indicates an increase in energy associated with the shell, and an increase in the distance of the shell from the nucleus. The shells are numbered ( n = 1,2,3 etc. ) outwards from the nucleus. ![]() No two electrons in an atom can have the same quantum numberĪ quantum number describes certain properties of a particle such as its charge and spin.Īn orbital or energy level cannot hold more than two electrons, one spinning clockwise, the other anti-clockwise.Įlectrons are grouped in shells, which contain orbitals.
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