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1. Experimental realization of an extended Fermi-Hubbard model using a 2D lattice of dopant-based quantum dots

   https://doi.org/10.1038/s41467-022-34220-w  

   Wang, Xiqiao; Khatami, Ehsan; Fei, Fan; Wyrick, Jonathan; Namboodiri, Pradeep; Kashid, Ranjit; Rigosi, Albert F.; Bryant, Garnett; Silver, Richard (December 2022, Nature Communications)

   Abstract The Hubbard model is an essential tool for understanding many-body physics in condensed matter systems. Artificial lattices of dopants in silicon are a promising method for the analog quantum simulation of extended Fermi-Hubbard Hamiltonians in the strong interaction regime. However, complex atom-based device fabrication requirements have meant emulating a tunable two-dimensional Fermi-Hubbard Hamiltonian in silicon has not been achieved. Here, we fabricate 3 × 3 arrays of single/few-dopant quantum dots with finite disorder and demonstrate tuning of the electron ensemble using gates and probe the many-body states using quantum transport measurements. By controlling the lattice constants, we tune the hopping amplitude and long-range interactions and observe the finite-size analogue of a transition from metallic to Mott insulating behavior. We simulate thermally activated hopping and Hubbard band formation using increased temperatures. As atomically precise fabrication continues to improve, these results enable a new class of engineered artificial lattices to simulate interactive fermionic models. 

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2. Low rank representations for quantum simulation of electronic structure

   https://doi.org/10.1038/s41534-021-00416-z  

   Motta, Mario; Ye, Erika; McClean, Jarrod R.; Li, Zhendong; Minnich, Austin J.; Babbush, Ryan; Chan, Garnet Kin-Lic (May 2021, npj Quantum Information)

   Abstract The quantum simulation of quantum chemistry is a promising application of quantum computers. However, forNmolecular orbitals, the$${\mathcal{O}}({N}^{4})$$ $O\left(N^4\right)$gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy allow one to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with$${\mathcal{O}}({N}^{3})$$ $O\left(N^3\right)$gate complexity in small simulations, which reduces to$${\mathcal{O}}({N}^{2})$$ $O\left(N^2\right)$gate complexity in the asymptotic regime; and unitary Coupled Cluster Trotter steps with$${\mathcal{O}}({N}^{3})$$ $O\left(N^3\right)$gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have$${\mathcal{O}}({N}^{2})$$ $O\left(N^2\right)$depth on a linearly connected array, an improvement over the$${\mathcal{O}}({N}^{3})$$ $O\left(N^3\right)$scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 10⁵non-Clifford rotations. We also apply our algorithm to iron–sulfur clusters relevant for elucidating the mode of action of metalloenzymes. 

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3. Numerical solution of large scale Hartree–Fock–Bogoliubov equations

   https://doi.org/10.1051/m2an/2020074  

   Lin, Lin; Wu, Xiaojie (May 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   The Hartree–Fock–Bogoliubov (HFB) theory is the starting point for treating superconducting systems. However, the computational cost for solving large scale HFB equations can be much larger than that of the Hartree–Fock equations, particularly when the Hamiltonian matrix is sparse, and the number of electrons N is relatively small compared to the matrix size N b . We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most 𝒪( N b 2 ) for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with N b up to 2.88 × 10 6 , and the wall clock time is less than 100 s using 17 280 CPU cores. This enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems. 

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4. Complexity Phase Transitions Generated by Entanglement

   https://doi.org/10.1103/PhysRevLett.131.030601  

   Ghosh, Soumik; Deshpande, Abhinav; Hangleiter, Dominik; Gorshkov, Alexey V.; Fefferman, Bill (July 2023, Physical Review Letters)

   Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this Letter, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of k-regular graph states on n qubits. We show that, as the regularity parameter is increased from 1 to n−1, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at k = 3, and a transition back to easy and low entanglement at k = n−3. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity. 

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5. Thermodynamic and kinetic studies of H ₂ and N ₂ binding to bimetallic nickel-group 13 complexes and neutron structure of a Ni(η ² -H ₂ ) adduct

   https://doi.org/10.1039/C9SC02018G  

   Cammarota, Ryan C.; Xie, Jing; Burgess, Samantha A.; Vollmer, Matthew V.; Vogiatzis, Konstantinos D.; Ye, Jingyun; Linehan, John C.; Appel, Aaron M.; Hoffmann, Christina; Wang, Xiaoping; et al (July 2019, Chemical Science)

   Understanding H 2 binding and activation is important in the context of designing transition metal catalysts for many processes, including hydrogenation and the interconversion of H 2 with protons and electrons. This work reports the first thermodynamic and kinetic H 2 binding studies for an isostructural series of first-row metal complexes: NiML, where M = Al ( 1 ), Ga ( 2 ), and In ( 3 ), and L = [N( o -(NCH 2 P i Pr 2 )C 6 H 4 ) 3 ] 3− . Thermodynamic free energies (Δ G °) and free energies of activation (Δ G ‡ ) for binding equilibria were obtained via variable-temperature 31 P NMR studies and lineshape analysis. The supporting metal exerts a large influence on the thermodynamic favorability of both H 2 and N 2 binding to Ni, with Δ G ° values for H 2 binding found to span nearly the entire range of previous reports. The non-classical H 2 adduct, (η 2 -H 2 )NiInL ( 3 -H 2 ), was structurally characterized by single-crystal neutron diffraction—the first such study for a Ni(η 2 -H 2 ) complex or any d 10 M(η 2 -H 2 ) complex. UV-Vis studies and TD-DFT calculations identified specific electronic structure perturbations of the supporting metal which poise NiML complexes for small-molecule binding. ETS-NOCV calculations indicate that H 2 binding primarily occurs via H–H σ-donation to the Ni 4p z -based LUMO, which is proposed to become energetically accessible as the Ni(0)→M( iii ) dative interaction increases for the larger M( iii ) ions. Linear free-energy relationships are discussed, with the activation barrier for H 2 binding (Δ G ‡ ) found to decrease proportionally for more thermodynamically favorable equilibria. The Δ G ° values for H 2 and N 2 binding to NiML complexes were also found to be more exergonic for the larger M( iii ) ions. 

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