Moire systems provide a rich platform for studies of strong correlation physics. Recent experiments on hetero-bilayer transition metal dichalcogenide (TMD) Moire systems are exciting in that they manifest a relatively simple model system of an extend
ed Hubbard model on a triangular lattice. Inspired by the prospect of the hetero-TMD Moire systems potential as a solid-state-based quantum simulator, we explore the extended Hubbard model on the triangular lattice using the density matrix renormalization group (DMRG). Specifically, we explore the two-dimensional phase space of the kinetic energy relative to the interaction strength $t/U$ and the further-range interaction strength $V_1/U$. We find competition between Fermi fluid, chiral spin liquid, spin density wave, and charge density wave. In particular, our finding of the optimal further-range interaction for the chiral correlation presents a tantalizing possibility.
It is imperative that useful quantum computers be very difficult to simulate classically; otherwise classical computers could be used for the applications envisioned for the quantum ones. Perfect quantum computers are unarguably exponentially difficu
lt to simulate: the classical resources required grow exponentially with the number of qubits $N$ or the depth $D$ of the circuit. Real quantum computing devices, however, are characterized by an exponentially decaying fidelity $mathcal{F} sim (1-epsilon)^{ND}$ with an error rate $epsilon$ per operation as small as $approx 1%$ for current devices. In this work, we demonstrate that real quantum computers can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer. Our algorithms compress the representations of quantum wavefunctions using matrix product states (MPS), which capture states with low to moderate entanglement very accurately. This compression introduces a finite error rate $epsilon$ so that the algorithms closely mimic the behavior of real quantum computing devices. The computing time of our algorithm increases only linearly with $N$ and $D$. We illustrate our algorithms with simulations of random circuits for qubits connected in both one and two dimensional lattices. We find that $epsilon$ can be decreased at a polynomial cost in computing power down to a minimum error $epsilon_infty$. Getting below $epsilon_infty$ requires computing resources that increase exponentially with $epsilon_infty/epsilon$. For a two dimensional array of $N=54$ qubits and a circuit with Control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours. For more complex gates such as a swap gate followed by a controlled rotation, the error rate increases by a factor three for similar computing time.
