Abstract
We construct entanglement renormalization schemes which provably approximate the ground states of non-interacting fermion nearest-neighbor hopping Hamiltonians on the one-dimensional discrete line and the two-dimensional square lattice. These schemes give hierarchical quantum circuits which build up the states from unentangled degrees of freedom. The circuits are based on pairs of discrete wavelet transforms which are approximately related by a "half-shift": translation by half a unit cell. The presence of the Fermi surface in the two-dimensional model requires a special kind of circuit architecture to properly capture the entanglement in the ground state. We show how the error in the approximation can be controlled without ever performing a variational optimization.
Highlights
Characterizing quantum phases of matter and computing their low-temperature physical properties are two of the central challenges of quantum many-body physics
We show that Eq (10) guarantees that the single-particle modes prepared by the multiscale entanglement renormalization ansatz (MERA) are approximate eigenmodes, and the boundedness of the scaling function ensures that the truncation error decreases exponentially with the number of layers of the tensor network
We showed how wavelet theory can be used to rigorously construct quantum circuits that approximate metallic states of noninteracting fermions
Summary
Characterizing quantum phases of matter and computing their low-temperature physical properties are two of the central challenges of quantum many-body physics. We provide quantum circuits that, when acting on a suitable unentangled state, prepare approximations to the metallic ground states of certain one- and twodimensional noninteracting fermion Hamiltonians This is a nontrivial task in part because these states of matter are highly entangled and violate the area law. The bottom of the figure corresponds to the state jψi, each layer of red and green blocks constitutes the quantum circuit implementing URG, the product states j1ij0i on half of the sites make up jχi, and the lines that go up into the layer correspond to jψi on the other half of the sites, which can be identified with the renormalized lattice To realize this approach, we still need to design finite-length filters hs, hw such that the wavelet transform. We discuss in detail how this can be done systematically and to arbitrarily high fidelity for two fundamental model systems
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