Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

Alexander M. Dalzell1 and Fernando G. S. L. Brandão1,2

1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
2Google LLC, Venice, CA 90291, USA

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A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length $N$ of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-$k$ segments of the chain, the reduced density matrices of our approximations are $\epsilon$-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like $(k/\epsilon)^{1+o(1)}$, and at the expense of worse but still poly$(k,1/\epsilon)$ scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension $\exp(O(k/\epsilon))$, which is exponentially worse, but still independent of $N$. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find $O(1)$-accurate local approximations to the ground state in $T(N)$ time implies the ability to estimate the ground state energy to $O(1)$ precision in $O(T(N)\log(N))$ time.

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[2] Karel Van Acoleyen, Andrew Hallam, Matthias Bal, Markus Hauru, Jutho Haegeman, and Frank Verstraete, "Entanglement compression in scale space: From the multiscale entanglement renormalization ansatz to matrix product operators", Physical Review B 102 16, 165131 (2020).

[3] Yichen Huang, "Matrix product state approximations: Bringing theory closer to practice", Quantum Views 3, 26 (2019).

[4] Zhi-Yuan Wei, Daniel Malz, Alejandro González-Tudela, and J. Ignacio Cirac, "Generation of photonic matrix product states with Rydberg atomic arrays", Physical Review Research 3 2, 023021 (2021).

[5] Yichen Huang, 2020 IEEE International Symposium on Information Theory (ISIT) 1927 (2020) ISBN:978-1-7281-6432-8.

[6] Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete, "Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems", arXiv:2011.12127.

[7] Zongping Gong, Naoto Kura, Masatoshi Sato, and Masahito Ueda, "Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology", arXiv:1904.12464.

[8] Anurag Anshu, Itai Arad, and David Gosset, "Entanglement subvolume law for 2D frustration-free spin systems", arXiv:1905.11337.

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