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

Dalzell, Alexander M. and Brandão, Fernando G. S. L. (2019) Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians. Quantum, 3 . Art. No. 187. ISSN 2521-327X. doi:10.22331/q-2019-09-23-187.

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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 ϵ-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/ϵ)^(1+o(1)), and at the expense of worse but still poly(k,1/ϵ) 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/ϵ)), 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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Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© 2019 This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. Published: 2019-09-23. We thank Thomas Vidick for useful discussions about this work and its algorithmic implications. AMD gratefully acknowledges support from the Dominic Orr Fellowship and the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1745301. This work was supported by NSF and Samsung. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Dominic Orr FellowshipUNSPECIFIED
NSF Graduate Research FellowshipDGE-1745301
Samsung ScholarshipUNSPECIFIED
NSF Physics Frontiers CenterUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Record Number:CaltechAUTHORS:20190801-134544835
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97602
Deposited By: George Porter
Deposited On:01 Aug 2019 21:37
Last Modified:16 Nov 2021 17:32

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