Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians
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 exponential...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2019-09-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2019-09-23-187/pdf/ |
| _version_ | 1818025700437786624 |
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| author | Alexander M. Dalzell Fernando G. S. L. Brandão |
| author_facet | Alexander M. Dalzell Fernando G. S. L. Brandão |
| author_sort | Alexander M. Dalzell |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-10T04:20:17Z |
| format | Article |
| id | doaj.art-7bf3afdb9896495ab8efa2e876f4cc6d |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-12-10T04:20:17Z |
| publishDate | 2019-09-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-7bf3afdb9896495ab8efa2e876f4cc6d2022-12-22T02:02:26ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2019-09-01318710.22331/q-2019-09-23-18710.22331/q-2019-09-23-187Locally accurate MPS approximations for ground states of one-dimensional gapped local HamiltoniansAlexander M. DalzellFernando G. S. L. BrandãoA 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.https://quantum-journal.org/papers/q-2019-09-23-187/pdf/ |
| spellingShingle | Alexander M. Dalzell Fernando G. S. L. Brandão Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians Quantum |
| title | Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians |
| title_full | Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians |
| title_fullStr | Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians |
| title_full_unstemmed | Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians |
| title_short | Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians |
| title_sort | locally accurate mps approximations for ground states of one dimensional gapped local hamiltonians |
| url | https://quantum-journal.org/papers/q-2019-09-23-187/pdf/ |
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