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arXiv:0910.5055 · Physical Review A

Efficient algorithm for approximating one-dimensional ground states

Dorit Aharonov, Itai Arad, Sandy Irani

Published Jul 16, 2010• 22 citations
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Abstract

The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension $D$ whose energy approximates the minimal energy such states can achieve. The running time is exponential in $D$, and so the algorithm can be considered tractable even for $D$, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.

DOI
10.1103/physreva.82.012315

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The density-matrix renormalization-group method is very effective at finding ground states of one-dimensional (1D) quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works. In this article we describe an efficient classical algorithm which provably finds a good approximation of the ground state of 1D systems under well-defined conditions. More precisely, our algorithm finds a matrix product state of bond dimension DDD whose energy approximates the minimal energy such states can achieve. The running time is exponential in DDD, and so the algorithm can be considered tractable even for DDD, which is logarithmic in the size of the chain. The result also implies trivially that the ground state of any local commuting Hamiltonian in 1D can be approximated efficiently; we improve this to an exact algorithm.
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. The problem is known to be QMA-complete, even for one-dimensional Hamiltonians [1]. This means that we do not even expect that there is a sub-exponential size description of the ground state that allows efficient computation of local observables such as the energy. In sharp contrast, the heuristic density matrix renormalization group (DMRG) algorithm invented two decades ago [5] has been remarkably successful in practice on one-dimensional problems. The situation is reminiscent of the unexplained success of the simplex algorithm before the advent of ellipsoid and interior-point methods. Is there a principled explanation for this, in the form of a large class of one-dimensional Hamiltonians whose ground states can be provably efficiently approximated? Here we give such an algorithm for gapped one-dimensional Hamiltonians: our algorithm outputs an (inverse-polynomial) approximation to the ground state, expressed as a matrix product state (MPS) of polynomial bond dimension. The running time of the algorithm is polynomial in the number of qudits n and the approximation quality δ, for a fixed local dimension d and gap Δ > 0.
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.
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. We give the first randomized polynomial-time algorithm for finding ground states of gapped one-dimensional Hamiltonians: it outputs an (inverse-polynomial) approximation, expressed as a matrix product state (MPS) of polynomial bond dimension. The algorithm combines many ingredients, including recently discovered structural features of gapped 1D systems, convex programming, insights from classical algorithms for 1D satisfiability, and new techniques for manipulating and bounding the complexity of MPS. Our result provides one of the first major classes of Hamiltonians for which computing ground states is provably tractable despite the exponential nature of the objects involved.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
We consider the problem of approximating ground states of one-dimensional quantum systems within the two most common variational ansatzes, namely, the mean-field ansatz and matrix product states. We show that both for mean field and for matrix product states of fixed bond dimension, the optimal solutions can be found in a way which is provably efficient (i.e., scales polynomially). This implies that the corresponding variational methods can be in principle recast in a way which scales provably polynomially. Moreover, our findings imply that ground states of one-dimensional commuting Hamiltonians can be found efficiently.
We study the problem of computing energy density in one-dimensional quantum systems. We show that the ground-state energy per site or per bond can be computed in time (i) independent of the system size and subexponential in the desired precision if the ground state satisfies area laws for the Renyi entanglement entropy (this is the first rigorous formulation of the folklore that area laws imply efficient matrix-product-state algorithms); (ii) independent of the system size and polynomial in the desired precision if the system is gapped. As a by-product, we prove that in the presence of area laws (or even an energy gap) the ground state can be approximated by a positive semidefinite matrix product operator of bond dimension independent of the system size and subpolynomial in the desired precision of local properties.