Abstract
One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an nO(logn) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is {tilde{O}(nM(n))} , where M(n) is the time required to multiply two n × n matrices.
Highlights
One of the central challenges in the study of quantum systems is their exponential complexity [14]: the state of a system on n particles is given by a vector in an exponentially large Hilbert space, so even giving a classical description of the state is a challenge
The approach was subsequently improved by White [33,34], to obtain the famous Density Matrix Renormalization Group (DMRG) algorithm [33,34], which is widely used as a numerical heuristic for identifying the ground and low energy states of 1D systems
It takes as input a local Hamiltonian satisfying assumptions (FF), (DG) or (LD) and a precision parameter δ, and returns matrix product states (MPS) representations for a viable set that is δ-close to the low-energy space T of H
Summary
One of the central challenges in the study of quantum systems is their exponential complexity [14]: the state of a system on n particles is given by a vector in an exponentially large Hilbert space, so even giving a classical description (of size polynomial in n) of the state is a challenge. This procedure is based on the construction of a suitable class of approximate ground state projections (AGSPs) [3,5]—spectral AGSPs—and improves the dimension-quality trade-off, at the cost of increasing the complexity of the underlying MPS representations Setting this last cost aside, the two procedures can be combined to achieve what we call viable set amplification: a reduction in the dimension of a viable set, while maintaining its viability parameter unchanged In the case of a frustration-free Hamiltonian with unique ground state we obtain a running time of O(2O(1/γ 2)n1+o(1)M(n)), where M(n) is the matrix multiplication time This has an exponentially better scaling in terms of the spectral gap γ (due to avoidance of the ε-net argument) and saves a factor of n/ log n (due to the logarithmic, instead of linear, number of iterations) as compared to an algorithm for the same problem considered in [18].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
