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Register a new userExploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions
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We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension D of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain of N sites and correlation length xi, the computational cost formally scales as g(D,xi /N)D^3, where g(D,xi /N) is a nontrivial function. For xi << N, this scaling reduces to D^3, independent of the system size N, making our algorithm N times faster than previous proposals. We apply the method to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/2 models as well as for the noncritical Heisenberg spin-1 model. In the critical case, for any chain length N, we find a model-dependent bond dimension D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.
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Using the algebraic Bethe ansatz, we derive a matrix product representation of the exact Bethe-ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-$frac{1}{2}$) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation and we further show by providing an additional example of the same matrix product state construction for a well known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries and thus suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.
We study transfer of a quantum state through XX spin chains with static imperfections. We combine the two standard approaches for state transfer based on (i) modulated couplings between neighboring spins throughout the spin chain and (ii) weak coupling of the outermost spins to an unmodulated spin chain. The combined approach allows us to design spin chains with modulated couplings and localized boundary states, permitting high-fidelity state transfer in the presence of random static imperfections of the couplings. The modulated couplings are explicitly obtained from an exact algorithm using the close relation between tridiagonal matrices and orthogonal polynomials [Linear Algebr. Appl. 21, 245 (1978)]. The implemented algorithm and a graphical user interface for constructing spin chains with boundary states (spinGUIn) are provided as Supplemental Material.
We present a method to apply the well-known matrix product state (MPS) formalism to partially separable states in solid state systems. The computational effort of our method is equal to the effort of the standard density matrix renormalisation group (DMRG) algorithm. Consequently, it is applicable to all usually considered condensed matter systems where the DMRG algorithm is successful. We also show in exemplary cases, that polymerisation properties of ground states are closely connected to properties of partial separability, even if the ground state itself is not partially separable.
We consider the tensors generating matrix product states and density operators in a spin chain. For pure states, we revise the renormalization procedure introduced by F. Verstraete et al. in 2005 and characterize the tensors corresponding to the fixed points. We relate them to the states possessing zero correlation length, saturation of the area law, as well as to those which generate ground states of local and commuting Hamiltonians. For mixed states, we introduce the concept of renormalization fixed points and characterize the corresponding tensors. We also relate them to concepts like finite correlation length, saturation of the area law, as well as to those which generate Gibbs states of local and commuting Hamiltonians. One of the main result of this work is that the resulting fixed points can be associated to the boundary theories of two-dimensional topological states, through the bulk-boundary correspondence introduced by Cirac et al. in 2011.
We detail techniques to optimise high-level classical simulations of Shors quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on $r$, the solution to the underlying order-finding problem of Shors algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shors algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.
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