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Unifying time evolution and optimization with matrix product states
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We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.
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“Without truncation (ε = 0), one can easily verify that, after applying the complete sweep (left-to-right and back) with a time step t to an MPS | [A] , the original physical state | [A] is obtained by reapplying the scheme with a time stept, although | [A] will now be encoded as an MPS with a larger bond dimension (and matrices l(n) or r(n), which do not have full rank).”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“That the adjoint method for the finite MPS integrator can be evaluated explicitly can be traced back to the fact that the individual terms obtained from splitting the tangent space projector are exactly integrable.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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“That these two gauge fixing conditions are related to a principal bundle connection implies that they transform covariantly under gauge transformations on the original MPS.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“This ensures that the overlap of two tangent vectors only contains diagonal contributions, as in [B 1 ]| [B 2 ] = N n=1 s n tr B s n 1 (n) † B s n 2 (n) .(A8) This corresponds to the standard Euclidean inner product and thus to the choice of an orthonormal basis.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“If the original MPS was already in the right canonical form, the singular values of C(n) correspond to the Schmidt coefficients of | for a bipartition of the lattice in [1 : n] and [n + 1 : N ].”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“As for the tangent space, the linear map from the representation {B s,t (n : n + 1); n = 1, . . .,N -1} to the physical states in T [2] | [A] M MPS has a nontrivial kernel corresponding to choices B s,t (n : n + 1) = A s L (n)X t (n + 1) -X s (n)A t R (n + 1) for any {X(n) ∈ C D n ×d×D n+1 ,n = 2, . . .,N -1}.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“That the adjoint method for the finite MPS integrator can be evaluated explicitly can be traced back to the fact that the individual terms obtained from splitting the tangent space projector are exactly integrable.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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“This freedom allows to impose a gauge-fixing condition of the form s A s L (n) † B s,t (n : n + 1) = 0, ∀n = 1, . . .,N -2.(C2)”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“Inserting this solution into | [B] gives immediate rise to | [B] = P T | [A] M MPS | = N n=1 P [1:n-1] L ⊗ 1n ⊗ P [n+1:N] R - N-1 n=1 P [1:n] L ⊗ P [n+1:N] R | ,(A12) where we can recognize the contents of the brackets as the decomposition of the tangent space projector into a sum of orthogonal projectors”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“In the case of a nearest-neighbor Hamiltonian, the projector error is absent as the action of Ĥ on | [A] is exactly contained in T [2] | [A L ] M MPS .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“The two-site evolution will take the state out of the manifold of MPS with fixed bond dimensions.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“The orthogonal projector PT [2] | [A] M MPS that maps any state | ∈ H onto the subspace T [2] | [A] M MPS can then be constructed and decomposed-using manipulations similar to those of the previous subsection-as a sum of orthogonal projections: PT [2] | [A L ] M MPS = N-1 n=1 P [1:n-1] L ⊗ 1n ⊗ 1n+1 ⊗ P [n+2:N] R - N-1 n=2 P [1:n-1] L ⊗ 1n ⊗ P [n+1:N] R .(C3)”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“To prevent this error from growing too large, the bond dimension can be dynamically increased using the two-site algorithm that is presented in the next section.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“This article overcomes such criticism by presenting an alternative integration scheme for the TDVP equations for finite MPS, based on a Lie-Trotter splitting of the tangent space projector.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“The proposed integration scheme contains evolution for the one-site and zero-site center blocks A C (n) and C(n) according to the respective effective Hamiltonian matrices H (n) or K(n), which are Hermitian.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“The key ingredient of TEBD-based methods is a Lie-Trotter-Suzuki [16,17] splitting of the Schrödinger equation according to the individual terms ĥi of the Hamiltonian Ĥ = i ĥi .”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“For the purpose of this paper, the TDVP can most easily be understood in a geometric fashion, namely as an orthogonal projection of the evolution vector of the Schrödinger equation (-iH | ) onto the tangent space of the MPS manifold M MPS at the current state”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
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“Equation ( 3 [27]), the tangent space projector can be decomposed as PT | (A) M MPS = N n=1 P [1:n-1] L ⊗ 1n ⊗ P [n+1:N] R - N-1 n=1 P [1:n] L ⊗ P [n+1:N] R , (4 )”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
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- Not checked yetcites [27] source
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“The TDVP equations define a simultaneous update for all MPS tensors and, as a consequence thereof, its original implementation depends on inverses of matrices conditioned by the Schmidt coefficients, similar to the original TEBD implementation.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“As this also has the disadvantage that the bond dimension cannot be dynamically altered in the standard scheme (similar to one-site DMRG), we also present an analogous two-site scheme that does enable a dynamical increase of the bond dimension.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“Upon applying PT | (A) M MPS onto Ĥ | [A] , we can write | [A] in the form of Eq. ( 1) or (2) with a properly chosen location for the center such that we can recognize the one-site or zero-site effective Hamiltonian H (n) or K(n).”
§ Unifying time evolution and optimization with matrix product states > Upon applying PT | (A) M MPS onto Ĥ | [A]
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“Similarly, a projector term of the form -P [1:n] L ⊗ P [n+1:N] R can be integrated explicitly by writing the | as in Eq. ( 2) and making C(n) time-dependent as C(n,t) = exp + iK(n)t C(n,0). (8 )”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“In particular, the time evolution algorithm presented below is compatible with any Hamiltonian for which groundstate DMRG can be executed efficiently, e.g., long-range Hamiltonians, which are written in the form of MPOs.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“More concretely, the TDVP equations suffer from what is known as stiffness in the numerical analysis literature [20] and this results in a severe stepsize restriction when using standard explicit time integrators.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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- Not checked yetcites [20] source
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“The key ingredient is to project the right-hand side of the Schrödinger equation, Ĥ | , onto the tangent space, so that the evolution never leaves the manifold.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“As this also has the disadvantage that the bond dimension cannot be dynamically altered in the standard scheme (similar to one-site DMRG), we also present an analogous two-site scheme that does enable a dynamical increase of the bond dimension.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“The crucial insight leading to our new integration scheme is that, while the TDVP equation ( 3 N n=1 H(n) -N -1 n=1 K(n) FIG. 4. Right-hand side (up to the factor -i) of the TDVP equation [Eq.(3)]. [corresponding to a single term from Eq. ( 4)] can be integrated exactly [27].”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“Completing a single left-to-right sweep is a first-order integrator that produces an updated | at time t + t with a local integration error of order O( t 2 ).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“It is also possible to obtain higher order integrators by applying composition schemes to this symmetric integrator [30].”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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- Not checked yetcites [30]
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“We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“This results in the paradoxical situation that the stability of this method deteriorates as the approximation of the exact state (corresponding to the value of the smallest Schmidt coefficient) is improved.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“The one-site DMRG method corresponds to an alternating least squares optimization and amounts to sequentially optimizing individual tensors in a number of sweeps until the whole process converges.”
§ Unifying time evolution and optimization with matrix product states > II. MATRIX PRODUCT STATES AND CANONICAL FORMS
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- Not checked yetcites [15] source
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“By gauging the MPS such that the current tensor being optimized is the center site A s C (n), this is transformed into a standard eigenvalue problem with an effective one-site Hamiltonian H (n) defined in Fig. 2(a).”
§ Unifying time evolution and optimization with matrix product states > II. MATRIX PRODUCT STATES AND CANONICAL FORMS
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“For the new representation | [B] , the parameters of B cannot easily be related to basis of partial derivatives with respect to all parameters A s (n), ∀n = 1, . . .,N.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“Put differently, the linear map from the parameters B i to the states B i |∂ i [A] has a nontrivial kernel.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“Every two-site block that is factored can increase the local bond dimension.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
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“The splitting scheme presented in the main text nevertheless allows us to update the MPS | using this representation.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“This should still hold for > 0, since any error of lower order in t should scale at least proportional to and is thus negligible in comparison to the contribution of O( ) that is introduced by the truncation anyway.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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- The authors state this as an assumption: The truncation scheme discards singular values smaller than a tolerance level ε.
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“Extending the left gauge fixing condition to n = N or the right gauge fixing condition to n = 1 is equivalent to restricting to those variations of | [A] that preserve norm and phase, i.e., those tangent vectors which are orthogonal to | [A] .”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“In conclusion, we have formulated a time integration scheme for the TDVP in the context of MPS, which is based on a Trotter decomposition of the tangent space projector rather than the Hamiltonian terms.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“The gauge redundancy of the MPS parametrization reflects itself in tangent space by the fact that not all linearly independent choices B i produce independent tangent vectors B i |∂ i [A] , i.e., the basis of partial derivatives is overcomplete.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“While one typically truncates back to the original manifold, the advantage is that such an approach allows to truncate to a different manifold with higher (or lower) bond dimension when this is still computationally feasible.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“The combined scheme I * t/2 • I t/2 is symmetric by construction.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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“This allows to simulate time evolution using an algorithm that is remarkably similar to DMRG and can deal with arbitrary long-range Hamiltonians.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“In the particular case of τ → ∞, the evolution of A C (n) will actually force it to be the lowest eigenvector of H (n), exactly as in the optimization step of the one-site DMRG algorithm.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“If I t is a first order integrator (meaning that the local error scales as O( t 2 ), the combined scheme becomes second order.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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“The difference between both schemes can be related to redundancy in the representation of states in T [2] | [A L ] M MPS .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“This integration scheme can also be combined with imaginary time evolution t → -iτ , e.g., for constructing (purifications of) thermal states or for finding ground states.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“The error analysis of this two-site integrator is more complicated, as there does not exist an underlying differential equation that describes the evolution projected down to T [2] | [A L ] M MPS .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“The stability of finite MPS methods originates from the fact that we can transform the original tensors A s (n) into, e.g., the left orthornormal representation without explicit computation of the inverses G(n) -1 .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“For general variational manifold, the TDVP principle of projecting the evolution vector onto the tangent space is the only strategy that ensures that the evolution can be described within the set of states that are assumed tractable, i.e., the manifold itself.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“In the proofs of Refs.[47,48], the mapping between the MPS parameters [the site dependent tensors A(n) ∈ C D n-1 ×d×D n ] and the state | [A] is identified as a principal fiber bundle, resulting from the gauge redundancy of MPS as discussed in the main text.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
What the sources say
- Not checked yetcites [47]
- Not checked yetcites [48]
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“If we separate the evolution in step 2 into two consecutive evolutions with time step t/2, referred to as step 2(a) and 2(b), respectively, then one can recognize the right-to-left sweep [step 2(b) and everything in step 3] as the adjoint method to the left-to-right sweep [everything in step 1 combined with step 2(a)].”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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“However, like most numerical integration schemes, the aforementioned algorithm is intrinsically discrete by choosing a time step, and it poses no problem to formulate an analogous two-site algorithm.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
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“It is straightforward to show that imaginary time evolution with a time step τ → ∞ gives rise to two-site DMRG.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
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“The long-range interactions of the Hamiltonian were approximated by an MPO [23] with a maximal absolute error smaller than 10 -8 .”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [23] source
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“In combination with the left, respectively right orthonormal form for the MPS tensors, these gauge fixing conditions express that the orthonormality constraint is preserved to first order.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“Matrix product states as variational ansatz are special in the sense that they have (a set of) refinement parameters corresponding to the virtual dimensions.There is thus not a single variational manifold, but rather a hierarchy of manifolds, where the ones with lower bond dimension actually correspond to the singular regions of the ones with higher bond dimension.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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- Rests on an assumption: The set of MPS with all possible bond dimensions constitutes a stratification.
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In the paper
“In contrast, our scheme is based on the splitting of the projector in Eq. (C3) and can be applied to arbitrary Hamiltonians (for which H (n) and H (n : n + 1) can be computed efficiently) with the same computational cost.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“With respect to the full Schrödinger equation, we expect a competition between the finite time step error, a truncation error ε, and a projection error that can be estimated in terms of ( 1 -PT [2] | [A L ] M MPS ) Ĥ | [A] .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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In the paper
“Therefore the resulting evolution will have exact norm and energy conservation (in case of a time-independent Hamiltonian), up to the errors in the iterative algorithm used to solve the local problem (which are typically close to machine precision and therefore negligible).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“Finite MPS for which some l(n) and/or r(n) do not have full rank, or uniform MPS which are not injective, correspond to singular points where the tangent space is not well defined.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“The TEBD algorithm also fits in this framework, by observing that for nearestneighbor interactions, the evolution vector Ĥ | is contained exactly in the space of two-site variations (no projection is necessary) by choosing, e.g., B s n ,s n+1 (n : n + 1) = t n ,t n+1 s n ,s n+1 | ĥn,n+1 |t n ,t n+1 A t n (n)A t n+1 (n + 1) and using a Trotter splitting based on these terms.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
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“At this point, our algorithm has become identical to variational MPS formulation of the one-site finite-size DMRG algorithm.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“The solution to this constrained optimization problem is given by B s (n) = t (δ s,t -A s L (n)A t L (n) † )F t (n) for all n = 1, . . .,N -1, where first factor is a projector that imposes the left-gauge fixing condition.Tensor B(N ) is not subject to any constraint and is given by B s (N ) = F s (N ).”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
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“In particular, by directly applying a nearest-neighbor Hamiltonian Ĥ = N-1 n=1 ĥ(n : n + 1) to | [A] , one obtains a state of the form of Eq. (C1), which is represented by B s 1 ,s 2 (n : n + 1) = t 1 ,t 2 s 1 ,s 2 | ĥ(n : n + 1)|t 1 ,t 2 A s 1 (n)A s 2 (n).This is different from the representation in Eq. (C4), even though both describe the same physical state Ĥ | [A] .”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“For α > 1, we observe a light cone and the leakage outside is in perfect agreement with the theoretical predictions of Ref. [38].”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [38] source
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In the paper
“The resulting virtual dimensions of the MPO are 28, 28, 24, and 12 for the cases α = 0.75, 1.5, 3.0 and 6.0, respectively.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
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“The light cone itself seems to be linear for α = 3.0 and α = 6.0, but the shape is less clear for α = 1.5.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
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“Absolute value of 0 |U † σ x r (t)U | 0 -0 |σ x r (t)| 0 with H the XY Hamiltonian in Eq. (9), | 0 the ground state of H and U = exp(iσ y 51 π/4) on a chain of N = 101 spins, as function of r and t for various values of α.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
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“For α = 0.75, there is no sharp wave front but it still seems to take a finite amount of time to signal over a finite distance, up to power law corrections.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
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“The underlying reason for this is that the effective Hamiltonian H (n) [or C(n)] for the one-site [or zero-site] center degrees of freedom A C (n) [or C(n)] does not depend on these degrees of freedom, but only on the tensors on the neighboring sites, so that the differential equations corresponding to the individual terms are first-order linear differential equations.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
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In the paper
“We furthermore relate this two-site approach to the more conventional time-dependent DMRG or TEBD methods, which are restricted to local (typically nearestneighbor) interactions.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“By construction, the resulting evolution will never leave the MPS manifold and can therefore be described by time-evolving parameters as | (A(t)) .”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
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“By construction, the TDVP projects the evolution onto the manifold of MPS with fixed bond dimension and avoids the need for a truncation step.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
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“We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency.”
§ Unifying time evolution and optimization with matrix product states > Abstract
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“Indeed, the TDVP can be recast into a complicated set of coupled nonlinear equations for all parameters in the MPS.”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
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“For this type of differential equation consisting of a sum of integrable parts, it is very natural to use a Lie-Trotter splitting approach [16,29,30], where one evolves according to each integrable part for a small time t.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“The algorithm presented in Ref. [18] used the explicit Euler integration scheme to simultaneously update all parameters, which was only possible by using explicit inverses of matrices containing very small singular values.”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
What the sources say
- Not checked yetcites [18] source
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In the paper
“As explained in Ref. [29], this source of error can be bound in terms of ( 1 -PT | [A] M MPS ) Ĥ | [A] , a quantity that can easily be evaluated [33] and monitored throughout the evolution.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
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“Having a Lanczos routine for computing the matrix exponential acting on a vector (e.g., Ref. [31]), this algorithm differs from one-site DMRG by replacing the optimization step for A C (n) (eigenvalue solver) with the evolution step, and by adding an extra evolution step for C(n) before absorbing it at the next site.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
What the sources say
- Not checked yetcites [31] source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“The individual optimization steps are homogeneous problems and give rise to a generalized eigenvalue equation.”
§ Unifying time evolution and optimization with matrix product states > II. MATRIX PRODUCT STATES AND CANONICAL FORMS
What the sources say
- Not checked yetcites [15] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, using the single projector P [1:n-1] L ⊗ 1n ⊗ P [n+1:N] R and writing | [A] as in Eq. ( 1) with center site n, the exact solution can be obtained by only making A C (n) timedependent and letting it satisfy ȦC (n,t) = -iH (n) A C (n,t) where a bold notation is used for the vector representation of A C (n).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Analogous to the eigenvalue problem in DMRG, both Eqs. ( 7) and ( 8) can be integrated using a Lanczos scheme [28] in order to keep the computational cost at O(D 3 ).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
What the sources say
- Not checked yetcites [28] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The resulting algorithm resembles DMRG so closely that it can be obtained by merely modifying two lines of code, and the one-site DMRG method is obtained as special case of imaginary time evolution with an infinite time step.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This approach is independent of the Hamiltonian and can be implemented efficiently for long-range Hamiltonians.”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“There is a natural order similar to the DRMG sweep, which optimizes the computational overhead (and results in robustness in the case of overapproximation [27]).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
What the sources say
- Not checked yetcites [27] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Completing the right-to-left sweep is equivalent to composing this integrator with its adjoint, resulting in a second-order symmetric method [32] so that the state at time t + 2 t has a more favourable error of order O( t 3 ).”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
What the sources say
- Not checked yetcites [32] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The evolution of C in Eq. ( 8) can be interpreted as an evolution backwards in time.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“These operators are independent of the unitary gauge freedom (basis rotation) that exists in the definition of | [1:n] L,α and | [n:N] R,β and correspond to orthogonal projectors onto the support of the reduced density matrix ρ of the state | [A] in the corresponding regions.”
§ Unifying time evolution and optimization with matrix product states > III. TANGENT SPACE PROJECTOR AND TIME-DEPENDENT VARIATIONAL PRINCIPLE
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“While K(n) serves no purpose in ground-state DMRG, it will appear in our time evolution algorithm below.”
§ Unifying time evolution and optimization with matrix product states > FIG. 2. (a)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“An alternative idea for time evolution, the Dirac-Frenkel time-dependent variational principle (TDVP), was formulated only recently for the variational class of MPS [18,19].”
§ Unifying time evolution and optimization with matrix product states > I. INTRODUCTION
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This allows to dynamically adapt the bond dimension to the a predefined precision or error criterion, and also allows to dynamically introduce new charge sectors (irreducible representations) when symmetries of the quantum state are explicitly encoded in the MPS representation.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“In Ref. [38], the time-dependent expectation value σ x r (t) = exp(-iH t)σ x r exp(iH t) was compared between a fully polarized product state | 0 = ⊗ n |↑ n and the perturbed state U | 0 with U = exp(iπσ y 0 /4) as function of r and t for various values α > 1.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [38] source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The representation B s n ,s n+1 (n : n + 1) is, however, nonunique, and our algorithm based on splitting the projector (even when it acts trivially) gives rise to a slightly different scheme.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“As two MPS with different bond dimensions do not belong to the same manifold [35], this idea cannot easily be captured in a smooth evolution described using a differential equation.”
§ Unifying time evolution and optimization with matrix product states > V. TWO-SITE ALGORITHM
What the sources say
- Not checked yetcites [35] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This result justifies the two-site integration scheme that was proposed in the main text.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
In the paper
“In accordance with the general result of that paper, there is a light cone and the leakage outside the light cone contains a contribution with exponential decay that is determined by the nearest-neighbor interactions and is thus independent of α, as well as a contribution with power-law decay that dominates at larger distances.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [38] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“This Hamiltonian was recently realized in an experiment with trapped 171 Yb + ions [39].”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [39] source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Finding a unique parametrization for every tangent vector is equivalent to specifying a complement of the vertical subspace (the so-called horizontal subspace), for which there is no unique prescription.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This approach generalizes neatly to tree tensor network states and will be equally relevant for continuous MPS [24,25], where algorithms based on a Trotter decomposition of the Hamiltonian terms are not applicable.”
§ Unifying time evolution and optimization with matrix product states > VII. CONCLUSIONS
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Hence, without the truncation, the integrator is still symmetric and one expects a finite time step error that scales as O( t 3 ).”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“One can easily verify that the C(n) that results from then factorizing A C (n) = A L (n)C(n) is then also an exact eigenstate of K(n), so that its backwards evolution has no further effect and can thus be omitted.”
§ Unifying time evolution and optimization with matrix product states > ONE-SITE INTEGRATION SCHEME
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“For this choice of | 0 , the quantum many-body problem can be reduced to a single-particle problem.”
§ Unifying time evolution and optimization with matrix product states > VI. EXAMPLE
What the sources say
- Not checked yetcites [38] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The tangent space projector is a generic concept that is independent of the TDVP and can be applied to arbitrary states | in the Hilbert space H.”
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
The paper relies on this without support.
The TEBD idea cannot be applied to long-range interactions.
“Unfortunately, the TEBD idea is not applicable to long-range interactions, which appear when using MPS for quasi-2D systems or quantum chemistry applications.”
The authors state this as an assumption.
The truncation scheme discards singular values smaller than a tolerance level ε.
“For the remainder, we assume that the truncation scheme is based on discarding singular values smaller than some tolerance level ε.”
The paper relies on this without support.
The set of MPS with all possible bond dimensions constitutes a stratification.
“The set of MPS with all possible bond dimensions constitutes a stratification .”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“In Ref. [20] a slightly different error was proposed that avoids this numerical inaccuracy.”
From Tangent-space methods for uniform matrix product states (2019) · cites this paper as [20]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Unifying time evolution and optimization with matrix product states > APPENDIX C: DERIVATION OF THE TWO-SITE INTEGRATION SCHEME
“B s,t (n : n + 1) = A s L (n)X t (n + 1) -X s (n)A t R (n + 1) for any {X(n) ∈ C D n ×d×D n+1 ,n = 2, . . .,N -1}.”
§ Unifying time evolution and optimization with matrix product states > II. MATRIX PRODUCT STATES AND CANONICAL FORMS
“A s (n) having site-dependent dimensions D n-1 × D n .”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
“(c) Absorb C(n,t + t/2) into A L (n,t + t/2) to obtain A C (n + 1,t + t/2).”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
“(d) Evolve A C (n,t + t/2) forward in time according to A C (n,t + t) = exp[-iH (n) t/2] A C (n,t + t/2)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
“(a) Perform an orthogonal decomposition of A C (n + 1,t + t) into C(n,t + t) and A R (n,t + t).”
§ Unifying time evolution and optimization with matrix product states > APPENDIX B: FULL ALGORITHM FOR THE SYMMETRIC ONE-SITE INTEGRATOR
“where K(n) is computed using A L (1,t + t/2),...,A L (n,t + t/2), A R (n + 1,t + t), ..., A R (N,t + t).”
Citations
Cites
Density matrix formulation for quantum renormalization groups
White, S. R. (1992) · Physical Review Letters · doi:10.1103/physrevlett.69.2863 · Cited by 7,702
The density-matrix renormalization group in the age of matrix product states
Schollwöck, U. (2011) · Annals of Physics · doi:10.1016/j.aop.2010.09.012 · Cited by 4,818
Solving Ordinary Differential Equations II
Hairer, E., & Wanner, G. (1996) · Springer Series in Computational Mathematics · doi:10.1007/978-3-642-05221-7 · Cited by 4,342
Density-matrix algorithms for quantum renormalization groups
White, S. R. (1993) · Physical Review B · doi:10.1103/physrevb.48.10345 · Cited by 3,244
Oseledets, I. V. (2011) · SIAM Journal on Scientific Computing · doi:10.1137/090752286 · Cited by 2,860
Self-citationCompletely positive linear maps on complex matrices
Choi, M.-D. (1975) · Linear Algebra and its Applications · doi:10.1016/0024-3795(75)90075-0 · Cited by 2,659
On the product of semi-groups of operators
Trotter, H. F. (1959) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1959-0108732-6 · Cited by 2,282
Verstraete, F., Murg, V., & Cirac, J. I. (2008) · Advances In Physics · doi:10.1080/14789940801912366 · Cited by 1,841
Self-citationEfficient Simulation of One-Dimensional Quantum Many-Body Systems
Vidal, G. (2004) · Physical Review Letters · doi:10.1103/physrevlett.93.040502 · Cited by 1,779
Linear transformations which preserve trace and positive semidefiniteness of operators
Jamiołkowski, A. (1972) · Reports on Mathematical Physics · doi:10.1016/0034-4877(72)90011-0 · Cited by 1,512
Cited by
Sorted by citations
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems
Cirac, J. I., Pérez-Garcı́a, D., Schuch, N., & Verstraete, F. (2021) · Reviews of Modern Physics · doi:10.1103/revmodphys.93.045003 · Cited by 1,025
Self-citationTime-evolution methods for matrix-product states
Paeckel, S., Köhler, T., Swoboda, A., Manmana, S. R., Schollwöck, U., & Hubig, C. (2019) · Annals of Physics · doi:10.1016/j.aop.2019.167998 · Cited by 768
Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms
Scholl, P., Schuler, M., Williams, H. J., Eberharter, A. A., Barredo, D., Schymik, K.-N., Lienhard, V., Henry, L.-P., Lang, T. C., Lahaye, T., et al. (2021) · Nature · doi:10.1038/s41586-021-03585-1 · Cited by 688
Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
Hauschild, J., & Pollmann, F. (2018) · SciPost Physics Lecture Notes · doi:10.21468/scipostphyslectnotes.5 · Cited by 606
Quantum Entanglement in Neural Network States
Deng, D.-L., Li, X., & Sarma, S. D. (2017) · Physical Review X · doi:10.1103/physrevx.7.021021 · Cited by 475
Metadata
- Authors
- Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B., & Verstraete, F.
- Affiliations
- Ghent University (BE)
- University of Tübingen (DE)
- Skolkovo Institute of Science and Technology (RU)
- Princeton University (US)
- University of Vienna (AT)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2016-10-10
- Volume
- 94
- Issue
- 16
- ISSN
- 2469-9950 (ISSN-L)
- 2469-9969
- OpenAlex
- W1580989833
- Cited by
- 804
- References
- 46
- Citations by year
- 2026: 88 · 2025: 148 · 2024: 106 · 2023: 73 · 2022: 79 · 2021: 91 · 2020: 89 · 2019: 56 · 2018: 38 · 2017: 28 · 2016: 7
- FWCI
- 20.33
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical review. B./Physical review. B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Ghent University Academic Bibliography (Ghent University) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Russian Science Foundation (14-11-00659)
- Deutsche Forschungsgemeinschaft (SPP 1324, GRK 1838)
- Austrian Science Fund (FoQuS, ViCoM)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Mathematics › Numerical Analysis › Numerical Solvers for Ordinary Differential Equations
- Keywords
- matrix product states, time-dependent variational principle, Dirac-Frenkel variational principle, time evolution, optimization, density matrix renormalization group, time-evolving block decimation, long-range interactions, imaginary time evolution, tangent space projection
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