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Unifying time evolution and optimization with matrix product states
Open accessAbstract
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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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
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- 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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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
What the sources say
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In the paper
“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
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- 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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In the paper
“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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably 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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIt 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.
Two-site gauge tensor dimensions are shifted by one site
MinorThe stated dimensions for X(n) match a tensor at site n + 1, but the displayed expression uses X(n) at site n. Under the paper's site-tensor dimensions, X(n) would need dimensions D n-1 × d × D n for that expression to be compatible.
“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}.”
“A s (n) having site-dependent dimensions D n-1 × D n .”
Reverse sweep gives the new center two different site indices
MinorStep 3(c) says absorbing the bond tensor into the tensor at site n produces a center at n + 1. Step 3(d) instead uses the resulting center at site n.
“(c) Absorb C(n,t + t/2) into A L (n,t + t/2) to obtain A C (n + 1,t + t/2).”
“(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)”
Reverse sweep assigns the right-hand tensor to the wrong site
MinorStep 3(a) decomposes a center tensor at site n + 1 but labels the resulting right-hand tensor as being at site n. The following step uses a right-hand tensor at site n + 1 instead.
“(a) Perform an orthogonal decomposition of A C (n + 1,t + t) into C(n,t + t) and A R (n,t + t).”
“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).”
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Stark many-body localization: Evidence for Hilbert-space shattering
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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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