Unifying time evolution and optimization with matrix product states

Without truncation (ε=0), after a complete sweep with time step Δt, the original state is recovered by reapplying the scheme with -Δt, though it is encoded with larger bond dimension and non-full-rank matrices.

“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).”

The individual terms from splitting the tangent space projector are exactly integrable.

“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.”

The left and right gauge fixing conditions transform covariantly under gauge transformations of the original MPS.

“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.”

Under the left gauge fixing condition, the overlap of two tangent vectors has only diagonal contributions, matching the standard Euclidean inner product and an orthonormal basis.

“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.”

If the original MPS is right canonical, the singular values of C(n) are the Schmidt coefficients for the bipartition [1:n] and [n+1:N].

“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 ].”

The linear map from the two-site representation to physical states has a nontrivial kernel corresponding to B^{s,t}(n:n+1)=A^s_L(n)X^t(n+1)-X^s(n)A^t_R(n+1).

“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}.”

The adjoint method for the finite MPS integrator can be evaluated explicitly.

“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.”

A gauge-fixing condition Σ_s A^s_L(n)† B^{s,t}(n:n+1)=0 can be imposed.

“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)”

The tangent space projector decomposes into a sum of orthogonal projectors as in Eq. (A12).

“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”

For a nearest-neighbor Hamiltonian, the projector error is absent because H|A> is exactly contained in T[2].

“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 .”

Two-site evolution takes the state out of the fixed-bond-dimension MPS manifold.

“The two-site evolution will take the state out of the manifold of MPS with fixed bond dimensions.”

The orthogonal projector onto the two-site variation subspace decomposes as a sum of orthogonal projections as in Eq. (C3).

“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)”

DMRG is a special case of the scheme for imaginary time evolution with infinite time step.

“In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.”

A new integration scheme for time evolution can handle arbitrary Hamiltonians, including long-range interactions.

“In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions.”

The bond dimension can be dynamically increased using the two-site algorithm to prevent the error from growing too large.

“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.”

The method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently.

“In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently.”

The paper presents an alternative integration scheme for finite-MPS TDVP based on a Lie-Trotter splitting of the tangent space projector.

“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.”

The proposed integration scheme evolves one-site and zero-site center blocks with Hermitian effective Hamiltonians.

“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.”

TEBD-based methods rely on a Lie-Trotter-Suzuki splitting of the Schrödinger equation according to the individual Hamiltonian terms.

“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 .”

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The method splits the projector onto the matrix product state tangent space as in the Dirac-Frenkel time-dependent variational principle, rather than splitting the Hamiltonian as in Suzuki-Trotter approaches.

“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.”

TDVP can be understood as an orthogonal projection of the Schrödinger evolution vector onto the tangent space of the MPS manifold at the current state.

“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”

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The tangent space projector can be decomposed as a sum of one-site projectors minus a sum of zero-site projectors.

“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 )”

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The original TDVP implementation depends on inverses of matrices conditioned by Schmidt coefficients, similar to the original TEBD implementation.

“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.”

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The standard one-site scheme cannot dynamically alter the bond dimension, similar to one-site DMRG.

“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.”

Applying the tangent space projector to the Hamiltonian acting on the MPS yields one-site or zero-site effective Hamiltonians.

“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).”

A zero-site projector term can be integrated explicitly by making the zero-site center time-dependent with an exponential of the zero-site effective Hamiltonian.

“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 )”

The time evolution algorithm is compatible with any Hamiltonian for which ground-state DMRG can be executed efficiently, including long-range Hamiltonians written as matrix product operators.

“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.”

The TDVP equations suffer from stiffness, causing a severe stepsize restriction with standard explicit time integrators.

“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.”

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TDVP projects the right-hand side of the Schrödinger equation onto the tangent space so that the evolution never leaves the manifold.

“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.”

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The two-site scheme enables a dynamical increase of the bond dimension.

“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.”

Each term in the TDVP equation corresponding to a single projector can be integrated exactly.

“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].”

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A single left-to-right sweep is a first-order integrator with local error of order O(dt^2).

“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 ).”

Higher-order integrators can be obtained by applying composition schemes to the symmetric integrator.

“It is also possible to obtain higher order integrators by applying composition schemes to this symmetric integrator [30].”

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The time-dependent variational principle provides a unifying framework for time-evolution and optimization methods for matrix product states.

“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.”

The stability of the original TDVP method deteriorates as the approximation of the exact state improves.

“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.”

One-site DMRG corresponds to an alternating least squares optimization that sequentially optimizes individual tensors in sweeps until convergence.

“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.”

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Gauging the MPS so that the current tensor is the center site transforms the problem into a standard eigenvalue problem with an effective one-site Hamiltonian.

“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).”

In the new representation |[B]>, the parameters B cannot easily be related to the basis of partial derivatives with respect to all A^s(n).

“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.”

The linear map from parameter variations to tangent vectors has a nontrivial kernel.

“Put differently, the linear map from the parameters B i to the states B i |∂ i [A] has a nontrivial kernel.”

Every factored two-site block can increase the local bond dimension.

“Every two-site block that is factored can increase the local bond dimension.”

The splitting scheme from the main text allows updating the MPS using the new representation.

“The splitting scheme presented in the main text nevertheless allows us to update the MPS | using this representation.”

The O(Δt^3) scaling should still hold for ε>0 because lower-order errors scale at least proportionally to ε and are negligible compared with the O(ε) truncation contribution.

“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.”

• AssumptionThe authors state this as an assumption: The truncation scheme discards singular values smaller than a tolerance level ε.

Extending the left gauge fixing condition to n=N or the right one to n=1 restricts to variations that preserve norm and phase, i.e., tangent vectors orthogonal to the state.

“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] .”

The paper formulated a time integration scheme for TDVP in MPS based on a Trotter decomposition of the tangent space projector rather than Hamiltonian terms.

“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.”

The basis of partial derivatives for the MPS tangent space is overcomplete because gauge redundancy makes some linearly independent parameter choices produce dependent tangent vectors.

“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.”

Truncation can allow moving to a different manifold with higher or lower bond dimension when computationally feasible.

“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.”

The combined scheme I*_{Δt/2} • I_{Δt/2} is symmetric by construction.

“The combined scheme I * t/2 • I t/2 is symmetric by construction.”

This allows simulation of time evolution with an algorithm similar to DMRG that can deal with arbitrary long-range Hamiltonians.

“This allows to simulate time evolution using an algorithm that is remarkably similar to DMRG and can deal with arbitrary long-range Hamiltonians.”

In the limit of infinite imaginary time, the evolution of the one-site center forces it to be the lowest eigenvector of the one-site effective Hamiltonian, as in one-site DMRG.

“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.”

If I_Δt is a first-order integrator, the combined scheme becomes second order.

“If I t is a first order integrator (meaning that the local error scales as O( t 2 ), the combined scheme becomes second order.”

The difference between TEBD and our scheme is related to redundancy in the representation of states in T[2].

“The difference between both schemes can be related to redundancy in the representation of states in T [2] | [A L ] M MPS .”

The scheme can be combined with imaginary time evolution for constructing thermal states or finding ground states.

“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.”

The error analysis of the two-site integrator is more complicated because no underlying differential equation describes the evolution projected to T[2].

“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 .”

The stability of finite MPS methods comes from transforming tensors to left orthonormal form without explicitly computing inverses G(n)^{-1}.

“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 .”

For a general variational manifold, projecting the evolution vector onto the tangent space is the only strategy that keeps the evolution within the manifold.

“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.”

The map from MPS parameters to the state is a principal fiber bundle because of MPS gauge redundancy.

“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.”

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The right-to-left sweep is the adjoint method to the left-to-right sweep.

“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)].”

Because the algorithm is intrinsically discrete through a time step, an analogous two-site algorithm can be formulated.

“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.”

Imaginary time evolution with infinite time step gives rise to two-site DMRG.

“It is straightforward to show that imaginary time evolution with a time step τ → ∞ gives rise to two-site DMRG.”

The long-range interactions of the Hamiltonian were approximated by a matrix product operator with maximal absolute error below 10^-8.

“The long-range interactions of the Hamiltonian were approximated by an MPO [23] with a maximal absolute error smaller than 10 -8 .”

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With left or right orthonormal MPS tensors, the gauge fixing conditions mean the orthonormality constraint is preserved to first order.

“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.”

MPS have refinement parameters (virtual dimensions), giving a hierarchy of manifolds in which lower bond dimension manifolds correspond to singular regions of higher bond dimension ones.

“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.”

• AssumptionRests on an assumption: The set of MPS with all possible bond dimensions constitutes a stratification.

Our scheme, based on splitting the projector in Eq. (C3), applies to arbitrary Hamiltonians with the same computational cost.

“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.”

One expects competition between finite time step error, truncation error ε, and projection error estimated by (1-P_T[2])H|A>.

“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] .”

• AssumptionThe authors state this as an assumption: The truncation scheme discards singular values smaller than a tolerance level ε.

The resulting evolution has exact norm and energy conservation for time-independent Hamiltonians, up to negligible iterative errors.

“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).”

MPS whose left or right density matrices are not full rank, or uniform MPS that are not injective, are singular points where the tangent space is not well defined.

“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.”

TEBD fits in this framework because for nearest-neighbor interactions the evolution vector is exactly contained in the space of two-site variations, with a specific choice of the two-site tensor and a Trotter splitting.

“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.”

In this limit the algorithm becomes identical to the variational MPS formulation of the one-site finite-size DMRG algorithm.

“At this point, our algorithm has become identical to variational MPS formulation of the one-site finite-size DMRG algorithm.”