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Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms
Garnet Kin-Lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. White
TL;DR
The paper addresses the lack of an explicit translation between renormalized-operator and MPO/MPS descriptions of ab initio DMRG. It explains implementations in both vocabularies and presents compression and parallelism as MPO-motivated formulations.
Problem
An explicit translation between the renormalized-operator and MPO/MPS descriptions of ab initio DMRG is lacking.
Method
The paper explains ab initio DMRG using MPO/MPS language, implements MPO/MPS algebra in a renormalized-operator code, and examines compression and parallelism.
Results
The paper describes Hamiltonian compression and a sum-over-operators representation as new formulations of the ab initio DMRG sweep.
Takeaways & Limitations
The described correspondences connect renormalized-operator implementations with MPO/MPS-based developments in ab initio DMRG.
Takeaways & Limitations
The renormalized-operator implementation is limited to algorithms whose final output is an MPO, which is rare in zero-
Abstract
from arXiv · showhide
Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past, and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.
I. INTRODUCTION
The paper translates ab initio DMRG between renormalized-operator and MPO/MPS languages, showing their computational equivalence and identifying MPO-motivated improvements to the sweep algorithm.
- Motivation and scope: MPO-based and pure renormalized-operator implementations use essentially the same ab initio DMRG computational steps.Explicit MPOs organize the notation, while efficient expectation-value calculations still build renormalized operators from MPO and bra/ket MPS.
- Motivation and scope: The paper addresses confusion caused by the different appearances of equivalent MPO/MPS and renormalized-operator quantities.The translation is especially relevant because optimized intermediates enter complicatedly in both implementation styles.
- Paper goals: It provides a pedagogical translation for efficiently implementing the ab initio DMRG sweep in an MPO-based setting.The discussion explicitly connects this implementation to the original renormalized-operator language.
- Paper goals: The paper also shows how relevant MPO/MPS algebra can be implemented in a pure renormalized-operator code without explicitly adding MPOs in most scalar-output or MPS-output cases.A lightweight layer over an existing renormalized-operator implementation supports operations used for perturbation and response calculations.
- MPO-motivated formulations: Hamiltonian compression can reduce the number of renormalized operators built during a sweep, potentially producing substantial speedups.The formulation is applied directly to ab initio Hamiltonian integrals and is illustrated using a linear hydrogen chain.
- MPO-motivated formulations: A sum-over-operators Hamiltonian representation provides perfect parallelization of the DMRG algorithm.Both this formulation and Hamiltonian compression can be implemented in pure renormalized-operator or MPO-based codes.
A. The DMRG in renormalization group language
In renormalization-group language, an ab initio DMRG sweep traverses orbitals on a one-dimensional lattice and repeatedly updates renormalized bases and operators while solving local ground-state problems.
- A. The DMRG in renormalization group language: The sweep maps K orbitals onto a one-dimensional lattice and iteratively solves one ground-state problem per orbital.A left-to-right sweep consists of K steps, with a corresponding return sweep completing the traversal.
- A. The DMRG in renormalization group language: Left and right orbital blocks carry renormalized bases and operators used to reconstruct the Hamiltonian and organize its normal/complementary decomposition.This decomposition is identified as a key step in efficient ab initio DMRG implementation.
- A. The DMRG in renormalization group language: Each sweep step performs blocking, solving, and decimation.Blocking updates bases and operators, solving obtains the renormalized wavefunction, and decimation returns to the renormalized representation.
- A. The DMRG in renormalization group language: A complete sweep updates renormalized bases and operators for every orbital partition.The same steps and quantities appear in an efficient MPO-based implementation.
- A. The DMRG in renormalization group language: The translation to MPO/MPS language targets correspondences among renormalized bases, wavefunctions, operators, and sweep operations.It also relates efficient energy and expectation-value calculations to the blocking, solving, and decimation organization.
B. Matrix product states
MPS represent wavefunction amplitudes as products of site tensors, whose recursive structure defines the renormalized bases used by DMRG; canonical gauges connect the tensor and renormalized descriptions.
- B. Matrix product states: An MPS writes each wavefunction amplitude as a product of site-associated matrices or tensors.The boundary tensors are vectors so the matrix product yields a scalar amplitude.
- B. Matrix product states: The bond dimension M controls representational flexibility and equals the number of renormalized states.Increasing M makes the MPS representation increasingly flexible.
- B. Matrix product states: MPS graphical notation represents amplitudes as tensors and contractions as summations across a connected set of site tensors.The notation is used to discuss the relationship between MPS and renormalized bases.
- B. Matrix product states: The DMRG sweep uses a mixed-canonical MPS, with left-canonical tensors left of the active site and right-canonical tensors to its right.The central tensor is the DMRG renormalized wavefunction.
- B. Matrix product states: MPS matrices recursively construct the left and right renormalized basis states used during a DMRG sweep.The transformation generated by the renormalization matrix is exactly the sweep’s blocking and decimation of states.
- B. Matrix product states: At any site bipartition, the MPS Schmidt decomposition contains at most M singular values.The renormalized wavefunction coefficients are the wavefunction coefficients in the DMRG renormalized space.
- B. Matrix product states: Renormalized representations and underlying matrix-product tensors are computationally indistinguishable because each defines the other and both coexist in DMRG implementations.This correspondence applies to left and right bases and renormalized wavefunctions.
C. Matrix product operators
An MPO represents a many-body operator as site-associated tensors or operator-valued matrices, with left-right decompositions that parallel DMRG block operators. Recursive MPO blocking is isomorphic to DMRG sweep blocking, while renormalized operator matrices enable efficient expectation-value computation.
- MPO formalism: An MPO generalizes matrix-product representations to operator space using connected site tensors with physical up and down legs.Its site tensors encode operator-matrix elements, while fermionic noncommutation is represented in the operator strings or tensor values.
- MPO formalism: Grouping each local operator with its tensor produces operator-valued matrices whose product represents the full operator.The resulting representation has the same gauge redundancy as an MPS and is not unique.
- Left-right decomposition: Partitioning the sites defines left and right operator sets, whose bond dimension equals the number of terms summed over the auxiliary index.The full operator is assembled from corresponding left and right operators across a partition.
- DMRG correspondence: The MPO left-right decomposition is isomorphic to the left-right decomposition used at each step of a DMRG sweep.Renormalized left- and right-block operators are the main computational intermediates in pure renormalized-operator implementations and play the same role in MPO implementations.
- DMRG correspondence: Recursive updates of neighboring left and right operator sets are vector-matrix or matrix-vector products that mirror DMRG blocking rules.Efficient expectation values use renormalized operator matrix representations rather than bare operators during blocking.
- DMRG correspondence: Element-wise DMRG blocking rules are equivalent to constructing the explicit MPO, with operator matrices replacing explicit operators for efficient computation.This establishes the computational equivalence between the two descriptions.
D. MPO representation of quantum chemistry Hamiltonians
The quantum-chemistry Hamiltonian is decomposed into left, right, normal, and complementary operators to obtain an efficient MPO. Its bond dimension scales as O(K^2), with the prefactor affected by operator splitting and minimized by switching complementary-operator blocks near the midpoint.
- Hamiltonian decomposition: The Hamiltonian MPO is obtained by identifying a left-right decomposition of the ab initio Hamiltonian and mapping its blocking rules to MPO tensors.Left and right Hamiltonians describe interactions between the corresponding blocks.
- Hamiltonian decomposition: Normal operators are not tied to electronic integrals, whereas complementary operators are associated with electronic integrals.This normal/complementary organization supplies the operator terms used in the quantum-chemistry Hamiltonian MPO.
- Bond dimension: O(K^2) is the total MPO bond-dimension scaling for the quantum-chemistry Hamiltonian.The prefactor depends on how normal and complementary operators are split and how integrals are distributed.
- Bond dimension: Switching from right-block to left-block complementary operators at k = K/2 minimizes the MPO bond dimension.Efficient implementations use the left-block representation for k > K/2.
- DMRG correspondence: The original DMRG blocking rule for a complementary operator carries out the MPO tensor matrix-vector multiplication element by element.This gives an explicit correspondence between renormalized-operator updates and MPO operations.
- Implementation equivalences: Fork-merge operations for reusing intermediate operators are equivalent to using complementary P and Q operators on the right and left blocks.The equivalence collects two-electron integrals into the MPO matrix at the relevant site.
E. Efficient implementation of expectation values
Efficient expectation values require contracting the bra MPS, MPO, and ket MPS site by site rather than forming the large intermediate H|Ψ⟩. Element-wise blocking and decimation recover the original DMRG cost through the same computational intermediates.
- Scaling: The quantum-chemistry DMRG sweep has O(K^4) cost, matching the number of fermionic terms in the Hamiltonian.A dense reconstruction of the MPO would instead appear to require O(K^5) operations.
- Expectation-value contraction: Site-by-site MPO-MPS contraction corresponds exactly to recursive renormalized-operator construction through DMRG blocking and decimation.The contraction order preserves the useful structure and sparsity of the MPO tensors.
- Efficient implementation: Separating blocking from decimation reduces dense transfer-matrix multiplication to O(M^3K^3) + O(M^2K^5).This is the generic dense cost for an MPO of bond dimension O(K^2) and an MPS of bond dimension O(M^2).
- Efficient implementation: O(M^3K^3) + O(M^2K^4) is recovered by incorporating element-wise blocking, decimation, sparsity, and symmetry.This matches the cost of the original ab initio DMRG algorithm and uses essentially identical computations.
III. MPO AND MPS ALGEBRA IN A RENORMALIZED OPERATOR-BASED IMPLEMENTATION
The paper shows that general MPO/MPS operations producing a scalar or MPS can be implemented in a pure renormalized-operator DMRG code. Only algorithms whose final output is itself an MPO require broader MPO functionality.
- The approach is motivated by mapping standard DMRG-sweep computations into MPO-based language while retaining expectation values as a natural target.
- MPO/MPS operations with scalar or MPS outputs can be implemented in a pure renormalized-operator DMRG implementation.
- The MPO × MPS product, followed by compression, can be recast as a sweep computation based on overlap maximization.
- Time evolution can produce the full MPS without compression by retaining bond dimension M × D, where D is the MPO bond dimension.
- Products of multiple MPOs acting on an MPS are evaluated by applying the overlap-maximization procedure repeatedly.
- Only algorithms whose final output is an MPO itself require a full MPO implementation beyond renormalized-operator computation.
A. Hamiltonian compression
Hamiltonian compression reduces the MPO representation by decomposing integral coefficients and truncating small singular values. In a 20-hydrogen-chain test, the compressed representation preserves energy efficiently and substantially lowers sweep cost.
- Hamiltonian compression uses a left-right decomposition whose coefficient matrix is factorized by singular value decomposition.The transformed left and right operators absorb the singular-vector factors, while small singular values are dropped.
- The compressed left-right decomposition retains standard DMRG summation structure with fewer summed indices.Existing parallelization over the left-right decomposition sum can therefore be used without modification.
- The energy error is proportional to the MPO truncation threshold and decreases exponentially with MPO bond dimension.
- For an energy error of 10^-6 Eh, an MPO bond dimension below 200 is sufficient, compared with a full dimension ranging from 43084 to 15096 across the tested bond lengths.
- Because each DMRG-sweep step scales with MPO bond dimension, the compressed representation yields a factor of 100 savings.
B. Efficient sum of operators representation
The paper introduces a sum-over-sub-Hamiltonians MPO representation that avoids incorrect dense-algebra scaling and recovers the correct quantum-chemistry scaling. Grouping terms by an index restriction gives each sub-Hamiltonian MPO bond dimension O(K).
- A naive dense-matrix MPO implementation gives an incorrect scaling algorithm, whereas the standard ab initio DMRG algorithm encodes sparse matrix multiplications for optimal scaling.
- The Hamiltonian can instead be written as a sum of sub-Hamiltonians, each represented separately by an MPO.
- This representation recovers the correct O(K^4) scaling even when naive dense matrix algebra is used.
- Restricting the first one- or two-electron integral index defines sub-Hamiltonians whose MPO bond dimension is O(K).
- Alternative groupings by the same rightmost operator preserve O(K) scaling while making bond dimensions to the right of the grouping site equal to 1.This lowers the average MPO bond dimension across the lattice.
Appendix V.)
The paper connects MPO/MPS descriptions of ab initio DMRG with renormalized-operator implementations, using this translation to motivate Hamiltonian compression and operator-level parallelization. These formulations preserve the quantum-chemistry scaling while reducing communication and exposing independent work units.
- Hamiltonian compression: Hamiltonian decomposition into K sub-Hamiltonians recovers the correct O(K^4) quantum-chemistry scaling, unlike the naive MPO representation’s O(K^5) scaling.Dense multiplication of the O(K) × O(K) MPO matrices costs O(K^3) per sub-Hamiltonian and O(K^4) over all K terms.
- Parallelization: The sum-over-operators formulation makes each sub-Hamiltonian and its renormalized operators independent, enabling parallelization up to O(K) processes.Compared with common parallelization, it communicates only the renormalized wavefunction rather than renormalized operators, while leading memory and computation requirements remain unaffected.
- Parallelization: The operator-level parallelization substantially decreases communication cost, but its scalability remains a subject for future investigation.The formulation can be further combined with other parallelization strategies, while communication steps remain in conventional approaches.
- Conclusions: The work translates ab initio DMRG between MPO/MPS and renormalized-operator languages, including sweep implementation and general MPO/MPS algebra.It explicitly targets the connection to the original sweep algorithm and implementation of broader MPO/MPS operations within renormalized-operator DMRG.
- Hamiltonian compression: Choosing intermediates reduces the bond dimensions of four two-body terms to O(K), while the full two-body Hamiltonian retains O(K^2) bond-dimension scaling.Different choices can redistribute leading bond dimensions across orbital partitions without changing the leading O(K^2) scaling.