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An Efficient Matrix Product Operator Representation of the Quantum-Chemical Hamiltonian
Sebastian Keller, Michele Dolfi, Matthias Troyer, Markus Reiher
TL;DR
Quantum-chemical DMRG needed a matrix-product formulation that avoided the performance disadvantage of straightforward MPO implementations while preserving their flexibility. The paper constructs the Hamiltonian efficiently as an MPO and implements a fully matrix-product DMRG framework. The resulting scheme achieves traditional DMRG scaling while supporting excited states, observables, models, and symmetries through a common formulation.
Problem
Straightforward matrix-product DMRG implementations had lower performance than traditional Hilbert-space-decimation DMRG, despite MPS/MPO flexibility for quantum chemistry.
Method
The paper constructs the quantum-chemical Hamiltonian as a compact MPO and uses it with MPS in an MPO-based DMRG implementation.
Results
The compact construction restores traditional DMRG scaling, with ground-state execution scaling as O(m^3L^3) + O(m^2L^4).
Takeaways & Limitations
The full matrix-product formulation supports efficient low-lying excited states, straightforward new observables, and multiple models and symmetries in one source code.
Abstract
from arXiv · showhide
We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications in a purely matrix product based framework. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from a viewpoint of Hilbert space decimation and attained a higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states (MPS), where operators are correspondingly represented as matrix product operators (MPO). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach; for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
I. INTRODUCTION
DMRG’s matrix-product formulation represents wave functions as MPS and operators as MPOs, offering flexible arithmetic and symmetry handling while retaining efficient quantum-chemical calculations. Limiting MPS dimensions converts the FCI ansatz from exponential to polynomial scaling and remains systematically improvable.
- DMRG selects states by reduced-density-matrix weight rather than Hamiltonian energy, substantially improving performance for one-dimensional systems.
- MPS encode wave functions as stand-alone objects that can be manipulated arithmetically and independently of sequential reduced-basis transformations.
- MPO arithmetic supports operations such as Hamiltonian squaring, while separating operators from contraction routines enables common treatment of abelian and non-abelian symmetries.
- The implementation is based on the ALPS MPS program and is called QCMaquis in its quantum-chemical version.
- Restricting MPS matrix dimensions to a maximum m changes the FCI ansatz from exponential to polynomial scaling and permits polynomial-time scalar products and expectation values.
- Even with limited m, MPS can accurately approximate ground and low-lying excited-state energies, and increasing m with extrapolation provides systematic improvement toward FCI.
III. MATRIX PRODUCT OPERATORS
MPOs represent operators as products of site-local operator-valued matrices whose auxiliary indices connect operator terms across sites. This construction accommodates the electronic Hamiltonian and supports compact operator representations for quantum-chemical calculations.
- An MPO generalizes the MPS concept to operators by factorizing operator coefficients into site-indexed tensors with auxiliary matrix indices.
- Each MPO tensor contains an additional bra-ket site index, while contraction over local indices produces operator-valued matrices linked across sites.
- The entries of MPO matrices are elementary single-site operators, including creation and annihilation operators, collected in matrix form.
- The electronic Hamiltonian can be arranged into MPO matrices, with a simple construction introduced before a second scheme designed to match traditional DMRG operational scaling.
A. Na¨ıve construction
The naïve Hamiltonian MPO enumerates Hamiltonian terms and distributes their elementary operators across sites. Although straightforward, its application scales as O(L^5), one factor of L worse than traditional quantum-chemical DMRG.
- The naïve scheme represents each Hamiltonian term as a product of site-local MPO matrices, using identities away from the term’s active sites.
- Two-electron terms are formed by replacing identity operators at two additional sites with creation or annihilation operators while preserving the MPO matrix shape.
- The construction encodes arbitrary sums of operator terms using diagonal intermediate matrices with row and column boundary vectors.
- The number of Hamiltonian terms determines the nonzero MPO elements, making single-site application scale as O(L^4) and the full H|ψ⟩ operation as O(L^5).This is an operational increase by a factor of L relative to traditional quantum-chemical DMRG.
B. Compact construction
The Hamiltonian MPO is compacted by identifying shared operator subsequences and collapsing equivalent bonds through coordinated fork and merge operations. This reduces construction cost while preserving the represented operator terms, subject to constraints that prevent unwanted cross terms.
- Shared subsequences: The optimized construction exploits identical operator subsequences by collapsing shared strings into common MPO substrings and forking them into distinct remainders.Forking collapses common left parts; merging analogously combines common right parts.
- Optimization constraints: The example's combined fork and merge optimization succeeds for shared operators, but an additional optimization exploiting a common identity generates unwanted cross terms.The compact panel C is valid only when the combined terms have the same matrix elements.
- Labeling: Fork labels identify operators already applied to the left, whereas merge labels identify operators that will be applied to the right.Labels encode operator-position pairs and determine where MPO bonds can be merged or forked.
- Connectivity: For two-electron terms, fork-fork-merge and fork-merge-merge connectivity patterns produce particularly compact MPO representations.The fork-fork-merge pattern combines terms sharing operators at sites i, k, and j while attaching the matrix element at the transition site.
- Scaling: The resulting MPO matrices have dimensions O(L^2) × O(L^2), and the contraction algorithm scales as O(L^4).Block D has O(L^2) rows and O(L) columns, yielding O(L^3) work per site across L sweep iterations.
IV. FERMIONIC ANTI-COMMUTATION
Fermionic anticommutation is incorporated by applying a Jordan-Wigner transformation to elementary creation and annihilation operators. Additional sign handling accounts for opposite-spin electrons occupying the same site.
- Jordan-Wigner transformation: A Jordan-Wigner transformation is applied to elementary creation and annihilation operators to enforce fermionic anticommutation across sites.The transformation introduces auxiliary sign operators into the MPO representation.
- Sign placement: For terms with an even number of fermionic operators, sign operators are needed only in sections containing an odd number of fermions to the left.This follows from the transformed sign operators becoming identities or commuting with operators away from the relevant site.
- Same-site anticommutation: Opposite-spin electrons on the same site require an additional sign operator to reproduce the correct anticommutation phase.The modification tracks the phase factor from same-site anticommutation between opposite spins.
V. MPS-MPO OPERATIONS
The matrix-product framework represents many-electron states as MPSs and operators as MPOs, enabling ground-state calculation followed by expectation-value evaluation.
- MPS-MPO workflow: The framework calculates ground states of specified Hamiltonians and then computes expectation values using MPS and MPO representations.This section follows the separate representations of states and arbitrary operators introduced earlier.
A. Expectation Values
Expectation values are evaluated by iteratively contracting MPS tensors with MPO tensors through left and right boundary objects. The final left boundary is a scalar equal to the desired expectation value, while regrouping avoids exponential evaluation cost.
- Expectation-value contraction: A transition matrix element is evaluated by introducing a second MPS and contracting its tensors with the MPO and the original MPS.The construction uses local tensor contractions and boundary objects rather than an explicit exponential sum.
- Computational cost: Suitable regrouping reduces the operation count for the expectation-value expression to O(L^4m^3).The unreorganized expression is exponentially expensive to evaluate.
- Boundary interpretation: The final boundary L_L is a scalar equal to the desired expectation value, and right boundaries provide an equivalent contraction direction for reusing objects during sweeps.The initial and final boundaries are scalar, while intermediate boundaries are vectors indexed by MPO bond indices whose elements are square matrices.
- Left-boundary iteration: Left boundaries are built iteratively by contracting the previous boundary, the local MPS tensor, and the local MPO tensor, followed by contraction with the conjugate MPS tensor.The four depicted steps include local contraction, summation over the operator input, boundary-index summation, and multiplication by the bra tensor.
B. Ground-state search
The MPS-based ground-state search variationally minimizes the energy with respect to local MPS tensors, producing the same local eigenvalue problem as traditional DMRG. Efficient MPO contractions enable ground-state calculations with quantum-chemical scaling O(m^3L^3) + O(m^2L^4).
- Traditional and MPS-based DMRG ultimately diagonalize the same local Hamiltonian matrices with sparse eigensolvers such as Jacobi-Davidson.
- The MPS formulation minimizes ⟨ψ|H|ψ⟩ with respect to one tensor under the normalization constraint ⟨ψ|ψ⟩ = 1.
- The resulting local optimization is expressed as a standard eigenvalue problem for the optimized MPS tensor.
- O(m^3L^3) + O(m^2L^4) is the optimal execution-time scaling obtained for the quantum-chemical Hamiltonian after MPO compression.
- A tiny ad-hoc noise term can improve convergence by avoiding local minima through reshuffling renormalized states.
C. Single-site and two-site DMRG
The implementation supports two-site DMRG by combining neighboring MPS and MPO sites into enlarged local objects. Its two-site optimization reuses the single-site evaluation routines after generating the two-site tensors.
- Two-site DMRG optimizes two sites simultaneously to achieve faster convergence in chemical applications.
- The two-site MPS and MPO tensors are constructed as the enlarged objects used in the joint optimization.
- Treating two neighboring four-dimensional local sites as one 16-dimensional space extends the single-site equations to two-site DMRG.
- After two-site tensor generation, the remaining evaluation steps are essentially unchanged and can use the same program routines.
D. Excited states
The state-specific MPS procedure computes excited states by enforcing orthogonality to previously supplied states during each Jacobi-Davidson micro-iteration. Updating the associated overlap objects during sweeps yields a converged state orthogonal to all supplied lower states.
- The excited-state algorithm repeatedly orthogonalizes the target state against supplied orthogonal states during Jacobi-Davidson iterations.
- Partial overlaps between the target wavefunction and each supplied state are represented through MPS contractions.
- Site-specific orthogonal vectors supplied to the Jacobi-Davidson eigensolver are calculated from these overlap representations.
- After left and right sweeps update the overlap objects with the optimized tensors, the converged state is orthogonal to every supplied state.
VI. CALCULATIONS
The calculations test the MPO implementation on delocalized graphene fragments using compact, sparse operator representations and carefully chosen orbital configurations. Ground- and excited-state energies are extrapolated with estimated per-carbon errors of ±0.05 mH for G2 and ±0.1 mH for G3.
- VI. CALCULATIONS: Compact MPO construction and sparsity exploitation are both required to achieve optimal scaling.
- VI. CALCULATIONS: The benchmark uses graphene fragments G2 and G3, whose long-range two-dimensional correlations make them difficult for DMRG.
- VI. CALCULATIONS: The active space contains π-orbitals in a minimal STO-3G basis, which excludes realistic energies but retains the active-space correlation problem.
- VI. CALCULATIONS: Non-abelian spin symmetry and mutual-information-based Fiedler orbital ordering were used to obtain accurate energies and improve performance.
- VI. CALCULATIONS: ±0.05 mH per carbon for G2 and ±0.1 mH per carbon for G3 are the estimated extrapolation errors.
- VI. CALCULATIONS: The study calculates ground- and excited-state energies for graphene fragments using the reported MPO implementation.