Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
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The density-matrix renormalization group in the age of matrix product states
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abstract
The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.
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representative citing papers
High-rate CSS codes compile nonlocal Clifford and non-Clifford logical circuits into onsite phases and classical permutations, so MPS bond dimension stays fixed by the encoder while logical entanglement, magic, and non-Gaussianity grow.
NGS-MPS simulations of Hubbard–Holstein models show soft phonons promote 1D phase separation and stabilize 2D stripe phases, including a novel bipolaronic stripe with enlarged unit cell.
Twisted multinomial coefficients factorize into a product of site-dependent q-binomials when inversion weights satisfy predecessor-uniformity, yielding exact MPS representations for pilot states in Hamiltonian Decoded Quantum Interferometry.
A matrix product operator construction using link-enhanced MPOs enables infinite-lattice simulations of (1+1)D gauge theories with manifest translation invariance and symmetry.
A multimode-cavity picture is introduced for non-Markovian waveguide QED via spatial decomposition, approximating dynamics with a finite and growing number of cavity modes.
The claimed intrinsic dipole moment at FQH edges is protected only at filling factor 1/3 and absent in other representative edge systems.
Numerical DMRG study of the anisotropic J1-J2 spin-1 chain uncovers two non-magnetic incommensurate floating Luttinger liquid phases emerging from the trimerized background, separated from the Haldane phase by a composite c=2 critical line.
Peaked quantum circuits claimed to show quantum advantage can be classically simulated in one hour on a GPU via mirrored MPO contraction and unswapping.
The Ising fusion category lattice model features a symmetric critical phase equivalent to the Ising model, a categorical ferromagnetic phase with threefold degeneracy, and a critical categorical antiferromagnetic phase with fourfold degeneracy described by an Ising CFT.
A linear MPS tangent-space method for open-boundary systems is derived, and a particle-resolved rank tomography is introduced to diagnose the tangent space's expressivity limits.
Introduces a parallelizable hybrid tensor network algorithm for time-evolving matrix product states that combines classical BUG integration with quantum methods without synchronization barriers.
A method using valence bond embeddings in hybrid encodings constructs shallow circuits that aim to extend VQE simulability to larger molecular systems.
Quantum kernels are spectral tensor networks because their Fourier coefficient tensors are matrix product operator factorizations, with kernel target alignment acting as Frobenius cosine similarity on frequency grids.
Introduces and tests a finite-volume scheme enabling stable continuum extrapolation of the step-scaling function in (2+1)D Hamiltonian U(1) gauge theory via matrix product states.
Linear-programming method for conjugating local fermionic unitaries with free evolution realizes arbitrary complex tunneling coefficients in fermionic lattice models constrained only by connectivity.
A compact neural statebank based on autoregressive Transformers simulates 34-qubit quantum circuits with ~0.01 infidelity using 0.3 million parameters, outperforming tested approximate simulators.
Uniform MPS simulations of dense 1+1D SU(2) gauge theory find Tomonaga-Luttinger liquid infrared behavior with central charge 1, density modulations at the predicted wavenumber, and a smooth crossover in the Luttinger parameter from K~1 to K~1/2 that realizes the quarkyonic picture with coexisting q
A variational quantum circuit trained solely on classical measurement outcomes reconstructs diverse quantum states including GHZ, spin-chain ground states, and random circuits with fidelities above 90% on simulators and real NISQ hardware.
Lower bounds on the best separable approximation distance for non-pure spin-squeezed states are obtained from the complete set of spin-squeezing inequalities, with symmetry-exploiting optimization for upper bounds, revealing finite-temperature entanglement in ordered phases of the XXZ model.
A variational quantum SVD framework with classical orthogonality correction enables high-precision extraction of Schmidt components from bipartite states using shallow circuits and classical tensor-network post-processing.
A gapped parent Hamiltonian built from two copies of a target Hamiltonian plus ultra-local inter-copy couplings allows adiabatic preparation of high-fidelity thermofield double states for ETH-obeying systems.
Numerical iPEPS with loop expansions indicates the SU(4) Heisenberg model on the hyperhoneycomb lattice has a gapless quantum spin-liquid ground state, consistent with prior variational Monte Carlo results.
Quantum simulation on IBM Nighthawk extracts attractive kink-antikink potential in large-N QCD2 mapped to XXZ chain, agreeing with exact diagonalization benchmarks.
citing papers explorer
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Renormalization flows for 1D mixed states and a quantum Goursat lemma
Convergent renormalization trajectories of Hopf-algebra boundary MPDOs under on-site noise are classified by finite *-quantum hypergroups via a new quantum Goursat lemma.
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Efficiently simulable quantum circuits with large entanglement, magic, and non-Gaussianity via code-compiled tensor networks
High-rate CSS codes compile nonlocal Clifford and non-Clifford logical circuits into onsite phases and classical permutations, so MPS bond dimension stays fixed by the encoder while logical entanglement, magic, and non-Gaussianity grow.
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Scalable Simulation of Strongly Correlated Electron-Phonon Systems via Non-Gaussian Matrix Product States
NGS-MPS simulations of Hubbard–Holstein models show soft phonons promote 1D phase separation and stabilize 2D stripe phases, including a novel bipolaronic stripe with enlarged unit cell.
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A Factorization Identity for Twisted Multinomial Coefficients with Application to Pilot States in Hamiltonian Decoded Quantum Interferometry
Twisted multinomial coefficients factorize into a product of site-dependent q-binomials when inversion weights satisfy predecessor-uniformity, yielding exact MPS representations for pilot states in Hamiltonian Decoded Quantum Interferometry.
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Infinite matrix product states for $(1+1)$-dimensional gauge theories
A matrix product operator construction using link-enhanced MPOs enables infinite-lattice simulations of (1+1)D gauge theories with manifest translation invariance and symmetry.
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Multimode-cavity picture of non-Markovian waveguide QED
A multimode-cavity picture is introduced for non-Markovian waveguide QED via spatial decomposition, approximating dynamics with a finite and growing number of cavity modes.
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Does a Fractional Quantum Hall Edge Have a Protected Intrinsic Dipole Moment?
The claimed intrinsic dipole moment at FQH edges is protected only at filling factor 1/3 and absent in other representative edge systems.
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Non-magnetic floating phases in frustrated Haldane chains with a single-ion anisotropy
Numerical DMRG study of the anisotropic J1-J2 spin-1 chain uncovers two non-magnetic incommensurate floating Luttinger liquid phases emerging from the trimerized background, separated from the Haldane phase by a composite c=2 critical line.
-
Efficient Classical Simulation of Heuristic Peaked Quantum Circuits
Peaked quantum circuits claimed to show quantum advantage can be classically simulated in one hour on a GPU via mirrored MPO contraction and unswapping.
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Symmetry breaking phases and transitions in an Ising fusion category lattice model
The Ising fusion category lattice model features a symmetric critical phase equivalent to the Ising model, a categorical ferromagnetic phase with threefold degeneracy, and a critical categorical antiferromagnetic phase with fourfold degeneracy described by an Ising CFT.
-
Excitation spectra and rank tomography of linear matrix product tangent spaces
A linear MPS tangent-space method for open-boundary systems is derived, and a particle-resolved rank tomography is introduced to diagnose the tangent space's expressivity limits.
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Time Evolution on Hybrid Tensor Networks -- A Novel and Parallelizable Algorithm
Introduces a parallelizable hybrid tensor network algorithm for time-evolving matrix product states that combines classical BUG integration with quantum methods without synchronization barriers.
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Shallow Quantum Circuits for Deep Chemistry via Valence Bond Embeddings
A method using valence bond embeddings in hybrid encodings constructs shallow circuits that aim to extend VQE simulability to larger molecular systems.
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Quantum Kernels are Spectral Tensor Networks
Quantum kernels are spectral tensor networks because their Fourier coefficient tensors are matrix product operator factorizations, with kernel target alignment acting as Frobenius cosine similarity on frequency grids.
-
A Finite-Volume Scheme for the Continuum Extrapolation of Lattice Step-Scaling in (2+1)D Hamiltonian U(1) Gauge Theory
Introduces and tests a finite-volume scheme enabling stable continuum extrapolation of the step-scaling function in (2+1)D Hamiltonian U(1) gauge theory via matrix product states.
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Fermionic Hamiltonian engineering with local control
Linear-programming method for conjugating local fermionic unitaries with free evolution realizes arbitrary complex tunneling coefficients in fermionic lattice models constrained only by connectivity.
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Simulating quantum circuits with a neural statebank
A compact neural statebank based on autoregressive Transformers simulates 34-qubit quantum circuits with ~0.01 infidelity using 0.3 million parameters, outperforming tested approximate simulators.
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Dense $\mathrm{QC_2D_2}$ with uniform matrix product states
Uniform MPS simulations of dense 1+1D SU(2) gauge theory find Tomonaga-Luttinger liquid infrared behavior with central charge 1, density modulations at the predicted wavenumber, and a smooth crossover in the Luttinger parameter from K~1 to K~1/2 that realizes the quarkyonic picture with coexisting q
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Quantum Machine Learning for State Tomography Using Classical Data
A variational quantum circuit trained solely on classical measurement outcomes reconstructs diverse quantum states including GHZ, spin-chain ground states, and random circuits with fidelities above 90% on simulators and real NISQ hardware.
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Estimating the best separable approximation of non-pure spin-squeezed states
Lower bounds on the best separable approximation distance for non-pure spin-squeezed states are obtained from the complete set of spin-squeezing inequalities, with symmetry-exploiting optimization for upper bounds, revealing finite-temperature entanglement in ordered phases of the XXZ model.
-
High-Precision Variational Quantum SVD via Classical Orthogonality Correction
A variational quantum SVD framework with classical orthogonality correction enables high-precision extraction of Schmidt components from bipartite states using shallow circuits and classical tensor-network post-processing.
-
Preparing High-Fidelity Thermofield Double States
A gapped parent Hamiltonian built from two copies of a target Hamiltonian plus ultra-local inter-copy couplings allows adiabatic preparation of high-fidelity thermofield double states for ETH-obeying systems.
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SU(4) Heisenberg model on the hyperhoneycomb lattice
Numerical iPEPS with loop expansions indicates the SU(4) Heisenberg model on the hyperhoneycomb lattice has a gapless quantum spin-liquid ground state, consistent with prior variational Monte Carlo results.
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Quantum Simulation of Nucleon-Antinucleon Interaction in Large-$N$ QCD$_2$ on an IBM Quantum Nighthawk Processor
Quantum simulation on IBM Nighthawk extracts attractive kink-antikink potential in large-N QCD2 mapped to XXZ chain, agreeing with exact diagonalization benchmarks.
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Entanglement is Half the Story: Post-Selection vs. Partial Traces
A hybrid tensor network framework interpolates between classical and quantum models via controllable post-selection, with a trainable hyperparameter that complements bond dimension to enhance quantum machine learning.
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Mixed-spin Heisenberg ladders in a magnetic field
DMRG study of (1/2,1) Heisenberg ladders finds 1/3 magnetization plateau for J_perp>0 (limited range for J_perp<0) and locates the KT critical point via finite-size transverse spin correlations.
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Lectures on Semiclassical Methods for Composite Operators
Lecture notes develop semiclassical methods to compute large-n scaling dimensions of composite operators in CFTs, recovering known results in free theory and deriving one-loop corrections at the Wilson-Fisher fixed point.
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Entanglement Certification $-$ From Theory to Experiment
Reviews paradigmatic entanglement quantifiers and state-of-the-art detection/certification methods, with emphasis on assumptions about states and measurements.
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