Minimally Entangled Typical Thermal State Algorithms

E. M. Stoudenmire; Steven R. White

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Minimally Entangled Typical Thermal State Algorithms

E. M. Stoudenmire, Steven R. White

arXiv:1002.1305v1Feb 5, 2010cond-mat.str-el

TL;DR

Finite-temperature quantum simulations need an efficient way to sample thermal behavior without representing prohibitively entangled states. The paper develops the METTS pure-state method and its MPS/MPO implementations, showing how METTS can support efficient sampling, reveal order and thermal fluctuations, and probe ensemble entanglement, while identifying open questions about optimal decompositions.

Problem
The paper addresses how finite-temperature quantum systems can be sampled efficiently while controlling the entanglement and computational cost of the represented states.
Method
The paper develops a pure-state Markovian sampling algorithm for METTS, together with implementation strategies for MPS/MPO calculations and flexible CPS measurement bases.
Results
METTS provide an efficient sampling basis whose individual properties expose order and thermal fluctuations, while their average entanglement exceeds the thermal-state benchmark E[ρ] at finite temperature.
Takeaways & Limitations
METTS can support finite-temperature simulations and analysis of thermal fluctuations, order, and entanglement properties in systems beyond the simplest one-dimensional setting.
Takeaways & Limitations
Whether optimal METTS decompositions with ideal sampling efficiency always exist and can be efficiently computed remains open.

Abstract

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We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.

I. INTRODUCTION

The paper develops METTS sampling as a finite-temperature method that represents thermal averages with efficiently generated quantum states, using CPS bases and a pure-state random walk. It details implementation strategies, demonstrates computational scaling comparable to ground-state DMRG, and applies the approach to characterize thermal systems.

Thermal sampling: Thermal expectation values can be estimated by sampling normalized quantum states |φ(i)⟩ with probability P(i)/Z and averaging their observable expectations.The states arise from expanding the thermal trace in an orthonormal basis.
Thermal sampling: Classical product states provide a zero-entanglement basis, helping control the computational cost of representing the sampled thermal states.The cost of quantum-state calculations increases with entanglement entropy, motivating CPS basis states.
Model systems: The paper uses the S = 1 bilinear-biquadratic chain, focusing on the Heisenberg antiferromagnet and AKLT model within the Haldane phase.The models have a gap to all excitations and a doubly degenerate entanglement spectrum at T = 0.
Pure-state algorithm: The pure-state algorithm repeatedly imaginary-time evolves a CPS into a METTS, measures observables, and collapses it into a new CPS for the next sample.The collapse probability is p(i → i′) = |⟨i′|φ(i)⟩|^2.
Pure-state algorithm: Detailed balance makes the CPS distribution P(i)/Z stationary, so deterministically constructed METTS are generated with the correct thermal distribution.The method uses a Markovian random walk and does not require rejection sampling.
Computational cost: The typical METTS bond dimension ranges from m = 1 at β = 0 to the ground-state DMRG dimension m0 at large β, giving production cost Nm3β versus Nm3 for ground-state DMRG.The CPS collapse step scales as Nm2.

A. Imaginary Time Evolution

Imaginary-time evolution uses Trotter-Suzuki gates, with swap gates extending the approach beyond nearest-neighbor interactions. The section also addresses cancellation strategies and their scaling costs.

For one-dimensional nearest-neighbor systems, second-order Trotter-Suzuki evolution applies local gates bond by bond, with finite timestep Trotter error as the approximation source.
Swap gates exchange lattice-site states so a gate can act on effective neighboring sites while evolving longer-range interactions.
The swap-gate method extends Trotter evolution to longer-range interactions, ladders, and systems whose interactions are not nearest-neighbor along the MPS path.
Because swap gates square to the identity, ordering Trotter gates can make repeated swaps cancel, producing a significant speedup.
For an Lx × Ly ladder, the resulting method scales as ym3 rather than proportionally to the number of sites as for a chain.

3. Imaginary Time Evolution with MPOs

MPO-based evolution requires restoring and controlling MPS orthogonality while truncating the enlarged product state. The fitting and zip-up algorithms provide complementary ways to apply MPOs efficiently.

Applying an MPO of bond dimension k to an MPS of bond dimension m initially produces a product MPS with bond dimension mk.
The naive truncation procedure scales as Nm3k3, making it highly inefficient when k is approximately 10–100.
The fitting algorithm minimizes || |ψB⟩−W|ψA⟩||2 by optimizing two-site tensors through sweeps and SVD splits.
The zip-up algorithm has similar scaling to fitting but avoids the sticking problem that can prevent reasonable convergence for non-nearest-neighbor interactions.
When k < m, zip-up multiplication scales as Nm3kd and truncates during successive SVDs, with truncation-error cutoffs preferred during the initial sweep.

4. Taylor Series Construction of the Time Evolution MPO

The Taylor-series construction builds a short imaginary-time evolution MPO and composes it through repeated multiplication. Timestep subdivision and truncation order control the approximation.

The Taylor approach constructs the evolution MPO from a Hamiltonian MPO, including by summing MPOs for individual Hamiltonian terms.
A timestep τ is divided into fractions ϵ = τ/n, and e−ϵH is approximated by a Taylor series truncated at order p with error controlled by ϵp.
The short-step operator is formed as Kϵ = 1 + Wp after recursively constructing the Taylor terms.
The full Kτ is obtained from the nth power of Kϵ using zip-up MPO multiplication, with powers of two enabling repeated squaring and fewer multiplications.

5. Trotter-Suzuki Construction of the Time Evolution MPO

The Trotter-Suzuki MPO construction combines local gates, using swaps when interactions are non-nearest-neighbor, and truncates the resulting MPO. Its errors are easier to identify and control than those of the Taylor construction.

The Trotter-Suzuki construction combines local gates Kij into an MPO rather than applying them directly to the system MPS.
Swap gates convert non-nearest-neighbor gates into effective local operations, after which redundant gates can be removed before constructing the evolution MPO.
The MPO can be built from an identity MPO by applying gates and truncating the MPO bond dimension after each gate, or with higher accuracy by repeated zip-up squaring.
Trotter-Suzuki errors are easier to identify and control because the gates act on pairs of sites and can be constructed exactly.
The remaining errors are finite timestep Trotter error and truncation error in constructing the MPO.

3. Estimation of Errors

METTS error estimates must account for correlations between sequential states and related observables. Binning handles sequential autocorrelation, while correlated bootstrap resampling is needed for cumulants and moment-based quantities.

METTS error analysis must account for correlations between sequential METTS and related observables within each METTS.
A properly chosen collapse basis can reduce the pure-state method’s autocorrelation time to less than 5 steps, but binning remains recommended for error estimation.Bins should exceed the autocorrelation time so their averages can be treated as statistically independent.
For specific heat, susceptibility, and similar cumulants, summing the errors of first and second moments greatly overestimates the true error because those moments are strongly correlated.A large value of ⟨A⟩ for one METTS tends to coincide with a large value of ⟨A^2⟩.
Correlated bootstrap resampling estimates cumulant uncertainty by resampling the same expectation-value subset for both moments before computing their difference.Repeating the procedure yields a distribution of second-cumulant estimates whose standard deviation gives the error.
CPS collapse can use an arbitrary local basis, and sequential sitewise projection ultimately produces a CPS with probability |⟨Ψ′|Ψ⟩|^2.The implementation permits basis choices at every site and algorithmic step.

2. Collapsing an MPS Into a CPS

The MPS-to-CPS collapse proceeds site by site using projector probabilities, with optimized updates that reduce the per-site cost. Choosing collapse bases strategically improves ergodicity and reveals temperature-dependent spin structure.

2. Collapsing an MPS Into a CPS: An MPS is collapsed site by site by computing local projector probabilities, sampling a basis state, and updating the state before proceeding to the next site.The orthogonality center makes the first-site projector expectations directly computable, after which the collapse advances along the chain.
2. Collapsing an MPS Into a CPS: Projector structure reduces the collapse cost from m^3d^2 to m^2d^2 per site.The optimization exploits the fact that repeated single-site projections produce a bond-dimension-one product state.
Choosing the CPS Basis for Spin Models: Random quantization axes generate CPS exploring many conserved-magnetization sectors without explicitly favoring one rotational axis.In practice, random projections produce a short autocorrelation time.
Choosing the CPS Basis for Spin Models: z-only collapses make CPS easy to visualize and reveal defects in the diluted Néel pattern, whose number decreases as temperature is lowered.For the AKLT model, the defects disappear entirely as T → 0.
Choosing the CPS Basis for Spin Models: z-only collapses create strong correlations and lengthen autocorrelation times, whereas alternating z-only and random-axis collapses maintains short autocorrelation times.A maximally mixed basis on odd steps reduces correlation effects even more effectively than random axes.

4. Diagonal Measurements with Product States

Diagonal observables can be measured efficiently from CPS collapsed from METTS rather than from the METTS wavefunctions themselves. This enables repeated CPS measurements while retaining correct thermal averages, alongside snapshot-based analysis of order and fluctuations.

4. Diagonal Measurements with Product States: Computing real-space spin correlations directly with MPOs or individual spin operators is costly because both MPO construction and expectation-value evaluation are expensive.
4. Diagonal Measurements with Product States: For rotationally invariant systems, only Sz correlations are needed because the other spin-component correlations are equal by symmetry.
4. Diagonal Measurements with Product States: If an observable is diagonal in the CPS collapse basis, its thermal average can be obtained from the collapsed CPS instead of the METTS.This applies directly to z-basis measurements of diagonal correlators such as Cz(∆).
4. Diagonal Measurements with Product States: Re-collapsing each METTS multiple times and measuring the resulting CPS gives the correct thermal average for diagonal observables and is more efficient at large β.
4. Diagonal Measurements with Product States: Alternating z-axis collapses with random or maximally mixed collapses restores ergodicity while permitting measurements such as C(∆) on every step in suitable rotationally invariant models.
4. Diagonal Measurements with Product States: Every METTS serves as a system snapshot for simultaneously measuring observables and identifying characteristic fluctuations or short-range order.METTS properties also help explain why relatively few samples can accurately estimate observables and motivate studying ensemble entanglement.

A. Observing Thermal Fluctuations With METTS

METTS reveal both ordered structure and thermal fluctuations in individual states, while ensemble-level energy fluctuations quantify sampling efficiency. The paper argues that METTS combine high practical efficiency with low production cost, though an ideal decomposition may be computationally inaccessible.

Observing Thermal Fluctuations With METTS: At T = 0.1, individual METTS show antiferromagnetic correlations alongside thermal fluctuations in the Heisenberg and AKLT chains.The AKLT example includes longitudinal spin and local bond-energy fluctuations away from the spin-liquid ground state.
Observing Thermal Fluctuations With METTS: Twists in the Heisenberg chain’s staggered magnetization mark thermal fluctuations, with shortened spin length and lower local bond energy nearby.A correlated-spin region persists for several correlation lengths before another twist occurs.
Observing Thermal Fluctuations With METTS: Regions that are more classical and have relatively higher bond energy exhibit lower entanglement entropy, indicating weaker quantum fluctuations.Because METTS are pure states, bipartition entanglement entropy is well defined for each bond.
Energy Measurements and the Efficiency of METTS: The difference between total and quantum specific heat equals, up to a constant factor, the ensemble variance of energy expectation values.A smaller difference means fewer states are needed to estimate the energy to a given accuracy.
Energy Measurements and the Efficiency of METTS: The sampled METTS decomposition is close to optimal: even near the specific-heat peak, quantum specific heat is only smaller than total specific heat by about a factor of 0.8.The paper attributes this efficiency to each METTS significantly overlapping with energy eigenstates having ϵ_i ≲ T.
Energy Measurements and the Efficiency of METTS: An alternative decomposition can achieve C_v^(q) = C_v exactly, but constructing one state requires diagonalizing the entire Hamiltonian.METTS therefore combine high sampling efficiency in practice with low production cost.

C. In What Sense are METTS Minimally Entangled?

The paper compares METTS average entanglement with the entanglement of formation in two-qubit systems, finding that even optimal METTS need not be minimally entangled. It also examines how optimal decompositions depend on temperature, fields, and symmetry.

Mixed-state entanglement: For mixed states, average decomposition entanglement depends on the chosen ensemble, whereas entanglement of formation is the minimum over all decompositions.For two qubits, Wootters’ exact expression permits an explicit comparison with METTS decompositions.
Two-qubit Heisenberg model: At βJ = ln(3), the two-site antiferromagnetic Heisenberg state reaches zero entanglement of formation and is separable for smaller βJ.At zero temperature, the singlet has entanglement entropy 1.
Two-qubit Heisenberg model: The XZ METTS have uniformly lower average entanglement than ZZ METTS and zero variance in both energy and entropy, making them optimal METTS up to global spin rotations.Their average entropy nevertheless remains above the entanglement of formation.
Two-qubit Heisenberg model: Even the optimal METTS decomposition exceeds the entanglement of formation at every finite temperature, so METTS are not minimally entangled in the strict sense.The average entanglement of each METTS decomposition goes smoothly to zero as β → 0.
Extended decompositions: Optimal extensions can be less entangled than optimal METTS at low temperatures, but eventually exceed them as temperature increases and increasingly resemble the optimal METTS ensemble.The paper conjectures that no extended decomposition can outperform optimal METTS at all temperatures.
Heisenberg model in a field: With a field, E[ρ] is zero for βJ ≤ ln(3), while for h > J it is no longer monotonic in temperature; XY METTS become optimal up to z-axis rotations.The change reflects that XZ METTS are no longer physically identical under the Hamiltonian’s symmetries and acquire finite entropy variance.
Open questions: The paper leaves open whether optimal METTS decompositions with ideal sampling efficiency always exist and can be efficiently computed for larger and more complex systems.Average METTS entanglement is not a strict entanglement measure because it need not vanish for separable ρ.

Matrix Product Operators

The paper introduces matrix product states and matrix product operators as tensor-network representations of wavefunctions and operators. MPS use products of site-dependent matrices, while MPOs use an analogous decomposition over local-operator bases.

Matrix Product States: An MPS represents a wavefunction by expressing amplitudes in a local basis as a product of matrices.The local basis labels take values s_i = 1, 2, . . . , d.
Matrix Product States: The bond dimension m is the range of the contracted index between neighboring MPS matrices and may vary across bonds.The discussion assumes a fixed bond dimension for a given MPS and open boundary conditions.
Matrix Product Operators: An MPO is defined analogously to an MPS, but decomposes an operator over a basis of local operators.For open boundaries, the first and last matrix dimensions are again 1 × m and m × 1.

1. Creating MPOs

This section constructs MPOs for products and translationally invariant sums, then develops addition procedures for MPS and MPOs with SVD truncation. These tools support Hamiltonians and operators such as H^2 and e^-τH.

Basic MPO constructions: Site-factorized operator products have MPOs whose local matrices are 1 × 1 elements ⟨t_i|Ô_i|s_i⟩.A single-site operator is obtained by setting all other local operators to the identity.
Basic MPO constructions: Translationally invariant one-dimensional operator sums can use identical lower-triangular bulk matrices, with boundary matrices enforcing the open-chain structure.Translationally invariant one-dimensional Hamiltonians admit lower-triangular MPOs of bond dimension 3 or higher.
General MPO algorithms: General MPO addition and multiplication algorithms address the lack of simple explicit prescriptions for operators such as two-dimensional Hamiltonians.Summing local-term MPOs constructs Hamiltonians, while multiplication forms H^2 and e^-τH.
Adding MPS and MPOs: MPS addition is exact before truncation, then sequential SVDs reduce the resulting bond dimension to a target comparable to the input dimensions.The same construction and truncation strategy extends to MPO addition by treating paired physical indices as a fat index.
Adding MPS and MPOs: During truncation, only the m_C largest singular values are retained at each step before proceeding recursively through the remaining tensors.This produces an MPS representation with controlled bond dimension.
Computational cost: MPS addition costs Nm^3d^2 for equal bond dimensions, while MPO addition scales as Nk^3d^4 with MPO bond dimension k.The MPO scaling reflects the d^2 local operator-index pairs being added.

3. Multiplying MPOs

The MPO multiplication algorithm builds a product MPO site by site, using reshaping, orthogonality-center placement, recursive contractions, and SVD truncation. Its dominant cost scales as Nk^4d^3.

Setup: To multiply MPOs, the method combines each operator’s local output and input indices into a fat index and maps the MPOs to MPS representations.The MPS are transformed so their orthogonality centers begin at the first site before mapping back to MPOs.
Site-by-site multiplication: The recursive construction starts from R_1, reshapes it as a matrix, and applies an SVD while retaining only the k largest singular values.The resulting factors define the first product-MPO matrix and seed the next recursion.
Site-by-site multiplication: Subsequent product-MPO tensors are obtained by recursively constructing R_2 and later R tensors from previously computed factors and the local W and X tensors.The same SVD-based truncation is applied at each site.
Computational cost: MPO multiplication scales as Nk^4d^3 when contractions are ordered as indicated, with construction of the R tensors as the most expensive step.The algorithm can also be carried out one site at a time, analogously to the zip-up algorithm.