Ground state approximation for strongly interacting systems in arbitrary dimension

F. Verstraete; H.-J. Briegel; M. B. Plenio; S. Anders; W. D\"ur

REVIEW

Ground state approximation for strongly interacting systems in arbitrary dimension

S. Anders, M. B. Plenio, W. Dür, F. Verstraete, H.-J. Briegel

Quantum Physics · Submitted 28 Feb 2006 · 15 references

Pith's verdict: A genuine new variational ansatz for higher-dimensional spin systems with efficient observable evaluation, but the Rayleigh-minimization step for phase gates ignores the unit-modulus constraint and the exponential-improvement claim needs a second look.

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arXiv IDquant-ph/0602230 (version 1)

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CitationF. Verstraete, H.-J. Briegel, M. B. Plenio, S. Anders, W. D\"ur. "Ground state approximation for strongly interacting systems in arbitrary dimension." arXiv:quant-ph/0602230 (2006). https://pith.science/paper/WLABWFWU

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Editor's note

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Colleague,

You should know two things about this paper. First, the deformed weighted graph state ansatz is genuinely new and the efficient evaluation of local observables via Eqs. (4)-(5) is a real, checkable contribution. Second, the Rayleigh-quotient optimization step for phase gates has a unit-modulus problem that the authors do not acknowledge, and it directly affects the exponential-improvement data in Fig. 1b.

The core idea: superpose m weight graph states that differ only in the complex deformation vector d, keep the graph Γ and the local unitaries U fixed, and show the reduced density matrices of one or two spins can be computed in O(m^2 N) time, giving O(m^2 K N) for a K-term two-body Hamiltonian. That is a solid algorithmic contribution, and the small-system comparisons against exact diagonalization (N=20 chain, 4x4 lattice) are reassuring. The generalization to continuous variables via Gaussian superpositions is only sketched, but plausible.

The soft spot: the variational section claims that with everything fixed except one phase gate U_ab(ϕ), the energy is a quotient of quadratic forms optimizable by Rayleigh. That is true as a statement about the state being linear in x=e^{-iϕ}, but the Rayleigh step minimizes over all complex x, not just |x|=1. The authors give no argument that the generalized eigenvector respects the phase constraint. For a generic two-state problem the unconstrained optimum lies off the unit circle, so the method can return a state outside the physical ansatz. Fig. 1b is explicitly computed with Rayleigh minimization without symmetrized phases, so the exponential improvement with m may be an artifact of this enlarged state space. This is the load-bearing concern; it does not invalidate the whole paper because the L-BFGS route is physical, but it does undercut the specific quantitative claim.

Other soft spots: the 2D and 3D large-system results lean on the Anderson bound, which the authors themselves note is loose in higher dimensions, and there is no direct benchmark against PEPS or DMRG. So the practical advantage in higher dimensions is unquantified. That is a minor complaint relative to the Rayleigh issue.

Bottom line: the ansatz is new, the observable-evaluation formulas are useful, and the paper deserves a serious referee. The Rayleigh claim needs a derivation and a fix for the unit-modulus constraint, and Fig. 1b should be re-examined or clearly labeled as applying to the unconstrained state family. I would not cite the exponential-improvement claim as it stands, but I would consider citing the formulas if I needed them. Bring it to reading group with a warning about the Rayleigh step.

Referee report

Major objections (2)

[Variational method] The claim that a single phase gate U_ab(φ) can be optimized by a generalized-eigenvalue Rayleigh step is not valid for the physical parameter domain. With the phase written as x=e^{-iφ}, the state is |A>+x|B>, and the energy is a Rayleigh quotient with v=(1,x)^T. The Rayleigh minimization over v∈C^2 solves an unconstrained problem over all complex x, whereas the phase gate only allows |x|=1. For a generic quotient (e.g., E(x)=|x-2|^2/(1+|x|^2)) the unconstrained minimizer lies off the unit circle, so the generalized eigenvector need not correspond to any phase gate. The paper provides no proof and no numerical check that the minimizing ratio has unit modulus. Consequently the alternating fast optimization can leave the manifold of Eq. (3), and the Fig. 1b data, which are stated to use 'Rayleigh minimization', may demonstrate a property of a larger state family rather than of the proposed ansatz. Please either prove the constraint is satisfied, replace the phase-gate update by a constrained optimization, or re-run the figure using only valid WGS states.
[Demonstration] The large-system 2D and 3D results are not validated quantitatively. Figure 2 compares the variational energy with mean field and with an Anderson bound computed from small clusters, but the text explicitly says this bound is not tight in 2D and 3D and 'may lead to a significantly underestimation of the precisions'. The gap between the variational upper bound and a loose lower bound gives no quantitative accuracy statement for the 900- and 27000-spin systems. The exact comparisons are only N=20 (1D) and 4x4 (2D). To substantiate the claim that the ansatz works in higher dimensions, the authors should benchmark against exact diagonalization for moderate 2D/3D clusters or against established tensor-network results, or report tight two-sided bounds; otherwise the higher-dimensional demonstration remains qualitative.

Minor objections (7)

[Variational method and Fig. 1b] The phrase 'Rayleigh minimization without symmetrized phases' is not defined; please state whether 'symmetrized phases' refers to translation invariance of the phase gates or to the phase-magnitude constraint, and clarify how the Rayleigh step handles the |x|=1 condition.
[Demonstration] The Anderson-bound subsystem sizes appear as 'N = 15, 32, 23'; this looks like a formatting artifact. Please restore the intended superscripts so the sizes are unambiguous.
[Continuous variables] The symbols '/CA' and '/CA 2N' are typesetting artifacts; they should be R and R^{2N}. Please correct the notation throughout that section.
[Generalizations] The extension to spin-n/2 systems is only sketched; no explicit generalized phase gate or deformation operator is given, and no numerical test is reported. Please either provide the explicit construction or clearly label the extension as an outlook.
[Properties of WGS] The claim that the states include diverging correlation length and unbounded block entanglement is imported from Refs [8,9] for WGS with d_a=0; the deformed WGS used here are not explicitly shown to retain these properties. A sentence justifying this would be helpful.
[Variational method] The statement that 'one does not get stuck in local minima' is made without supporting statistics; please report the number of random starts and the distribution of final energies.
[Eq. (5)] The displayed formula is hard to parse: the notation '∑_{rs,t}|s><t|' should be written as '∑_{s,t} r_{s,t}|s><t|' to avoid ambiguity.

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The paper

The paper claims

Superposed weighted graph states allow efficient, geometry-agnostic variational ground-state approximation for strongly interacting spin systems, demonstrated on Ising models in one, two, and three dimensions.

The reading

This paper introduces a variational method for approximating ground states of strongly interacting spin systems in arbitrary geometries and dimensions. The ansatz states are superpositions of weighted graph states that share the same phase graph and local unitaries but differ in complex single-qubit deformations. The central claim is that for such superpositions, the expectation value of any local observable—in particular the Hamiltonian—can be computed in $O(m^2 K N)$ steps, where $N$ is the number of spins, $K$ the number of terms in the observable, and $m$ the number of superposed states. Because the underlying graph can be chosen to fit any geometry and the states can carry volume-proportional entanglement and diverging correlation length, the method offers a route to strongly correlated systems in two and three dimensions that one-dimensional variational methods handle poorly. The authors demonstrate the approach on the transverse-field Ising model for chains, square lattices, and cubic lattices, and outline extensions to higher spins and continuous-variable systems.

What carries the argument

The central object is the deformed weighted graph state, generated by applying controlled-phase gates $U_{ab}(\phi_{ab})$ with phases from a symmetric adjacency matrix $\Gamma$ to a product of $\sigma_x$ eigenstates, then applying single-qubit filters $D_a = \mathrm{diag}(1, e^{d_a})$ with complex $d_a$ and local unitaries $U_a$. In the computational basis the amplitudes read $e^{-i s^T \Gamma s/2 + d^T s}$. The load-bearing identity is the reduced-density formula (Eq. (5)), which expresses two-qubit density matrix elements as a product over the remaining $N-2$ spins of factors $(1 + e^{d_c + d_c^* - i \sum_{e=1}^{2}(s_e - t_e)\Gamma_{ec}})$, so that each element is evaluated in $O(N)$ time instead of the exponential time needed for a generic state. Superposing $m$ states with common $\Gamma$ and $U$ but different $d$ gives reduced density matrices in $O(m^2 N)$ time, and energy estimates in $O(m^2 K N)$ time. This factorization is what turns the variational search into a polynomial-time procedure.

Extended reading notes

Core claim

The paper's central claim is that the family $|\Psi\rangle \propto \sum_{i=1}^{m} \alpha_i |\Psi_{\Gamma, d^{(i)}, U}\rangle$, composed of weighted graph states with a fixed adjacency matrix $\Gamma$ and fixed local unitaries $U$ but different deformation vectors $d^{(i)}$, is a practical variational class for ground-state approximation. For a single weighted graph state, the two-spin reduced density matrix can be obtained in $O(N)$ operations because the sum over the environment factorizes into a product over individual spins; carrying this through for a superposition of $m$ states raises the cost to $O(m^2 N)$. Consequently, energy expectation values for a two-body Hamiltonian with $K$ terms require $O(m^2 K N)$ steps. The states are not tied to a particular geometry—$\Gamma$ can be chosen to match short-range or long-range couplings, chains, or higher-dimensional lattices—and they can encode diverging correlation length and block entanglement that grows with block volume. Numerical tests on the Ising model in one, two, and three dimensions show accurate ground-state energies and locate the critical fields close to established values, supporting the method's usefulness.

Load-bearing premise

The fast alternating optimization presumes, without proof, that when all parameters except one (a local unitary, a phase gate, or a single deformation parameter) are held fixed, the energy is a quotient of quadratic forms in the free variables; if that structure ever fails, the Rayleigh step loses its foundation and only the slower general-purpose optimizer remains.

Editorial extensions

If this is right

For short-range Hamiltonians, energy expectation values can be computed in $O(m^2 N)$ time, so the method can reach systems of tens of thousands of spins (the paper treats a $30\times30\times30$ cubic lattice with $N=27{,}000$).
The ansatz reproduces qualitative features of critical ground states, including a diverging correlation length and block-wise entanglement growing with block size, suggesting applicability near quantum phase transitions.
By choosing the adjacency matrix appropriately, the same formulation handles arbitrary geometries and spatial dimensions without changing the computational scheme.
The continuous-variable generalization, using superpositions of Gaussian states, keeps the characteristic function a sum of Gaussians, allowing efficient evaluation of position-momentum polynomial observables and hence a route to lattice field theories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The numerically observed exponential improvement in accuracy with the number of superposed states $m$ is suggestive but unproven; if it holds generally, even small $m$ would suffice for high precision in strongly correlated regimes.
The factorization behind the reduced-density formula likely extends to $k$-body reduced density matrices with cost $O(m^2 N^{k-1})$, which would provide direct access to higher-order correlation functions and multipartite entanglement measures.
Because the deformation parameters appear as complex phases, the ansatz may be naturally suited to imaginary-time evolution or open-system dynamics, where complex weights arise; the paper notes such generalizations but does not develop them.
The reduction to translation-invariant weight phases with only $O(N)$ parameters appears to preserve accuracy for the Ising model; testing it on other universality classes would reveal whether the method is generally scalable at critical points in two and three dimensions.

Original abstract

We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions.	contribution 0.99	0.95	0.22	asserted 1.00
s1	The approach is based on weighted graph states and superpositions thereof.	method 0.98	0.56	0.06	asserted 1.00
s2	These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement.	method 0.49	0.65	0.23	asserted 1.00
s3	As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices.	contribution 0.54	0.95	0.06	asserted 1.00
s4	We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.	contribution 0.93	0.88	0.10	asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.84.

What was examined, and what was not

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arXiv listing abstract v1

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: (Color online.) (a) Relative deviation from exact

**Figure 2.** FIG. 2: (Color online.) Ising model in 1D (blue) with

Figures reproduced from arXiv:quant-ph/0602230 by the authors.

References

Works cited (15)

[1]

S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993)

View on Pith 1992
[2]

U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005)

View on Pith 2005
[3]

G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); 93, 040502 (2004)

View on Pith 2003
[4]

F. Verstraete, D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)

View on Pith 2004
[5]

M. Fannes, D. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992)

View on Pith 1992
[6]

F. Verstraete and J. I. Cirac, e-print cond-mat/0407066

View on Pith Pith review arXiv
[7]

G. Vidal, e-print cond-mat/0512165

View on Pith Pith review arXiv
[8]

J. Calsamiglia et al., Phys. Rev. Lett. 95, 180502 (2005)

View on Pith 2005
[9]

W. D¨ ur, et al., Phys. Rev. Lett. 94, 097203 (2005)

View on Pith 2005
[10]

H.-J. Briegel and R. Raußendorf, Phys. Rev. lett. 86, 910 (2001)

View on Pith 2001
[11]

We remark that for 1D systems, the accuracies appear to scale less well in the resources as for DMRG methods. However, our approach yields accurate results also for 2D and 3D systems

View on Pith
[12]

R. H. Byrd, P. Lu, and J. Nocedal, SIAM J. Sci. Stat. Comp. 16, 1190 (1995)

View on Pith 1995
[13]

H.-X. He, C. J. Hamer and J. Oitmaa, J. Phys. A 23, 1775 (1990); Z. Weihong, J. Oitmaa, and C. J. Hamer, J. Phys. A 27, 5425 (1994)

View on Pith 1990
[14]

K. Audenaert et al., Phys. Rev. A 66, 042327 (2002); M. B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005)

View on Pith 2002
[15]

J. Eisert and M. B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)

View on Pith 2003

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993)

View on Pith 1992

[2] [2]

U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005)

View on Pith 2005

[3] [3]

G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); 93, 040502 (2004)

View on Pith 2003

[4] [4]

F. Verstraete, D. Porras and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004)

View on Pith 2004

[5] [5]

M. Fannes, D. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992)

View on Pith 1992

[6] [6]

F. Verstraete and J. I. Cirac, e-print cond-mat/0407066

View on Pith Pith review arXiv

[7] [7]

G. Vidal, e-print cond-mat/0512165

View on Pith Pith review arXiv

[8] [8]

J. Calsamiglia et al., Phys. Rev. Lett. 95, 180502 (2005)

View on Pith 2005

[9] [9]

W. D¨ ur, et al., Phys. Rev. Lett. 94, 097203 (2005)

View on Pith 2005

[10] [10]

H.-J. Briegel and R. Raußendorf, Phys. Rev. lett. 86, 910 (2001)

View on Pith 2001

[11] [11]

We remark that for 1D systems, the accuracies appear to scale less well in the resources as for DMRG methods. However, our approach yields accurate results also for 2D and 3D systems

View on Pith

[12] [12]

R. H. Byrd, P. Lu, and J. Nocedal, SIAM J. Sci. Stat. Comp. 16, 1190 (1995)

View on Pith 1995

[13] [13]

H.-X. He, C. J. Hamer and J. Oitmaa, J. Phys. A 23, 1775 (1990); Z. Weihong, J. Oitmaa, and C. J. Hamer, J. Phys. A 27, 5425 (1994)

View on Pith 1990

[14] [14]

K. Audenaert et al., Phys. Rev. A 66, 042327 (2002); M. B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005)

View on Pith 2002

[15] [15]

J. Eisert and M. B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)

View on Pith 2003