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Alternating tensor optimization and effective eigenproblems — Examples

1 · Prerequisites

2 · Summary

These finite examples compute a two-site spin-chain local solve, an exact entangled ground state whose rank-one TT rounding raises energy, and a deterministic choice in a degenerate local ground eigenspace. Each verification states its basis, Hamiltonian and truncation data explicitly.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A two-site spin-chain local solve reaches the exact ground state

Example

Take two spin- $1 / 2$ sites with computational basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ , Pauli matrix $Z = diag (1, - 1)$ , and finite Hamiltonian $H = - Z \otimes I - I \otimes Z - Z \otimes Z .$ Supply the initial product state $∣ 11 ⟩$ by the rank-one open-boundary TT cores $G_{k} (1, i_{k}, 1) = 1_{{i_{k} = 2}}$ for $k = 1, 2$ . Starting from this presentation, one completed width-two exact sweep of Finite alternating tensor optimization with effective eigenproblems returns $∣ 00 ⟩$ and lowers the energy from $1$ to $- 3$ .

Facts & Assumptions

Given: The two-site state, Hamiltonian, and width-two exact sweep just specified.

[F1]

Finite matrix-product states and Hermitian Hamiltonians: a normalized finite state is a tensor array, and a finite Hamiltonian input is Hermitian.

[F2]

Finite alternating tensor optimization with effective eigenproblems: each window is updated by the exact lowest eigenvector of its effective Hermitian matrix; with two sites the width-two window is the entire coefficient space.

Verification

1.1givenF1algebra

In the displayed basis, $H$ is the diagonal matrix $diag (- 3, 1, 1, 1)$ : the three summands contribute respectively $(- 1, - 1, 1, 1)$ , $(- 1, 1, - 1, 1)$ , and $(- 1, 1, 1, - 1)$ . Thus $∣ 00 ⟩$ is the unique unit ground vector with energy $- 3$ , while $∣ 11 ⟩$ has energy $1$ .

2.1givenF2step 1.1algebra∎

For the only two-site window, the fixed environments are identities, so the embedding $E$ and effective Hamiltonian are $I_{4}$ and $H$ . The exact solve in [F2] therefore selects $∣ 00 ⟩$ . The backward pass sees the same $H$ and leaves this unique ground vector unchanged, and its exact SVD split has rank one and no discarded tail. Hence the completed sweep returns $∣ 00 ⟩$ with energy $- 3 < 1$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A degenerate local ground eigenspace has a deterministic tie choice

Example

For two spin- $1 / 2$ sites let $Z = diag (1, - 1)$ and $H = Z \otimes Z$ . Use the initial state $∣ 00 ⟩$ with rank-one open-boundary TT cores $G_{k} (1, i_{k}, 1) = 1_{{i_{k} = 1}}$ for $k = 1, 2$ , and run one width-two sweep without rank caps. The two-site ground eigenspace is $span {∣ 01 ⟩, ∣ 10 ⟩}$ . The deterministic standard-basis projection rule in Finite Hermitian ground vectors and a deterministic tie rule selects $∣ 01 ⟩$ .

Facts & Assumptions

Given: The ordered two-site computational basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ , the initial $∣ 00 ⟩$ core list and width-two no-cap sweep specified above, and the Hamiltonian $Z \otimes Z$ .

[F1]

Finite matrix-product states and Hermitian Hamiltonians: a finite Hermitian Hamiltonian acts on the finite tensor-array coefficient space.

[F2]

Finite Hermitian ground vectors and a deterministic tie rule: among lowest eigenvectors, the rule projects standard basis vectors in order and takes the first nonzero projection.

[F3]

Finite alternating tensor optimization with effective eigenproblems: the exact local solve uses that deterministic ground-vector rule.

Verification

1.1givenF1algebra

In the ordered basis, $H = diag (1, - 1, - 1, 1)$ . Its least eigenvalue is $- 1$ and its eigenspace is exactly $span {∣ 01 ⟩, ∣ 10 ⟩}$ .

2.1F2F3step 1.1algebra∎

The orthogonal projection onto that eigenspace sends $∣ 00 ⟩$ to zero and sends $∣ 01 ⟩$ to itself. Hence the first standard basis vector with nonzero projection is $∣ 01 ⟩$ ; it is already unit length and has positive first nonzero coordinate. The local solve therefore returns $∣ 01 ⟩$ deterministically, with energy $- 1$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

A two-site TT truncation can increase energy after an exact solve

Example

Let $ψ = \frac{2 \sqrt{2}}{3} ∣ 00 ⟩ + \frac{1}{3} ∣ 11 ⟩, H = I_{4} - 2 ψ ψ^{*} .$ Take the initial state $∣ 00 ⟩$ with rank-one open-boundary TT cores $G_{k} (1, i_{k}, 1) = 1_{{i_{k} = 1}}$ for $k = 1, 2$ , and run one width-two sweep with rank cap $c_{1} = 1$ . The exact two-site solve returns the deterministically selected unit ground vector $ψ$ with energy $- 1$ . The post-sweep rank-one TT rounding keeps the larger Schmidt term, returns the normalized state $∣ 00 ⟩$ , and raises the energy to $- 7 / 9$ , an increase of $2 / 9$ .

Facts & Assumptions

Given: The target state $ψ$ , initial $∣ 00 ⟩$ core list, Hermitian reflection Hamiltonian, and one width-two sweep with rank cap $c_{1} = 1$ specified above.

[F1]

Finite matrix-product states and Hermitian Hamiltonians: states use the Frobenius norm and a finite Hamiltonian input is Hermitian.

[F2]

Finite alternating tensor optimization with effective eigenproblems: an exact width-two update solves the full local Hermitian problem, then rank-cap rounding may be applied after the completed sweep.

[F3]

Finite Hermitian ground vectors and a deterministic tie rule: the exact ground rule projects standard basis vectors in order and returns the normalized first nonzero projection.

[F4]

Tensor-train rounding: TT rounding applies its exact truncated SVDs at the prescribed rank caps and records each discarded singular-value tail.

[F5]

TT discarded weights bound the post-sweep DMRG energy increase: when the total TT tail $τ < 1$ , the energy increase from the unrounded exact-sweep output to the normalized rounded output is at most $4 ∥ H ∥_{o p} τ$ .

Verification

1.1givenF1algebra

The squared norm of $ψ$ is $8 / 9 + 1 / 9 = 1$ . Since $ψ ψ^{*}$ is the orthogonal projection onto $span {ψ}$ , $H ψ = - ψ$ and $H x = x$ for every $x ⊥ ψ$ . Thus $H$ is Hermitian, its unique ground energy is $- 1$ , and $∥ H ∥_{o p} = 1$ . Also $E_{H} (∣ 00 ⟩) = 1 - 2 (8 / 9) = - 7 / 9$ .

2.1givenF2F3F4step 1.1algebra

The width-two center spans all four coordinates, so $H_{e f f} = H$ . Its ground eigenspace is $span {ψ}$ , and its ground projection sends $∣ 00 ⟩$ to $(2 \sqrt{2} / 3) ψ$ ; [F3]'s rule therefore returns exactly $ψ$ . The backward pass leaves it unchanged. Reshaping $ψ$ into a $2 \times 2$ matrix gives $diag (2 \sqrt{2} / 3, 1 / 3)$ ; its singular values are distinct and ordered as written, so rank-one rounding in [F4] retains $(2 \sqrt{2} / 3) ∣ 00 ⟩$ , records discarded tail $τ = 1 / 3$ , and after normalization gives $∣ 00 ⟩$ .

3.1F5step 1.1step 2.1algebra∎

The rounded energy is the value from step 1.1, so its increase from the exact-sweep energy is $- 7 / 9 - (- 1) = 2 / 9 > 0$ . The bound in [F5] permits at most $4 \cdot 1 \cdot (1 / 3) = 4 / 3$ , which indeed exceeds $2 / 9$ . This exhibits a possible post-truncation increase while satisfying the discarded-weight estimate.