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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.

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