Alternating tensor optimization and effective eigenproblems

Let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, let $m\ge 1$, and let $A\in {\mathbb{F}}^{m\times m}$ satisfy $A^{\ast }=A$. Define $q_A(x):=\frac{x^{\ast }Ax}{x^{\ast }x}(x\in {\mathbb{F}}^m\setminus \{0\})\mathrm{.}$ Then $q_A$ has a minimum ${\lambda }_{\min }$, equal to the least eigenvalue of $A$, and its minimizers are precisely the nonzero vectors in the lowest eigenspace $E_{\min }=\ker (A-{\lambda }_{\min }I)$. There is a deterministic choice of a unit minimizer: let $P_{\min }$ be the orthogonal projection onto $E_{\min }$, scan the standard basis $e_1,\dots ,e_m$, and take the first $j$ with $P_{\min }{e}_j\mathrm{\ne }0$. Normalize $P_{\min }{e}_j$; over $\mathbb{C}$, if $z$ is its first nonzero coordinate, multiply by $\bar{z}\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$, and over $\mathbb{R}$ multiply by $z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$.

Let $\mathcal{H}={\mathbb{F}}^{n_1\times \cdots \times n_d}$ be a finite tensor-array space over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, let $H:\mathcal{H}\to \mathcal{H}$ be self-adjoint, and let $\psi ,\varphi \in \mathcal{H}$ satisfy $\mathrm{\parallel }\psi {\mathrm{\parallel }}_F=\mathrm{\parallel }\varphi {\mathrm{\parallel }}_F=1$. Then $\mid ⟨\psi ,H\psi {⟩}_F-⟨\varphi ,H\varphi {⟩}_F\mid \le 2\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{\parallel }\psi -\varphi {\mathrm{\parallel }}_F\mathrm{.}$

Fix a field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, an order $d\ge 2$, and positive physical dimensions $n_1,\dots ,n_d$. The finite coefficient space is the tensor-array space $\mathcal{H}={\mathbb{F}}^{n_1\times \cdots \times n_d}$ with the Frobenius pairing and lexicographic index conventions of Finite tensor arrays and the Frobenius inner product. A normalized finite state is an array $\psi \in \mathcal{H}$ with $\mathrm{\parallel }\psi {\mathrm{\parallel }}_F=1$.

An open-boundary matrix-product state (MPS), or tensor train (TT), presentation is a TT core list from Tensor train cores, bond ranks, and canonical gauges with $r_0=r_d=1$ whose contraction is $\psi$. In particular, each core has shape $r_{k-1}\times n_k\times r_k$ and the shared internal indices are summed.

A periodic MPS presentation consists of positive integers $r_0,\dots ,r_d$ with $r_d=r_0$ and matrices $A_k^{(i_k)}\in {\mathbb{F}}^{r_{k-1}\times r_k}$ for each $1\le i_k\le n_k$. It represents the array ${\psi }_{i_1,\dots ,i_d}=\mathrm{tr}\text{\hspace{0.17em}⁣}(A_1^{(i_1)}{A}_2^{(i_2)}\cdots A_d^{(i_d)})\mathrm{.}$ A finite MPS/TT state is a normalized array admitting at least one of these exact presentations; a presentation need not be unique or canonical.

A finite Hermitian Hamiltonian input is a linear endomorphism $H:\mathcal{H}\to \mathcal{H}$ whose matrix in the standard lexicographic basis satisfies $H^{\ast }=H$, where $\ast$ is conjugate transpose as in Finite tensor arrays and the Frobenius inner product. This page uses only that finite-dimensional Hermitian input and makes no claim about how a physical model assembles it. For nonzero $\psi$, its Rayleigh energy is ${\mathcal{E}}_H(\psi )=\frac{⟨\psi ,H\psi {⟩}_F}{\mathrm{\parallel }\psi {\mathrm{\parallel }}_F^2};$ for a normalized state this equals $⟨\psi ,H\psi {⟩}_F$.

The diagram records the open-boundary chain with a two-site center. Each box is a core or the contracted center tensor, each horizontal wire is a summed virtual bond index, and each vertical open leg is a free physical index. Cores to the left and right of the center are isometric in their respective matricizations when the chain is in mixed canonical gauge. These wire conventions abbreviate the exact finite index sums in the definition and are not used as a proof.

Let a periodic MPS presentation have order $d\ge 2$, positive cyclic bond sizes $r_0,\dots ,r_d$ with $r_d=r_0$, and matrices $A_k^{(i_k)}\in {\mathbb{F}}^{r_{k-1}\times r_k}$. It has an exactly equal open-boundary TT presentation with internal bond sizes $s_k=r_k{r}_0$ for $1\le k<d$. In the pair index convention $(a_k,a_0)$, define $G_1(1,i_1,(a_1,a_0))=(A_1^{(i_1)}{)}_{a_0,a_1},$ $G_k((a_{k-1},b_0),i_k,(a_k,c_0))={\delta }_{b_0{c}_0}(A_k^{(i_k)}{)}_{a_{k-1},a_k}(2\le k<d,1\le b_0,c_0\le r_0),$ where ${\delta }_{b_0{c}_0}=1$ for $b_0=c_0$ and $0$ otherwise, so every middle-core entry with unequal carried indices is zero, and $G_d((a_{d-1},a_0),i_d,1)=(A_d^{(i_d)}{)}_{a_{d-1},a_0}\mathrm{.}$ For $d=2$ the middle-core clause is empty.

Let a nonzero finite open-boundary TT have order $d\ge 2$, and choose a center window $[s,t]$ with $1\le s\le t\le d$ and $t-s+1\in \{1,2\}$. There is an exact rank-revealing QR transformation of its cores that preserves the represented array, makes every core $k<s$ left-isometric and every core $k>t$ right-isometric, and removes exactly the dependent coordinates on each swept bond. No independent column direction is truncated; only exactly dependent directions are removed.

For each reshaped matrix $M$, the rank-revealing QR used here scans its columns in their fixed order, retaining a column exactly when it increases the span. It applies Gram–Schmidt to the retained columns, writes all other columns in that orthonormal span, and records the resulting permutation and coefficient factor.

Let a finite open-boundary TT have order $d\ge 2$ and be in the following mixed gauge about a one-site or two-site center window $J=[k,k+w-1]$, where $w\in \{1,2\}$: every core strictly left of $J$ is left-isometric and every core strictly right of $J$ is right-isometric. The center coefficient array has shape $r_{k-1}\times n_k\times \cdots \times n_{k+w-1}\times r_{k+w-1}$, with boundary ranks interpreted as one, and $x$ is its vectorization in the fixed lexicographic order. Hold every core outside $J$ fixed. Contracting the fixed prefix and suffix defines a linear embedding $E$ from center coefficients to the full tensor space. Then $E^{\ast }E=I$, so $\mathrm{\parallel }Ex{\mathrm{\parallel }}_F=\mathrm{\parallel }x{\mathrm{\parallel }}_2$. For every finite Hermitian Hamiltonian input $H$, the effective matrix $H_{\mathrm{e}\mathrm{f}\mathrm{f}}:=E^{\ast }HE$ is Hermitian and satisfies $⟨Ex,HEx{⟩}_F=x^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}x\mathrm{.}$

Let ${\psi }_0=E{x}_0$ be a normalized finite MPS in mixed canonical gauge about a one-site or two-site center, with $\mathrm{\parallel }{x}_0{\mathrm{\parallel }}_2=1$. Let $H_{\mathrm{e}\mathrm{f}\mathrm{f}}=E^{\ast }HE$ be its effective Hermitian matrix, and replace the center by the deterministic unit ground vector $x_{\ast }$ of $H_{\mathrm{e}\mathrm{f}\mathrm{f}}$, giving ${\psi }_{\ast }=E{x}_{\ast }$. Then ${\psi }_{\ast }$ is normalized and ${\mathcal{E}}_H({\psi }_{\ast })\le {\mathcal{E}}_H({\psi }_0)\mathrm{.}$ This claim is before any bond truncation.

Work in exact arithmetic over a fixed field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and order $d\ge 2$. The inputs are:

• a normalized finite MPS/TT state $\psi \in {\mathbb{F}}^{n_1\times \cdots \times n_d}$ together with one concrete presentation of $\psi$, either an open-boundary TT core list or periodic trace matrices; the supplied presentation must contract to $\psi$;
• a finite Hermitian Hamiltonian $H$ on that same tensor-array coefficient space;
• a positive integer sweep budget $M$;
• a window width $w\in \{1,2\}$; and
• optionally, positive integer rank caps $c_1,\dots ,c_{d-1}$.

The procedure is the following.

1. If the supplied presentation is in periodic trace form, convert it exactly to the open-boundary TT form using A periodic finite MPS has an exact open-boundary TT conversion. Otherwise take the supplied open-boundary core list. Keep the resulting open-boundary core list.

2. For each of $M$ full sweeps, perform one forward pass and one backward pass. For width $w$, the windows are the $d$ single sites when $w=1$, or the $d-1$ adjacent site pairs when $w=2$. Visit them in increasing order in the forward pass and decreasing order in the backward pass.

3. Before each window update, place the TT in mixed canonical gauge about that window using the deterministic rank-revealing QR sweeps of Rank-revealing QR sweeps give exact mixed canonical gauge. Contract the fixed left and right cores to form the isometric center embedding $E$ and compute the finite effective matrix $H_{\mathrm{e}\mathrm{f}\mathrm{f}}=E^{\ast }HE$.

4. Apply the exact ground-vector rule of Finite Hermitian ground vectors and a deterministic tie rule to $H_{\mathrm{e}\mathrm{f}\mathrm{f}}$. Let $x$ be the resulting unit eigenvector and replace the center coefficient vector by $x$. Record its local Rayleigh energy $x^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}x={\lambda }_{\min }$ and residual $\mathrm{\parallel }{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}x-{\lambda }_{\min }x{\mathrm{\parallel }}_2$.

5. For a two-site window, reshape the updated center as a matrix with row index (left bond, first physical index) and column index (second physical index, right bond). Compute an exact compact SVD $X=U\mathrm{\Sigma }{V}^{\ast }$. In a forward pass replace the first core by the reshape of $U$ and absorb $\mathrm{\Sigma }{V}^{\ast }$ into the second core. In a backward pass absorb $U\mathrm{\Sigma }$ into the first core and replace the second core by the reshape of $V^{\ast }$. Retain every positive singular direction, so this split changes no tensor entries. For a one-site window, keep that site as the center.

6. After each complete forward/backward sweep, if rank caps were supplied, apply Tensor-train rounding to the current TT and record its cut tails ${\delta }_1,\dots ,{\delta }_{d-1}$ and $\tau =({\sum }_{k=1}^{d-1}{\delta }_k^2{)}^{1\mathrm{/}2}$. If $\tau <1$, normalize the rounded tensor and continue from it. If $\tau \ge 1$, discard that rounded candidate, retain the last normalized unrounded tensor, and record status rounding-skipped. If no rank caps were supplied, do not round.

Return the final normalized TT state, the energy and local-residual records, all accepted discarded-tail records, and status budget-complete or rounding-skipped according to whether any requested rounding was skipped.

The local eigenproblem is specified as an exact finite Hermitian ground solve. The procedure makes no finite-precision stability, polynomial-cost, spectral-gap, global-minimizer, or convergence claim. A periodic input is first converted exactly to an open-boundary TT; the canonical environment argument is applied only to that open-boundary representation.

Let ${\psi }_0$ be a normalized finite MPS/TT state with a supplied concrete open-boundary TT core list or periodic trace presentation contracting to ${\psi }_0$, and let $H$ be a finite Hermitian Hamiltonian on the same coefficient space. Run Finite alternating tensor optimization with effective eigenproblems for one full forward/backward sweep of a fixed width $w\in \{1,2\}$, with $M=1$ and no rank caps, from that presentation using exact local solves and exact SVD splits; call its normalized unrounded output ${\psi }_{\ast }$. Apply exact TT rounding with rank caps $c_1,\dots ,c_{d-1}\in {\mathbb{N}}_0$, obtaining $\tilde{\psi }$ and discarded tails ${\delta }_1,\dots ,{\delta }_{d-1}$. Put $\tau =({\sum }_{k=1}^{d-1}{\delta }_k^2{)}^{1\mathrm{/}2}$. If $\tau <1$, define $\varphi =\tilde{\psi }\mathrm{/}\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F$. Then ${\mathcal{E}}_H(\varphi )\le {\mathcal{E}}_H({\psi }_0)+4\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\tau \mathrm{.}$ In particular, ${\mathcal{E}}_H(\varphi )-{\mathcal{E}}_H({\psi }_{\ast })\le 4\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\tau \mathrm{.}$ No normalized-output claim is made when $\tau \ge 1$.

Fix a finite Hermitian Hamiltonian $H$ on the coefficient space of the states. Consider an infinite sequence obtained by repeatedly applying one full no-truncation sweep of Finite alternating tensor optimization with effective eigenproblems, starting from a supplied concrete open-boundary TT core list or periodic trace presentation representing a normalized state, with fixed bond dimensions and a fixed finite list of one-site or two-site update windows. Assume the canonical core arrays at every stage of each sweep are bounded and converge componentwise in one consistent gauge to a single normalized canonical core tuple. Then the limit is coordinatewise stationary, meaning that for each update window, its limiting center vector minimizes the Rayleigh quotient of the effective Hermitian Hamiltonian over all unit center vectors with the other limiting cores fixed. In particular, each limiting center is a lowest eigenvector of its limiting effective Hamiltonian. This is conditional on the stated convergence; it asserts neither convergence of DMRG nor global minimality.