Isometric Tensor Product States (isoTPS)

• IsoTPS are higher-dimensional tensor networks defined by local isometries arranged toward an orthogonality hypersurface, enabling efficient contraction of complex quantum states.
• They bridge the gap between intractable PEPS and tractable MPS by reducing double-layer contractions to identities, thereby simplifying the evaluation of local observables.
• IsoTPS support practical algorithms like Moses Moves, TEBD², and DMRG², and extend to fermionic, thermal, and circuit-based implementations in quantum many-body simulations.

Isometric tensor product states (isoTPS), often called isometric tensor network states (isoTNS) in the same literature, are a constrained class of higher-dimensional tensor networks that generalize the canonical or isometric form of one-dimensional matrix product states to lattices in two and higher dimensions. Their defining structure is a directed arrangement of local isometries toward an orthogonality hypersurface and an orthogonality center, so that large portions of double-layer contractions collapse to identities and local observables reduce to lower-dimensional contractions. In this sense, isoTPS occupy an intermediate position between generic projected entangled pair states (PEPS), which respect two-dimensional geometry but are difficult to contract, and one-dimensional MPS constructions, which are tractable but geometrically mismatched to genuinely two-dimensional entanglement (Zaletel et al., 2019).

1. Canonical structure, isometries, and orthogonality hypersurfaces

The basic local notion is the standard isometry condition. For a tensor reshaped into a matrix $W \in \mathbb{C}^{n\times p}$ with $n \ge p$, the defining relations are

$$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$

with $\mathcal{P}_n$ a projector onto the image of the isometry. In PEPS-like notation, a local tensor is isometric when contraction over its incoming legs yields the identity on its outgoing legs; a representative form is

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$

These conditions are the higher-dimensional analogue of left- and right-canonical MPS tensors (Sappler et al., 10 Jul 2025).

In two dimensions, the single MPS orthogonality center is replaced by an orthogonality hypersurface. In the original square-lattice construction, this hypersurface is a distinguished row and column, or effectively a central column in algorithmic formulations, toward which all tensor arrows point. Any region whose boundary has only outgoing arrows defines an isometric boundary map, while the hypersurface carries the nontrivial variational data. A compact formulation is

$$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$

so the norm tensor at the orthogonality center is the identity. Consequently,

$$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$

where $\Lambda$ is the effective one-dimensional state on the orthogonality hypersurface (Lin et al., 2021).

This causal organization is the core algorithmic reason isoTPS are tractable. A generic PEPS cannot generally be brought exactly into a globally isometric canonical form because of closed loops, whereas isoTPS enforce a restricted geometry in which the tensors outside the hypersurface contract away orthogonally. Later variants preserve this principle while modifying the geometry. In the diagonal isometric form, for example, the orthogonality hypersurface is represented not by physical tensors but by a chain of auxiliary tensors $W_j$ between columns of physical tensors $T_j$, producing an explicitly MPS-like backbone inside a two-dimensional network (Sappler et al., 10 Jul 2025).

2. Algorithms: Moses moves, TEBD$n \ge p$0, DMRG$n \ge p$1, and diagonal updates

The original algorithmic program for isoTPS had two complementary aims: to convert an existing representation of a two-dimensional state into an isometric form, and to optimize directly within the isometric manifold. The first route starts from a strip represented as a one-dimensional MPS over composite row sites and iteratively “peels off” columns. This requires the Moses Move (MM), an approximate higher-dimensional analogue of the QR/SVD step used to shift the MPS orthogonality center. In the 2019 construction, the essential factorization is

$n \ge p$2

where $n \ge p$3 is an isometric column tensor and the residual state remains efficiently representable as an MPS because the gauge is chosen to minimize the newly exposed entanglement. The tripartite splitting subproblem is explicitly related to minimizing entanglement across a cut and hence to the entanglement of purification (Zaletel et al., 2019).

This structure supports a nested time-evolution strategy, TEBD$n \ge p$4, in which ordinary one-dimensional TEBD acts on the active row or column while Moses Moves shift the orthogonality hypersurface across the lattice. In the same framework, DMRG$n \ge p$5 becomes possible because the norm tensor at the orthogonality center is exactly the identity, so the local variational problem is an ordinary eigenvalue problem rather than a generalized one. In the original cost estimates, a one-dimensional TEBD update on a column costs $n \ge p$6, shifting the orthogonality center with the MM costs $n \ge p$7, and this was contrasted with $n \ge p$8 for a full PEPS update. The 2021 refinement analyzed disentanglers in detail, rederived TEBD$n \ge p$9 for real-time evolution, and introduced DMRG$$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$0 for ground states and dynamical structure factors on the square and honeycomb lattices (Lin et al., 2021).

A more local alternative is the diagonal or YB-isoTPS form. Here the lattice is rotated by $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$1, the orthogonality hypersurface is an auxiliary chain, and its motion is implemented by a Yang–Baxter move. The local step contracts a physical tensor $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$2 with two neighboring auxiliary tensors $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$3, then approximates the result by a constrained tripartite decomposition. The local optimization targets

$$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$4

subject to isometry and norm constraints, and because only a small patch changes, the overlap reduces to a contraction of six tensors. With an approximate truncated SVD in the Rényi-entropy-based disentangling stage, the YB move cost is reduced from $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$5 to $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$6, while the local TEBD gate application costs $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$7. This formulation was benchmarked on the two-dimensional transverse-field Ising model on square lattices of up to 1250 sites and was also extended to honeycomb and kagome geometries (Sappler et al., 10 Jul 2025).

A separate line of work addresses approximate isometrization of a generic tensor network rather than optimization inside a preexisting isoTPS ansatz. Gauge-propagation methods fix an orthogonality center and sweep gauge factors inward. In one dimension the procedure is exact because the non-isometric remainder after QR or SVD lives on a single bond. In higher dimensions the obstruction is local: the residual generally lives on several outgoing legs and is not separable into single-bond factors. The proposed remedy is a structured approximation

$$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$8

with the isometric factor $$W^\dagger W = \mathbb{I}_p,\qquad WW^\dagger = \mathcal{P}_n,$$9 retained locally and the leg factors absorbed into neighbors. Benchmarks on random tensors and the loop-gas tensor for the Kitaev spin liquid showed that retaining two structured terms reduced the local residual to numerical precision, and enlarging the local cluster from 2-in-2-out to 4-in-2-out and 6-in-2-out reduced both local and accumulated propagation errors (Jiang et al., 22 Jun 2026).

3. Expressive power, topological phases, and exact representability

A central early question was whether the isometric restriction excludes intrinsically topological or long-range-entangled phases. The decisive result is that two-dimensional string-net liquids admit exact finite-bond-dimension isoTNS representations. The construction begins from Levin–Wen string-net PEPS data encoded by fusion rules and $\mathcal{P}_n$0-symbols, then rewrites the local tensor structure so that the network becomes manifestly isometric without changing the state. The key technical step is to extend the fusion-constrained $\mathcal{P}_n$1-symbols to full-rank $\mathcal{P}_n$2-symbols satisfying

$\mathcal{P}_n$3

The resulting four-region decomposition around the orthogonality hypersurface gives an exact isoTNS for every string-net fixed point in scope, with the hypersurface movable anywhere (Soejima et al., 2019).

This established that long-range entanglement is not, by itself, an obstruction to an isometric tensor-network representation. The same work also showed that the isometric form can be preserved after applying a finite-depth local quantum circuit, with bond dimension growing roughly as

$\mathcal{P}_n$4

for a depth-$\mathcal{P}_n$5 circuit in the geometry considered. The paper therefore suggested, carefully and not as a theorem, that all two-dimensional gapped phases with gappable edges may admit isoTNS representations. At the same time, it explicitly did not cover chiral topological phases or, more generally, phases with ungappable edges (Soejima et al., 2019).

Subsequent work sharpened the picture rather than weakening it. A complexity-theoretic analysis described isoTNS as a variational ansatz that is much more expressive than MPS, often easier to optimize and contract than generic PEPS, and capable of capturing some string-net fixed points, certain constant-depth circuit states, thermal states, and some critical states. This description does not amount to a classification theorem, but it places isoTPS squarely among nontrivial two-dimensional ansätze rather than as a merely weakly entangled subclass (Malz et al., 2024).

An additional topological development is the construction of parametrized isometric tensor-network “skeletons” for Abelian string-net models. These preserve the relevant virtual symmetries and local isometry constraints while generating finite-correlation-length deformations of fixed points,

$\mathcal{P}_n$6

and connect distinct topological phases through shared critical points. For normalized single-line tensors, the local amplitudes satisfy

$\mathcal{P}_n$7

which permits a mapping to a one-dimensional stochastic automaton. The same framework yields efficient classical evaluation of generalized Pauli strings of arbitrary weight and gives analytically tractable examples of continuous topological phase transitions beyond anyon condensation (Boesl et al., 17 Nov 2025).

4. Circuit mappings, complexity, and sampling

The isometric constraint does not imply classical easiness in general. A key conceptual result is a mapping from two-dimensional isoTNS to $\mathcal{P}_n$8-dimensional open quantum dynamics of virtual ancillas. In one dimension the reduced state of an ancilla evolves under a CPTP map

$\mathcal{P}_n$9

and in two dimensions the same logic produces a sequential picture with two stacks of ancillas whose worldlines cross at isometric tensors. Contracting the two-dimensional tensor network is then equivalent to simulating a one-dimensional circuit of quantum channels acting on the ancillas. This correspondence implies that computing local expectation values in isoTNS is $$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$0-complete (Malz et al., 2024).

The same paper introduced injective isoTNS, defined by adapting injectivity from PEPS and parameterized by the smallest singular value

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$1

For isoTNS this has a dynamical consequence: each ancilla channel contains an unavoidable depolarizing component,

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$2

Weakly injective isoTNS, where $$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$3 and hence $$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$4 are small, remain $$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$5-complete by fault-tolerant embedding arguments. Strongly injective isoTNS admit an efficient classical algorithm for local observables when

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$6

because the contraction reduces to a subcritical percolation problem with exponentially decaying cluster-size distribution (Malz et al., 2024).

Sampling exhibits an analogous division between hard and easy regimes. Multiplicative-precision sampling remains hard even in the maximally injective case. For additive-error sampling, however, the monitored-circuit interpretation becomes central: sampling from an isoTNS can be viewed as monitoring the ancilla dynamics while the circuit runs, and a family of examples exhibits a measurement-induced phase transition from a hard regime to an easy phase where the relevant entangled clusters are subcritical (Malz et al., 2024).

This theoretical picture has been complemented by explicit two-dimensional sampling algorithms. One algorithm performs independent sampling of a single configuration together with its probability; the other uses a greedy search to identify $$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$7 high-probability configurations and their probabilities. Both generalize one-dimensional MPS perfect-sampling ideas by using the orthogonality center to compute local conditionals exactly within each row, while inter-row contractions are approximated by MPO–MPS routines. The total complexity is

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$8

for independent sampling and

$$(T^{i_v}_{\alpha \beta \gamma' \varepsilon'})^* T^{i_v}_{\alpha \beta \gamma \varepsilon} = \delta_{\gamma,\gamma'}\delta_{\varepsilon,\varepsilon'}.$$9

for the top-$$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$0 variant. Benchmarks on GHZ, W, and random circuit states showed the expected $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$1 decay of the empirical KL divergence in the independent-sampling setting; for the top-$$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$2 method, the GHZ distribution was recovered to machine precision with $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$3 up to $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$4, and the W-state distribution was recovered to machine precision with $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$5 up to $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$6 (Dektor et al., 2 Feb 2026).

5. Fermionic, thermal, and holographic generalizations

The fermionic generalization, usually called fisoTNS, retains the orthogonality hypersurface and orthogonality center but requires $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$7-graded tensors and fermionic sign handling via either the swap-gate convention or the virtual fermion ansatz. Tensors are parity-even,

$$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$8

and in the upper-right quadrant the isometry is parity-corrected: $$\left(T^{V\leftarrow \partial V}\right)^\dagger T^{V\leftarrow \partial V}=\mathbb{1}_{\partial V},$$9 The central cancellation theorem has the same structure as in the bosonic case: if two fisoTNS differ only at the orthogonality center, then their overlap reduces to a single local contraction,

$$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$0

On this basis, a fermionic TEBD algorithm was developed for both real-time and imaginary-time evolution, with computational bottleneck $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$1 when the orthogonality-hypersurface bond dimension is $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$2 (Dai et al., 2022).

The benchmarks for fisoTNS show the intended niche of isoTPS particularly clearly. Imaginary-time evolution produced good ground-state energies for gapped systems, systems with a Dirac point, and systems with gapless edge modes. Real-time simulations captured the scattering of two fermions on a $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$3 lattice and the chiral edge dynamics of a Chern insulator. In the scattering example, MPS methods such as TDVP and $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$4 failed early because snake ordering obstructed correct two-dimensional transport, while fisoTNS with $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$5 and $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$6 maintained errors within about $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$7 by the end. For the chiral-edge benchmark, the excitation propagated along the edge without leaking into the bulk, with error remaining within about $$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$8 near the end (Dai et al., 2022).

Finite-temperature extensions use a purification ansatz, adding an ancilla leg to each local tensor and representing

$$\langle \Psi | O | \Psi \rangle = \langle \Lambda | O | \Lambda \rangle,\qquad \|\Psi\|=\|\Lambda\|,$$9

The infinite-temperature initial state is a product of local Bell pairs between physical and ancilla spins, and imaginary-time evolution generates

$\Lambda$0

so that tracing out ancillas yields the thermal state. For purified isoTNS, the reported costs were $\Lambda$1 for the Moses Move and $\Lambda$2 for TEBD with purification. In the two-dimensional transverse-field Ising model, the method reproduced quantum Monte Carlo energy densities and specific heat well away from criticality, while long-range correlations and critical regions remained difficult at fixed bond dimension (Kadow et al., 2023).

A conceptually different extension brings isoTNS ideas back to one-dimensional physics through a holographic $\Lambda$3-dimensional network. The horizontal axis encodes physical space and the vertical axis a holographic direction; contractions remain efficient because only vertical bonds along the orthogonality surface are truncated. This geometry permits volume-law entanglement in one dimension, with the averaged second Rényi entropy of random states growing linearly with $\Lambda$4 and approaching the Page value for sufficiently large $\Lambda$5. The ansatz exactly embeds MPS, exactly represents all fermionic Gaussian states with $\Lambda$6, and variationally captures random Clifford states with $\Lambda$7. At the same time, the present TEBD implementation accumulates surface-shifting error and spuriously suppresses entanglement at longer times, so algorithmic limitations rather than representability are the dominant bottleneck in the reported dynamics (Kobayashi et al., 12 Dec 2025).

6. Physical realizations and circuit-oriented interpretations

IsoTPS are not only a classical variational ansatz. A concrete circuit-QED proposal showed how to generate photonic tensor-network states deterministically. For a single cavity–transmon pair, a cavity ancilla of dimension $\Lambda$8 stores the virtual state and photons are emitted sequentially, producing a photonic MPS. For an array of $\Lambda$9 coupled cavity–transmon pairs with tunable bilinear couplers

$W_j$0

the emitted photons acquire a genuinely two-dimensional entanglement structure described as a radial plaquette PEPS,

$W_j$1

The paper states explicitly that rp-PEPS include isoTNS as a subclass, and it provides the isoTNS isometry condition

$W_j$2

together with an $W_j$3-shaped unitary representation

$W_j$4

For plaquette size $W_j$5, the preparation depth scales as $W_j$6, and for two-dimensional cluster states as $W_j$7 (Wei et al., 2021).

The same circuit-oriented viewpoint appears in purely algorithmic work. Purified isoTNS were described as directly realizable by sequential unitary gates on a quantum computer, with circuit depth $W_j$8 and total number of two-qubit gates $W_j$9 for an $T_j$0 lattice. More broadly, the isometric constraint makes the network resemble a causal circuit flowing opposite to the arrow direction, which explains why local observables depend only on restricted causal regions and why hardware-oriented constructions frequently parallel classical contraction algorithms (Kadow et al., 2023).

Taken together, these developments define the current profile of isoTPS. They are computationally efficient because of their canonical isometric geometry, expressively nontrivial because they exactly capture string-net liquids and related topological structures, and physically interpretable because they map naturally to sequential circuits, open-system dynamics, and concrete photonic-generation protocols. Their main limitations are equally well defined: moving the orthogonality hypersurface is approximate in higher dimensions, long-time dynamics and critical long-range correlations remain difficult at moderate bond dimension, and worst-case contraction or local-observable evaluation is still quantum-computationally hard.