Tensor Train Optimizer (TTOpt)

• TTOpt is a derivative-free optimization approach that leverages low-rank tensor train representations and maximum-volume sampling to reduce high-dimensional search spaces.
• It employs adaptive sampling and candidate pruning strategies to efficiently solve black-box function minimization and reinforcement learning tasks.
• Empirical results demonstrate that TTOpt outperforms traditional methods in applications like adsorption energy minimization, quantum variational simulations, and adversarial attacks.

Tensor Train Optimizer (TTOpt) is presented in the literature as a procedure for optimization based on the combination of efficient quantized tensor train representation and a generalized maximum matrix volume principle, as a new algorithm for TT-tensor optimization, and, in adsorption studies, as a classical, quantum-inspired algorithm for the global optimization of higher-order unconstrained binary optimization (HUBO) problems (Sozykin et al., 2022, Chertkov et al., 2022, Do et al., 28 Jul 2025). Across these formulations, the objective is treated as an implicitly defined high-dimensional tensor, and the search for a minimum or maximum is carried out through low-rank tensor-train structure, adaptive sampling, and maximum-volume or related candidate-selection procedures rather than exhaustive enumeration (Do et al., 28 Jul 2025, Sozykin et al., 26 Jan 2026).

1. Origins and scope

The modern TTOpt literature emerged from several closely related lines of work. One line formulates TTOpt as a maximum-volume quantized tensor train-based optimization method for multidimensional function minimization and reinforcement learning (Sozykin et al., 2022). A second line studies TT-tensor optimization directly for functions already given in the tensor-train format, with sequential tensor multiplications of the TT-cores and an intelligent selection of candidates for the optimum (Chertkov et al., 2022). A third line develops TTOpt as an algebraic method based on maximum volume sampling for molecular and materials problems, including atomic clusters and adsorption configurations on alloy surfaces (Sozykin et al., 26 Jan 2026, Do et al., 28 Jul 2025).

This scope is unusually broad. In the cited works, TTOpt is applied to black-box function minimization, reinforcement learning, variational quantum eigensolvers, black-box adversarial attacks on computer vision models, global optimization of atomic clusters, and adsorption-energy minimization on alloy surfaces (Sozykin et al., 2022, Paradezhenko et al., 2023, Chertkov et al., 2023, Sozykin et al., 26 Jan 2026, Do et al., 28 Jul 2025). A plausible implication is that TTOpt is best understood not as a single domain-specific solver, but as a family of TT-based derivative-free optimization procedures whose common premise is that the sampled objective admits a useful low-rank tensor structure.

2. Tensor-train representation of objective functions

The fundamental abstraction is a tensorization of the objective. In the general discrete setting, a $d$-dimensional function $f(n_1,\dots,n_d)$ is approximated in tensor-train form as

$$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$

with $R_0=R_d=1$. The total storage is $\sum_{i=1}^d R_{i-1}N_iR_i$, which for bounded ranks avoids the exponential curse of dimensionality (Sozykin et al., 26 Jan 2026). In the TT-tensor-optimization formulation, storage is correspondingly reduced from $O(N^d)$ to $O(d\cdot N\cdot R^2)$ when the ranks remain bounded (Chertkov et al., 2022).

In adsorption optimization, the tensorized object is explicit. Let $\sigma=(\sigma_1,\sigma_2,\dots,\sigma_N)$ with $\sigma_i\in\{0,1\}$ denote occupancy of adsorption sites. The adsorption energy is written as

$$E^{ads}(\sigma)=\sum_i U_{iii}\sigma_i +6\sum_{i<j}U_{ijj}\sigma_i\sigma_j +6\sum_{i<j<k}U_{ijk}\sigma_i\sigma_j\sigma_k,$$

and TTOpt defines an $f(n_1,\dots,n_d)$0-dimensional tensor $f(n_1,\dots,n_d)$1 of size $f(n_1,\dots,n_d)$2 by $f(n_1,\dots,n_d)$3 (Do et al., 28 Jul 2025). The global minimization problem is then reframed as finding the tensor entry with minimal value, without enumerating all $f(n_1,\dots,n_d)$4 possibilities.

A related construction appears in quantized TT (QTT) formulations. When each mode size $f(n_1,\dots,n_d)$5 is large, one may decompose it into $f(n_1,\dots,n_d)$6 sub-modes of size $f(n_1,\dots,n_d)$7, reshape the original tensor into a $f(n_1,\dots,n_d)$8-way tensor, and then apply TT decomposition to the quantized object (Sozykin et al., 2022). In practice, the cited work often chooses $f(n_1,\dots,n_d)$9, so that a mode of length $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$0 becomes a chain of $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$1 binary modes.

3. Core algorithmic mechanisms

A central mechanism in several TTOpt realizations is the maximum-volume cross-approximation principle. For a tensor unfolding $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$2, one seeks row and column index sets such that the corresponding square submatrix has large volume, defined as the absolute value of its determinant. The maxvol algorithm iteratively swaps rows to increase this quantity, and forward and backward sweeps across tensor unfoldings extract index sets that identify informative fibers of the tensor (Sozykin et al., 26 Jan 2026). In the cluster-optimization formulation, one full TTOpt sweep alternates forward and backward passes until index sets stabilize or a maximum number of black-box calls is exceeded.

The adsorption formulation instantiates this logic as an adaptive cross-approximation method in the TT format combined with a maximum-volume selection procedure. Rather than performing a full decomposition, TTOpt samples carefully chosen slices of the energy tensor and builds the TT cores on the fly. During the forward and backward passes, it applies the mapping

$$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$3

so that maxvol emphasizes low-energy entries, and it updates the current best energy and configuration as evaluations proceed (Do et al., 28 Jul 2025). With TT ranks bounded by $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$4, each pass costs $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$5 function evaluations, and the total arithmetic sampling cost is $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$6.

A distinct TTOpt variant operates directly on a TT-encoded tensor and uses sequential tensor multiplications with candidate pruning. In that formulation, the method treats $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$7 as an unnormalized probability mass function on multi-indices and greedily tracks the $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$8 most likely partial assignments, effectively performing a beam search over prefixes (Chertkov et al., 2022). The resulting complexity is $$f(n_1,\dots,n_d)\approx \sum_{r_0=1}^{R_0}\sum_{r_1=1}^{R_1}\cdots\sum_{r_d=1}^{R_d} G_1[r_0,n_1,r_1]\cdot G_2[r_1,n_2,r_2]\cdots G_d[r_{d-1},n_d,r_d],$$9, and the same work provides a worst-case guarantee for $R_0=R_d=1$0, although the bound is explicitly described as pessimistic.

4. Higher-order structure and constraint encoding

The role of higher-order structure is particularly explicit in adsorption optimization. The cited adsorption study formulates the search for the most stable configuration as a HUBO problem and reports that including interaction terms up to third order may be sufficient to approximate adsorption energies within chemical accuracy and to identify optimal configurations (Do et al., 28 Jul 2025). It also reports that TTOpt performs better with the HUBO formulation than with QUBO, suggesting that third-order terms help preserve correlations between adsorption sites. In that interpretation, quadratic truncation neglects three-body couplings, artificially flattening or distorting the energy landscape, whereas the HUBO tensor preserves true many-body correlations.

In molecular structure prediction, the analogous issue is not interaction order but physical admissibility of the discretization. The cluster-optimization framework introduces physically-constrained encoding schemes that incorporate molecular constraints directly into the discretization process (Sozykin et al., 26 Jan 2026). These include simple relative encoding in Z-matrix style, constant-distance relative encoding, angle-restricted relative encoding, and bit-coding with $R_0=R_d=1$1-ary digits. Every sampled multi-index is decoded into atomic positions through the chosen encoding before the potential energy is evaluated, so the algorithm never visits unphysical or symmetry-redundant regions.

Continuous-domain applications use the same tensorization principle after discretization. In black-box adversarial attacks, each coordinate perturbation $R_0=R_d=1$2 is discretized on $R_0=R_d=1$3 levels in $R_0=R_d=1$4, and the tensor entry is the classifier score at the perturbed image (Chertkov et al., 2023). In VQE, each circuit parameter $R_0=R_d=1$5 is placed on an $R_0=R_d=1$6-point grid so that the variational energy becomes a $R_0=R_d=1$7-dimensional array approximated in TT format (Paradezhenko et al., 2023). These constructions preserve the derivative-free character of TTOpt while making the objective compatible with TT-cross or related low-rank procedures.

5. Empirical results and application domains

In surface chemistry, TTOpt is used with an adsorption-energy model based on multi-adsorbate interaction terms evaluated by the in-house trained machine learning interatomic potential MACE-Osaka24 (Do et al., 28 Jul 2025). On binary alloy slabs with $R_0=R_d=1$8 sites and CO and NO adsorbates, HUBO+TTOpt recovered adsorption energies within chemical accuracy $R_0=R_d=1$9 up to monolayer coverage, whereas QUBO errors grew beyond chemical accuracy at coverages $\sum_{i=1}^d R_{i-1}N_iR_i$0. On an IrPdPtRhRu high-entropy alloy nanoparticle with $\sum_{i=1}^d R_{i-1}N_iR_i$1 sites, HUBO errors remained $\sum_{i=1}^d R_{i-1}N_iR_i$2 across all coverages, while QUBO errors rose to $\sum_{i=1}^d R_{i-1}N_iR_i$3 at high coverage. Under the same evaluation budget of $\sum_{i=1}^d R_{i-1}N_iR_i$4, TTOpt+HUBO consistently found equal or lower energies than a Genetic Algorithm.

In atomic-cluster optimization, the physically-constrained TT framework identifies global minima of Lennard-Jones clusters containing up to 45 atoms and optimizes 20-atom carbon clusters using a machine-learned Moment Tensor Potential (Sozykin et al., 26 Jan 2026). The reported numerical experiments state that TTOpt with direct encoding locates the global minimum of $\sum_{i=1}^d R_{i-1}N_iR_i$5 up to $\sum_{i=1}^d R_{i-1}N_iR_i$6 with 100% reliability, requiring on the order of $\sum_{i=1}^d R_{i-1}N_iR_i$7–$\sum_{i=1}^d R_{i-1}N_iR_i$8 energy evaluations, and that even for the double-funnel $\sum_{i=1}^d R_{i-1}N_iR_i$9 it achieves 100% success at $O(N^d)$0 calls averaged over 30 runs. When combined with physically-constrained relative encoding, TTOpt and PROTES reduce calls for small-to-medium clusters $O(N^d)$1 to $O(N^d)$2.

In parametrized quantum circuits, TTOpt is examined as a derivative-free optimizer for VQE with hardware-efficient and Hamiltonian variational ansätze (Paradezhenko et al., 2023). For shallow circuits, the reported study states that TTOpt consistently finds lower energies than BFGS from random initializations, and under depolarizing noise $O(N^d)$3 it is nearly unaffected relative to the degradation observed for BFGS. The same study also emphasizes the computational cost: TTOpt uses up to $O(N^d)$4 evaluations in the numerical studies, with wall-clock times of $O(N^d)$5–$O(N^d)$6, while BFGS runs in $O(N^d)$7–$O(N^d)$8 per VQE run.

In black-box adversarial attacks, a TT-based optimizer combined with attribution from an auxiliary white-box model is reported to outperform three popular baselines on five modern DNNs on the ImageNet dataset (Chertkov et al., 2023). With a fixed query budget of $O(N^d)$9, the reported attack success rates exceeded 97% on all but ResNet-152, where approximately 85% was achieved. In reinforcement learning, TTOpt is evaluated on Swimmer-v3, LunarLanderContinuous-v2, InvertedPendulum-v2, and HalfCheetah-v3, with comparisons against GA, OpenAI-ES, and CMA-ES under matched call budgets and bound constraints (Sozykin et al., 2022). In TT-tensor optimization benchmarks, the algorithm is tested on random tensors and benchmark functions with up to 100 input dimensions; the abstract reports solutions close to the exact optimum for all model problems, with running time no more than 50 seconds on a regular laptop (Chertkov et al., 2022).

6. Practical considerations, limitations, and related methods

Practical performance depends on rank, sampling budget, and ordering heuristics. In the adsorption benchmarks, the maximum TT rank $O(d\cdot N\cdot R^2)$0 was varied between 2 and 50, the best run per instance was selected post-hoc, the evaluation budget was set to $O(d\cdot N\cdot R^2)$1 energy calls, the penalty weight for the fixed coverage constraint was $O(d\cdot N\cdot R^2)$2, and ACAT’s spatial ordering yielded better performance than random shuffles (Do et al., 28 Jul 2025). The same study argues that TTOpt benefits more from HUBO than from QUBO because third-order terms preserve many-body correlations, and it explicitly contrasts this with quantum and digital annealers that are restricted to cost functions with at most quadratic terms in similar global optimization tasks. It also states that the method does not require specialized hardware.

The limitations are equally clear in the cited literature. In the physically-constrained cluster formulation, TTOpt is guaranteed to converge in the sense that index sets eventually stabilize, but the quality depends on how low-rank the objective actually is (Sozykin et al., 26 Jan 2026). In the TT-tensor-optimization formulation, the worst-case bound is described as very weak and performance depends on ordering of modes, with mode permutation and two-way passes proposed as heuristics (Chertkov et al., 2022). In VQE, increasing ansatz depth and parameter count slows convergence, and gradient-based methods eventually outperform TTOpt for deeper circuits when gradients can still be estimated reliably (Paradezhenko et al., 2023). The relation to adjacent TT-based optimizers is also explicit: the cluster work presents TTOpt and PROTES as complementary strategies, with TTOpt using maximum volume sampling and PROTES using generative sampling (Sozykin et al., 26 Jan 2026). This suggests that the main methodological distinction within the recent literature is not whether TT structure is used, but whether the search is driven by algebraic cross-approximation or by probabilistic sampling over TT-parameterized distributions.