Quantics Representation

• Quantics representation is a mathematical framework that encodes high-dimensional vectors and functions by mapping them into low-mode, binary-encoded tensors.
• It employs tensor train (TT) decomposition to factorize multi-index tensors, drastically reducing storage and computational complexities compared to traditional grid-based methods.
• Adaptive algorithms like Tensor Cross Interpolation enable efficient construction of TT approximations, finding applications in quantum many-body theory, PDE solvers, and high-resolution data compression.

A quantics representation, in its modern technical sense, refers to a class of mathematical encodings and computational techniques that express high-dimensional vectors, functions, or operators via binary-encoded multi-index tensors, subsequently factorized into low-rank tensor trains (TT) or matrix product states (MPS). This representation is central to the compact parameterization and efficient manipulation of objects arising in fields such as quantum many-body theory, computational chemistry, machine learning, and optimization, enabling exponential reduction in storage and computation relative to conventional grid- or basis-based approaches. The quantics framework has spawned a variety of algorithmic developments—most notably Quantics Tensor Train (QTT) representations and associated learning algorithms such as Tensor Cross Interpolation (TCI)—and now underpins quantum-inspired solvers for PDEs, high-resolution diagrammatic simulations, and data compression for multi-variable functions (Ishida et al., 2024, Waintal et al., 6 Jan 2026, Ritter et al., 2023, Fernández et al., 2024, Takahashi et al., 2024, Shinaoka et al., 2022, Soley et al., 2021, Bou-Comas et al., 3 Jul 2025, Niedermeier et al., 6 Jul 2025, Murray et al., 2023, Środa et al., 2024).

1. Binary Tensorization and Quantics Mapping

The defining step of a quantics representation is the mapping—termed binary tensorization or quantization—of a long one-dimensional array or a multivariate function's sampled grid onto the bits of its integer indices. For an array $x \in \mathbb{C}^N$ with $N = 2^d$ (or a function sampled at $N$ points), the quantics map expresses each index $i$ by its $d$-bit binary expansion: $$i = \sum_{k=1}^d i_k 2^{k-1}, \quad i_k \in \{0,1\},$$ and defines a $d$-way tensor

$X(i_1, i_2, ..., i_d) = x(i),$

where the value $x(i)$ is associated with the multi-index $(i_1, ..., i_d)$. For multivariate functions, each coordinate or variable is quantized independently, such that the overall tensor has order equal to the total number of quantized bits (across all variables) (Ishida et al., 2024, Ritter et al., 2023, Waintal et al., 6 Jan 2026).

This folding step transforms exponentially large data structures into high-order but mode-size-2 (or low-mode) tensors, which can then be efficiently factorized under suitable low-rankness or separation of variables assumptions.

2. Tensor-Train Decomposition and Low-Rank Structure

Once binary tensorization is performed, the next crucial stage is the Tensor Train (TT) or MPS decomposition. For a $d$-way binary tensor $X$, the TT form reads

$$X(i_1, ..., i_d) \approx \sum_{\alpha_0, ..., \alpha_d} G^{(1)}_{\alpha_0, i_1, \alpha_1} G^{(2)}_{\alpha_1, i_2, \alpha_2} \cdots G^{(d)}_{\alpha_{d-1}, i_d, \alpha_d},$$

with auxiliary (bond) indices $\alpha_k$ of dimension $\chi_k$ and boundary conditions $\alpha_0 = \alpha_d = 1$. The maximal bond dimension $\chi = \max_k \chi_k$ governs both storage, $O(d \chi^2)$, and computational complexity, $O(d \chi^2)$ for contractions (Ishida et al., 2024, Waintal et al., 6 Jan 2026, Bou-Comas et al., 3 Jul 2025).

In practical applications—such as Feynman diagram integrands or multiscale Green's functions—empirical studies have shown that the quantics tensorization followed by TT decomposition reveals unexpectedly low ranks, even in high-dimensional problems or for target errors as low as $10^{-10}$. The physical explanation lies in the hierarchical (scale-separated) structure of the problems, leading to `scale-wise entanglement' among bits of time, frequency, or spatial coordinates and among discrete degrees of freedom (orbitals, flavors, spins) (Ishida et al., 2024, Shinaoka et al., 2022, Takahashi et al., 2024).

3. Quantics Tensor Cross Interpolation (TCI) and Adaptive Learning

A fundamental practical advance in quantics representations is the use of adaptive, sampling-efficient learning algorithms, such as Tensor Cross Interpolation (TCI), to construct the TT representation from black-box function evaluations rather than full grid data. In the QTT setting, TCI proceeds as follows (Ishida et al., 2024, Fernández et al., 2024, Ritter et al., 2023):

• Initialization with small sets of pivot multi-indices for each bond.
• Local updates: Partial rank-revealing LU (prrLU) decompositions on two-site slices to reveal low TT ranks, adaptively expanding the pivots so that the local approximation residual falls below a specified tolerance.
• Global updates: Greedy search over multi-index space to find the location of maximum local error, ensuring ergodic exploration of discrete flavor or mode spaces.
• Alternating left-to-right and right-to-left sweeps continue until the global supremum norm error $\|f - \tilde{f}\|_\infty / \|f\|_\infty < \epsilon$.

This approach constructs TT approximations requiring $O(d \chi^2)$ function evaluations (where $d$ is the total number of bits and $\chi$ the bond dimension), achieving guaranteed error bounds in the chosen norm.

Empirically, the maximal error decays exponentially with $\chi$, so to reach error $\epsilon$ one needs $\chi = O(\log(1/\epsilon))$, with total cost $O(d (\log(1/\epsilon))^2)$ (Ishida et al., 2024, Fernández et al., 2024, Ritter et al., 2023).

4. Computational Complexity and Compression Properties

Quantics representations provide exponentially compressed storage and evaluation for structured functions on large grids. For an underlying $N$-point grid with $N = 2^d$, the QTT + TT representation requires $O(d \chi^2)$ storage. For multivariate functions on $D$ variables with $2^{d_k}$ points each, the total parameter count is $O((\sum_k d_k) \chi^2)$. In comparison, a naive tabulation would store $O(2^{\sum_k d_k})$ values.

Table: Scaling Properties

Representation	Storage	Contraction Cost	Construction (TCI) Cost
Full Grid	$O(2^d)$	$O(2^d)$	$O(2^d)$
Quantics TT (QTT)	$O(d \chi^2)$	$O(d \chi^2)$	$O(d \chi^2)$ (TCI)

For fixed target accuracies in realistic correlated electron and lattice models, as well as PDE applications and quantum impurity models, observed $\chi$ values are in the range 10–50 even for grid sizes $N \sim 2^{30}$ or higher, enabling efficient resolution of phenomena across many scales (Bou-Comas et al., 3 Jul 2025, Niedermeier et al., 6 Jul 2025, Takahashi et al., 2024).

5. Applications in Physics, Chemistry, and Numerical Analysis

Quantics representations have been successfully applied in several domains:

• Diagrammatic Quantum Many-Body Theory: Construction of Feynman self-energies, vertex corrections, and multi-orbital or multi-flavor objects in DMFT, GW, and NEGF approaches is dramatically accelerated; QTT enables faithful storage and manipulation of two- and three-time Green's functions, overcoming the memory bottleneck inherent to direct discretization (Ishida et al., 2024, Murray et al., 2023, Shinaoka et al., 2022, Środa et al., 2024).
• Solvers for Nonlinear and Linear PDEs: The time-dependent and stationary Gross-Pitaevskii equation, Schrödinger-type equations, and general Poisson/diffusion problems are addressed via quantics TT representations, employing TCI-learned operator MPOs (for differentiation, convolution, and QFT) and achieving exponential reduction in both memory and runtime, even for grids $>2^{40}$ points (Bou-Comas et al., 3 Jul 2025, Niedermeier et al., 6 Jul 2025, Waintal et al., 6 Jan 2026, Fernández et al., 2024).
• High-Resolution Numerical Integration: Physical integrals (e.g., Berry curvature and Chern number over Brillouin zones) and high-frequency oscillatory integrands are exploited with the `superhigh-resolution' quantics approach, allowing evaluation with controlled uniform $\infty$-norm error at polylogarithmic complexity (Ritter et al., 2023, Fernández et al., 2024).
• Statistical and Optimization Algorithms: Global optimization routines, such as the Iterative Power Algorithm, operate on QTT-represented probability densities and potential energy surfaces to locate global minima in prohibitively large search spaces (Soley et al., 2021).

6. Limitations, Bond-Dimension Scaling, and Regimes of Compressibility

Quantics representations are most effective for objects exhibiting hierarchical separability' orlow entanglement' in the binary-factored variable space. For imaginary-time propagators, Green's functions, and self-energies, empirical and theoretical evidence shows that TT bond-dimensions saturate (or increase extremely slowly) at low temperatures and for large domains, especially under Frobenius-norm truncation. For one-time functions, $\chi$ becomes essentially constant with increasing $\beta$, and for two-time objects, the increase is slower than predicted dimensionality arguments; physical and random-pole models both confirm this behavior (Takahashi et al., 2024, Shinaoka et al., 2022).

The primary failure mode arises with functions possessing strong discontinuities, turbulence, or chaos at all scales, which can force bond-dimensions to scale as $O(2^{d/2})$, negating compression. In practice, adaptive monitoring of rank growth and truncation error is employed, and physical models (correlated electron systems, nonlinear condensates) generally remain tractable due to inherent smoothness or multiscale organization (Niedermeier et al., 6 Jul 2025).

7. Extensions, Quantum Information, and Unified Symbolic Frameworks

The concept of quantics also encompasses phase-space and probability representations of quantum states (“star-product formalisms”), where finite-dimensional density matrices and observables are encoded as vectors of classical numbers (“symbols”) with explicit quantizer and dequantizer operator bases. Here, ordinary functions on a finite set replace operator algebra, and all quantum mechanical structure—including non-commutativity and entanglement—are captured via associative star products and duality relations (Adam et al., 2019).

Additionally, the general quantics approach subsumes sparse distributed representations for quantum-like computation, graph/operator encodings in quantum embeddings (QuOp), and tensor network analogs of graphical models for representing and manipulating joint quantum probabilities (Rinkus, 2017, Vlasic et al., 2024, Loeliger et al., 2015).

In summary, the quantics representation unifies a family of quantum-inspired and quantum-adjacent computational frameworks in which binary tensorization and low-rank tensor decomposition enable polynomially- or even logarithmically-scaling algorithms for storing, integrating, and evolving high-dimensional objects that were previously out of reach by direct methods. This structure is foundational for the current generation of efficient tensor-network algorithms in quantum simulation, high-dimensional numerical analysis, and quantum-inspired computational science.