Quantics Tensor Cross Interpolation (QTCI)

• Quantics Tensor Cross Interpolation (QTCI) is a method that applies tensor cross interpolation to quantics-encoded tensor trains for efficient, high-dimensional function approximation.
• It employs binary multiscale representations to handle exponentially large grids with minimal samples, achieving high-resolution computations without exponential memory growth.
• The algorithm uses rank-revealing techniques such as prrLU to adaptively select pivotal tensor slices, ensuring stable convergence and accurate low-rank approximations.

Quantics Tensor Cross Interpolation (QTCI) is the application of tensor cross interpolation (TCI) to a quantics encoding of a function or tensor. In this construction, continuous or discrete variables are first rewritten through binary digits, producing a quantics tensor train (QTT), and that tensor train is then learned adaptively from sampled entries rather than from a full dense grid. The method was introduced to reconcile high resolution with parsimonious memory usage for multivariate functions, and later systematized as part of a broader TCI family for tensor-train / matrix product state (MPS) learning (Ritter et al., 2023, Fernández et al., 2024).

1. Definition and formal setting

QTCI sits at the intersection of two ideas. The first is the tensor-train or MPS factorization of a high-dimensional tensor,

$$F_{\sigma_1\ldots\sigma_d}\approx M_1^{\sigma_1}M_2^{\sigma_2}\cdots M_d^{\sigma_d},$$

where the three-leg tensors $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$ have bond dimensions $\chi_\ell$, with $\chi=\max_\ell \chi_\ell$ the tensor-train rank. The second is cross interpolation, which constructs such a factorization from selected tensor entries rather than from the full object. In the TCI form emphasized in the 2024 algorithmic treatment,

$$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$

the tensors $T_\ell$ and matrices $P_\ell$ are slices of the original tensor $F$, so the approximation is an interpolation built directly from sampled values (Fernández et al., 2024).

In this sense, QTCI is not a distinct tensor format but a construction procedure: it learns a QTT representation by querying only a small number of values of the target function. The method is explicitly described as rank-revealing. If the target is low rank, TCI rapidly finds an accurate TT/MPS; if it is not, the method does not manufacture a misleading low-rank surrogate, but instead converges slowly or fails to satisfy the prescribed tolerance (Fernández et al., 2024).

The theoretical backbone of this viewpoint predates the specific QTCI terminology. TT-cross interpolation with maximum-volume interpolation sets was shown to be quasioptimal, with an error factor that does not grow exponentially with dimension in the balanced-tree case. The same analysis also established the importance of nested interpolation sets for exact interpolation on sampled subtensors and for practical greedy algorithms (Savostyanov, 2013).

2. Quantics representation and multiscale structure

The “quantics” component is the binary multiscale encoding of variables. For a one-dimensional variable on a dyadic grid,

$x\in[0,1), \qquad x(m)=\frac{m}{2^R},$

with

$$m(\sigma_1,\ldots,\sigma_R)=\sum_{r=1}^{R}\sigma_r 2^{R-r},$$

the sampled function becomes an $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$0-way tensor

$[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$1

For $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$2 variables, one may either interleave the bits from different variables by scale or fuse the bits at a given scale into a single local variable of dimension $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$3 (Ritter et al., 2023, Fernández et al., 2024).

This representation is called “superhigh-resolution” because $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$4 bits per variable access a grid of size $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$5, while the number of tensor legs grows only linearly with $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$6. The central claim of QTCI is that many multiscale functions remain low rank in this quantics basis, so exponentially fine resolution need not imply exponential storage (Fernández et al., 2024). In the original QTCI exposition, this was interpreted as weak coupling between scales: if coarse and fine features do not mix strongly, the QTT bond dimensions remain moderate (Ritter et al., 2023).

The ordering of the quantized legs is consequential. Interleaving is often preferable when different variables are strongly coupled across scales, and the 2024 TCI review explicitly stresses that interleaving is often better in that regime (Fernández et al., 2024). By contrast, later real-frequency vertex calculations found that a fused representation was needed for stable convergence in a demanding Keldysh setting, while Matsubara data were successfully treated in an interleaved form (Frankenbach et al., 16 Jun 2025). A plausible implication is that the best quantics layout depends on whether cross-scale entanglement is dominant within variables, across variables, or concentrated near discontinuities.

The lecture treatment of quantics makes the compressibility criterion more explicit through exact low-rank examples. Exponentials have rank $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$7; $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$8, $[M_\ell]^{\sigma_\ell}_{a_{\ell-1}a_\ell}$9, $\chi_\ell$0, and $\chi_\ell$1 have rank $\chi_\ell$2; and a polynomial of degree $\chi_\ell$3 has exact quantics rank $\chi_\ell$4 (Waintal et al., 6 Jan 2026). These examples are not special to QTCI itself, but they clarify why binary multiscale encoding can be effective before any cross interpolation is performed.

3. Cross interpolation, pivots, and algorithmic refinements

The local mechanism of QTCI comes from matrix cross interpolation. For a matrix $\chi_\ell$5 with pivot row and column sets $\chi_\ell$6 and $\chi_\ell$7,

$\chi_\ell$8

and the error is the Schur complement of the pivot block. TCI generalizes this recursively to tensors by selecting informative slices and enforcing nesting relations between left and right pivot sets (Fernández et al., 2024).

The practical algorithm is typically phrased as a sweeping two-site update. At bond $\chi_\ell$9, one inspects a local slice

$\chi=\max_\ell \chi_\ell$0

views it as a matrix, and applies a rank-revealing decomposition. Provided the nesting conditions hold, the local interpolation error on $\chi=\max_\ell \chi_\ell$1 matches the global tensor error on the corresponding configurations. This is why QTCI is often described as a deterministic, active-learning-like scheme: it samples the function at pivot locations expected to be maximally informative for the unresolved Schur complement (Fernández et al., 2024).

A major algorithmic advance was the replacement of explicit cross interpolation by partially rank-revealing LU decomposition (prrLU). The 2024 TCI paper shows that matrix CI and prrLU are mathematically equivalent, but prrLU is numerically superior because it avoids explicit formation and inversion of the pivot matrix $\chi=\max_\ell \chi_\ell$2. The needed solves become forward and backward substitutions through triangular factors, which are more stable than direct inversion. The same paper highlights further advantages: prrLU is more stable than direct CI, more flexible than the older QR-stabilized approach, can remove bad pivots, supports reset mode, and can be combined with rook or full pivot search (Fernández et al., 2024).

Later applications refined the pivot strategy further. In multiorbital electron-phonon diagrammatics, purely local two-site updates could miss important sectors of the discrete orbital/phonon space, causing an ergodicity problem. That work therefore combined local prrLU-based updates with global searches started from random multi-indices and greedy exploration of the full space (Ishida et al., 2024). This does not change the QTCI formalism, but it shows that pivot geometry can be as important as tensor rank when the target mixes quantized continuous variables with highly structured discrete sectors.

4. Accuracy, complexity, and failure modes

The attraction of QTCI is that a tiny adaptively chosen training set can determine a very large low-rank object. The 2024 TCI review states that TCI needs only $\chi=\max_\ell \chi_\ell$3 function evaluations rather than all $\chi=\max_\ell \chi_\ell$4 entries, because the tensor train is fully determined by $\chi=\max_\ell \chi_\ell$5 pivot indices (Fernández et al., 2024). In the quantics setting this leverage is especially pronounced: one of the examples discussed there represents a function on $\chi=\max_\ell \chi_\ell$6 grid points with only about $\chi=\max_\ell \chi_\ell$7 numbers in the MPS (Fernández et al., 2024).

The original QTCI paper gave a canonical benchmark on a strongly multiscale one-dimensional oscillatory function. With $\chi=\max_\ell \chi_\ell$8 bits and tolerance $\chi=\max_\ell \chi_\ell$9, QTCI found a representation with $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$0 using only $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$1 samples, even though the full tensor has $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$2 entries. The resulting integral was evaluated in $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$3 ms, compared with $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$4 hours for adaptive Gauss–Kronrod quadrature on the same CPU (Ritter et al., 2023). This example became emblematic because it isolates the core QTCI promise: exponentially large grids can be treated as long as the quantics ranks remain modest.

That promise is conditional. The 2023 introduction is explicit that QTCI works best when the function exhibits scale separation and is compressible in QTT form; very irregular or effectively random functions are not expected to have low rank (Ritter et al., 2023). A complementary 2026 study sharpened this point for binary tensors. When the target is sparse, discontinuous, or defined by a Boolean support predicate, standard cross interpolation may fail to converge because the data are non-smooth and important nonzero regions can be missed. That paper therefore advocated a hybrid analytical-numerical rank-product construction as a complement to QTCI rather than a replacement for it (Haubenwallner et al., 3 Jun 2026).

Noise introduces a different failure mode. Since TCI is an interpolation method, it reproduces noisy sampled values at its interpolation points. For noisy functions, a 2024 extension therefore used QTCI only as an adaptive sampler, then compressed the resulting TT by SVD and optimized the cores with nonlinear least squares over all measured points. In the reported sine and two-time-correlation examples, the optimized QTT was more robust against noise than raw QTCI, and in the pseudo-imaginary-time application it produced ground-state energies more accurate than QTCI or Monte Carlo (Sakaue et al., 2024). This suggests that, in noisy settings, QTCI is most naturally interpreted as an adaptive design-of-experiments layer rather than as the final estimator.

5. Applications across physics and computation

By 2026, QTCI had become a reusable compression primitive across several branches of computational physics. In some works it is the main numerical engine; in others it is the input-generation stage that preserves an all-compressed workflow. The range of applications documented in the literature is broad.

Domain	Role of QTCI	Representative source
Multiscale functions and Brillouin-zone integrals	High-resolution QTT construction and fast integration	(Ritter et al., 2023)
Diagrammatic many-body integrals	Low-rank interpolation of Feynman-diagram integrands, sign-problem-free integration	(Ishida et al., 2024)
Self-consistent two-particle equations	Compression of three-frequency vertices and parquet building blocks	(Rohshap et al., 2024)
Nonequilibrium Green’s-function methods	Preparation of QTT-compressed input functions for all-compressed $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$5 workflows	(Środa et al., 2024)
Impurity solvers and DMFT	Efficient evaluation of weak- and strong-coupling diagram weights, including real-frequency formulations	(Matsuura et al., 22 Jan 2025, Kim et al., 2024, Geng et al., 27 Jul 2025, Matsuura et al., 1 Jul 2026)
Local vertex compression from mpNRG	Compression of imaginary- and real-frequency four-point vertices	(Frankenbach et al., 16 Jun 2025)
Large real-space operators and materials simulations	MPO compression for billion-site non-Hermitian systems and million-site super-moiré models	(Sun et al., 15 Jun 2026, Fumega et al., 2024)

In condensed-matter applications, the earliest showcase was the computation of Brillouin-zone integrals and the Chern number in the Haldane model. Near the topological transition, a fused quantics representation with $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$6 allowed the Berry-curvature mesh to be resolved on a $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$7 grid; for tolerance $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$8, QTCI required about $$F_{\sigma_1\ldots\sigma_d}\approx T_1^{\sigma_1}P_1^{-1}T_2^{\sigma_2}P_2^{-1}\cdots P_{d-1}^{-1}T_d^{\sigma_d},$$9 samples, ran in about $T_\ell$0 s on a single Apple M1 core, and produced a Chern number within $T_\ell$1 of the expected integer value $T_\ell$2 (Ritter et al., 2023).

In many-body diagrammatics, QTCI has been used as a deterministic alternative to Monte Carlo. For weak-coupling multiorbital electron-phonon diagrams, the method exposed low-rank structure in second-order self-energy integrands, gave exponential resolution in time and exponential convergence of error with respect to computational cost, and required bond dimensions around $T_\ell$3 for practical construction at tolerance $T_\ell$4 (Ishida et al., 2024). In weak-coupling impurity solvers, the same low-rank TT structure allowed high-order perturbative terms to be evaluated without stochastic sampling and with direct access to the free energy (Matsuura et al., 22 Jan 2025). Strong-coupling impurity solvers then incorporated QTCI in the one-crossing approximation and in third-order real-frequency formulations, making direct real-axis spectral calculations feasible without analytic continuation (Kim et al., 2024, Geng et al., 27 Jul 2025).

For multivariate self-consistent equations, QTCI has been applied to parquet calculations. There the three-frequency vertices are represented as QTTs, affine channel transformations are implemented as low-bond-dimension MPOs, and repeated application of Bethe–Salpeter, parquet, and Schwinger–Dyson steps was reported not to accumulate error beyond the specified tolerance (Rohshap et al., 2024). A related nonequilibrium $T_\ell$5 study used QTCI to prepare the noninteracting Green’s function $T_\ell$6 for driven calculations, thereby preserving an all-QTT workflow from the input stage onward. The paper quantified the stakes: a decompressed representation for $T_\ell$7 and $T_\ell$8 would require $T_\ell$9 exabytes for the retarded and lesser components, or $P_\ell$0 terabytes even with a coarser conventional discretization $P_\ell$1, whereas the QTT-compressed Green’s function required only $P_\ell$2 gigabytes (Środa et al., 2024).

QTCI has also moved beyond frequency-space many-body methods. In mpNRG-based local vertex calculations, the full imaginary-frequency vertex was represented to an accuracy on the order of $P_\ell$3 with maximum bond dimensions not exceeding $P_\ell$4, while the more complex full real-frequency vertex required maximum bond dimensions not exceeding $P_\ell$5 for an accuracy of $P_\ell$6 (Frankenbach et al., 16 Jun 2025). In real-space operator compression, QTCI-based MPO representations were coupled to kernel polynomial methods to study non-Hermitian systems with $P_\ell$7 sites (Sun et al., 15 Jun 2026). In super-moiré materials, QTCI compressed the self-consistent mean-field map so effectively that systems with millions of atoms could be treated in less than one day, whereas the pure KPM benchmark was reported to require around $P_\ell$8 days (Fumega et al., 2024).

6. Software, scope, and relation to adjacent methods

The maturation of QTCI is closely tied to software. The 2024 TCI paper introduced two open-source libraries implementing the improved prrLU-based algorithms: xfac, a C++ library with Python bindings, and TensorCrossInterpolation.jl, a Julia implementation. The former includes classes for TCI, continuous TCI, quantics TCI, tensor trains, and auto-MPO construction, with support for rook/full pivoting, global pivots, $P_\ell$9-site/$F$0-site/$F$1-site sweeps, compression, and environment-weighted integration. The Julia ecosystem includes QuanticsGrids.jl and QuanticsTCI.jl, as well as conversion to ITensors.jl objects (Fernández et al., 2024).

The practical workflow is correspondingly standardized: provide a function $F$2, choose a grid or quantics mapping, let TCI sample informative pivots, and obtain a compact TT/MPS/QTT representation for subsequent integration, Fourier transforms, PDE solvers, contractions, or MPO construction (Fernández et al., 2024). The lecture notes on tensor-network numerical analysis place QTCI in this broader operator-algebra setting by giving explicit MPO constructions for differentiation, indefinite integration, convolution, and the quantum Fourier transform in the quantics representation (Waintal et al., 6 Jan 2026).

A recurrent misconception is that every TCI-based compression of a high-dimensional function is automatically QTCI. The literature is more specific. In the nonequilibrium $F$3 work, QTCI is explicitly the procedure that prepares certain QTT input functions; the self-consistent Dyson iterations are then performed with QTT-compressed objects, so QTCI is the initialization/compression stage rather than the main solver (Środa et al., 2024). Conversely, some later applications are methodologically adjacent but not quantics-specific. The framework for computing quantum resources via TCI compresses tensors defined over Pauli strings or basis configurations and cites the 2024 LU-based TCI method, but the paper explicitly states that its implementation is not a QTCI-specific quantization of continuous variables (Kožić et al., 10 Feb 2025). Likewise, TCI learning of entanglement features and purities is “QTCI-style” in the sense of adaptive recovery from a binary domain, yet it is presented as TCI on a discrete tensor of bipartition labels rather than as a dedicated quantics construction (Kolisnyk et al., 21 Mar 2025).

The relationship to exact structural tensor construction is similarly complementary rather than competitive. For binary tensors defined by Boolean predicates, a 2026 factorization scheme used analytical hyperplane decomposition and rank products to build exact QTT representations where standard cross interpolation may struggle (Haubenwallner et al., 3 Jun 2026). This suggests that QTCI is best understood not as a universal black-box replacement for all tensorization problems, but as one part of a larger tensor-network toolkit whose effectiveness depends on compressibility, smoothness, and the availability of a suitable multiscale encoding.