Tensor networks for quantum machine learning

Abstract: Once developed for quantum theory, tensor networks have been established as a successful machine learning paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum machine learning to assess problems that classical computers are unable to solve efficiently. Their nature at the interface between physics and machine learning makes tensor networks easily deployable on quantum computers. In this review article, we shed light on one of the major architectures considered to be predestined for variational quantum machine learning. In particular, we discuss how layouts like MPS, PEPS, TTNs and MERA can be mapped to a quantum computer, how they can be used for machine learning and data encoding and which implementation techniques improve their performance.

• The paper reviews tensor networks (TNs) as structured, interpretable architectures bridging classical machine learning, quantum many-body physics, and variational quantum machine learning (QML).
• Key methodologies include TM-specific inductive biases, capacity control via bond dimensions, and systematic mapping of classical TNs to quantum circuits.
• The primary TN implementations distinguish between classical TN methods, quantum TN (QTN) methods specifically for NISQ hardware, and hybrid methods combining quantum circuits with classical pretraining.

Scope and central thesis

“Tensor networks for quantum machine learning” is a review of tensor-network (TN) architectures as a bridge between classical machine learning, many-body quantum physics, and variational quantum machine learning (QML) (2303.11735). Its central thesis is that TNs provide more than a compression formalism: they define structured, interpretable ansätze whose correlation geometry, bond dimensions, and canonical forms can be translated directly into quantum circuits. This correspondence supports a unified treatment of data encoding, model construction, parameter initialization, optimization, and hybrid classical–quantum execution.

The review distinguishes three settings. Classical TN methods are evaluated entirely on conventional hardware; quantum TNs (QTNs) are architectures designed for quantum computers but may use classical optimization; and hybrid methods combine quantum circuits with classical preprocessing, postprocessing, or pretraining. This terminology is important because many methods described as “quantum machine learning” in the literature are quantum-inspired classical algorithms rather than algorithms requiring quantum hardware.

The paper is primarily architectural and methodological. It does not present a single new algorithm or a unified benchmark demonstrating quantum advantage. Instead, it synthesizes established TN theory and surveys how MPS, PEPS, TTNs, and MERA can be adapted to QML under NISQ constraints.

Tensor-network foundations and inductive bias

A tensor network decomposes a high-order tensor into locally connected lower-rank tensors. The internal indices have bond dimension $\chi$, which controls the amount of information and correlation that can be transmitted through the network. This factorization avoids explicitly storing an exponentially large tensor when the target object has a compatible low-complexity correlation structure.

The review emphasizes that TNs are nonunique because of gauge freedom on internal bonds. Gauge transformations can be exploited to obtain canonical forms in which tensors are isometric or unitary. Canonicalization is not merely a numerical convenience: it supplies the normalization properties required by local optimization and is a prerequisite for mapping classical tensors to quantum gates. Singular-value decompositions provide both the factorization mechanism and a principled way to truncate bonds by removing small singular values.

The architecture determines the correlation and entanglement structure that can be represented efficiently. MPS and PEPS obey area-law restrictions, with MPS supporting constant entanglement across a one-dimensional cut and PEPS supporting boundary-scaling entanglement in higher-dimensional geometries. TTNs represent hierarchical correlations efficiently but do not transmit arbitrary long-range correlations between individual leaves. MERA supplements the hierarchy with disentanglers, increasing its capacity to represent long-range and scale-dependent correlations at greater contraction cost.

This architecture–correlation correspondence becomes an inductive bias in machine learning. A one-dimensional sequence may be naturally represented by an MPS, whereas image data may favor patchwise MPS, PEPS, MERA, or hierarchical constructions. The relevant criterion is not simply the number of features but the scaling of mutual information between subsets of features. The review cites evidence that image data can exhibit approximate boundary-law behavior, while text exhibits substantially steeper scaling and is therefore poorly matched to simple one-dimensional TNs (Lu et al., 2021). The implication is that TN architecture selection should be data-structural rather than generic.

Classical TN machine learning

The basic supervised-learning construction embeds an input $x$ into a feature tensor $\Phi(x)$ and contracts it with a trainable weight tensor. Replacing the full weight tensor by an MPS, TTN, or another TN produces a model whose parameter count and computational cost are governed by local tensor dimensions and bond dimensions rather than by the full Cartesian product of feature dimensions.

The bond dimension is consequently a direct capacity-control parameter. Increasing $\chi$ enlarges the representable function class, while truncating $\chi$ regularizes the model and reduces computational cost. This provides a more structurally explicit capacity control than merely increasing or decreasing the width of a conventional dense layer. Density-matrix formulations additionally permit missing or incomplete data to be handled by contracting over absent feature bonds.

The review surveys discriminative, generative, and hybrid TN models. Classifiers and regressors use TNs as compressed weight tensors; generative TNs represent probability distributions or amplitudes and generate samples through measurement-like procedures. MPS Born machines, for example, use the squared amplitudes of an MPS to define a distribution and can be trained for unsupervised modeling (Paron et al., 2018). TTNs and MERA-like networks can perform hierarchical feature extraction, while TN layers can also replace or compress dense layers in neural networks.

A significant structural limitation is the no-cloning constraint inherited from quantum tensor constructions. Ordinary TNs cannot freely copy an intermediate feature into multiple branches. In image processing, this means that adjacent or overlapping observation windows may be needed to expose the same local information to multiple parts of the model. Generalized TNs that permit controlled information reuse address this limitation and can improve image and sound classification (Glasser et al., 2018). Thus, the strict TN formalism provides interpretability and contraction structure at the cost of reduced architectural flexibility.

Optimization methods divide into TN-specific local methods and global gradient-based methods. DMRG and related renormalization procedures optimize local tensors against effective environments and are particularly effective for MPS. They exploit locality and canonical gauges but are difficult to integrate into generic differentiable machine-learning software and do not generalize efficiently to higher-dimensional networks. Global automatic differentiation is easier to combine with neural components but can suffer from exploding or vanishing gradients and high-dimensional ill-conditioning. Riemannian optimization on manifolds of isometric tensors offers a geometrically consistent alternative (Luchnikov et al., 2020, Charalampidis et al., 2021).

Mapping TNs to quantum circuits

The review’s main technical contribution is its systematic account of the TN-to-circuit correspondence. An isometric tensor can be embedded into a unitary gate by introducing ancilla qubits for missing input dimensions and discarding surplus output qubits. A bond of dimension $\chi$ is represented by approximately $\log_2 \chi$ virtual qubits. Physical indices become data or output qubits, while virtual indices carry correlations between successive circuit blocks.

This mapping is exact only under appropriate boundary conditions. In particular, discarded qubits must be post-selected onto the reference state dual to the initialized ancillas if the quantum circuit is to reproduce the contraction of the original classical TN. Without post-selection, the circuit generally implements a related but different operation. The review notes two ways to remove this restriction: optimize the quantum circuit directly so that discarded degrees of freedom no longer affect the measured output, or reset and reuse qubits during execution.

The distinction between tensor contraction and circuit execution is operationally important. A classical TN must explicitly contract tensors, whereas a QTN realizes the contraction through unitary evolution and measurement. However, this does not remove the underlying costs. Larger bond dimensions require more virtual qubits, larger multi-qubit gates, or deeper decompositions into one- and two-qubit gates. PEPS and MERA mappings also require nonlocal connectivity or swap networks, making their hardware realization substantially more demanding than their graphical descriptions suggest.

MPS is the most hardware-compatible layout. Its sequential structure permits qubit reuse, so the number of simultaneously active qubits can remain constant with respect to the number of input sites, depending on the physical input dimension and bond dimension. A TTN requires a number of active qubits that scales logarithmically with the number of inputs. By contrast, brickwall, MERA, and PEPS circuits generally sacrifice this qubit efficiency for greater entanglement capacity or more favorable correlation geometry.

Circuit compression and hardware-aware implementation

The review identifies multi-qubit gate decomposition as a central implementation problem. A tensor node associated with $\log_2\chi$ virtual qubits may be a large unitary, but NISQ hardware generally supports only a restricted native gate set. Decomposing a general unitary can produce circuits that are too deep and noise-sensitive. Structured decompositions of MPS tensors into layers of two-qubit gates preserve the relevant connectivity while reducing gate arity (Ran, 2019).

Simplified two-qubit ansätze reduce parameter count and circuit depth but can lower expressivity. The review reports a notable asymmetry: for quantum inputs, restricted ansätze can sometimes approach the performance of general unitaries, whereas performance degradation is more pronounced for classical inputs. This result cautions against treating hardware-efficient ansätze as universally interchangeable with full tensor-node unitaries.

Several implementation strategies are highlighted:

• Virtual-qubit reuse: discarded qubits are reset and reused for subsequent sites, giving MPS implementations with constant active-qubit requirements.
• Central gauges: a central MPS gauge can approximately halve circuit depth relative to left- or right-canonical sequential implementations.
• Two-qubit decompositions: multi-qubit tensor nodes are replaced by structured layers of local gates.
• Error-correction reuse: qubits that would otherwise be discarded may carry information useful for error correction.
• Hardware-aware compilation: TN structure can be compiled directly to the target connectivity rather than through a generic circuit compiler.

The review also states that the consequences of qubit-efficient execution for trainability are not fully established. The “no signalling” argument suggests that discarded qubits should not alter optimized model parameters when the circuit is trained for the retained outputs, but this does not settle effects on gradient variance, noise accumulation, or optimization dynamics.

Variational QML with structured TN architectures

A QTN with trainable gates is a structured variational quantum circuit. Unlike a generic layered VQC, its topology imposes a predefined hierarchy or locality pattern. This creates a specific inductive bias and enables capacity tuning through bond dimensions and network connectivity.

The paper presents several advantages over unstructured VQCs. First, classical TN theory provides tools for initialization, compression, interpretability, and optimization. Second, the TN topology can be selected to match the correlation structure of the data. Third, the bond dimension gives an explicit control over expressivity and therefore overfitting. These advantages remain conditional: the expressivity of QTN architectures relative to generic layered VQCs is not yet characterized in a general way, even though mappings between the two classes exist (Cecconi et al., 2020).

The reviewed QTN applications are predominantly classification and generation. In supervised classification, feature maps encode data into physical qubits, the QTN represents trainable weights, and an exit node produces a class distribution or decision statistic. Generative QTNs reverse the network direction and begin from computational-basis states, labels, or noise variables. They can be trained by sampling from the circuit or by converting a trained discriminative model into a generative network.

The numerical evidence is promising but heterogeneous. Demonstrative studies report classification accuracies approximately between $0.85$ and $0.95$ on classical and quantum datasets, including grayscale images with dimensions as high as $x$0 pixels. In quantum phase recognition, a checkerboard tensor-network-inspired classifier achieved reported accuracies of $x$1 for the transverse-field Ising model and $x$2 for the XXZ model (Uvarov et al., 2019). Other multi-class studies report a broader range, including $x$3–$x$4 for reduced MNIST tasks and $x$5–$x$6 for XXZ phase recognition, depending on architecture and preprocessing (Lazzarin et al., 2021). These figures establish feasibility rather than quantum advantage: the datasets, preprocessing, parameter budgets, and classical baselines differ substantially across studies.

The review also discusses noise. Low noise need not immediately destroy classification performance, and noisy ancillas can in some cases act as a stochastic resource that improves optimization or model flexibility. Excessive decoherence, however, eventually destroys the encoded correlations. Numerical results on decohered TTNs even suggest that adding ancillas can compensate for noise under particular conditions (Liao et al., 2022). This is a model- and noise-dependent observation, not a general claim that decoherence is beneficial.

Trainability and hybrid pretraining

Trainability is presented as a decisive advantage that TN structure might provide over generic variational circuits. Random or overly expressive circuits can exhibit barren plateaus, where gradient magnitudes decrease exponentially with system size (Gupta et al., 2018). Hierarchical and local architectures can mitigate this concentration phenomenon. The review cites theoretical results showing that tree-structured quantum neural networks can have gradients that vanish at most polynomially with the number of qubits (2011.06258).

The strongest qualification concerns the loss function. For MPS-based learning, global losses can still produce barren plateaus, while local losses yield gradients that remain nonvanishing near the corresponding local observables (Liu et al., 2021). This implies that locality of the architecture alone is insufficient; the measurement strategy and objective function must also preserve locality. Related work on quantum convolutional architectures establishes the absence of barren plateaus under specific locality and initialization assumptions (Csóka et al., 2020).

Hybrid pretraining addresses both trainability and hardware cost. A classical TN can first be trained using DMRG, density-matrix methods, or other TN-specific optimizers. Its canonical tensors are then mapped to quantum gates, after which the quantum circuit is refined or expanded. MPS pretraining can initialize the diagonal or embedded portion of a brickwall circuit while leaving other gates at identity. This avoids beginning quantum optimization in a random high-dimensional parameter region and can substantially reduce the amount of quantum hardware training required (Barratt et al., 2022).

A related strategy prepares a classical prior distribution with a homogeneous MPS and then uploads data through a second MPS-like circuit. This reduces the bond dimension and gate complexity required for data-dependent processing. The broader implication is that quantum hardware need not carry the entire learning pipeline: classical TNs can perform compression and coarse optimization, while the quantum component supplies a restricted, hardware-aware variational refinement.

TN-based data encoding

The review treats data encoding as an architectural problem rather than a preprocessing detail. Direct amplitude or basis encoding is often infeasible for high-dimensional classical data because current devices have limited qubit counts and circuit depths. TNs provide a compression mechanism by truncating low-weight singular directions or learning a fixed-bond-dimension latent representation.

Several encoding strategies are described. A classical tensor or quantum state can be approximated by an MPS or TTN and then compiled into a circuit. Alternatively, a trainable TN can map data into a lower-dimensional latent space before a variational classifier. Multiple independent MPS encoders can preserve distinct feature groups, while an output node can improve multi-class decision performance.

For images, flattening a two-dimensional grid into a one-dimensional MPS introduces an ordering problem: pixels that are adjacent in the image may become distant along the chain. Patchwise encoding reduces this distortion by applying separate MPS representations to local regions, at the cost of additional qubits and storage. The review therefore treats patching as a correlation-preserving compromise rather than a universally optimal solution.

The relationship between data correlations and TN geometry is central. MPS are suitable for data with one-dimensional or sequential correlations; TTNs offer logarithmic hierarchical connectivity; and two-dimensional PEPS or MERA are better candidates when correlations follow image-like spatial structure. For text, the review notes that near-volume-law mutual-information scaling makes simple one-dimensional TN representations inefficient (Lu et al., 2021). This limits claims that TN encoding generically solves the classical-data loading bottleneck.

Limitations and open questions

The review is explicit that the field lacks a general demonstration of quantum advantage. Most QTN studies are feasibility experiments on small, structured datasets, often with substantial classical preprocessing. Reported accuracies therefore cannot be interpreted as evidence that QTN models outperform classical TNs, neural networks, or generic VQCs at comparable end-to-end cost.

The architecture imposes substantial constraints. A fixed TN topology may be well matched to one correlation structure and poorly matched to another. PEPS and MERA have attractive representational properties but are difficult to contract, compile, and execute on hardware with limited connectivity. Variable bond dimensions, which are standard in classical TN algorithms for allocating capacity adaptively, had not been fully incorporated into QTN implementations at the time of the review.

Optimization remains unresolved. Classical local methods such as DMRG are highly effective for MPS but lack a direct and general quantum analogue. Quantum optimization is also affected by shot noise, hardware noise, measurement overhead, and potentially hard nonconvex landscapes. The review’s claims about improved trainability rely on particular architectures, local losses, initialization procedures, or error-correction mechanisms and therefore do not generalize automatically to all QTNs.

Data encoding presents a separate bottleneck. TN compression is effective only when the data have a compatible correlation structure and when the truncation error is acceptable. A quantum circuit may represent a compressed state with few qubits, but preparing that state can itself require substantial classical computation, circuit depth, or post-selection. The resulting end-to-end resource advantage remains dataset- and hardware-dependent.

The paper identifies several unresolved theoretical questions: how QTN expressivity scales with bond dimension and depth; when QTN trainability is superior to generic VQCs; how noise and qubit reuse alter optimization; which data distributions admit useful TN compression; and whether any of these properties yield a practical computational advantage after including encoding, compilation, sampling, and classical optimization costs.

Conclusion

The review presents tensor networks as a coherent design framework for QML rather than as a single model class. Their principal strengths are structured inductive bias, explicit capacity control through bond dimension, compatibility with canonical and isometric representations, qubit-efficient circuit mappings, and natural interfaces between classical pretraining and quantum execution.

The evidence surveyed supports practical feasibility: QTN classifiers have achieved accuracies from roughly $x$7 to $x$8 in several demonstrations, with higher task-specific results such as $x$9 and $\Phi(x)$0 in quantum phase-recognition benchmarks. These results do not establish quantum advantage, and the review appropriately frames the field as methodologically immature. The decisive open problem is a theory and benchmark methodology that compares QTN architectures with classical TNs and generic VQCs under equal end-to-end resource accounting.