Though there has been substantial progress in developing quantum algorithms to study classical datasets, the cost of simply \textit{loading} classical data is an obstacle to quantum advantage. When the amplitude encoding is used, loading an arbitrary classical vector requires up to exponential circuit depths with respect to the number of qubits. Here, we address this ``input problem'' with two contributions. First, we introduce a circuit compilation method based on tensor network (TN) theory. Our method -- AMLET (Automatic Multi-layer Loader Exploiting TNs) -- proceeds via careful construction of a specific TN topology and can be tailored to arbitrary circuit depths. Second, we perform numerical experiments on real-world classical data from four distinct areas: finance, images, fluid mechanics, and proteins. To the best of our knowledge, this is the broadest numerical analysis to date of loading classical data into a quantum computer. The required circuit depths are often several orders of magnitude lower than the exponentially-scaling general loading algorithm would require. Besides introducing a more efficient loading algorithm, this work demonstrates that many classical datasets are loadable in depths that are much shorter than previously expected, which has positive implications for speeding up classical workloads on quantum computers.
Though there has been substantial progress in developing quantum algorithms to study classical datasets, the cost of simply \textit{loading} classical data is an obstacle to quantum advantage. When the amplitude encoding is used, loading an arbitrary classical vector requires up to exponential circuit depths with respect to the number of qubits. Here, we address this ``input problem'' with two contributions. First, we introduce a circuit compilation method based on tensor network (TN) theory. Our method -- AMLET (Automatic Multi-layer Loader Exploiting TNs) -- proceeds via careful construction of a specific TN topology and can be tailored to arbitrary circuit depths. Second, we perform numerical experiments on real-world classical data from four distinct areas: finance, images, fluid mechanics, and proteins. To the best of our knowledge, this is the broadest numerical analysis to date of loading classical data into a quantum computer. The required circuit depths are often several orders of magnitude lower than the exponentially-scaling general loading algorithm would require. Besides introducing a more efficient loading algorithm, this work demonstrates that many classical datasets are loadable in depths that are much shorter than previously expected, which has positive implications for speeding up classical workloads on quantum computers.
Abstract Quantum machine learning (QML) is an emerging field that investigates the capabilities of quantum computers for learning tasks. While QML models can theoretically offer advantages such as exponential speed-ups, challenges in data loading and the ability to scale to relevant problem sizes have prevented demonstrations of such advantages on practical problems. In particular, the encoding of arbitrary classical data into quantum states usually comes at a high computational cost, either in terms of qubits or gate count. However, real-world data typically exhibits some inherent structure (such as image data) which can be leveraged to load them with a much smaller cost on a quantum computer. This work further develops an efficient algorithm for finding low-depth quantum circuits to load classical image data as quantum states. To evaluate its effectiveness, we conduct systematic studies on the MNIST, Fashion-MNIST, CIFAR-10, and Imagenette datasets. The corresponding circuits for loading the full large-scale datasets are available publicly as PennyLane datasets and can be used by the community for their own benchmarks. We further analyze the performance of various quantum classifiers, such as quantum kernel methods, parameterized quantum circuits, and tensor-network classifiers, and we compare them to convolutional neural networks. In particular, we focus on the performance of the quantum classifiers as we introduce nonlinear functions of the input state, e.g. by letting the circuit parameters depend on the input state.
Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose tensor network-based quantum data encoding (TNQE), a circuit-efficient framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as 0.04× the depth of amplitude encoding, while naturally scaling to high-resolution images (256×256) and demonstrating practical feasibility on real quantum hardware.
We introduce AQCtensor , a novel algorithm to produce short-depth quantum circuits from Matrix Product States (MPS). Our approach is specifically tailored to the preparation of quantum states generated from the time evolution of quantum many-body Hamiltonians. This tailored approach has two clear advantages over previous algorithms that were designed to map a generic MPS to a quantum circuit. First, we optimise all parameters of a parametric circuit at once using Approximate Quantum Compiling (AQC)—this is to be contrasted with other approaches based on locally optimizing a subset of circuit parameters and “sweeping” across the system. We introduce an optimisation scheme to avoid the so-called “orthogonality catastrophe”—i.e., the fact that the fidelity of two arbitrary quantum states decays exponentially with the number of qubits—that would otherwise render a global optimisation of the circuit impractical. Second, the depth of our parametric circuit is constant in the number of qubits for a fixed simulation time and fixed error tolerance. This is to be contrasted with the linear circuit Ansatz used in generic algorithms whose depth scales linearly in the number of qubits. For simulation problems on 100 qubits, we show that AQCtensor thus achieves at least an order of magnitude reduction in the depth of the resulting optimized circuit, as compared with the best generic MPS to quantum circuit algorithms. We demonstrate our approach on simulation problems on Heisenberg-like Hamiltonians on up to 100 qubits and find optimized quantum circuits that have signficantly reduced depth as compared with standard Trotterized circuits.
Abstract Tensor networks (TNs) are a family of computational methods built on graph-structured factorizations of large tensors, which have long represented state-of-the-art methods for the approximate simulation of complex quantum systems on classical computers. The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed TN algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The "quantum-inspired" nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum-classical training frameworks which benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, something which remains an unresolved question. In this work, we compare a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimension into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition scheme with constrained optimization of circuit unitaries, and all possess efficient classical runtimes. Our experimental results reveal one particular protocol, involving sequential growth and optimization of the quantum circuit, to outperform all other methods, with even greater benefits seen in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, in turn further unlocking the rich and complementary benefits of classical and quantum computation.
Digital quantum computing promises to offer computational capabilities beyond the reach of classical systems, yet its capabilities are often challenged by scarce quantum resources. A critical bottleneck in this context is how to load classical or quantum data into quantum circuits efficiently. Approximate quantum loaders (AQLs) provide a viable solution to this problem by balancing fidelity and circuit complexity. However, most existing AQL methods are either heuristic or provide guarantees only for specific input types, and a general theoretical framework is still lacking. To address this gap, here we reformulate most AQL methods into a unified framework and establish information-theoretic bounds on their approximation error. Our analysis reveals that the achievable infidelity between the prepared state and target state scales linearly with the total entanglement entropy across subsystems when the loading circuit is applied to the target state. In light of this, we develop AQER, a scalable AQL method that constructs the loading circuit by systematically reducing entanglement in target states. We conduct systematic experiments to evaluate the effectiveness of AQER, using synthetic datasets, classical image and language datasets, and a quantum many-body state datasets with up to 50 qubits. The results show that AQER consistently outperforms existing methods in both accuracy and gate efficiency. Our work paves the way for scalable quantum data processing and real-world quantum computing applications.
Efficient encoding of classical data into quantum circuits is a critical challenge that directly impacts the scalability of quantum algorithms. In this work, we present an automated compilation framework for resource-aware quantum data loading tailored to a given input vector and target error tolerance. By explicitly exploiting the trade-off between exact and approximate state preparation, our approach systematically partitions the total error budget between precision and approximation errors, thereby minimizing quantum resource costs. The framework supports a comprehensive suite of state-of-the-art methods, including multiplexer-based loaders, quantum read-only memory (QROM) constructions, sparse encodings, matrix product states (MPS), Fourier series loaders (FSL), and Walsh transform-based diagonal operators. We demonstrate the effectiveness of our framework across several applications, where it consistently uncovers non-obvious, resource-efficient strategies enabled by controlled approximation. In particular, we analyze a computational fluid dynamics workflow where the automated selection of MPS state preparation and Walsh transform-based encoding, combined with a novel Walsh-based measurement technique, leads to resource reductions of over four orders of magnitude compared to previous approaches. We also introduce two independent advances developed through the framework: a more efficient circuit for d-diagonal matrices, and an optimized block encoding for kinetic energy operators. Our results underscore the indispensable role of automated, approximation-aware compilation in making large-scale quantum algorithms feasible on resource-constrained hardware.