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Scalable Generic State Preparation through Tensor Cross Interpolation and Binary Trees

Vittorio Pagni, Kevin Lively, Peter Ken Schuhmacher, Michael Felderer

Published May 18, 2026• 0 citations
OpenAlex

Abstract

Efficient preparation of generic quantum states remains a major computational bottleneck in many quantum algorithms and protocols. In this work, we employ a maximally expressive recursive circuit capable of representing arbitrary pure quantum states exactly. We then introduce two complementary methods, both based on an intermediate binary-tree representation, that systematically exploit structure in the target data across multiple scales to reduce circuit depth under realistic connectivity constraints. The first method uses a boolean representation of the circuit parameters and connects the optimization problem to classical logic simplification techniques. The second method, based on tensor networks, is inherently scalable to large system sizes and particularly effective for structured high-dimensional functions. In particular, it uses tensor cross interpolation to construct tensor-train representations of queryable functions beyond direct memory storage and maps them onto sequences of single-qubit and controlled rotations. This framework offers a transparent, deterministic, highly tunable, scalable and general purpose data encoding method, which outperforms other methods in literature, and offers many distinct paths forward in this critical bottleneck on the road to quantum advantage.

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arXiv:2410.03572•SciPost Physics•Oct 15, 2026•2 citations
Efficient preparation of generic quantum states remains a major computational bottleneck in many quantum algorithms and protocols. In this work, we employ a maximally expressive recursive circuit capable of representing arbitrary pure quantum states exactly. We then introduce two complementary methods, both based on an intermediate binary-tree representation, that systematically exploit structure in the target data across multiple scales to reduce circuit depth under realistic connectivity constraints. The first method uses a boolean representation of the circuit parameters and connects the optimization problem to classical logic simplification techniques. The second method, based on tensor networks, is inherently scalable to large system sizes and particularly effective for structured high-dimensional functions. In particular, it uses tensor cross interpolation to construct tensor-train representations of queryable functions beyond direct memory storage and maps them onto sequences of single-qubit and controlled rotations. This framework offers a transparent, deterministic, highly tunable, scalable and general purpose data encoding method, which outperforms other methods in literature, and offers many distinct paths forward in this critical bottleneck on the road to quantum advantage.
Quantics tensor trains are attracting intense interest for quantum-inspired computing and quantum state preparation. These methods, which approximate continuum functions by representing their amplitude encoding as a matrix product state (MPS), are exceedingly powerful for univariate functions but rapidly become challenging when handling multivariate functions, since the linear chain topology leads to a large distance between highly-entangled qubits. We overcome this limitation by introducing SCENT (Spectral Clustering for Entanglement miNimizing Trees). SCENT is a protocol that utilizes efficiently-computable pairwise entanglement metrics to determine a suitable tree tensor network (TTN) structure, which can then be efficiently approximated using tensor cross-interpolation (TCI); we find that it substantially outperforms MPS methods and improves upon previous TTN methods. We then apply these optimized TTNs to state preparation, introducing an approximate circuit compilation method based on environment-tensor methods without significantly conceding overall accuracy. Importantly, the inherent gauge freedom of TTNs can be directly exploited in this method, resulting in higher fidelity at a given circuit depth. We demonstrate this quantum state-preparation pipeline on archetypal state-preparation problems in quantum chemistry and financial portfolio optimization. In our flagship demonstration, we encode a 20-variable probability distribution with long-ranged, non-nearest neighbor inter-variable correlations in a 200-qubit state-preparation circuit with infidelity 7.44×10−97.44\times 10^{-9}7.44×10−9 using only 43284 CNOTs; depth and fidelity can be traded, allowing the same distribution to be prepared to infidelity 10−310^{-3}10−3 with as few as 5584 CNOTs.
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 0.04×0.04\times0.04× the depth of amplitude encoding, while naturally scaling to high-resolution images (256×256256 \times 256256×256) and demonstrating practical feasibility on real quantum hardware.
Quantum state preparation is a fundamental component of quantum algorithms, particularly in quantum machine learning and data processing, where classical data must be encoded efficiently into quantum states. Existing amplitude encoding techniques often rely on recursive bipartitions or tensor decompositions, which either lead to deep circuits or lack practical guidance for circuit construction. In this work, we introduce Tucker Iterative Quantum State Preparation (Q-Tucker), a novel method that adaptively constructs shallow, deterministic quantum circuits by exploiting the global entanglement structure of target states. Building upon the Tucker decomposition, our method factors the target quantum state into a core tensor and mode-specific operators, enabling direct decompositions across multiple subsystems.
Abstract Quantum algorithms and simulations often require the preparation of complex states through sequences of 2-qubit gates. For a generic quantum state, the number of required gates grows exponentially with the number of qubits, becoming unfeasible on near-term quantum devices. Here, we aim at creating an approximate encoding of the target state using a limited number of gates. As a first step, we consider a quantum state that is efficiently represented classically, such as a one-dimensional matrix product state. Using tensor network techniques, we develop and implement an efficient optimization algorithm that approaches the optimal implementation, requiring a polynomial number of iterations. We, next, consider the implementation of the proposed optimization algorithm directly on a quantum computer and overcome inherent barren plateaus by employing a local cost function. Our work offers a universal method to prepare target states using local gates and represents a significant improvement over known strategies.
Efficient state preparation is a challenging and important problem in quantum computing. In this work, we present a recursive state preparation algorithm that combines logarithmic-depth Dicke state circuits with Hamming weight encoders for efficiently preparing ``leaf-separable" quantum states. The algorithm is built on binary partition trees, generalized weight distribution blocks (gWDBs), and leaf-level encoders. We evaluate the performance of the algorithm by numerically simulating it on randomly generated target states with between 4 and 15 qubits. Compared to general state preparation approaches which require O(2n)O(2^n)O(2n) CX gates, our algorithm achieves a circuit depth of O(klog⁡nk+2k)O(k\log\frac{n}{k} + 2^k)O(klogkn​+2k) and uses O(n(k+2k))O(n(k+2^k))O(n(k+2k)) two-qubit gates, where k<nk < nk<n denotes the subtree size. We also compare implementations of the algorithm with and without the use of ancilla qubits, providing a detailed analysis of the trade-offs in circuit depth and two-qubit gate counts. These results contribute to scalable state preparation for quantum algorithms that require structured inputs such as Dicke or near-Dicke states.
Tensor networks are a compressed format for multi-dimensional data. One dimensional tensor networks—often referred to as tensor trains (TT) or matrix product states (MPS)—are increasingly being used as a numerical Ansatz for continuum functions by ``quantizing’’ the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks (TTNs) for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient Ansatz than the commonly used tensor train. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.