Shorter Quantum Circuits Unlock Faster Data Processing for Future Computers

Scientists have long sought efficient methods for preparing quantum states, a crucial step in many quantum algorithms and data processing applications. Carsten Blank and Israel F. Araujo, both from data cybernetics ssc GmbH, alongside colleagues, present a new approach called Tucker Iterative State Preparation (Q-Tucker) that addresses limitations in existing amplitude encoding techniques. Q-Tucker constructs shallow, deterministic circuits by leveraging the global entanglement structure of target states, offering a practical guide for circuit construction unlike methods reliant on recursive bipartitions or tensor decompositions.

This research is significant because it provides a novel pathway towards more readily implementable quantum circuits, potentially accelerating progress in various quantum computing applications. Keywords include quantum state preparation, entanglement, circuit decomposition, tensor decomposition, and Tucker decomposition. NISQ era limitations and quantum data encoding challenges Scientists have observed rapid advancements in both the theoretical and experimental domains of quantum computing. Current quantum hardware exists in the Noisy Intermediate-Scale Quantum (NISQ) era, characterised by devices with a limited number of qubits susceptible to noise and decoherence. These limitations present substantial challenges in the engineering, control, and deployment of quantum systems. NISQ devices lack full fault tolerance and require sophisticated quantum error mitigation or correction strategies. Complications also arise in the design and implementation of quantum circuits, particularly concerning the reliable execution of quantum gates. A critical bottleneck in practical quantum computation is the task of encoding classical data into quantum states, a process referred to as quantum state preparation, or more generally, data encoding and loading. Quantum state preparation involves encoding classical data into quantum states, serving as inputs for quantum algorithms in applications such as quantum machine learning and data processing. Multiple approaches have been developed to address the quantum state preparation problem. This technique aims to provide a deterministic approach to state preparation by generating quantum circuits whose depth is adaptively determined by the entanglement structure of the target quantum state. Entanglement has long been recognised as a fundamental resource in quantum computing, enabling capabilities unattainable in classical computation. Nevertheless, quantum features such as entanglement and superposition can, to a certain extent, be simulated on classical hardware. The preparation of an n-qubit quantum state, represented by a complex vector of N = 2n amplitudes, for execution on current NISQ-era quantum processing units (QPUs) often depends on classical pre-processing. Delegating these preparatory computations to classical resources becomes increasingly challenging as the dimensionality of the quantum state grows, a scenario commonly encountered in quantum machine learning and quantum data processing. A well-established strategy for mitigating this complexity involves decomposing the target state into approximately low-entangled subsystems. Even approximate decompositions can offer substantial advantages, such as reduced quantum circuit depth and, in many cases, improved fidelity when compared to exact state initialisation. Consequently, efficiently characterising and exploiting the entanglement structure of quantum states is a critical objective in the design of scalable quantum algorithms and state preparation protocols. Several previous studies have explored low-rank quantum state preparation by leveraging hierarchical decompositions of multipartite quantum systems. Araujo et al. introduced an iterative scheme based on successive applications of the Schmidt decomposition to identify and disentangle bipartitions within a quantum system. In their method, at each iteration, a suitable bipartition is selected, and the entanglement is reduced until no further disentanglement is possible, yielding a compact, low-rank representation of the original quantum state. Regarding quantum state encoding, a variety of approaches have been proposed to map the N complex-valued amplitudes of an arbitrary quantum state onto a physical quantum circuit. One of the earliest deterministic techniques was presented by Grover et al., who proposed a method to construct quantum superpositions corresponding to efficiently integrable probability distributions. While conceptually straightforward, this method requires O(N) quantum operations, rendering it impractical for large-scale quantum systems. An alternative family of methods exploits tensor network structures, particularly the matrix product state (MPS) representation, to facilitate efficient state encoding. Schön et al. proposed a sequential scheme to construct MPS representations via one- and two-qubit quantum gates, enabling scalable circuit generation for a broad class of quantum states. Owing to their strong approximation capabilities, MPS methods have become widely adopted in quantum many-body physics. Building upon this foundation, subsequent works have integrated parameterised tensor-network models to reduce quantum circuit depth and to support hybrid quantum-classical training schemes. In some instances, iterative application of so-called disentangler operators to MPS representations has been proposed to further improve state compression and circuit efficiency. A conceptually distinct approach based on the Tucker decomposition has been proposed by Protasov et al. In contrast to methods relying on the Schmidt decomposition or singular value decomposition (SVD), the Tucker decomposition generalises these techniques by enabling one-versus-all decompositions across multiple modes. While the authors advocate for the use of the Tucker decomposition in quantum state preparation, their approach does not specify a selection criterion for determining the appropriate decomposition among the many non-unique possibilities. Furthermore, the work provides no concrete procedure for mapping the decomposed representation into a quantum circuit, leaving a critical gap between theoretical formulation and practical implementation. The primary objective of this work is to address the limitations of existing state preparation techniques by introducing a method based on tensor decomposition that is naturally suited for multipartite quantum systems. Unlike prior approaches that rely on recursive bipartitions, often resulting in complex hierarchical structures, the method circumvents these constraints by directly exploiting the global structure of the quantum state. The remainder of the paper is organised as follows. Section 2 introduces the Tucker Iterative Process, detailing its decomposition strategy, fidelity control mechanisms, and the circuit synthesis procedure. Section 3 presents an analysis of the method’s convergence. Section 5 reports the experimental results. Finally, Section 6 concludes the paper with a summary of the findings and future research directions. This section introduces a hardware-efficient method for encoding classical data into a quantum device using an iterative tensor decomposition strategy. Using a multipartite-specific decomposition based on the Tucker tensor representation, the method addresses the limitations of previous methods and eliminates the need for complex hierarchical bipartitions. The quantum circuit is constructed from the Tucker representation, with a performance metric guiding the iterative refinement process (see Fig0.0.1). Any vector v ∈ CN can be represented as a m-tensor T ∈ Cd1 ⊗. ⊗Cdm, for an arbitrary m 1, with the constraint Σi di = N. Given such a tensor T, a representation or decomposition can be determined. One such representation is the Tucker decomposition: T = G ×1W ×2W ×3 . ×m W (m), where ×i denotes the mode-i product, G ∈ Cp1 ⊗. ⊗Cpm is the core tensor (with pi ≤di), and the factor matrices W (i) ∈Cdi×pi. This decomposition is naturally suited for quantum computing since the tensor decomposition of T directly corresponds to a quantum state decomposition: |ψ⟩= (W ⊗. ⊗W (m))|G⟩, where the operators W (i) are denoted using the same symbol as the factor matrices, even if their dimensions do not match (pi = di), in which case columns are padded with basis vectors. The core tensor G is interpreted as a quantum state |G⟩via vectorisation. A key feature is that the factor matrices can be interpreted as unitary operators acting on separate subspaces (hence, the Tucker format is also known as the subspace format). From a quantum computing perspective, these unitary operations can be executed in parallel, if hardware permits, making the depth of any resulting quantum circuit determined by the longest individual factor circuit W (i). The geometric entanglement of the core tensor is never greater than that of the original state and can, in many cases, be significantly lower. This property enables a recursive or iterative approach that terminates. If recursion halts after r steps, the resulting decomposition is: |ψ⟩= W1 . Wr|0⟩. Each iteration j of the process yields a factorization Wj = W(j,1) ⊗. ⊗ W(j,m). If the resulting core tensor, or quantum state |Gr⟩, can be expressed as a product state (i.e., composed solely of operations that can be absorbed into the adjacent factors), or if the fidelity between the core state and the true ground state exceeds a predefined threshold, then the core state is adopted as the ground state. In this case, it requires no additional quantum operations. Each factor W(j,i) is implemented as a unitary or isometry operator W (i) ∈Cdi×pi. If a full unitary operator is not required (pi It is well known that the tensorisation of a vector v ∈ C into a tensor T ∈ Cd1 ⊗. ⊗Cdm is not unique. Prior work has shown that different sequences of logical swap operations can lead to different entanglement behaviours. This result extends to tensorisation itself, resulting in a search space defined by the number of modes m, the dimension tuple d = (di1, . , dim), and a permutation π ∈Sym([m]) that encodes the swap pattern. Each tensorisation configuration is thus characterised by a tuple (m, d, π). Since the optimal configuration (m, d, π) is not unique, it is possible to seek those that balance approximation quality with implementation complexity. In general, there exists a configuration (m, d, π) such that the entanglement in the core tensor is smaller than in the original tensor T. This allows the method to be applied recursively until a product state is reached, which can be trivially initialised. Alternatively, methods such as that of Rudolph & Grover, or their concrete implementations, can be employed. The search can be simplified by taking into account the coupling map limitations of the target device. On currently available quantum hardware, efficient two-qubit operations are typically restricted to neighbouring qubits. Under this constraint, m = n/2 can be fixed, leading to di = Since the dimension of these subsystems is fixed, the cost of each SVD is constant. This eliminates the scalability bottleneck often associated with tensor decompositions such as Tucker, where SVD costs scale cubically with the input dimension. Even with such simplifications, the search process still depends on logical qubit swaps, which are costly. This work introduces a technique that adaptively builds circuits by exploiting the global entanglement structure of target states, offering an alternative to existing amplitude encoding methods reliant on recursive bipartitions or tensor decompositions. The research demonstrates the ability to factor target states into a core tensor and mode-specific operators, enabling direct decompositions across multiple subsystems. The method utilizes a multipartite-specific decomposition based on the Tucker tensor representation, circumventing the need for complex hierarchical bipartitions typically found in prior approaches. Any vector v ∈CN can be represented as a m-tensor T ∈Cd1 ⊗. ⊗Cdm, where the constraint Q i di = N is maintained throughout the process. This decomposition directly corresponds to a quantum state decomposition, expressed as |ψ⟩= (W ⊗. ⊗W (m))|G⟩, where W(i) represents unitary operators acting on separate subspaces. Geometric entanglement of the core tensor remains no greater than that of the original state, and can often be significantly lower, enabling a recursive or iterative approach. If recursion halts after r steps, the resulting decomposition is |ψ⟩= W1 . Wr|0⟩, with the fidelity F(|corej⟩, |0⟩) evaluated at each iteration to determine if the core can be approximated by the ground state. When the fidelity exceeds a threshold of F ≥1 −ε, the core is removed, requiring no additional quantum operations. This method leverages the Tucker decomposition to factor complex states into core tensors and mode-specific operators, allowing for direct decomposition across multiple quantum subsystems. By exploiting the global entanglement structure of these states, Q-Tucker adaptively builds deterministic circuits that require fewer computational steps than existing amplitude encoding methods. Empirical tests demonstrate that Q-Tucker consistently delivers efficient decompositions, even when full convergence is not achieved, and provides high-quality approximations within a defined classical computational budget. The algorithm’s reliance on singular value decompositions performed on two-qubit subsystems, which have a fixed computational cost, avoids the scalability issues common in other tensor decomposition techniques. Furthermore, the method incorporates a convergence detection mechanism that falls back on a Schmidt decomposition for partial disentanglement, ensuring a result is always obtained, even for complex states. The authors acknowledge that this fallback method remains relatively slow, indicating a need for faster alternatives. Future improvements in unitary synthesis techniques could further enhance the convergence speed of the iterative Tucker decomposition, while the number of iterations can be tuned to suit the constraints of specific quantum devices and applications. 👉 More information 🗞 Tucker iterative quantum state preparation 🧠 ArXiv: https://arxiv.org/abs/2602.09909 Tags: