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arXiv:2606.24475

Exact log-depth preparation of highly entangled matrix product states

Keisuke Murota, Frédéric Sauvage, Marco Ballarin, Gabriel Matos, Enrico Rinaldi

quant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations
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Abstract

Preparing matrix product states (MPS) on a quantum device is a key subroutine in many quantum algorithms. The most competitive methods, based on the renormalisation group, prepare translationally invariant MPS of size $L$ and bond dimension $χ$, up to an error $\varepsilon$, in circuit depth $\tilde O(χ^{4}\log(L/\varepsilon))$ or $\tilde O(χ^{6}\log\log(L/\varepsilon))$. We improve multiple aspects of these methods. First, using block-encoded correction maps, whose post-selection succeeds with constant probability, we render the preparation exact without sacrificing the scaling in $L$. Second, through a generalisation of oblivious amplitude amplification to isometries, we reduce the bond-dimension dependence, improving the depth to $\tilde O(χ^{2}\log L + χ^{4})$ or $\tilde O(χ^{2}\log\log L + χ^{4})$, and even to $\tilde O(χ^{3}\log L)$ for incoherent preparations. Finally, we extend the framework to non-translationally invariant MPS and prove logarithmic-depth exact preparation for independent and identically distributed random tensor sequences. Confirmed by numerical studies, these results constitute, to the best of our knowledge, the most efficient exact MPS preparation protocols in the relevant parameter regimes.

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Preparing matrix product states (MPS) on a quantum device is a key subroutine in many quantum algorithms. The most competitive methods, based on the renormalisation group, prepare translationally invariant MPS of size LLL and bond dimension χχχ, up to an error ε\varepsilonε, in circuit depth O~(χ4log⁡(L/ε))\tilde O(χ^{4}\log(L/\varepsilon))O~(χ4log(L/ε)) or O~(χ6log⁡log⁡(L/ε))\tilde O(χ^{6}\log\log(L/\varepsilon))O~(χ6loglog(L/ε)). We improve multiple aspects of these methods. First, using block-encoded correction maps, whose post-selection succeeds with constant probability, we render the preparation exact without sacrificing the scaling in LLL. Second, through a generalisation of oblivious amplitude amplification to isometries, we reduce the bond-dimension dependence, improving the depth to O~(χ2log⁡L+χ4)\tilde O(χ^{2}\log L + χ^{4})O~(χ2logL+χ4) or O~(χ2log⁡log⁡L+χ4)\tilde O(χ^{2}\log\log L + χ^{4})O~(χ2loglogL+χ4), and even to O~(χ3log⁡L)\tilde O(χ^{3}\log L)O~(χ3logL) for incoherent preparations. Finally, we extend the framework to non-translationally invariant MPS and prove logarithmic-depth exact preparation for independent and identically distributed random tensor sequences. Confirmed by numerical studies, these results constitute, to the best of our knowledge, the most efficient exact MPS preparation protocols in the relevant parameter regimes.
We consider the preparation of matrix product states (MPS) on quantum devices via quantum circuits of local gates. We first prove that faithfully preparing translation-invariant normal MPS of N sites requires a circuit depth T=Ω(logN). We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error ε in depth T=O[log(N/ε)], which is optimal. We also show that measurement and feedback leads to an exponential speedup of the algorithm to T=O[loglog(N/ε)]. Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range non-normal ones, in the same depth. Finally, the algorithm naturally extends to inhomogeneous MPS.
Matrix product states provide efficient classical descriptions of quantum systems that may be useful as reference states for quantum algorithms such as quantum phase estimation and quantum-selected configuration interaction. Shallow circuit constructions for loading matrix product states onto quantum computers is necessary for this to be practical on near-term hardware. We present a decomposition of matrix product states to log-depth quantum circuits via a simple tree tensor network renormalisation procedure. Our method exposes an explicit parameter which can be used to trade a small amount of fidelity for large savings in circuit depth. We extend this decomposition to the case of matrix product operators allowing us to construct log-depth and ancilla-free circuits to calculate overlaps of the form ∣⟨φ∣U∣ψ⟩∣2\left |\langleφ|U|ψ\rangle\right |^2∣⟨φ∣U∣ψ⟩∣2. In particular, we demonstrate an interpretation of these circuits as \emph{verifier circuits} with application to circuit-level device calibration.
Efficient encoding of classical information plays a fundamental role in numerous practical quantum algorithms. However, the preparation of an arbitrary amplitude-encoded state has been proven to be time-consuming, and its deployment on current noisy devices can be challenging. In this work, we propose an improved Matrix Product State(MPS) method preparation protocol with an exponential reduction on the circuit depth, as well as topological adaptability. By refined utilization of the disentangling principle, we also reduce approximately 33% two-qubit gate count. To validate our method, we study various families of functions and distributions with provably bounded MPS rank. Numerical experiments show that our method significantly reduces circuit depth while achieving higher fidelity for states arising in financial and other applications.
Simulating quantum error correction (QEC) circuits including non-Clifford gates at scale is important to accelerate progress toward fault-tolerant quantum computing. Here we demonstrate that matrix product state (MPS) techniques can handle many QEC circuits exactly and without restriction on gate types. Crucially, we find that MPS efficiency depends sensitively on implementation choices, and we introduce a series of targeted optimizations that reduce bond dimensions and simulation time by several orders of magnitude compared to naive approaches. We illustrate this with examples including: (a) a rotated surface code quantum memory up to distance 11, (b) logical Bell-state preparation up to distance 9, (c) a 15-to-1 magic-state distillation circuit including hundreds of QEC rounds that we optimize to be simulated with only 11 logical qubits (187 physical qubits) and a maximal bond dimension of 64 in under 40 seconds, and (d) a narrow, deep random circuit that scales linearly with the number of T gates. These results demonstrate the importance of circuit-level optimizations and position MPS as a valuable complement to near-Clifford simulators for QEC circuits.
Learning the closest matrix product state (MPS) representation of a quantum state enables useful tools for quantum machine learning and analysis of complex quantum systems. In this work, we study the problem of learning MPS in the following setting: given many copies of an input MPS, the task is to recover a classical description of the state. The best known polynomial-time algorithm, introduced by [LCLP10, CPF+10], requires linear circuit depth and O~(n5)\widetilde O(n^5)O(n5) samples, and has seen no improvement in over a decade. These costs, neither known to be optimal, renders existing algorithms impractical for near-term quantum devices with limited resources. We introduce parallel disentangling algorithms for MPS learning. For exact MPS learning, our algorithm runs in polynomial time and uses circuit depth O(log⁡n)O(\log n)O(logn) and sample complexity O~(n3)\widetilde O(n^3)O(n3), improving both the depth and the dependence on the system size nnn. The key idea is to exploit the bounded-rank structure of reduced states on middle blocks of an MPS and organize the disentangling operations in a tree structure. We further extend the algorithm to closest MPS learning, improving the sample complexity dependence on nnn from n9n^9n9 to n7n^7n7 and complement the algorithms with an Ω(n)Ω(n)Ω(n) product-state lower bound. We also investigate MPS learning under hardware constraints, including restricted measurements and geometric connectivity. Under the Learning Parity with Noise (LPN) assumption, we show computational hardness for learning an MPS(2) family with non-adaptive single-qubit measurements. Finally, we show that our algorithm can be implemented with depth O(qn1/q)O(q n^{1/q})O(qn1/q) on a qqq-dimensional hypercubic lattice, giving an asymptotic reduction in depth. Together, our work provides a complete characterization of the quantum resources needed for efficient MPS learning.
Multipartite entanglement offers a powerful framework for understanding the complex collective phenomena in quantum many-body systems that are often beyond the description of conventional bipartite entanglement measures. Here, we propose a class of multipartite entanglement measures that incorporate the matrix product state (MPS) representation, enabling the characterization of the optimality of quantum circuits for state preparation. These measures are defined as the minimal distances from a target state to the manifolds of MPSs with specified virtual bond dimensions χ, and thus are dubbed as χ-specified matrix product entanglement (χ-MPE). We demonstrate superlinear, linear, and sublinear scaling behaviors of χ-MPE with respect to the negative logarithmic fidelity F in state preparation, which correspond to excessive, optimal, and insufficient circuit depth D for preparing χ-virtual-dimensional MPSs, respectively. Specifically, a linearly growing χ-MPE with F suggests H_{χ}≃H_{D}, where H_{χ} denotes the manifold of the χ-virtual-dimensional MPSs and H_{D} denotes that of the states accessible by the D-layer circuits. We provide an exact proof that H_{χ=2}≡H_{D=1}. Our results establish tensor networks as a powerful and general tool for developing parametrized measures of multipartite entanglement. The matrix product form adopted in χ-MPE can be readily extended to other tensor network Ansätze, whose scaling behaviors are expected to assess the optimality of quantum circuit in preparing the corresponding tensor network states.