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arXiv:2506.02105

Learning Circuits with Infinite Tensor Networks

Joe Gibbs, Łukasz Cincio

Published Jun 2, 2025• 0 citations
arXivHTMLPDF

Abstract

Hamiltonian simulation on quantum computers is strongly constrained by gate counts, motivating techniques to reduce circuit depths. While tensor networks are natural competitors to quantum computers, we instead leverage them to support circuit design, with datasets of tensor networks enabling a unitary synthesis inspired by quantum machine learning. For a target simulation in the thermodynamic limit, translation invariance is exploited to significantly reduce the optimization complexity, avoiding a scaling with system size. Our approach finds circuits to efficiently prepare ground states, and perform time evolution on both infinite and finite systems with substantially lower gate depths than conventional Trotterized methods. In addition to reducing CNOT depths, we motivate similar utility for fault-tolerant quantum algorithms, with a demonstrated $5.2\times$ reduction in $T$-count to realize $e^{-iHt}$. The key output of our approach is the optimized unit-cell of a translation invariant circuit. This provides an advantage for Hamiltonian simulation of finite, yet arbitrarily large, systems on real quantum computers.

DOI
10.48550/arxiv.2506.02105

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Hamiltonian simulation on quantum computers is strongly constrained by gate counts, motivating techniques to reduce circuit depths. While tensor networks are natural competitors to quantum computers, we instead leverage them to support circuit design, with datasets of tensor networks enabling a unitary synthesis inspired by quantum machine learning. For a target simulation in the thermodynamic limit, translation invariance is exploited to significantly reduce the optimization complexity, avoiding a scaling with system size. Our approach finds circuits to efficiently prepare ground states, and perform time evolution on both infinite and finite systems with substantially lower gate depths than conventional Trotterized methods. In addition to reducing CNOT depths, we motivate similar utility for fault-tolerant quantum algorithms, with a demonstrated 5.2×5.2\times5.2× reduction in TTT-count to realize e−iHte^{-iHt}e−iHt. The key output of our approach is the optimized unit-cell of a translation invariant circuit. This provides an advantage for Hamiltonian simulation of finite, yet arbitrarily large, systems on real quantum computers.
Abstract Hamiltonian simulation, i.e. simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation algorithm.
As the number of qubits in a sensor increases, the complexity of designing and controlling the quantum circuits grows exponentially. Manually optimizing these circuits becomes infeasible. Optimizing entanglement distribution in large-scale quantum circuits is critical for enhancing the sensitivity and efficiency of quantum sensors (Yi, in Quantum Sci Technol, 045001, 2022, Thomas et al, in 2021). This paper presents an engineering integration of reinforcement learning with tensor-network-based simulation (MPS) for scalable circuit optimization for optimizing quantum sensor circuits with up to 60 qubits. To enable efficient simulation and scalability, we adopt tensor network methods—specifically, the Matrix Product State (MPS) representation—instead of traditional state vector or density matrix approaches. Our reinforcement learning agent learns to restructure circuits to maximize Quantum Fisher Information (QFI) and entanglement entropy while reducing gate counts and circuit depth. Experimental results show consistent improvements, with QFI values approaching 1, entanglement entropy in the 0.8–1.0 range, and up to 90% reduction in depth and gate count. These results highlight the potential of combining quantum machine learning and tensor networks to optimize complex quantum circuits under realistic constraints.
Quantum computers promise an exciting path to the scalable simulation of quantum matter. However, in their nascent state, the accessible gate depth of quantum computation on available hardware remains highly constrained. In this thesis, I present a suite of hybrid quantum-classical algorithms that aid Hamiltonian simulation across both the Noisy Intermediate-Scale Quantum era and future fault-tolerant architectures. By treating classical computation as an active resource to refine quantum algorithms, I develop automated, variational methods to learn hardware-efficient circuits. This work aims to overcome traditional trainability barriers, such as barren plateaus, by embedding physical symmetries directly into the circuit design. Central to the thesis is the synergistic use of tensor networks to classically pre-compute weakly entangled stages of a quantum simulation. This enables the compression of deep circuits, drastically reducing both the accumulation of gate errors on near-term physical qubits and the prohibitive overhead of operations like magic state distillation on error-corrected hardware, reserving the quantum processor strictly for the most challenging, highly entangled dynamics. These methodologies are ultimately directed toward the ab-initio simulation of subatomic physics, providing a scalable approach for the first-principles study of strongly correlated systems, including nuclear structure and lattice gauge theories.
Abstract The simulation of time evolution of large quantum systems is a classically challenging and in general intractable task, making it a promising application for quantum computation. A Trotter–Suzuki approximation yields an implementation thereof, where a higher approximation accuracy can be traded for an increased gate count. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of translationally invariant quantum systems. Our strategy can improve upon the Trotter–Suzuki accuracy by several orders of magnitude. It translates into a reduction in gate count and hence gain in overall fidelity at the same algorithmic accuracy. This is important in noisy intermediate scale quantum-applications where the fidelity of the output state decays exponentially with the number of gates. The performance advantage of our classical assisted strategy can be extended to open boundaries with translational symmetry in the bulk. We can extrapolate our method to beyond classically simulatable system sizes, maintaining its total fidelity advantage over a Trotter–Suzuki approximation making it an interesting candidate for beyond classical time evolution.
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.
Quantum Circuit Pre-Synthesis: Learning Local Edits to Reduce TTT-count
Compiling quantum circuits into Clifford+TTT gates is a central task for fault-tolerant quantum computing using stabilizer codes. In the near term, TTT gates will dominate the cost of fault tolerant implementations, and any reduction in the number of such expensive gates could mean the difference between being able to run a circuit or not. While exact synthesis is exponentially hard in the number of qubits, local synthesis approaches are commonly used to compile large circuits by decomposing them into substructures. However, composing local methods leads to suboptimal compilations in key metrics such as TTT-count or circuit depth, and their performance strongly depends on circuit representation. In this work, we address this challenge by proposing \textsc{Q-PreSyn}, a strategy that, given a set of local edits preserving circuit equivalence, uses a RL agent to identify effective sequences of such actions and thereby obtain circuit representations that yield a reduced TTT-count upon synthesis. Experimental results of our proposed strategy, applied on top of well-known synthesis algorithms, show up to a 20%20\%20% reduction in TTT-count on circuits with up to 25 qubits, without introducing any additional approximation error prior to synthesis.