Simulating quantum circuits using tree tensor networks

Philipp Seitz1, Ismael Medina2, Esther Cruz3, Qunsheng Huang1, and Christian B. Mendl1

1Technical University of Munich, Department of Informatics, Boltzmannstraße 3, 85748 Garching, Germany
2University of Göttingen, Campus Institute Data Science
3Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

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We develop and analyze a method for simulating quantum circuits on classical computers by representing quantum states as rooted tree tensor networks. Our algorithm first determines a suitable, fixed tree structure adapted to the expected entanglement generated by the quantum circuit. The gates are sequentially applied to the tree by absorbing single-qubit gates into leaf nodes, and splitting two-qubit gates via singular value decomposition and threading the resulting virtual bond through the tree. We theoretically analyze the applicability of the method as well as its computational cost and memory requirements, and identify advantageous scenarios in terms of required bond dimensions as compared to a matrix product state representation. The study is complemented by numerical experiments for different quantum circuit layouts up to 37 qubits.

Classical simulations of quantum systems lie at the heart of the quantum supremacy discussion, with tensor network methods being one of the most competitive classical simulation approaches.

In this work, we perform quantum circuit simulations by representing quantum states as tree tensor networks. Our algorithm clusters qubits based on the expected entanglement between them, reducing the computational cost. Two-qubit quantum gates are applied by threading their introduced entanglement, represented as a "virtual bond", through the tree structure.

We conduct theoretical analysis by comparing the computational cost and memory consumption with traditional statevector based simulations under different scenarios. In favorable scenarios, our simulator outperforms the baseline implementation significantly; we see a simulation speedup of up to two magnitudes and a memory reduction by a factor of up to 32x.

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[2] Yaling Ke, "Tree tensor network state approach for solving hierarchical equations of motion", The Journal of Chemical Physics 158 21, 211102 (2023).

[3] Kouichi Okunishi, Hiroshi Ueda, and Tomotoshi Nishino, "Entanglement bipartitioning and tree tensor networks", Progress of Theoretical and Experimental Physics 2023 2, 023A02 (2023).

[4] Toshiya Hikihara, Hiroshi Ueda, Kouichi Okunishi, Kenji Harada, and Tomotoshi Nishino, "Automatic structural optimization of tree tensor networks", Physical Review Research 5 1, 013031 (2023).

[5] Marcel Niedermeier, Jose L. Lado, and Christian Flindt, "Tensor-Network Simulations of Noisy Quantum Computers", arXiv:2304.01751, (2023).

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