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On the Satisfiability of Quantum Circuits of Small Treewidth

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9139)

Abstract

It has been known since long time that many NP-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit C with n uninitialized inputs, \( poly (n)\) gates, and treewidth t, one can compute in time \((\frac{n}{\delta })^{\exp (O(t))}\) a classical assignment \(y\in \{0,1\}^n\) that maximizes the acceptance probability of C up to a \(\delta \) additive factor. In particular our algorithm runs in polynomial time if t is constant and \(1/poly(n) < \delta < 1\). For unrestricted values of t this problem is known to be hard for the complexity class QCMA, a quantum generalization of NP. In contrast, we show that the same problem is already NP-hard if \(t=O(\log n)\) even when \(\delta \) is constant. Finally, we show that for \(t=O(\log n)\) and constant \(\delta \), it is QMA-hard to find a quantum witness \(|\varphi \rangle \) that maximizes the acceptance probability of a quantum circuit of treewidth t up to a \(\delta \) additive factor.

Keywords

  • Treewidth
  • Satisfiability of quantum circuits
  • Tensor networks

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Fig. 1.
Fig. 2.

Notes

  1. 1.

    In the case of classical circuits, it is assumed that each variable labels a unique input of unbounded fan-out.

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Acknowledgements

The author gratefully acknowledges financial support from the European Research Council, ERC grant agreement 339691, within the context of the project Feasibility, Logic and Randomness (FEALORA).

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Correspondence to Mateus de Oliveira Oliveira .

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de Oliveira Oliveira, M. (2015). On the Satisfiability of Quantum Circuits of Small Treewidth . In: Beklemishev, L., Musatov, D. (eds) Computer Science -- Theory and Applications. CSR 2015. Lecture Notes in Computer Science(), vol 9139. Springer, Cham. https://doi.org/10.1007/978-3-319-20297-6_11

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  • DOI: https://doi.org/10.1007/978-3-319-20297-6_11

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