On the Satisfiability of Quantum Circuits of Small Treewidth

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.