The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in . Among many implications, we show efficient simulations for quantum formulas, defined and studied by Yao (for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188-5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph.
For the quantum circuit model, it was shown by Markov & Shi REF that a quantum circuit with n gates whose underlying graph has treewidth tw can be simulated in O(poly(n)exp(tw)) time.
Simulating quantum computation by contracting tensor networks
SIAM Journal on Computing
Mathematics Physics Computer Science