Efficient quantum circuit simulations using magic state injected Stabilizer Tensor Networks

Simulation protocols utilising both tensor network methods and stabilizer tableaus have garnered significant interest in recent years as they enable the simulation of both highly magical states with low entanglement, as well as highly entangled states with low magic. In this work, we introduce the use of magic state injection to one of these protocols, the Stabilizer Tensor Networks (STN) protocol [Masot-Llima et al, arXiv:2403.08724], finding that for random Clifford + T gate simulations the computational cost of circuits prepared with magic state injection using this protocol is such that one can simulate circuits of N qubits with O(N) T gates in polynomial time. Simulating systems with greater than N T gates, we show that there is a phase transition with respect to the T-gate density. Our work finds that in this intermediate-depth case, circuits prepared with state injection continue to outperform standard STN and conventional MPS simulations until a saturation depth is reached. We can extend these results beyond T gates to other single-qubit non-Clifford operations, yielding similar results. These findings may present a significant advantage in the simulation of many-qubit systems where system size grows much faster than the number of non-Clifford gates in the quantum circuit, for example in Quantum Error Correction simulations.