One Polynomial Strategy for Computing Local Projections on Square-Lattice Cluster States

Quantum computing has attracted a lot of attention in recent years. It is one of the promising candidates for the next-generation computing paradigms. Basically, there are two technical lines to realize quantum computing. One is composing the unitary operators of a few qubits to achieve general unitary operators on an arbitrary number of qubits, known as the approach of quantum circuits. The other one focuses on preparing quantum cluster states and performing the computation by measuring the states with a particular basis, known as measurement-based quantum computing or one-way quantum computing. The two strategies have been proven to be equivalent to each other. This note aims to discuss the strategies for computing the local projections on square-lattice cluster states. Seemingly, one strategy for the computation could require both polynomial steps and memories. In particular, if the number of qubits in a square-lattice is denoted by $N$, the step number for computing an arbitrary local projection on the state could be proportional to a polynomial of $N$ under a bounded cost of memory. Consider that the square-lattice cluster states are one kind of universal computing resource, the results might be helpful for understanding the computational advantages of quantum algorithms, as well as the limits of the numerical analysis on other relevant quantum models. Although the results in the note are not peer reviewed, we would like to make them public because they are quite easy to check.