Joseph Cunningham

We present a framework for quantum computation, similar to Adiabatic Quantum Computation (AQC), that is based on the quantum Zeno effect. By performing randomised dephasing operations at intervals determined by a Poisson process, we are able to track the eigenspace associated to a particular eigenvalue. We derive a simple differential equation for the fidelity leading to general theorems bounding the time complexity of a whole class of algorithms. We also use eigenstate filtering to optimise the scaling of the complexity in the error tolerance ε. In many cases the bounds given by our general theorems are optimal, giving a time complexity of O(1/Δ) with Δ the minimum of the gap. This allows us to prove optimal results using very general features of problems, minimising the amount of problem-specific insight necessary. As two applications of our framework we obtain optimal scaling for the Grover problem (i.e. O(N^1/2) where N is the database size) and the Quantum Linear System Problem (i.e. O(κłog(1/ε)) where κ is the condition number and ε the error tolerance) by direct applications of our theorems.