Eigenpath traversal by Poisson-distributed phase randomisation

• URL: http://arxiv.org/abs/2406.03972v1
• Date: Thu, 6 Jun 2024 11:33:29 GMT
• Title: Eigenpath traversal by Poisson-distributed phase randomisation
• Authors: Joseph Cunningham, Jérémie Roland,
• Abstract summary: We present a framework for quantum computation, similar to Adiabatic Quantum Computation (AQC) 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 class of algorithms.
• Score: 0.08192907805418585
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\epsilon$. In many cases the bounds given by our general theorems are optimal, giving a time complexity of $O(1/\Delta_m)$ with $\Delta_m$ the minimum of the gap. This allows us to prove optimal results using very general features of problems, minimising the problem-specific insight necessary. As two applications of our framework, we obtain optimal scaling for the Grover problem (i.e.\ $O(\sqrt{N})$ where $N$ is the database size) and the Quantum Linear System Problem (i.e.\ $O(\kappa\log(1/\epsilon))$ where $\kappa$ is the condition number and $\epsilon$ the error tolerance) by direct applications of our theorems.

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