Geometric quantum control and the random Schrödinger equation

Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very fundamental, but hard to analyze (evolution by general dynamical map) and others perhaps too simplistic to represent physical reality. In this paper, we present an intermediate approach, introducing the random Schrödinger equation, with a noise term given by a time-varying random Hermitian matrix as a means to model noisy quantum systems. We derive bounds on the error of the synthesized unitary in terms of bounds on the norm of the noise, and show that for certain noise processes these bounds are tight. We then show that in certain situations, minimizing the error is equivalent to finding a geodesic on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>SU</a:mi> <a:mo>(</a:mo> <a:mi>n</a:mi> <a:mo>)</a:mo> </a:mrow> </a:math> with respect to a Riemannian metric encoding the coupling between the control pulse and the noise process, thus connecting our work to the complexity geometry pioneered by Michael Nielsen.