Geometric quantum control and the random Schrödinger equation

Abstract

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 SU(n) 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.

Figures (4)

• Figure 1: Plots of 10 pathwise solutions to the random Schrödinger equation for a single qubit with $H_{0,S}(t) = cos(t)\sigma_y$ and $H_{1,S}(t,\omega) = \frac{1}{2} M_\nu(t, \cdot)$, where $M_\nu (t, \cdot)$ is a Gaussian process with Matérn covariance. The parameter $\nu$ controls the smoothness of sample paths. We take to $\nu$ to be $0.2, 0.6$ and $1.0$ from left to right. A Gaussian process with Matérn covariance is in general $\lceil \nu \rceil -1$ times $L^2$-differentiable. Trajectories were sampled using sci-kit learn and plotted using QuTiP qutip.
• Figure 2: Schematic of a deformation of $SU(n)$ corresponding to penalising one direction. The picture on the left represents (a hyperplane section of) $SU(n)$ with the usual round metric, showing two "equatorial" points joined by a geodesic. The middle picture shows a deformed $SU(2)$, with the blue line representing the geodesic with respect to the warped metric, and the orange line shows the image of the geodesic with respect to the round metric. The right hand picture shows the limiting case, where one direction has been "infinitely penalised" effectively reducing the number of tangent directions (and hence the dimension) by one. Note that this diagram is merely a schematic: each $g_\Lambda$ is a homogeneous metric, meaning the curvature should be identical at each point. Unfortunately, it is not possible to satisfactorily depict this in a sketch. For a more detailed discussion, see PhysRevD.100.046020.
• Figure 3: Violin plots of the error, accounting for the noise (i.e. minimising \ref{['eq:noisecost']}) and not accounting for the noise (i.e. minimsing \ref{['eq:nonoisecost']}). Each distribution consists of 100 random samples for the control functions, with the dashed lines indicating the mean. We see an approximately 2.6-fold reduction in operator-norm error.
• Figure 4: A plot of the average unitary error accounting for the noise (orange circles) and not accounting for the noise (blue triangles), for 100 random control functions.

Theorems & Definitions (39)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof