The Complexity Geometry of a Single Qubit

TL;DR

The paper investigates Nielsen's complexity geometry for a single qubit, distinguishing unitary and state complexity and contrasting with gate-counting approaches. By deriving explicit metrics on SU(2) (unitary) and on CP^1 (state space) using anisotropic penalty tensors I_{IJ}, it reveals how a tiny system already exhibits features central to multi-qubit complexity geometry, such as right-invariance without left-invariance, geodesics requiring time-dependent Hamiltonians, and regions of negative curvature (especially in easy-easy directions). The Berger-sphere and Bloch-sphere deformations illustrate how penalties reshape geometry, induce cut loci, and produce large complexity-to-inner-product distance ratios, while remaining continuous and, in principle, exact in this geometric framework. While enlightening, the single-qubit model cannot capture large-N phenomena like exponential scaling of hard directions or fractal complexity frontiers, but it serves as a concrete, tractable laboratory for foundational concepts with holographic relevance. The work thereby strengthens the case for complexity geometry as a robust, mathematically tractable notion of quantum complexity, with clear connections to rigid-body dynamics and potential implications for holography and quantum gravity.

Abstract

The computational complexity of a quantum state quantifies how hard it is to make. `Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we demonstrate many of the attractive features of complexity geometry using the example of a single qubit, which turns out to be rich enough to be illustrative but simple enough to be illuminating.

Figures (8)

• Figure 1: According to the inner-product metric, these two states are as far apart as any two states can be: they are orthogonal. The 'complexity distance' captures the sense in which they are nevertheless close.
• Figure 2: The 'gate' definition of complexity imagines implementing $U$ by building a quantum circuit out of primitive gates. In this example, the primitive gates each act only on one or two qubits at a time, but in combination they may approximate any $N$-qubit unitary with arbitrary accuracy. The computational complexity is then defined as the total number of gates in the smallest circuit that approximates the desired unitary.
• Figure 3: Top left: the state of the qubit can be represented as a unit vector $\vec{\psi}$ on the two-sphere. Top right: to transform from $|\vec{\psi} \rangle$ to $|\vec{\psi} + {\color{red} \vec{ d \psi} }\rangle$ we can rotate around any axis ${\color{blue} \vec{r}}$ that is orthogonal to the change in state, ${\color{blue} \vec{r}} \cdot {\color{red} \vec{ d \psi }} =0$. There is a one-parameter family of possible rotation axes that all satisfy ${\color{blue} \vec{r}} \cdot {\color{red} \vec{ d \psi }} =0$ and therefore any of which can be used to implement the transformation; these lie on a great circle. Bottom left: in order to make the rotation angle $d \theta$ as small as possible, we should rotate around an axis orthogonal to the initial state, $\vec{\psi} \cdot {\color{blue} \vec{r} } = 0$; this is the appropriate rotation axis when $\mathcal{I}_{zz} = 1$. Bottom right: in order to make the penalty factor as small as possible, we should rotate around an axis orthogonal to the penalized direction, ${\color{green} \vec{p}} \cdot {\color{blue} \vec{r} } = 0$; this is the appropriate rotation axis when $\mathcal{I}_{zz} \rightarrow \infty$, as in Sec. \ref{['subsec:infiniteIzz']}. For intermediate values, $1<\mathcal{I}_{zz}<\infty$, the optimal rotation axis, given by Eq. \ref{['eq:optimalalphaaxis']}, lies between these two extremes.
• Figure 4: Deformed Bloch spheres with various penalty factors $\mathcal{I}_{zz}$. Unlike the 'squashed' three-spheres of Sec. \ref{['sec:unitaryinEuler']}, these two-spheres are squashed in the same sense that a beachball gets squashed when you sit on it. For $\mathcal{I}_{zz}=1$ every direction is punished equally and we have the Bloch sphere with the standard inner-product metric. The Bloch spheroid is prolate for $\mathcal{I}_{zz}<1$ and oblate for $\mathcal{I}_{zz}>1$. For $\mathcal{I}_{zz}>3/2$ the Bloch spheroid is negatively curved at the poles and cannot be embedded in $\mathbb{R}^3$. For very large $\mathcal{I}_{zz}$ the Bloch spheroid becomes two back-to-back negatively curved spaces glued together by positive curvature at the equator.
• Figure 5: The curvature $\mathcal{R}$ of the deformed Bloch sphere for different values of $\mathcal{I}_{zz}$, given by Eq. \ref{['eq:curvatureofBloch']}. For $\mathcal{I}_{zz} < 1$ the curvature is everywhere positive and largest at the poles. For $\mathcal{I}_{zz}=1$ the curvature is uniformly positive and the Bloch sphere is spherical. For large $\mathcal{I}_{zz}$, the curvature is like two negatively curved disks stuck together by a thin very positively curved wall.