This exercise develops the Riemannian geometry of quantum state space from first principles, connecting the Fubini-Study metric, Berry phase, and quantum geometric tensor into a unified framework that underpins the trainability analysis of Part 7 and the metrology of quantum scrambling.
Background
Physics does not see the overall phase of a quantum state, so two vectors that differ only by a phase describe the same physical situation. Removing that redundancy from the space of qubit states leaves the complex projective line \(\mathbb{CP}^1\), which is the familiar Bloch sphere. To do geometry on this space we need a way to measure distance between nearby states, and the right one is the Fubini-Study metric, built so that it ignores the phase that physics discards.
A single object packages this metric together with the geometry of how states respond to changing a parameter. It is the quantum geometric tensor, a Hermitian object whose real part is the Fubini-Study metric and whose imaginary part is the Berry curvature. The metric tells us how far apart two states are, while the Berry curvature records the geometric phase that accumulates when a state is carried around a loop in parameter space.
The diagonal of this tensor has a direct operational meaning, since it gives the quantum Fisher information. That quantity sets the ultimate precision for estimating a parameter through the Cramer-Rao inequality, and the same Fisher information later governs how easily variational circuits can be trained. For a qubit swept across the Bloch sphere the real part reproduces the metric of a sphere of radius \(1/2\), while the imaginary part integrates to the flux of a single monopole. The Fisher information read from the diagonal drops to zero when the driving axis lines up with the initial Bloch vector.
The notebook evaluates the quantum geometric tensor for a qubit over the Bloch sphere and separates its metric and curvature parts.
1. Physical Motivation
A quantum state \(|\psi\rangle\) lives in the Hilbert space \(\mathcal{H} \cong \mathbb{C}^D\), but physical observables are blind to the overall phase: \(|\psi\rangle\) and \(e^{i\alpha}|\psi\rangle\) represent the same physics. The true arena of quantum mechanics is therefore the projective Hilbert space\(\mathbb{CP}^{D-1}\), obtained by quotienting out the \(U(1)\) phase.
This quotient endows \(\mathbb{CP}^{D-1}\) with a natural Riemannian metric, the Fubini-Study metric, that provides a gauge-invariant notion of distance between quantum states. This metric is not merely an abstract mathematical structure: it appears directly as the Quantum Fisher Information (QFI) in parameter estimation, as the Fubini-Study speed in quantum dynamics, and as the quantum geometric tensor controlling Berry phases and adiabatic transport.
Understanding this geometry is essential for: - Quantum metrology: The QFI bounds the precision of parameter estimation via the Cramér-Rao bound. - Variational algorithms: The Fubini-Study metric determines the natural gradient on the variational manifold. - Scrambling diagnostics: The QFI concentration bounds the trainability of deep circuits (Part 7).
2. Mathematical Toolbox
Tool 1, Quantum Geometric Tensor (QGT): For a family of states \(|\psi(\boldsymbol{\theta})\rangle\) parameterized by \(\boldsymbol{\theta} = (\theta_1, \ldots, \theta_p)\), the QGT is: \[
Q_{\mu\nu} = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle.
\] Its real part is the Fubini-Study metric\(g_{\mu\nu}^{\mathrm{FS}} = \mathrm{Re}\, Q_{\mu\nu}\); its imaginary part is proportional to the Berry curvature\(\Omega_{\mu\nu} = -2\, \mathrm{Im}\, Q_{\mu\nu}\).
Tool 2, Fubini-Study distance: For two pure states, \[
d_{\mathrm{FS}}(|\psi\rangle, |\phi\rangle) = \arccos |\langle \psi | \phi \rangle|.
\]
Tool 3, QFI and Cramér-Rao bound:\(F_Q(\theta) = 4\, g_{\theta\theta}^{\mathrm{FS}}\). For unitary encoding \(|\psi(\theta)\rangle = e^{-i\theta G}|\psi_0\rangle\): \(F_Q = 4\,\mathrm{Var}(G)\).
Tool 5, Sectional curvature: For \(X, Y \in \mathfrak{su}(D)\) with \(\langle X, Y \rangle = -\mathrm{Tr}(XY)\): \(K(X,Y) = \frac{1}{4}\|[X,Y]\|^2\).
3. Exercise
(a) Fubini-Study metric for a qubit. Consider a qubit state \(|\psi(\theta, \phi)\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle\). Compute the full QGT and show: \[ds^2_{\mathrm{FS}} = \frac{1}{4}\left(d\theta^2 + \sin^2\theta\, d\phi^2\right).\]
(b) Berry curvature as monopole. Extract \(\Omega_{\theta\phi}\) and show it corresponds to a magnetic monopole. Compute the total Berry flux.
(c) QFI for a rotation. For \(|\psi(\theta)\rangle = e^{-i\theta \sigma_z/2}|+\rangle\), verify \(F_Q = 4\,\mathrm{Var}(\sigma_z/2) = 1\).
(d) QFI for general SU(2). Show \(F_Q = 1 - (\hat{n} \cdot \langle \boldsymbol{\sigma} \rangle_0)^2\). When does \(F_Q\) vanish?
(e) Geodesic distances. Compute \(d_{\mathrm{FS}}(|0\rangle, |+\rangle)\) and \(d_{\mathrm{FS}}(|0\rangle, |1\rangle)\).
Total flux: \(\Phi = \int_0^{2\pi}d\phi \int_0^\pi d\theta\,(-\sin\theta/2) = -2\pi\). This is half the solid angle \(4\pi\) of the sphere, the flux of a monopole of charge \(-1/2\) at the degeneracy \(|\boldsymbol{B}| = 0\); the sign depends on the orientation chosen for \((\theta,\phi)\).
Parts (c)-(d): QFI
For \(G = \sigma_z/2\) on \(|+\rangle\): \(\langle G\rangle = 0\), \(\langle G^2\rangle = 1/4\), so \(F_Q = 4(1/4 - 0) = 1\).
General: \(G = \hat{n}\cdot\boldsymbol{\sigma}/2\), \(G^2 = I/4\), \(\langle G\rangle = \hat{n}\cdot\langle\boldsymbol{\sigma}\rangle_0/2\). Therefore \(F_Q = 1 - (\hat{n}\cdot\langle\boldsymbol{\sigma}\rangle_0)^2\). Vanishes when \(|\psi_0\rangle\) is eigenstate of \(G\) (\(\hat{n}\) aligned with Bloch vector).
The qubit quantum geometric tensor has real part \(\mathrm{diag}(1/4,\sin^2\theta/4)\), the Fubini-Study metric of a sphere of radius \(1/2\), and imaginary part the Berry curvature \(-\sin\theta/2\), whose total flux \(-2\pi\) is half the solid angle of the sphere. The quantum Fisher information \(F_Q=1-(\hat n\cdot\langle\boldsymbol\sigma\rangle_0)^2\) vanishes exactly when the generator’s axis aligns with the initial Bloch vector.
