Topics#

Key Concepts#

• Qubit: A two-dimensional Hilbert space \(\mathbb{C}^2\). Every pure state is \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) with \(\vert\alpha\vert^2 + \vert\beta\vert^2 = 1\).

• Global phase: The overall phase \(\mathrm{e}^{\mathrm{i}\phi}\vert\psi\rangle\) is unobservable — it drops out of all measurement probabilities.

• Bloch sphere: A qubit state maps one-to-one to a point \((\theta,\varphi)\) on the unit sphere. The Bloch vector \(\boldsymbol{n} = \langle\hat{\boldsymbol{\sigma}}\rangle\) gives spin expectation values.

• Hermitian operators: Observables satisfy \(\hat{O} = \hat{O}^\dagger\), guaranteeing real eigenvalues and orthonormal eigenstates.

• Spectral decomposition: Every Hermitian operator resolves as \(\hat{O} = \sum_m m\vert m\rangle\langle m\vert\).

• Pauli operators: \(\hat{X}, \hat{Y}, \hat{Z}\) form a basis for traceless \(2\times 2\) Hermitian matrices with commutation relations \([\hat{\sigma}^i, \hat{\sigma}^j] = 2\mathrm{i}\epsilon^{ijk}\hat{\sigma}^k\) and anticommutation \(\{\hat{\sigma}^i, \hat{\sigma}^j\} = 2\delta^{ij}\hat{I}\).

Learning Objectives#

• Represent a qubit state as a vector in \(\mathbb{C}^2\) and as a Bloch sphere point; explain unobservable global phase.

• Compute eigenvalues, eigenstates, and spectral decompositions of Pauli matrices.

• Evaluate commutators and anticommutators of Pauli operators.

• Calculate expectation values \(\langle\hat{\sigma}^i\rangle\) for general qubit states and relate them to the Bloch vector.

Project: Quantum State Tomography via Classical Shadow Measurements#

Objective: Implement classical shadow tomography to reconstruct arbitrary quantum states from weak, non-invasive measurements.

Background: Shadow tomography uses randomized measurements followed by efficient classical post-processing. The approach scales logarithmically with state dimension and is robust to noise — key advantages for near-term quantum devices.

Suggested Approach:

• Literature survey on classical shadow formalism (Huang et al., Nature Phys. 2020)

• Implementation: simulate quantum device, apply random Clifford rotations, measure computational basis, reconstruct fidelity \(F = \langle\psi_{\text{true}}\vert\rho_{\text{reconstructed}}\vert\psi_{\text{true}}\rangle\)

• Benchmark against standard tomography for various qubit states

• Discuss noise robustness and connection to Bloch sphere geometry

Expected Deliverable: Research report (6–8 pages) including literature review, protocol description, numerical results on 1–2 qubit systems, comparison with alternative methods, and near-term applicability discussion.