The Geometry of Clifford Algorithms: Bernstein-Vazirani as Classical Computation in a Rotated Basis

TL;DR

This paper offers a geometric perspective on the Bernstein-Vazirani algorithm, showing it as a classical linear computation in a rotated basis, which clarifies its simplicity and pedagogical value in quantum information theory.

Contribution

It introduces a formal geometric framework distinguishing between classical, rotated, and entangled quantum circuits, enhancing understanding of quantum algorithms and the Gottesman-Knill theorem.

Findings

Reveals BV as a classical linear computation in a rotated basis

Provides a pedagogical taxonomy of quantum circuits

Extends understanding of entanglement as topological twists

Abstract

The Bernstein-Vazirani (BV) algorithm is frequently taught as a canonical example of quantum parallelism, yet the standard interference-based explanation often obscures its underlying simplicity. We present a geometric reframing in which the Hadamard gate "wrapping" acts as a global basis rotation rather than a generator of computational complexity. This perspective reveals that the algorithm is effectively a classical linear computation over GF(2) performed in the conjugate Fourier basis, with the apparent parallelism arising from coordinate transformation. Building on Mermin's earlier pedagogical shortcut, which presented a 'classical' circuit equivalent but stopped short of explicitly labeling it as such, we elevate this to a formal geometric framework. In the extension, we distinguish between globally rotated circuits -- which we reveal as classical linear computations -- and topologically twisted circuits that generate quantum entanglement. We introduce a pedagogical taxonomy distinguishing (1) pure computational-basis circuits, (2) globally rotated circuits (exemplified by Bernstein-Vazirani), and (3) topologically twisted circuits involving non-aligned subsystem bases. This framework allows viewing the Gottesman-Knill theorem from a new angle, extends students' understanding of phase kickback and the 'Ricochet Property'. Furthermore, it provides a more intuitive starting point for explaining Bell-pair extensions through concrete circuit derivations and Qiskit simulations suitable for undergraduate quantum information courses. The outlook explores how this geometric view paves the way for understanding entanglement as topological twists.