Topological sum rule for geometric phases of quantum gates

Abstract

We establish a topological sum rule, $ν_U = \frac{1}{2π}\sum_nγ_n = 2mν_H$, connecting the geometric phases accumulated by a two-qubit system over a complete basis of initial states to the winding number $ν_H$ classifying its Hamiltonian. Implementations of the same gate from different topological classes must distribute these phases differently, making their distinction measurable through the Wootters concurrence. As a corollary, nontrivial topology is a necessary condition for entanglement: only Hamiltonians with access to $ν_H \neq 0$ can generate it.

Figures (8)

• Figure 1: Topological correspondence $\pi_0(H) = \pi_1(U)$. (a) Flattened eigenvalue spectra of two-qubit Hamiltonians: each column shows how the four eigenvalues ($\pm 1$) distribute, with the imbalance $\nu_H = (n_+ - n_-)/2$ labeling the three disconnected classes for Hamiltonians of the type \ref{['eq:He']}. (b) Phase evolution of $\det U(t) = e^{-2i\nu_H t}$ as a spiral, projected to a circle when taken modulo $2\pi$. Each $\nu_H$ produces $\nu_U = 2m\nu_H$ windings (shown for $m=1$), making the correspondence between Hamiltonian classes and loop topology visually manifest.
• Figure 2: A quantum circuit implementing SWAP using three CNOT operators, where for the first and last operators qubit A controls qubit B, and for the second qubit B controls qubit A (reversed).
• Figure 3: Geometric phase $\gamma/\pi$ as a function of concurrence $C_{2q}$ for two implementations of the SWAP$^2$ gate applied to symmetric initial states \ref{['eq:psi_s']}. The Heisenberg-based SWAP$_1$ (solid blue) yields a constant phase $\gamma=\pi$, independent of entanglement. The 3-CNOT-based SWAP$_2$ (dashed red) decreases from $2\pi$ at $C_{2q}=0$ to $\pi$ at maximal entanglement. The shaded region highlights the measurable difference between two implementations of the same gate.
• Figure 4: Two cyclic paths on the Schmidt sphere, both starting and ending at the same initial state $|\psi_0\rangle$, corresponding to two distinct implementations of the SWAP gate. The Heisenberg-based SWAP$_1$ (dashed blue) traces a smooth great-circle arc. The 3-CNOT SWAP$_2$ (solid purple) follows three distinct segments, with circles marking the corners where the Hamiltonian switches; these segments correspond to the three CNOT operations in \ref{['eq:SWAP2']}. The arrows on the SWAP$_1$ curve show how the path continues from one side of the sphere to the other. Although both paths implement the same gate, they enclose different solid angles and thus acquire different geometric phases; this is a direct manifestation of their different topological classes.
• Figure 5: A quantum circuit implementing CNOT (qubit A controls qubit B) using Hadamard operators and CNOT$_\leftrightarrow$ (qubit B controls qubit A).