Comment on "Nongeometric Conditional Phase Shift via Adiabatic Evolution of Dark Eigenstates: A New Approach to Quantum Computation"

TL;DR

This paper clarifies that Zheng's proposed quantum phase gate is indeed geometric, involving noncyclic Hamiltonians, and emphasizes that such gates require similar precision as cyclic geometric gates, challenging previous claims of their non-geometric nature.

Contribution

The authors demonstrate that noncyclic Hamiltonians can induce geometric operations and clarify that Zheng's phase gate is fundamentally geometric, countering prior assertions.

Findings

Zheng's gate is geometric despite noncyclic Hamiltonian.

Nontrivial loops are present in the phase shift process.

Precision requirements for noncyclic geometric gates are comparable to cyclic ones.

Abstract

In [Phys. Rev. Lett. 95, 080502 (2005)], Zheng proposed a scheme for implementing a conditional phase shift via adiabatic passages. The author claims that the gate is "neither of dynamical nor geometric origin" on the grounds that the Hamiltonian does not follow a cyclic change. He further argues that "in comparison with the adiabatic geometric gates, the nontrivial cyclic loop is unnecessary, and thus the errors in obtaining the required solid angle are avoided, which makes this new kind of phase gates superior to the geometric gates." In this Comment, we point out that geometric operations, including adiabatic holonomies, can be induced by noncyclic Hamiltonians, and show that Zheng's gate is geometric. We also argue that the nontrivial loop responsible for the phase shift is there, and it requires the same precision as in any adiabatic geometric gate.