Abstract
Geometric quantum gates are often expected to be more resilient than dynamical gates against certain types of error, which would make them ideal for robust quantum computing. However, this is still up for debate due to seemingly conflicting results in the literature. Here we use dynamical invariant theory in conjunction with filter functions in order to analytically characterize the noise sensitivity of an arbitrary quantum gate. For any control Hamiltonian that produces a geometric gate, we find that, under certain common conditions, one can construct another control Hamiltonian that produces an equivalent dynamical gate with identical noise sensitivity (as characterized by the filter function). Our result holds for a Hilbert space of arbitrary dimensions, but we illustrate our result by examining experimentally relevant single-qubit scenarios and providing explicit examples of equivalent geometric and dynamical gates.
- Received 19 May 2021
- Revised 23 June 2022
- Accepted 30 June 2022
DOI:https://doi.org/10.1103/PRXQuantum.3.030310
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
If one slowly changes the control knobs of a quantum system in such a way that at the end they all return to their starting positions, the system itself also returns to its original state except for one subtle property, its phase, which acquires a shift. In 1984, Berry discovered that this phase shift has two contributions, called dynamical and geometric. Whereas the dynamical phase shift generally depends on the duration over which the changes are made, the geometric phase shift depends only on the geometry of the cyclic trajectory that the nonphase degrees of freedom of the system traverse. Because of this property, it is widely believed that unitary quantum evolution with only a geometric phase shift (i.e., a “geometric gate”) is insensitive to certain noise sources and is, therefore, ideal for quantum computing. However, the actual extent of any such insensitivity remains up for debate and is the subject of many studies. Here we show that for any geometric gate it is possible to find, under certain common conditions, an equivalent dynamical gate with identical noise sensitivity.
In other words, we show that noise sensitivity and phase-shift type are generally independent properties. We explicitly demonstrate our result in experimentally relevant single-qubit cases, and our analysis applies equally to the original case of slow changes and an extension to rapid changes. This contradicts any expectation that geometric gates should generally be superior to dynamical gates. We also discuss how the presence of control constraints can break this equivalence of geometric and dynamical phase robustness capability and give rise to a preferred robust phase type.
