A note on the non-adiabatic geometric phase and quantum computation Alexandre Blais and Andr´e-Marie S. Tremblay D´epartement de Physique and Centre de Recherche sur les Propri´et´es ´ Electroniques de Mat´eriaux Avanc´es, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1, Canada (May 2, 2001) We consider the non-adiabatic, or Aharonov-Anandan, geometric phase as a tool for intrinsically fault-tolerant quantum computation. While this phase seems to answer many of the issues related to the adiabatic version of the geometric gate, we show that it is not straightforward to implement and that it is sensitive to small errors. In the quest for the realization of a low noise quantum computing device, geometric phases [1] are now getting considerable attention because of their intrinsic tolerance to area preserving noise [2–5]. So far, only the adiabatic geometric phase (i.e. Berry phase) was discussed in this context. While it is indeed useful because of it’s intrinsic tol- erance to noise, the application of adiabatic geometric phases to quantum computing has several drawbacks. First, while adiabaticity does not necessarily mean slow evolutions, it would nevertheless be advantageous, while retaining the tolerance to noise, to remove the adiabatic- ity constraint in order to take full advantage of the short coherence times of the envisioned quantum computers. Moreover, another drawback of the adiabatic phase gate is that during the adiabatic evolution, both geo- metric and dynamic phases are acquired. The later is not tolerant to area preserving noise and must be re- moved. This could be done using spin-echo like refocus- ing schemes which require going over the adiabatic evo- lution twice [2,3,5]. However, if the second pass does not retrace exactly the first one, the dynamic phase will not completely cancel, thereby introducing errors. In other words, leaving aside experimental errors, the ‘random’ noise in the classical fields controlling the quantum evo- lution should be the same on each pass for the tolerance to noise to be preserved. A third difficulty is that adiabatic geometric phases are only possible if non-trivial loops are available in the space of parameters controlling the qubit’s evolution. In other words, the single qubit Hamiltonian must be of the form H = − 1 2 B x (t) σ x − 1 2 B y (t) σ y − 1 2 B z (t) σ z , (1) where external control over all three (effective) fields B i (t) is possible. Such control is not possible in most of the current quantum computer architectures propos- als. Control over only two fields, B x and B z , is usually the norm. In this case, all loops in parameter space are limited to the x − z plane and the (relative) Berry phase is limited to integer multiples of 2π. It therefore can- not be observed and is of no use for computation. Note that control over fields in all three directions is possible in NMR where the Berry phase as been observed experi- mentally [2]. More recently, Falci et al. [5], extended the original charge qubit proposal [6] from a symmetric to an asymmetric dc-SQUID design to allow a non-zero B y and therefore non-trivial closed paths in parameter space. As we shall see, all of the above issues, namely slow evolutions, need of refocusing and control over many fields, seems to be resolved when one considers the non- adiabatic generalization of Berry’s phase, the Aharonov- Anandan (AA) phase [7]. This was noticed very recently by Xiangbin and Keiji [8]. In this note, we point out that, while being an attractive idea, the application of the AA phase to quantum computation is not straightforward. Let us start by recalling the main ideas related to the AA phase and then comment on it’s use as a geometric phase gates for quantum computation. Consider a system whose Hamiltonian H is controlled by a set of external parameters R(t ). Upon varying adi- abatically the control parameters R(t ) around a closed loop C such that R(τ )= R(0), if the system is initially in an eigenstate of H it will remain, by adiabaticity, an eigenstate of the instantaneous Hamiltonian. The final state will therefore differ only by a phase factor from the initial state. Berry showed that this phase factor has a dynamic and geometric contribution, the later depend- ing solely on the loop C in parameter space [1]. For a Hamiltonian (which is non-degenerate on C), starting the evolution with a superposition of eigenstates, each eigen- state will acquire it’s corresponding Berry phase and it can be observed by interference. Adiabaticity was invoked by Berry to ensure that the evolution of each eigenstate is cyclic, i.e. that the final and initial state differ only by a phase factor: |ψ(τ )〉 = U (τ )|ψ(0)〉 = e iφ |ψ(0)〉, (2) for some real phase φ. It is then possible to generalize Berry’s phase to non-adiabatic evolutions by choosing, for a given H (t), the particular initial states for which eq. (2) holds. For non-adiabatic evolutions, these so- called cyclic initial states [9] are generally not eigenstates of the system’s Hamiltonian. Aharonov and Anandan [7] showed that the total phase acquired by such a cyclic initial state in the interval [0,τ ] on which it is cyclic is given by the sum of a dynamic δ = −  τ 0 dt 〈ψ(t)|H (t)|ψ(t)〉, (3) 1 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by CERN Document Server