Logic Gate Analysis for Quantum Phase Differences in Systems with Multiple Degenerate Points

The field of quantum information has been extensively researched, and an important concept here is the phase difference between the two basis states (|0&gt; and |1&gt;) in a qubit, which is an important concept used to distinguish quantum logic from classical logic. Hence, manipulating this phase difference is a prerequisite for quantum arithmetic. Consequently, various methods for quantum (phase) operations have been proposed, and one such method uses the non-Abelian Berry phase [1]. Because the non-Abelian Berry phase has a characteristic structure, particularly at <i>quasi</i>-degenerate points (QDPs) in the parameter space, combining QDPs with quantum operations becomes important. However, a QDP exists alone in the parameter space, complicating its combination with quantum operations. Nevertheless, heavy-mass holes (HHs) in SiGe two-dimensional quantum wells, where Rashba and Dresselhaus spin–orbit interactions (SOIs) coexist, feature multiple spin QDPs near point Γ in the Brillouin zone (BZ) [2]. In the vicinity of these QDPs, a characteristic structure is observed in the non-Abelian Berry phase when the wave vectors are chosen as parameters. Because these QDPs lie close to each other, they can realize novel quantum operations.<br/><br/>Here, we focus on the HH with multiple QDPs in the BZ and discuss the features of quantum operations based on the non-Abelian Berry phase near these QDPs. We choose the |0&gt; and |1&gt; states as the SOI-stabilized (+) and -destabilized (-) HH (HH+/-) states, respectively, including the dynamical phases, and discuss the change in hybridization and phase difference between the |0/1&gt; states. A Bloch sphere is used to visualize the hybridization ratio and phase difference. Further, we construct a time-evolution operator that reproduces the motion on the Bloch sphere and decompose it into quantum logic gates to interpret changes in the state vector as quantum operations. We assume that the wave vector moves under the energy conservation law and recreate this motion as “cyclotron motion” under a perpendicular weak magnetic field. The time evolution of the state and wave vectors is determined by the simultaneous rate equation and semi-classical equation of motion.<br/><br/>In addition to point Γ, an HH has the following QDPs: M₁ and M₂ (16 meV) along the [1-10] and [-110] directions and D₁ and D₂ (24 meV) along the [-1-10] and [110] directions, respectively. To discuss such QDP-induced modulation, we consider the lower-energy region away from M and D (10 meV), and just above M (16 meV) and D (24 meV). We set the initial states of the wave vector along the [100] direction and the state vector to the |0&gt; state (HH+).<br/><br/>At 10 meV, the state barely moves from the |0&gt; state (+z direction on the Bloch sphere), and no operation is apparently performed. However, the rate-equation-based time-evolution operator is equivalent to the R_z gate, and its rotation angle is 2π. This rotation angle corresponds to the (Abelian) Berry phase difference of HH+/- at 10 meV (+/-π). This Berry phase can be attributed to the QDP at point Γ. This implies that even in the region away from the QDP, the state vector is operated by R_z corresponding to the Berry phase of the QDP surrounded by the parameter (wave vector) trajectory. By contrast, at 16 meV (point M), the state flip-flops during the passage of M, and the states |0&gt; and |1&gt; change completely to |1&gt; and |0&gt;, respectively. Through logic gate decomposition, we obtain the R_y gate corresponding to M, whose rotation angle is π. At 24 meV, the R_y gate corresponding to D is also obtained. However, the rotation angle is less than π, and the state transition is incomplete. Therefore, the state is operated by the R_y gate when it passes just above the QDP, and the rotation angle depends on the QDP.<br/><br/>[1] J. Pachos <i>et al.</i>, Phys. Rev. A <b>61</b>, 010305(R) (1999).<br/>[2] T. Tojo and K. Takeda, Phys. Status Solidi B <b>260</b>, 220031 (2023).