Noncyclic geometric quantum computation with shortcut to adiabaticity

Fast and robust quantum gates are the cornerstones of fault-tolerance quantum computation. In this paper, as opposed to the previous proposals which use cyclic geometric phases, we propose to realize quantum computation based on noncyclic geometric evolution. The superiority of the noncyclic scheme is that the operation time is proportional to the rotation angle of the quantum state. Therefore, the noncyclic scheme becomes fairly fast in the case of quantum gates with small rotation angle, which will be more insensitive to the decoherence and the leakage to the states outside the computational basis. Similar to the cyclic version of geometric quantum computation, dynamical phases here during the evolution are canceled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabaticity. The proposed scheme is robust against random noise due to the geometric characteristic of projective Hilbert space. Since the proposed scheme is fast and robust, it is a particularly suitable way to manipulate the physical systems with weak nonlinearity, such as superconducting systems.

Figure 1

Evolution paths of adiabatic eigenstates and the control field in the noncyclic evolution. (a), (b) ${\sigma }_y$ gate. (c), (d) $\pi$-phase gate. The effective magnetic fields $\mathbf{B}$ that control pseudospins evolve along the path $\mathrm{A}1\mathrm{B}1-\mathrm{B}1\mathrm{C}1-\mathrm{C}1\mathrm{A}1(\mathrm{A}2\mathrm{B}2-\mathrm{B}2\mathrm{C}2-\mathrm{C}2\mathrm{A}2)$. ${\mathbf{P}}_{-}$ denotes the polarization vector of eigenstates $|{\lambda }_{-}\rangle$ which evolve along $\mathrm{a}1\mathrm{b}1-\mathrm{b}1\mathrm{c}1(\mathrm{a}2\mathrm{b}2-\mathrm{b}2\mathrm{c}2)$. The dashed lines B1C1 and B2C2 indicate that the $\pi$-phases flipping of $\mathbf{B}$ are sudden quenches which will not induce evolutions of eigenstates. Here the eigenstates have not evolved cyclically and will not acquire geometric phase as the evolutions are along the longitude line.

Figure 2

Dynamics of the geometric gates. Population dynamics (blue solid line) and the relative phases (red dashed line) of geometric gates based on noncyclic evolution. The initial state is $|\psi \left(0\right)\rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. (a) ${\sigma }_y$ gate. (b) $\pi$-phase gate. The relative phase is in units of $\pi$.

Figure 3

Performance of the geometric gates. (a) Robustness of the proposed ${\sigma }_y$ gate against the variation of Rabi frequency. The variation is introduced by ${\mathrm{\Omega }}^{\prime }=\eta \mathrm{\Omega }$. The plot of the ${\sigma }_y$ gate based on Rabi oscillations and composite pulses is used as a comparison. Red dashed line shows the noncyclic scheme. Blue solid line shows the Rabi oscillation. Black solid line shows the composite pulses. The proposed scheme has been sped up by STA and the evolution time is five times as long as for Rabi oscillation. It can be seen that the robustness of the noncyclic scheme is much stronger than the for Rabi oscillation and is comparable with the composite pulses. (b) Comparison of the $\pi /8$ rotation gate (along the $y$ axis) realized by noncyclic or cyclic schemes under decay process. $\pi /8$ rotation gate in the cyclic scheme is achieved by SGQG with three stages. The red dashed line is the ideal case of the noncyclic scheme while the black solid line is the one that introduces a decay rate of $\mathrm{\Gamma }=2\pi \times 100$ kHz. The blue dot-dashed line is the result of SGQG with the same decay rate. As shown in the inset, the cyclic scheme will suffer more from the decay since its operation time is eight times longer than that of the noncyclic scheme. The evolution time has been normalized for the convenience of comparison. (c) Comparison of leakage between the proposed scheme (red dashed line) and SGQG (blue solid line). (d) Comparison of the ${\sigma }_y$ gate realized by noncyclic or cyclic scheme process against random noise. The random noise has a zero mean value over the evolution with an amplitude of $\gamma \mathrm{\Omega }$. The red solid line and the black dot-dashed line are the numerical result of the noncyclic scheme and the cyclic scheme with same frequency of noise, respectively. The blue dashed line is the result of the cyclic scheme with two times lower frequency. The robustness of the noncyclic scheme is as good as the cyclic scheme with noise of the same frequency. The data have been averaged 50 times.

Figure 4

Waveforms of Rabi frequencies and detuning to realize geometric quantum gates. Red solid lines show detuning. Blue dashed lines show the Rabi frequency. Black dot-dashed lines show the auxiliary Rabi frequency. (a) ${\sigma }_y$ gate with noncyclic scheme. (b) ${\sigma }_y$ gate with cyclic scheme of SGQG. (c) $\pi /8$ rotation gate with noncyclic scheme. (d) $\pi /8$ rotation gate with cyclic scheme of SGQG. All Rabi frequencies in the plots are in units of $\mathrm{\Omega }$, and $t$ is in units of $T$. The relative phases of Rabi frequencies (in arbitrary units) are also shown in the plots. It can be seen that the noncyclic scheme has the virtue of operating faster in a small rotation angle.

Figure 5

Fidelity of ${\sigma }_y$ gate versus flipping rate of the control phase. The fidelity approaches 1 when $M$ is larger than 100.