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Analysis of the action of conventional trapped-ion entangling gates in qudit space

Pavel Kamenskikh, Nikita Semenin, Ilia Zalivako, Vasiliy Smirnov, Ilya Semerikov, Ksenia Khabarova, Nikolay Kolachevsky

Abstract

Qudits, or multi-level quantum information carriers, present a promising path for scaling quantum computers. However, their use introduces increased complexity in quantum logic, necessitating careful control of relative phases between different qudit levels. In trapped-ion systems, entangling operations accumulate phases on specific levels that are no longer global, unlike in qubit architectures. Furthermore, the structure of multi-level gates becomes increasingly intricate with higher-dimensional Hilbert spaces. This work explores the theory of these additional entangling and non-entangling phases, accumulated in Mølmer--Sørensen and Light-shift gates. We propose methods to actively compensate for these phases, enhance gate robustness against parameter fluctuations, and simplify native gates for more efficient circuit decomposition. Our results pave the way toward the practical and scalable implementation of qudit-based quantum processors.

Analysis of the action of conventional trapped-ion entangling gates in qudit space

Abstract

Qudits, or multi-level quantum information carriers, present a promising path for scaling quantum computers. However, their use introduces increased complexity in quantum logic, necessitating careful control of relative phases between different qudit levels. In trapped-ion systems, entangling operations accumulate phases on specific levels that are no longer global, unlike in qubit architectures. Furthermore, the structure of multi-level gates becomes increasingly intricate with higher-dimensional Hilbert spaces. This work explores the theory of these additional entangling and non-entangling phases, accumulated in Mølmer--Sørensen and Light-shift gates. We propose methods to actively compensate for these phases, enhance gate robustness against parameter fluctuations, and simplify native gates for more efficient circuit decomposition. Our results pave the way toward the practical and scalable implementation of qudit-based quantum processors.
Paper Structure (16 sections, 77 equations, 9 figures, 1 table)

This paper contains 16 sections, 77 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. General description of entangling gates acting on trapped-ion qudits
  3. Mø lmer--Sø rensen gate
  4. Light-shift gate
  5. Robustness to fluctuations of experimental parameters
  6. Reduction of the qudit LS gate with spin-echo sequences
  7. Type (a) sequence, zero-order case
  8. Type (b) sequence, zero-order case
  9. Type (c) sequence, general case for even $d$
  10. Non-entangling phase compensation with spin echo
  11. Discussion and conclusion
  12. Magnus expansion for MS and LS gates. Decoupling condition.
  13. Multi-tone pulse shaping of the MS gate
  14. The existence of the solution for compensation of the single-qudit phases
  15. Native qudit gate representation of the cyclic shift $\hat{X}_m^d$
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Schematic representation of the Mø lmer--Sø rensen gate, acting on levels $\ket{0}$ and $\ket{1}$. Red and blue lines represent the spectral components of the bichromatic laser beam with frequencies $\omega_r$ and $\omega_b$. The figure also shows the motional sidebands and indicates the phonon number $n_l$ in the $l^\mathrm{th}$ motional mode.
  • Figure 2: (a) An exemplary configuration of the LS gate, where two counter-propagating laser beams with frequencies $\omega_\mathrm{r}$ and $\omega_\mathrm{b}$ form a walking wave. (b) In this example state-depended Stark shifts on qudit levels arise from their off-resonant coupling to a single auxiliary level $\ket{e}$ by these laser fields. The frequency difference between two beams is chosen close to the secular frequency ($\omega_\mathrm{r}-\omega_\mathrm{b}=\mu\approx\omega_l$) to excite motion. Several motional levels with different phonon numbers $n_l$ are also depicted. The qudit states are numbered here by their proximity to the auxiliary state.
  • Figure 3: Sensitivity of the phases $\chi^{\mathrm{MS}}_{jk}$ to drifts of motional modes $\omega_l$. Fluctuations are calculated for multi-tone MS gate.
  • Figure 4: A general spin-echo sequence of length $L$. Each rectangle represents a qudit basis state, and the red arrows correspond to the echo pulses
  • Figure 5: Spin-echo sequences for the LS gate. (a) The spin-echo sequence from Ref. hrmo2023Native. (b) The spin-echo sequence that is valid for the zero-order LS gate. (c) The spin-echo sequence that can be used for a general LS gate with an even qudit dimension $d$.
  • ...and 4 more figures