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Scalable global entangling gates on arbitrary ion qubits

Yao Lu, Shuaining Zhang, Kuan Zhang, Wentao Chen, Yangchao Shen, Jialiang Zhang, Jing-Ning Zhang, Kihwan Kim

arXiv:1901.03508v1quant-phphysics.atom-phphysics.optics

TL;DR

Replacing multiple two-qubit gates with one global entangling gate could make quantum-circuit construction more efficient, but scaling beyond a single motional mode is difficult. This work drives multiple motional modes with modulated fields and independently controlled ion beams; it demonstrates GHZ-state preparation up to four ions, with fidelity above 93%.

  • Problem

    Global gates based on a single center-of-mass mode are difficult to scale as ion number increases, while multi-mode schemes had been demonstrated for two-qubit gates but not global multi-qubit gates.

    Source: p. 1

  • Method

    The authors derive the multi-mode decoupling and pairwise-coupling constraints and implement modulated external fields with independently controlled parameters on each ion.

    Source: pp. 1–2

  • Results

    The global gates prepare GHZ states up to four ions, with fidelities of (93.4 ± 2.0)% for four qubits, (94.2 ± 1.8)% for three, and (95.1 ± 1.6)% for two.

    Source: p. 4

  • Takeaways & Limitations

    The experiment demonstrates global multi-qubit entangling gates on a trapped-ion platform, with theoretical pulse schemes also optimized for five and six qubits.

    Source: pp. 1, 4

    Gate-state fidelities are mainly limited by fluctuations of tightly focused individual beams and Raman-beam path jitter, with motional-frequency drift and beam crosstalk also contributing.

    Source: p. 4

Abstract

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A quantum algorithm can be decomposed into a sequence consisting of single qubit and 2-qubit entangling gates. To optimize the decomposition and achieve more efficient construction of the quantum circuit, we can replace multiple 2-qubit gates with a single global entangling gate. Here, we propose and implement a scalable scheme to realize the global entangling gates on multiple $\yb$ ion qubits by coupling to multiple motional modes through external fields. Such global gates require simultaneously decoupling of multiple motional modes and balancing of the coupling strengths for all the qubit-pairs at the gate time. To satisfy the complicated requirements, we develop a trapped-ion system with fully-independent control capability on each ion, and experimentally realize the global entangling gates. As examples, we utilize them to prepare the Greenberger-Horne-Zeilinger (GHZ) states in a single entangling operation, and successfully show the genuine multi-partite entanglements up to four qubits with the state fidelities over $93.4\%$.

Author information

Yao Lu, Shuaining Zhang, and Kuan Zhang are identified as equal contributors. The accompanying figure caption describes a four-ion global gate and its GHZ-state results.

Author information

  • Yao Lu, Shuaining Zhang, and Kuan Zhang are identified as equal contributors.The author note states that these three authors contributed equally.Source: Author information · p. 4
  • The four-ion pulse scheme uses matching modulation patterns for outer qubits {1, 4} and inner qubits {2, 3}.The coupling strengths for all qubit pairs converge to π/4 at the gate's end.Source: Figure 4 · p. 5
  • The prepared 4-, 3-, and 2-qubit GHZ states have fidelities of (93.4 ± 2.0)%, (94.2 ± 1.8)%, and (95.1 ± 1.6)%, respectively.The caption says parity-oscillation frequency proportional to the number of addressed qubits reveals the GHZ state.Source: Figure 4 · p. 5

Expressions of Constraints under Discrete Phase Modulation

The constraints are expressed for segmented, discretely phase-modulated pulses, including residual mode displacements and pairwise coupling strengths. The pulse amplitude is shaped to ramp at the gate boundaries.

Expressions of Constraints under Discrete Phase Modulation

  • The gate divides duration τ into K equal segments, with discrete phase modulation on each ion and amplitude shaped to turn on and off slowly at the edges.The segment duration is τ_s = τ/K, and the amplitude envelope has a sin² profile.Source: Expressions of Constraints under… · p. 8
  • The residual displacement α_j,m(τ) and pairwise coupling θ_j,j′(τ) define the constraints that must hold at gate time.The passages identify α as the residual displacement of qubit j and mode m, and θ as the coupling strength between qubits j and j′.Source: Expressions of Constraints under… · pp. 8–9
  • Substituting the segmented pulse scheme rewrites the residual displacements using phase-dependent terms and scaled displacements.The passage introduces d_j,m as the scaled residual displacement of α_j,m(τ).Source: Expressions of Constraints under… · p. 8
  • The pairwise coupling-strength constraints are also rewritten and arranged in matrix form.The matrix-form passages define the coupling matrices and the column vectors used in the rearrangement.Source: Expressions of Constraints under… · p. 9

Pulse Scheme Optimization

Because the constraints are nonlinear, the pulse scheme is constructed through optimization. Symmetries in the Lamb-Dicke parameters reduce the four-ion coupling constraints, while a preset phase symmetry further structures the search.

Pulse Scheme Optimization

  • The optimization minimizes residual-displacement terms subject to nonlinear constraints, which become more numerous as qubit count grows.The text identifies the coupling-strength constraints as growing quadratically with the number of qubits.Source: Pulse Scheme Optimization · pp. 9–10
  • Using η_j,m = ±η_N−j+1,m, the four-ion coupling-strength constraints reduce by equating symmetric ion-pair couplings.The text notes θ_1,2(τ) = θ_3,4(τ) and θ_1,3(τ) = θ_2,4(τ), reducing the constraint set.Source: Pulse Scheme Optimization · pp. 9–10
  • The phase patterns are preset with the time-reversal symmetry φ_j(t) = −φ_j(τ − t) before optimization.For discrete segments, the text gives φ_j,k = −φ_j,K−k+1.Source: Pulse Scheme Optimization · p. 10

Experimental Setup

The experiment uses a blade trap and a picosecond-pulse Raman laser, with fluorescence detection imaged onto an EMCCD. The side-view description specifies the field direction and distinct beam paths.

Experimental Setup

  • The ion chain is held in a blade trap, and Raman beams come from a 377 nm picosecond-pulse laser operating at approximately 76 MHz.Fluorescence is collected through the top re-entrant viewport and imaged to the EMCCD.Source: Experimental Setup · p. 10
  • The magnetic field is around 6 Gauss, while the cover-all beam enters through the side viewport and individual beams through the bottom re-entrant viewport.The individual beams achieve a focused waist of approximately 1 µm; the Raman-beam effective wave-vector is nearly along x.Source: Figure 5 · p. 10

Experimental Parameters

The experimental pulse schemes specify optimized phase and amplitude values for the 3- and 4-qubit gates. The realized Rabi amplitudes exceeded theoretical calculations because the Lamb-Dicke parameters were overestimated.

Experimental Parameters

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