Inertial geometric quantum logic gates

D. Turyansky, O. Ovdat, R. Dann, Z. Aqua, R. Kosloff, B. Dayan, and A. Pick

Phys. Rev. Applied 21, 054033 – Published 17 May 2024

Abstract

We present rapid and robust protocols for stimulated rapid adiabatic passage and quantum logic gates. Our gates are based on geometric phases acquired by instantaneous eigenstates of a slowly accelerating “inertial” Hamiltonian. To begin, we establish the criteria for inertial evolution and subsequently engineer pulse shapes that fulfill these conditions. These tailored pulses are then used to optimize geometric logic gates. We analyze a realization of our protocols with $^{87} Rb$ atoms, resulting in gate fidelity that approaches the current state of the art, with marked improvements in robustness.

Received 14 December 2023
Revised 5 March 2024
Accepted 25 April 2024

DOI:https://doi.org/10.1103/PhysRevApplied.21.054033

Physics Subject Headings (PhySH)

Coherent control Geometric & topological phases Quantum control Quantum gates Quantum information with atoms & light Quantum protocols Quantum tomography Stimulated Raman adiabatic passage

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

D. Turyansky¹, O. Ovdat², R. Dann^3,4, Z. Aqua², R. Kosloff⁴, B. Dayan², and A. Pick ^1,*

¹Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel
²Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
³Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, Garching D-85748, Germany
⁴The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

^*Corresponding author: adi.pick@mail.huji.il

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Vol. 21, Iss. 5 — May 2024

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Images

Figure 1
(a) STIRAP infidelity versus effective pulse area ( $Ω_{max} \times t_{f}$ ) for four pulse shapes [Eq. (14a)]: Gaussian (green), sinSQ (blue), cubic (red curve), and numerically optimized (red triangles). Parameters: $Δ = 0 MHz, Ω_{max} = 50 MHz$ , and $γ_{31} / 2 π = γ_{32} / 2 π = 3 MHz$ , accounting for decay from $| 3 ⟩$ into $| 1 ⟩$ and $| 2 ⟩$ . Parameters of the optimization code: $λ_{1} = 0.1, λ_{2} = 0, λ_{3} = 1 \times 10^{- 7}$ (see Sec. 6). (b) Infidelity, $1 - F$ , versus $Δ$ , for $t_{f} = 0.25 μ s$ . (c) Effective pulse area, $Ω_{max} t_{f}$ , (relative to cubic pulse) for protocols generated with constrained QOCT versus weight of inertial constraint, $λ_{3}$ . (d) Pulse shapes generated with QOCT with a cubic guess pulse for different values of $λ_{3}$ , which achieve fidelity of $0.99$ .
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Figure 2
Inertial geometric gates (a1),(a2) Single-qubit phase gate, using a tripod-level atom (inset) with control pulses $Ω_{1} (t)$ and $Ω_{2} (t)$ that transfer population from $| 1 ⟩$ to $| 2 ⟩$ and back to $| 1 ⟩$ , while flipping the phase of $Ω_{2}$ in the middle of the protocol [17, 18]. (a1) Gaussian (green), sinSQ (blue), and quartic (red) pulses are shown. (a2) Gate infidelity versus $Ω_{max}$ for the Gaussian, sinSQ, and quartic pulses. (b1),(b2) Hadamard gate, using $Ω_{0}$ , $Ω_{1}$ , and $Ω_{2}$ and gate infidelity. (c1),(c2) CZ gate, using four-level atoms with Rydberg-induced phase shifts (inset). Pulse shapes and gate infidelity versus $Ω_{max}$ are shown. Parameters of single-qubit gates: $γ / 2 π = 6 MHz$ , $Δ = 0$ . Phase gate: $Ω_{1}^{max} = Ω_{2}^{max} = 2 π \times 50 MHz$ . Hadamard gate: $Ω_{1}^{max} = 2 π \times 50 MHz$ , $Ω_{2}^{max} = (1 - \sqrt{2}) Ω_{1}^{max}$ , $(Ω_{0}^{max})^{2} = (Ω_{1}^{max})^{2} + (Ω_{2}^{max})^{2}$ . Two-qubit CZ gate: $Ω_{max} / 2 π = Δ = 50 MHz$ , $V_{R} = 4 / 2 π MHz$ , $γ_{p} / 2 π = 6 MHz$ , $γ_{r} / 2 π = 1 kHz$ , $γ_{dep} / 2 π = 10 kHz$ .
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Figure 3
Infidelity of the CZ gate protocol [Fig. 2] performed with Gaussian, sinSQ, and quartic pulses, for 100 realizations of noisy model parameters. The detuning, maximum Rabi frequency, and deviation in atomic positions along and perpendicular to the tweezers’ direction are Gaussian random variables, with average values listed in Table 1 and standard deviations: $σ_{Δ} = 14 kHz$ , $σ_{Ω} = 2.5 MHz$ , $σ_{d z} = 0.2 μ m$ , and $σ_{d x} = σ_{d y} = 0.07 μ m$ , respectively [6, 24]. Realization are shown by thin lines and their average is shown by the thick dashed lines.
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Figure 4
Single-qubit inertial gates with Rb atoms. (a),(b) Two possible realizations of a tripod-level system using the D2 line of $^{87} Rb$ . Desired transitions (solid lines) are shown with the corresponding polarizations: circular ( $σ_{\pm}$ ) and $π$ . Additional transitions are shown by gray arrows. (c) Infidelity of the phase gate [Fig. 3] versus polarization error, $ε$ , defined as the fraction of undesired polarization in each STIRAP pulse. Nonlinear (Paschen-Back) corrections are not included in our model (see text).
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Figure 5
Quantum gate tomography: visualization of target (top) and simulated (bottom) single- and two-qubit gates. The plots show the matrix $χ$ , which expresses the expansion coefficients of the gate transformations in terms of Pauli basis operators, ${\tilde{E}}_{i}$ [Eq. (C2)] for single-qubit gates and Kronecker products of ${\tilde{E}}_{i}$ for two-qubit gates, respectively. (a)–(d) The matrix $χ$ , calculated using Eq. (C4), for Hadamard and Pauli- $Z$ single-qubit gates. (e),(f) The matrix $χ$ , calculated via Eq. (C6) for a two-qubit CZ gate.
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