[Paper Review] Further quantum-gate methods using selective displacement of trapped ions

Marek Šašura, Andrew Steane|arXiv (Cornell University)|Feb 6, 2004

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper extends the state-selective displacement method for trapped-ion quantum gates to enable high-fidelity two- and three-qubit operations, including non-neighboring qubit gates and Toffoli gates, even when ions are in the same trap. It shows that anharmonicity from Coulomb interactions remains negligible, preserving gate robustness against thermal motion.

ABSTRACT

We consider quantum gates for trapped ions using state-selective displacement of the ions. We generalize earlier work in order to treat arbitrary separations between the traps. This requires the impact of anharmonicity arising from the Coulomb interaction to be estimated. We show that its effects are always small enough to allow high fidelity. In particular, the method can be applied to two ions in the same trap. We also show that gates between non-neighbour ions, and hence a Toffoli (three-qubit controlled-NOT) gate, can be achieved. We discuss how the gate can be applied to logical qubits encoded in the decoherence-free-subspace {|01>,|10>}, where each pair of ions stores a single qubit. We also suggest alternatives to the spin-echo method to suppress unwanted terms in the evolution.

Motivation & Objective

• Develop high-fidelity quantum gates for trapped ions using state-selective displacement, extending beyond nearest-neighbor interactions.
• Enable implementation of the Toffoli (three-qubit controlled-NOT) gate using three ions in separate microtraps without ion swapping or reconfiguration.
• Analyze and mitigate the impact of anharmonicity in the effective trapping potential due to Coulomb interactions, especially in close-proximity or same-trap configurations.
• Propose alternatives to the spin-echo method for suppressing unwanted evolution terms in gate operations.
• Apply the gate to logical qubits encoded in decoherence-free subspaces, such as the {|01⟩, |10⟩} subspace, to enhance fault tolerance.

Proposed method

• Use state-selective laser forces to displace ions based on their internal states, inducing state-dependent phases via Coulomb interaction.
• Implement two-qubit phase gates (CZ or CP(θ)) via a sequence of local single-qubit rotations (S) and a global evolution (G), satisfying phase conditions A₀ + B₀ + Θ₀₀ = 0, etc.
• Decompose the three-qubit CCZ gate into five two-qubit gates, including one between non-neighboring ions, using the pushing method without intermediate qubit swaps.
• Model the effective trapping potential as anharmonic due to Coulomb interaction, and quantify its impact on gate fidelity using perturbative analysis.
• Use analytical expressions for the overall gate phase ϑ to compute infidelity contributions from thermal averaging over dynamic phases and force profiles.
• Propose alternative suppression techniques to spin-echo for unwanted evolution terms, improving gate robustness.

Experimental results

Research questions

• RQ1 Can the state-selective displacement method be extended to implement a Toffoli gate between three ions in separate microtraps without ion rearrangement or quantum information swapping?
• RQ2 How does Coulomb-induced anharmonicity in the trapping potential affect gate fidelity when two ions are confined in the same trap?
• RQ3 What is the relative contribution of thermal averaging over dynamic phases versus force profile to gate infidelity in the ε = 2 regime?
• RQ4 Can the gate be applied to logical qubits encoded in the decoherence-free subspace {|01⟩, |10⟩} to enhance fault tolerance?
• RQ5 Are there viable alternatives to the spin-echo method for suppressing unwanted evolution terms in the gate sequence?

Key findings

• The Toffoli (CCZ) gate can be implemented with three ions in separate microtraps using five two-qubit gates, including a non-neighboring gate, without requiring ion movement or qubit swapping.
• The gate remains robust even when two ions are in the same trap, as Coulomb-induced anharmonicity has a negligible effect on fidelity.
• For ε = 2, the infidelity from thermal averaging over dynamic phases scales as (a/d)⁴ and is comparable in magnitude to the contribution from force profile averaging, both remaining small.
• The total infidelity is dominated by thermal effects, with the anharmonicity contribution scaling as (a/d)⁴ and not dominating the overall error budget.
• The method achieves high fidelity even at the Doppler temperature (T ≈ 538 μK) for ⁴⁰Ca⁺ ions, with gate times τ_L = 5/ω.
• The analytical expression for the phase ϑ in the ε = 2 regime confirms that anharmonicity does not significantly degrade gate performance, supporting scalability to dense ion arrays.