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Towards fault-tolerant quantum computing with trapped ions
J. Benhelm, G. Kirchmair, C. F. Roos, R. Blatt
TL;DR
Fault-tolerant quantum computing needs highly accurate entangling gates, but trapped-ion implementations had not reached the required error levels. This paper demonstrates an adiabatically controlled Molmer-Sorensen gate on calcium ions with 99.3(1)% fidelity, supporting universal quantum logic and prospects for multi-qubit operations.
Problem
The paper addresses whether trapped-ion entangling gates can reach error levels below those required by some fault-tolerant quantum-computing models.
Method
The paper implements a Molmer-Sorensen-type gate and describes its operation using a propagator for the ion dynamics.
Results
99.3(1)% fidelity is obtained for the Bell state Ψ1.
Takeaways & Limitations
The gate makes quantum algorithms with tens of entangling operations realistic and opens perspectives for generating multi-particle entanglement with a single laser.
Takeaways & Limitations
The overhead implied by some fault-tolerance models remains considerable.
Abstract
from arXiv · showhide
Today ion traps are among the most promising physical systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics. The standard circuit model of quantum computing requires a universal set of quantum logic gates for the implementation of arbitrary quantum operations. As in classical models of computation, quantum error correction techniques enable rectification of small imperfections in gate operations, thus allowing for perfect computation in the presence of noise. For fault-tolerant computation, it is commonly believed that error thresholds ranging between 10^-4 and 10^-2 will be required depending on the noise model and the computational overhead for realizing the quantum gates. Up to now, all experimental implementations have fallen short of these requirements. Here, we report on a Molmer-Sorensen type gate operation entangling ions with a fidelity of 99.3(1)% which together with single-qubit operations forms a universal set of quantum gates. The gate operation is performed on a pair of qubits encoded in two trapped calcium ions using a single amplitude-modulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states is achieved by adiabatically switching the laser-ion coupling on and off. We analyse the performance of a single gate and concatenations of up to 21 gate operations. The gate mechanism holds great promise not only for two-qubit but also for multi-qubit operations.
denotes a collective atomic operator, and σ(i)
The gate is modeled as collective spin dynamics coupled transiently to the shared motional mode, with smooth laser shaping suppressing non-resonant excitations and phase sensitivity. Repeated bichromatic pulses map product states to maximally entangled states at specified gate instances.
- Gate mechanism: The propagator includes collective spin flips and a displacement operator describing transient entanglement between the qubits and harmonic oscillator.The displacement becomes the identity after the gate operation, removing the residual motional entanglement.
- Gate mechanism: Smoothly switching the laser intensity on and off suppresses fast non-resonant carrier excitations and makes the collective spin-flip operator independent of ζ.Adiabatic following is achieved by switching on the laser within 2.5 trap cycles, with pulse slopes of duration τr = 2 µs.
- Experimental implementation: The experiment initializes two 40Ca+ ions close to the motional ground state and prepares |SS⟩ with a probability of more than 99.8%.The measured axial-mode occupations satisfy n̄com, n̄stretch < 0.05(5).
- Gate action: Repeated bichromatic pulses ideally map |SS⟩ to an entangled state up to global phases, with maximally entangled states at τm = m · τgate for m = 1, 3, . . .The same type of mapping between product states and Bell states occurs when starting from |SD⟩.
states reveal that ρexp
The section introduces measurements for different values of φ using the operator σφ, defined as a φ-dependent combination of σx and σy.
- For different values of φ, the passage uses σφ = σx cos φ+σy sin φ.Here, σφ is defined in terms of σx and σy.
by applying ( π
The gate produced Bell states with 99.3(1)% fidelity and maintained predicted entangling dynamics across repeated operations. Its imperfections indicate both a per-gate population loss and noise sources, while the approach supports realistic multi-operation algorithms and tunable multi-qubit entanglement.
- Repeated gate dynamics: Ions were consecutively entangled and disentangled up to nine times, with populations closely following predicted unitary evolution.The dynamics were measured for pulse lengths equivalent to up to 17 gate times.
- Gate imperfections: Gaussian parity-amplitude decay was consistent with low-frequency magnetic-field and laser-frequency noise.The decay was measured at odd integer multiples of τgate.
- Fault-tolerant prospects: 7 · 10−3 Bell-state infidelity suggests an operation infidelity below thresholds required by some fault-tolerant computation models.Further experimental advances remain necessary because the overhead of these models is considerable.
- Multi-qubit prospects: The gate makes algorithms with tens of entangling operations realistic and enables multi-particle entanglement using a single laser interacting with more than two qubits.Interleaving global entangling pulses with focused single-qubit phase shifts could refocus unwanted interactions and engineer varied multi-qubit interactions.
Methods · AC-Stark-shift compensation
The bichromatic light field produces dynamic ac-Stark shifts, while equal first-order sideband intensities cancel corresponding contributions. Residual shifts and coupling imbalance are addressed through additional compensation or intensity-ratio tuning, with off-resonant light also inducing spin flips.
- AC-Stark-shift compensation: Bichromatic light causes dynamic ac-Stark shifts through non-resonant excitation on the carrier.
- AC-Stark-shift compensation: Equal corresponding laser intensities make the first-order sideband shifts exactly cancel each other.
- AC-Stark-shift compensation: Residual ac-Stark shifts arise from other Zeeman transitions and far-detuned dipole transitions.
- AC-Stark-shift compensation: 7 kHz is the remaining ac-Stark shift for a gate time τgate = 50 µs.
- AC-Stark-shift compensation: The shifts can be compensated using an additional far-detuned light field or by setting the intensity ratio I+/I−.
- Methods: I+ and I− are slightly unequal, but the coupling-strength error is insignificant at ΩSD↔DS/ΩSS↔DD −1 = 4 · 10−3.
- Methods: Incoherent off-resonant bichromatic light induces spin flips.
Sources of gate infidelity
Gate infidelity arises from laser-frequency and intensity noise, spin flips near motional sidebands, and imperfect phase-space closure. The measured frequency uncertainty causes a 0.1% single-gate fidelity loss, while several errors can be reduced through filtering, higher trap frequencies, or longer gates.
- Laser-induced errors: 2 · 10−7 of total laser power lies within a 20 kHz bandwidth B around the carrier transition, contributing to spin-flip errors near motional sidebands.The model predicts pflip = (πγ|ν −δ|)/(2η2B).
- Laser-induced errors: 8 · 10−4 is the predicted spin-flip probability, whereas measured state populations are consistent with pflip = 2 · 10−3.Spin-flip errors could be reduced by two orders of magnitude through spectral filtering and increasing ν above 2 MHz.
- Laser-frequency noise: 180 Hz is the full-width at half maximum δω/(2π) of Gaussian laser-frequency fluctuations, consistent with 160 Hz from Ramsey measurements.For a single gate, this frequency uncertainty causes a fidelity loss of 0.1%.
- Phase-space closure: Nonclosing phase-space trajectories from time-dependent bichromatic forces can be made negligibly small by slightly increasing the gate time.The effect is negligible for the short rise times used in the experiments after this adjustment.