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A quantum information processor with trapped ions
Philipp Schindler, Daniel Nigg, Thomas Monz, Julio T. Barreiro, Esteban Martinez, Shannon X. Wang, Stephan Quint, Matthias F. Brandl, Volckmar Nebendahl, Christian F. Roos, Michael Chwalla, Markus Hennrich, Rainer Blatt
TL;DR
The paper addresses how to build and scale a trapped-ion quantum information processor while understanding and reducing operation errors. It presents a 40Ca+ processor with coherent and non-coherent operations, characterizes noise and its algorithmic effects, and demonstrates implementations of the quantum Fourier transform and order finding. The dominant error source depends on the operation sequence, while the full operation set supports efficient order finding with a single qubit for the QFT register.
Problem
Scaling trapped-ion quantum processors requires characterizing memory and operational noise because error correction depends on understanding these error sources.
Method
The paper develops a trapped-40Ca+ processor, analyzes its coherent and non-coherent operations and noise sources, simulates algorithmic effects, and demonstrates QFT and order-finding implementations.
Results
The dominant error source depends on the operation sequence; the QFT simulation predicts 92.6% fidelity versus 81(3)% experimental overlap, while the processor realizes order finding using a single qubit for the QFT register.
Takeaways & Limitations
The demonstrated operation set provides building blocks for scalable implementations of Shor's factoring algorithm.
Takeaways & Limitations
Inferring a noise spectrum from measured coherence decay is not unique, and optimizing MS gates for each rotation angle is time-consuming.
Abstract
from arXiv · showhide
Quantum computers hold the promise to solve certain problems exponentially faster than their classical counterparts. Trapped atomic ions are among the physical systems in which building such a computing device seems viable. In this work we present a small-scale quantum information processor based on a string of $^{40}$Ca${^+}$ ions confined in a macroscopic linear Paul trap. We review our set of operations which includes non-coherent operations allowing us to realize arbitrary Markovian processes. In order to build a larger quantum information processor it is mandatory to reduce the error rate of the available operations which is only possible if the physics of the noise processes is well understood. We identify the dominant noise sources in our system and discuss their effects on different algorithms. Finally we demonstrate how our entire set of operations can be used to facilitate the implementation of algorithms by examples of the quantum Fourier transform and the quantum order finding algorithm.
1. Tools for quantum information processing in ion traps
The processor combines optical and ground-state qubits with state preparation, cooling, measurement, universal coherent gates, optimized decompositions, and controlled dissipative operations. These tools support arbitrary Markovian processes while introducing practical calibration and scalability constraints.
- 1.2. The qubit - 40Ca+: The processor uses optical or ground-state qubits in trapped 40Ca+ ions, with the optical qubit commonly encoded in 4S1/2 and 3D5/2 states.The 3D5/2 lifetime is 1.1 s, while the ground-state encoding uses two Zeeman substates without spontaneous decay.
- 1.2. The qubit - 40Ca+: Projective measurement uses 397 nm excitation and electron shelving, distinguishing the qubit states through the presence or absence of fluorescence.An 866 nm repumping beam prevents population trapping in 3D3/2 during detection.
- 1.2. The qubit - 40Ca+: Initialization combines optical pumping into a selected Zeeman state with Doppler and sideband cooling to prepare the ion string's motional ground state.Sideband cooling can use the narrow qubit transition or a Raman process, with 854 nm repumping used to adjust the effective linewidth in one approach.
- 1.3. The universal set of gates: The universal coherent gate set combines arbitrary single-qubit operations with a Mølmer-Sørensen entangling operation implemented by bichromatic light.The MS interaction couples |SS, n⟩ and |DD, n⟩ through motional sideband states with detuning δ.
- 1.4. Optimized sequences of operations: Gate sequences are optimized numerically by adjusting operation order and rotation angles, but the search space grows exponentially with qubit number.Larger rotation angles may require concatenating instances optimized for the smallest occurring angle, and complex algorithms may require subset-specific decompositions.
- 1.5. Tools beyond coherent operations: The extended operation set includes individual-qubit decoupling and controlled amplitude- and phase-damping processes, enabling realization of any completely positive Markovian map.The operations use population transfers and optical pumping to implement controlled dissipative behavior beyond coherent gates.
2. Experimental setup
The processor combines a macroscopic linear Paul trap, multiple laser systems, focused addressing, FPGA-based timing and feedback, and photon-counting detection. Automated calibration and spatially resolved fluorescence support preparation, manipulation, measurement, and conditional operations on ion strings.
- 2.1. The linear Paul trap: The experiment uses a macroscopic linear Paul trap operated near 3MHz radial and 1MHz axial motional frequencies, adjusted with ion number.The apparatus is enclosed in a magnetic shield to reduce magnetic-field fluctuations.
- 2.2. Optical setup: The processor uses multiple laser sources for photo-ionization, qubit manipulation, cooling, optical pumping, Raman cooling, and detection.The 729nm Titanium-Sapphire manipulation laser addresses the 4S1/2 ↔ 3D5/2 qubit transition, while the required wavelengths are listed in table 2.
- 2.2. Optical setup: The optical setup accommodates constrained vacuum-vessel access, magnetic-field coils, and beams transferred between separate optical tables through fibers.The 397nm, 866nm, and 854nm beams are overlapped for detection, while the magnetic-field orientation can be adjusted without moving mechanical components.
- 2.2. Optical setup: Addressed single-qubit control uses a focused beam positioned by a motorized lens and electro-optic deflector, resolving ions separated by approximately 5µm.The addressed beam has η_add = 2.5%; switching between neighboring ions takes approximately 15µs, with a 30µs delay sufficient for strings of up to 8 ions.
- 2.3. Experiment control: An FPGA controls experiment timing from binary sequences generated by a custom LabView program, with DDS units producing RF signals for coherent manipulation.Measurement outcomes can be counted and used to condition subsequent operations, enabling feed-forward sequences.
- 2.4. State detection: Quantum-state detection combines PMT photon counting with CCD imaging: thresholded PMT counts estimate the number of bright ions, whereas the CCD determines each ion’s state separately.The PMT histogram shown for a 4-ion string uses 21,200 measurements with a 5ms detection time.
3. Error sources
The processor’s errors arise from qubit-memory noise and operation-specific imperfections, whose effects depend on physical mechanisms, correlations, and algorithm structure. Characterizing these sources reveals both constraints and mitigation strategies, including alternative encodings, echo techniques, and compensation.
- Qubit-memory errors: Qubit-memory errors combine amplitude damping, caused by excited-state decay, and phase damping, which destroys superposition phase without changing populations.The measured 3D5/2 lifetime is τ1 = 1.13(5)s, close to the natural lifetime of 1.168(7)s.
- Qubit-memory errors: Phase damping is assessed through Ramsey contrast decay and mainly arises from laser-frequency and magnetic-field fluctuations.The coherence decay deviates from a simple exponential because technical noise is temporally correlated.
- Qubit-memory errors: Correlated laser and magnetic-field noise can accelerate decoherence for states with large total energy differences but also support decoherence-free subspaces.A logical qubit can be encoded in two physical qubits whose states acquire the same total phase.
- Operation errors: Initialization and motional imperfections affect operation quality: optical pumping exceeds 99%, optical-transition sideband cooling reaches ⟨n⟩ = 0.05(3), and motion remains coherent for 110(20) ms.Raman sideband cooling is faster but produces a higher steady-state phonon number; incomplete cooling damps Rabi-oscillation contrast.
- Operation errors: Addressing crosstalk is typically below 3% for up to eight ions and can be compensated when its addressing matrix is known and stable.In the illustrated three-ion case, the maximum crosstalk is ϵmax = 22/121 = 18%.
- Operation errors: Measurement errors are dominated by spontaneous decay and stray light, although bright-dark distribution overlap can yield detection errors below 10^-3.CCD-camera detection agrees with photomultiplier-tube outcomes at better than 99.3%.
- Algorithm-level impact: Noise sources produce algorithm-dependent effects: QFT simulation predicts 92.6% fidelity, experiment obtains 81(3)%, and crosstalk is its largest simulated contribution.The comparison uses the same operation parameters as the simulated algorithm sequence.
4. Example algorithms
The processor’s operation toolbox is demonstrated on the quantum Fourier transform and order-finding algorithm, including semiclassical variants that use measurement and classical control. The three-qubit QFT reaches 72% process fidelity and 87% average SSO, while the two-qubit order-finding demonstration reaches 80.7% average SSO.
- Quantum Fourier transform: The three-qubit QFT maps a quantum state through the classical discrete Fourier transform and can be implemented using an optimized decomposition of 18 operations.The direct unitary implementation is described as straightforward but not necessarily most effective.
- Quantum Fourier transform: 72% process fidelity was measured for the fully coherent three-qubit QFT relative to the ideal QFT.The fidelity comes from full three-qubit quantum process tomography.
- Quantum Fourier transform: 87% average SSO was obtained on representative QFT inputs, exceeding the quantum process fidelity because the algorithm’s classical output probabilities are the relevant benchmark.The inputs cover all possible periods and are chosen for comparison with earlier benchmarking.
- Semiclassical QFT: The Kitaev QFT replaces quantum-controlled rotations with measurement and classically controlled rotations, allowing the algorithm to reuse a single physical qubit.This version cannot generate entangled input states and requires in-sequence measurement and reset.
- Order finding: The order-finding experiment applies two-qubit permutation operations with orders from 2 to 4 and compares experimental output probabilities with ideal and classical-simulation predictions.The algorithm is compatible with the Kitaev single-qubit QFT because its output is classical.
- Order finding: 80.7% average SSO was achieved for the semiclassical order-finding implementation across the tested permutation operations.The experiment benchmarks the classical output probabilities using SSO and distinguishability measures.
5. Conclusion and Outlook
The paper presents a trapped-40Ca+ quantum processor with non-coherent operations, analyzes operation-dependent noise, and demonstrates an efficient order-finding implementation aimed toward scalable factoring.
- The processor is based on trapped 40Ca+ ions and includes operations beyond coherent operations for implementing arbitrary Markovian processes.
- The study analyzes major noise sources affecting qubit memory and operations, showing that the dominant error source depends on the operation sequence.
- The full operation set enabled an efficient order-finding implementation using a single qubit for the entire QFT register.
- The authors propose these techniques as building blocks for scalable Shor factoring and improved noise-reduction strategies toward fault-tolerant quantum computation.
6. Appendix
The appendix catalogs operation sequences used for an open-system simulator, a three-qubit coherent QFT, and several controlled permutation operations.
- Table 10 gives the operation sequence for the algorithm used as an open-system quantum simulator.
- Table 11 gives the operation sequence for a fully coherent QFT operation on three qubits.
- Tables 12–17 give sequences for controlled π1(y), π2(y), π3(y), π2 3(y), π4(y), and π2 4(y) permutation operations.