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Error correction of a logical grid state qubit by dissipative pumping
Brennan de Neeve, Thanh Long Nguyen, Tanja Behrle, Jonathan Home
TL;DR
The paper develops a dissipative correction procedure using finite-state measurements, feedback, and ancilla reset for GKP states. Stabilization enables logical readouts, while added diffusion eventually drives logical values to zero.
Problem
Finite GKP-state correction requires a procedure that combines measurement with feedback while accounting for realistic ancilla operations.
Method
The procedure uses ancilla-assisted finite-state measurements, Kraus-operator descriptions, state-dependent displacements, feedback, and optical repumping to manipulate GKP states.
Results
97% success probability is achieved for a disentangling measurement without photon recoils, while stabilization data include logical X, Y, and Z readouts for square and hexagonal encodings.
Takeaways & Limitations
Stabilization preserves access to logical readouts, but the correction procedure introduces additional diffusion that eventually brings logical values to zero.
Abstract
from arXiv · showhide
Stabilization of encoded logical qubits using quantum error correction is key to the realization of reliable quantum computers. While qubit codes require many physical systems to be controlled, oscillator codes offer the possibility to perform error correction on a single physical entity. One powerful encoding for oscillators is the grid state or GKP encoding, which allows small displacement errors to be corrected. Here we introduce and implement a dissipative map designed for physically realistic finite GKP codes which performs quantum error correction of a logical qubit implemented in the motion of a single trapped ion. The correction cycle involves two rounds, which correct small displacements in position and momentum respectively. Each consists of first mapping the finite GKP code stabilizer information onto an internal electronic state ancilla qubit, and then applying coherent feedback and ancilla repumping. We demonstrate the extension of logical coherence using both square and hexagonal GKP codes, achieving an increase in logical lifetime of a factor of three. The simple dissipative map used for the correction can be viewed as a type of reservoir engineering, which pumps into the highly non-classical GKP qubit manifold. These techniques open new possibilities for quantum state control and sensing alongside their application to scaling quantum computing.
Supplementary Information
The supplementary calculations derive Kraus-operator descriptions for finite GKP measurements and stabilization, then evaluate their action using displacement-operator expansions and finite-state approximations.
- Kraus operators: The measurement process is represented by Kraus operators obtained from ancilla preparation, the unitary e^{iαqX}e^{iϵpY}, and projection in the Z or Y basis.The resulting operators are expressed as products of displacement operators.
- Kraus operators: The stabilization Kraus operators modify the measurement operators with feedback-controlled displacements and can be arranged as e^{iAq}e^{iBp}e^{iφAB}.Here A and B are real displacement amplitudes, while φAB is the phase from operator reordering.
- Finite GKP state calculations: The finite GKP-state calculations use separated squeezed states with negligible neighboring overlap, reducing the evaluation to displacement-operator terms on the oscillator ground state.The approximation is applied to the finite-state representation and to the small displacements used in the measurement sequence.
- Finite GKP state calculations: Each term is evaluated using the ground-state expectation value of a displacement operator, with A and B supplied by the relevant Kraus or measurement operators.The full logical-state evaluation then consists of a double sum over these terms.
C. Preservation fidelity
The preservation-fidelity calculation compares a finite GKP state before stabilization with its state after one stabilization round using the fidelity definition.
- Preservation fidelity: The fidelity in equation 2 measures the overlap between the pre-stabilization finite GKP state and the state produced after one stabilization round.For the input ρm = |0L⟩⟨0L|, the stabilization map produces a density matrix whose overlap is evaluated with the initial logical state.
A. Experimental sequence
The experimental sequence is organized conceptually into state preparation and GKP stabilization sub-sequences.
- A. Experimental sequence: The experiments are summarized by grouping their sub-sequences into state preparation and GKP stabilization.These two groups define the main stages of the experimental protocol.
1. State preparation
State preparation combines motional cooling, squeezed-state and finite-GKP preparation pulses, and a final internal-state detection that disentangles the ancilla with high success probability.
- State preparation: The preparation sequence cools a single trapped 40Ca+ ion, further cools its axial mode, and creates a squeezed state before GKP preparation.The sequence uses Doppler cooling, EIT cooling, sideband cooling, and squeezed-Fock-basis preparation.
- State preparation: The GKP preparation uses state-dependent displacement pulses whose amplitudes and phases are selected to prepare logical eigenstates of square and hexagonal codes.The pulse sequence can be adjusted by motional-phase rotation and parameters listed for the different code states.
- State preparation: 97% success probability is achieved for a single dark-state measurement that disentangles the internal state from the motion without photon recoils from repumping.The remaining entanglement before detection is approximately 3%.
- State preparation: A square-code stabilization cycle lasts approximately 150 µs, with each correction round taking approximately 75 µs.The step durations are constrained by laser power, off-resonant excitation, and smoothly shaped displacement pulses.
B. Logical state initialization
Logical GKP eigenstates are prepared by measuring a logical operator through state-dependent displacements and applying feedback conditioned on the ancilla state. The pulse sequence uses three state-dependent displacements, one global displacement, and calibrated displacement parameters.
- Logical state preparation: State-dependent displacements perform a finite logical measurement on the ancilla and apply feedback toward the desired GKP eigenstate.The ancilla is rotated differently for the two logical eigenstates, enabling Y-basis-conditioned feedback.
- Pulse sequence: The projective preparation sequence contains three state-dependent displacements and one global displacement.The sequence is specified for preparing a GKP eigenstate |ψL⟩.
- Pulse parameters: The sequence uses δ = √π/2 and ϵ ≈0.06√π.
- Global displacement: A resonant electrode tickle produces a global phase-space displacement whose distance scales with pulse duration and whose direction is controlled by phase.The rotating-frame Hamiltonian is HE = eE(ae−iϕm + a†eiϕm).
- Calibration: Trap-frequency calibration uses two equal, opposite-phase tickle pulses, which return the ion to the ground state when resonant.Off-resonant pulses leave a displacement proportional to τ^2δ.
2. State-dependent displacement strength
State-dependent displacement strength is calibrated through the motional state's characteristic function using a Fock state with a distinct feature. Maximizing the bright-state signal determines the displacement amplitude.
- Calibration method: A state-dependent displacement reads out the motional state's characteristic function and is calibrated using the Fock state |n = 1⟩.The chosen Fock state provides a distinct feature that is not correlated with finite thermal occupancy.
- Calibration signal: For α = √π/2, the internal state is maximally disentangled from motion and ends in the bright state for detection.Maximizing this signal provides the displacement-amplitude calibration.
3. Global displacement
The global displacement is calibrated relative to a state-dependent displacement by testing whether opposite-phase pulses cancel and return the oscillator to its motional ground state.
- Calibration sequence: The calibration replaces the first displacement in a trap-frequency sequence with a state-independent laser displacement.The ion is prepared in a superposition before the state-dependent displacement and tickle pulse.
- Cancellation test: Opposite-phase state-dependent and tickle pulses of equal magnitude should cancel, which is tested by probing return to the motional ground state.A red sideband pulse probes whether the ion has returned to that state.
D. Minimization of photon recoils
Photon-recoil minimization uses carefully chosen repumping polarization and measures the resulting stabilizer-readout degradation. Numerical simulations are used to analyze relevant experimental error channels.
- Repumping design: Photon recoil during optical repumping can affect stabilization, so polarization is chosen to minimize scattered photons.The ancilla uses two Zeeman sub-levels of a ^40Ca+ ion.
- Repumping design: Primarily π-polarized repumping directly returns the ion to |0⟩S with only 2 photons scattered per repump event.Additional repumping addresses decay into the 2D3/2 manifold.
- Measured recoil effect: ⟨Sx⟩ decreases from 0.68(1) to 0.58(1) after excitation and repumping because of photon recoils.The effect is measured through stabilizer readout after GKP state preparation.
- Error analysis: Numerical simulations are used to understand the experiment's relevant error channels.
1. Photon recoils
Photon recoils are modeled by tracking internal-state repumping and converting recoil kicks into phase-space displacements.
- 1. Photon recoils: Photon-recoil simulations follow the ion’s internal state during repumping using a rate-equation approach.Each transition updates the internal state and generates a recoil kick based on beam geometry and dipole emission.
- 1. Photon recoils: Recoil kicks are mapped into oscillator displacements in phase space.
- 1. Photon recoils: Initialization trajectories begin with a global displacement, followed by logical-state measurement and feedback along eigenstate-dependent paths.
2. Error channels
The experiments are limited by motional heating and trap-frequency fluctuations, which are incorporated into trajectory simulations alongside photon recoils.
- 2. Error channels: Heating from the quantum ground state occurs at about 10 quanta/s in the motional mode.It is modeled with loss and gain Lindblad processes having γ1 ≈ γ2 = 10 s−1.
- 2. Error channels: Trap-frequency fluctuations include a component arising from mains noise measured relative to the line cycle.
- 2. Error channels: Monte-Carlo wavefunction simulations sample non-Markovian noise and photon recoils using oscillator–ancilla evolution blocks.Mains noise and the static frequency offset are randomly drawn at the beginning of each trajectory.
F. Additional data
Additional datasets compare stabilized and unstabilized logical readouts for square and hexagonal finite-GKP encodings, with simulations reproducing the experimental trends qualitatively.
- F. Additional data: Logical readouts ⟨XL⟩, ⟨YL⟩ and ⟨ZL⟩ are provided for both square and hexagonal encodings with and without stabilization.The full datasets include exponential fits for both conditions.
- F. Additional data: Stabilization introduces additional diffusion that eventually drives the logical values to zero.
- F. Additional data: Figure 6 compares trap-frequency noise, motional coherence, individual error-channel decay simulations and logical-readout dynamics.The panels cover Markovian dephasing, 50 Hz noise, heating, and both finite-GKP geometries.
- F. Additional data: The simulations reproduce the experimental data qualitatively for both square and hexagonal finite-GKP encodings.