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Experimental Quantum Computations on a Topologically Encoded Qubit
Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt
TL;DR
The paper addresses how to protect quantum information from noise by experimentally implementing a seven-qubit error-correcting code. It encodes one qubit in seven trapped-ion qubits, tests error detection and logical Clifford operations, and demonstrates the smallest fully functional color-code instance.
Problem
Quantum computation requires fault-tolerant methods because quantum states and operations are subject to unavoidable noise.
Method
The experiment encodes one logical qubit in entangled states distributed across 7 trapped-ion qubits using a complete CSS code and applies logical single-qubit Clifford operations.
Results
The code detects one bit flip, phase flip, or combined error on any physical qubit and supports the generating logical Clifford gates ZL, XL, HL, and KL.
Takeaways & Limitations
The seven-qubit implementation constitutes the smallest fully functional 2D color code and a building block for topological fault-tolerant quantum computing.
Abstract
from arXiv · showhide
The construction of a quantum computer remains a fundamental scientific and technological challenge, in particular due to unavoidable noise. Quantum states and operations can be protected from errors using protocols for fault-tolerant quantum computing (FTQC). Here we present a step towards this by implementing a quantum error correcting code, encoding one qubit in entangled states distributed over 7 trapped-ion qubits. We demonstrate the capability of the code to detect one bit flip, phase flip or a combined error of both, regardless on which of the qubits they occur. Furthermore, we apply combinations of the entire set of logical single-qubit Clifford gates on the encoded qubit to explore its computational capabilities. The implemented 7-qubit code is the first realization of a complete Calderbank-Shor-Steane (CSS) code and constitutes a central building block for FTQC schemes based on concatenated elementary quantum codes. It also represents the smallest fully functional instance of the color code, opening a route towards topological FTQC.
APPENDICES
The appendices provide theoretical background, experimental details, and supplementary data on state characterization, error detection, and logical Clifford operations.
- Supplementary data include the complete table of all 21 single-qubit error syndromes and tomography confirming local-order absence and global-order presence.
- The appendices describe the trapped-ion setup, encoding procedure, state-fidelity and entanglement characterization, error-detection methods, and logical Clifford sequences.
Appendix A: Experimental system and techniques
The experimental system uses seven trapped calcium ions in a linear Paul trap, with optical control and fluorescence-based state detection.
- All experiments use a linear string of 7 40Ca+ ions confined in a Paul trap.
- Each physical qubit is encoded in two electronic states, and coherent operations are driven on an optical transition using a narrow-linewidth 729 nm laser.
- Each experimental cycle initializes the qubits, cools the ion string to the motional ground state, applies coherent operations, and detects the final state through fluorescence and electron shelving.
1. Coherent gate operations
The setup combines global optical control, entangling operations, and focused single-qubit addressing to realize coherent gate operations.
- A spatially wide 729 nm beam illuminates the entire ion string to implement collective operations.
- The apparatus implements a Mølmer-Sørensen-type entangling operation for multiqubit control.
- The global beam is shaped elliptically to equalize illumination across the ions, with measured Rabi-frequency inhomogeneity of about 1%.
- A focused addressing beam performs single-ion rotations, including off-resonant AC-Stark Z rotations and resonant operations with adjustable angle and phase.
2. Spectroscopic decoupling and recoupling of ions
Spectroscopic decoupling and recoupling adapt global entangling operations so they act on selected ions, enabling encoded-state preparation and related gate sequences.
- 2. Spectroscopic decoupling and recoupling of ions: The encoded |0⟩L state is initialized using an entangling operation acting on a subset of four of the seven qubits.
- 2. Spectroscopic decoupling and recoupling of ions: Selective entangling dynamics can be produced by interspersing global Mølmer-Sørensen operations with single-qubit AC-Stark shifts that decouple chosen ions.
- 2. Spectroscopic decoupling and recoupling of ions: The four-ion entangling operation achieves a fidelity of 88.5(5)% based on tomography of the generated four-qubit GHZ state.
- 2. Spectroscopic decoupling and recoupling of ions: The four-ion operation has higher fidelity than the global seven-ion entangling operation, whose fidelity is estimated at about 84%.
- 2. Spectroscopic decoupling and recoupling of ions: Addressing errors on neighboring ions are reduced because off-resonant AC-Stark effects scale quadratically with the addressing error, while resonant-pulse errors cancel to leading order.
- 2. Spectroscopic decoupling and recoupling of ions: Decoupling and recoupling one qubit across both computational basis states requires 9 single-qubit operations, with approximately 10 µs per pulse.
Appendix B: Details on encoding of the logical qubit
The logical |0⟩L state is encoded through sequential plaquette entanglement, using decoupling to restrict each operation to active qubits and producing the required stabilizer pattern.
- The code space is the simultaneous +1 eigenspace of six stabilizer operators.
- The initialization first converts the optically pumped |1111111⟩ state into |1010101⟩, satisfying the three Z-type stabilizer constraints.
- Three plaquette subsets are sequentially entangled to impose the remaining X-type stabilizer constraints.
- Spectroscopic decoupling protects inactive ions while an MS gate creates four-qubit GHZ-type entanglement among active plaquette qubits.
- The completed encoding is signaled by positive values of all six stabilizers and the logical ZL operator.
- Choosing |1010101⟩ provides a period in which the seven-qubit state lies in a decoherence-free subspace insensitive to global phase noise.
1. Quantum state fidelities and overlap with the code space
The encoded states are characterized efficiently through stabilizer-based measurements rather than full density-matrix reconstruction, separating total fidelity, code-space overlap, and within-code-space fidelity.
- The measured encoding proceeds from the product state through plaquette entanglement, with expected populations and stabilizer coherences emerging stepwise.
- For seven-qubit stabilizer states, measuring 128 Pauli expectation values exactly determines quantum-state fidelity without reconstructing the full density matrix.
- The fidelity decomposes into code-space population multiplied by fidelity with the target state projected within the code space.
- The characterization uses local basis-changing unitaries before fluorescence measurements of the required Pauli operators.
2. Entanglement properties of the encoded logical qubit
The encoded logical states exhibit genuine multipartite entanglement: fidelities above a witness threshold certify entanglement across every 2-versus-5 bipartition and thereby all seven qubits.
- A fidelity above 25% makes the entanglement witness negative and rules out separability across any 2-versus-5 bipartition.
- The resulting entanglement involves at least six qubits and implies mutual entanglement of all three plaquettes.
- The six logical states are locally equivalent through single-qubit rotations associated with transversal Clifford operations.
- The measured fidelities for |0⟩L, |1⟩L, and |+x⟩L are 32.7(8)%, 28(1)%, and 33(1)%, respectively, exceeding 25%.These exceedances correspond to more than 9, 3, and 8 standard deviations, respectively.
Appendix D: Absence of local order and presence of global order in the color code state
The seven-qubit color-code state lacks local order but retains global quantum order, observed through nonzero three-qubit string correlations and reduced-state measurements.
- The seven-qubit code has local stabilizers and global logical operators, including a three-qubit string operator equivalent to logical ZL within the code space.
- The seven-qubit realization is the smallest fully functional triangular 2D color-code lattice and has distance d = 3, correcting one arbitrary physical error.
- Two-qubit tomographies across all 21 ion pairs yield an average fidelity of 98.3(2)% with the completely mixed state.
- The global correlation ⟨Z1Z4Z7⟩ = −0.46(6) demonstrates non-vanishing three-qubit Z-type order.
- For |+x⟩L, the corresponding X-type string correlation is ⟨X1X4X7⟩ = 0.40(5).
- These measurements demonstrate absence of local order alongside global quantum order in the experimental encoded state.
Appendix E: Quantum error detection and complete experimental syndrome table
The 7-qubit CSS code has distance d = 3, allowing correction of one physical error while exposing the limits of correcting multiple errors. Syndrome classification approaches perfect assignment after about 20 measurement cycles.
- Code capacity: The 7-qubit CSS code has logical distance d = 3 and can correct one physical error on any qubit.Larger color-code lattices support correction of more single-qubit errors, including d = 5 with n = 2 and d = 11 with n = 5.
- Syndrome classification: The syndrome-assignment success rate converges rapidly to 100% after about 20 measurement cycles.The simulation uses 5000 Monte-Carlo samples per data point; above 20 cycles, syndromes can be clearly distinguished and associated with the induced error.
- Syndrome measurement: 1000 measurement cycles are used experimentally to measure the stabilizers needed for error-syndrome identification.The syndrome-classification method compares sampled and reference stabilizer distributions using classical trace distance.
- Failure cases: Two phase-flip errors can produce syndromes indistinguishable from single-qubit errors, causing an effective logical phase flip.For Z errors on qubits 5 and 2 or 5 and 3, mistaken correction applies operators equivalent to ZL.
Appendix F: Encoded Clifford quantum gate operations on the logical qubit
The color-code implementation processes information within the code space and supports transversal implementation of the full Clifford group. Its syndrome table records the error signatures used to distinguish physical errors.
- Color-code operation: Topological color codes process quantum information directly within the code space, where stabilizer violations indicate errors.This contrasts with topological schemes based on controlled generation and manipulation of quasiparticle excitations.
- Color-code operation: Two-dimensional color codes enable transversal, bit-wise implementation of the entire Clifford gate group.This gate capability is a distinguishing feature of the color-code approach described in the paper.
- Syndrome table: The syndrome table records the error syndromes produced by all 21 single-qubit errors and highlights indistinguishable two-error cases.Single-qubit Z errors on qubits 1 or 4 share syndromes with particular pairs of Z errors.
1. Implementation of logical single-qubit Clifford gate operations
The encoded logical qubit implements logical Pauli, Hadamard, phase, and Y operations while preserving the code space. The phase gate is realized transversally using bit-wise physical operations.
- Logical operators: The logical operators XL and ZL act on all seven qubits, anticommute, and commute with the six stabilizers defining the code space.Their seven-qubit support gives the required anticommutation relation while preserving encoded states.
- Logical Pauli gates: The logical YL operation is implemented sequentially as YL = +iXLZL = −Y1Y2Y3Y4Y5Y6Y7.The logical ZL gate uses single-ion Z rotations, while XL uses a collective local X rotation.
- Hadamard gate: The logical Hadamard gate is implemented transversally as HL = H1H2H3H4H5H6H7 using collective Y rotation and single-ion Z rotations.The addressed Z-rotation pulses can introduce decoherence and small addressing errors during implementation.
- Phase gate: The logical phase or KL gate is implemented transversally, without magic-state injection or multi-qubit entangling operations.The paper identifies this transversal KL implementation as a distinguishing feature of the 2D color-code architecture.
- Phase gate: The KL gate can be implemented by bit-wise Z rotations across the seven physical qubits.The physical implementation uses KL = Q7 operations as described for the encoded qubit.
2. Preparation of the six eigenstates of the logical operators XL, YL and ZL
Starting from |0⟩L, up to three logical Clifford gates prepare the six eigenstates along the logical Bloch-sphere axes. The resulting states retain measurable stabilizer order and Bloch-vector length.
- State preparation: Up to three Clifford gates prepare the six logical eigenstates of XL, YL, and ZL from the initial state |0⟩L.The required circuits and resulting stabilizer and Bloch-vector values are tabulated for each state.
- Z-axis states: The prepared |1⟩L state has average stabilizer expectation ⟨Si⟩ = 0.54(1) and Bloch-vector length L = 0.57(6).It requires one XL gate operation.
- X-axis states: The prepared |+x⟩L state has ⟨Si⟩ = 0.52(1) and L = 0.48(2), requiring one HL gate operation.The state is obtained from the initial encoded state using a single logical Hadamard.
- Y-axis states: The prepared |+y⟩L and |−y⟩L states have (⟨Si⟩, L) = (0.39(1), 0.42(2)) and (0.42(1), 0.38(2)), respectively.The |+y⟩L state uses HL and KL, whereas |−y⟩L additionally uses XL.
3. Longer sequences of encoded gate operations and decay of coherences of the logical qubit
Repeated logical X_L operations produce an average logical Y_L decay of 3.8(5)% per gate, consistent with imperfections in collective seven-ion rotations. Full characterization of longer gate sequences or quantum processes is left for future work.
- Longer encoded gate sequences: 3.8(5)% per logical gate is the measured decay rate of the average ⟨Y_L⟩ expectation value over up to 10 additional X_L operations.The decay was obtained by fitting the expectation values with A exp(−n_gate/B).
- Decay mechanisms: A single collective resonant π-rotation fidelity of about 99.6% per ion would already produce an approximately 3.8% loss per gate.This connects the observed logical decay to the accuracy of seven-ion collective rotations.
- Decay mechanisms: Global-beam intensity inhomogeneity and thermal occupation of higher motional modes dominate the measured fidelity loss per gate.Beam inhomogeneity lowers Rabi frequencies at ions near the string edges, while motional excitation decoheres global Rabi oscillations.
- Scope: Characterization through longer gate sequences or other quantum-process techniques remains outside the scope of the work.The authors identify these measurements as a direction for future research.