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Entangling logical qubits with lattice surgery
Alexander Erhard, Hendrik Poulsen Nautrup, Michael Meth, Lukas Postler, Roman Stricker, Martin Ringbauer, Philipp Schindler, Hans J. Briegel, Rainer Blatt, Nicolai Friis, Thomas Monz
TL;DR
Fault-tolerant quantum computing requires practical methods for operating on encoded qubits despite the overhead of quantum error correction. This work experimentally realizes lattice surgery between two topologically encoded qubits in a 10-qubit ion-trap processor, demonstrating logical entanglement and state teleportation.
Problem
Fault-tolerant quantum computing needs resource-efficient logical operations on error-corrected qubits, whose implementation carries substantial overhead.
Method
The experiment performs lattice surgery between two topologically encoded qubits using merging and splitting operations with post-selected stabilizer measurements.
Results
58(2) % (rough) and 64(3) % (smooth) measured Bell-state fidelities demonstrate logical entanglement, while the experiment also generates Bell states conditioned on measurement outcomes.
Takeaways & Limitations
The results experimentally demonstrate lattice surgery as a means of entangling logical qubits and teleporting logical quantum states.
Abstract
from arXiv · showhide
Future quantum computers will require quantum error correction for faithful operation. The correction capabilities come with an overhead for performing fault-tolerant logical operations on the encoded qubits. One of the most resource efficient ways to implement logical operations is lattice surgery, where groups of physical qubits, arranged on lattices, can be merged and split to realize entangling gates and teleport logical information. Here, we report on the experimental realization of lattice surgery between two topologically encoded qubits in a 10-qubit ion trap quantum information processor. In particular, we demonstrate entanglement between two logical qubits and we implement logical state teleportation.
APPENDIX: SUPPLEMENTAL INFORMATION
The supplemental information develops the theoretical background for quantum error correction and surface-code lattice surgery, then details the experimental realization and analysis procedures.
- The appendices cover stabilizer-formalism theory, surface codes, lattice surgery, experimental circuits, ancilla readout, survival probabilities, error detection, and logical Bell-state fidelity estimates.
A.I. Stabilizer Quantum Error Correction
The stabilizer formalism protects encoded quantum information by measuring commuting operators that reveal error syndromes without directly measuring the logical state. Logical operators are defined modulo stabilizers and determine the code’s protected degrees of freedom and distance.
- Quantum error correction encodes logical basis states into redundant physical-qubit states so errors can be detected and corrected.
- Commuting stabilizers preserve encoded information while their measurement outcomes provide syndromes for identifying errors.
- Logical operators commute with all stabilizers but are not stabilizers, and they form equivalence classes under multiplication by stabilizers.
- The code distance is determined by the minimum weight of a nontrivial logical operator, setting the error tolerance before a logical error occurs.
A.II. Surface Code
The surface code places data qubits on a bicolorable lattice and uses face stabilizers to encode one logical qubit. Logical operators are boundary-connecting Pauli strings, while lattice surgery enables measurement-based logical operations.
- Surface-code stabilizers are assigned to colored plaquettes and commute because neighboring plaquettes share two vertices.
- The lattice has n − 1 independent stabilizers, leaving one encoded qubit with logical X and Z operators represented by boundary-connecting Pauli strings.
- Continuous stabilizer measurements detect sign changes in syndromes, and corrections recover the +1 stabilizer state when fewer than (d−1)/2 errors occur.
- Lattice surgery supports measurement-based logical CNOT, state teleportation, and Hadamard operations using joint Pauli measurements and outcome-conditioned corrections.
A.III. Lattice Surgery
Lattice surgery implements logical joint measurements by merging two surface-code lattices and then splitting them, projecting the codes onto a joint logical-Pauli eigenstate. The procedure supports fault-tolerant logical operations and state teleportation.
- Lattice surgery projects two stabilizer codes onto a joint eigenstate of logical Pauli operators, with the eigenvalue determined by merging-measurement outcomes.
- Merging: The operation begins by merging separate codes through boundary stabilizer measurements that create a combined code containing the desired joint logical operator.
- Merging: Measurement errors are addressed by repeating merging-stabilizer measurements d times and comparing results across time.
- Splitting: Splitting measures stabilizers of the separated codes, discards anticommuting merging stabilizers, and recovers the original codes while preserving the joint-eigenstate condition.
- Logical operations: Joint Pauli measurements can implement logical CNOT, Hadamard, and code teleportation, while full fault tolerance requires quantum error correction at the end.
A.IV. Results Smooth Lattice Surgery
Smooth lattice surgery merges and splits two surface codes along smooth boundaries, producing a logical Bell state through joint stabilizer measurements.
- Smooth lattice surgery: Smooth lattice surgery is equivalent to rotating both codes by 90 degrees and merging or splitting along their upper and lower boundaries.
- Merged code: The merged code becomes a 4 × 2 asymmetric surface code after introducing the merging stabilizers.
- Splitting: The merged code can be split by discarding the merging stabilizer, restoring two logical qubits while preserving the relevant logical eigenstate.
- Smooth lattice surgery: The procedure measures a Z-type merging stabilizer to project the two logical qubits onto a joint eigenstate.
A.V. Experimental Circuits
The experimental circuits implement lattice surgery with ion-trap gates, decoupling and recoupling operations, and in-sequence ancilla measurements; measured Bell fidelities match the expected order of magnitude.
- Circuit implementation: The rough-boundary lattice-surgery circuit uses local rotations, spectroscopic decoupling and recoupling, and multi-qubit Mølmer-Sørensen gates.
- Experimental results: Measured Bell-state fidelities are 58(2) % for rough surgery and 64(3) % for smooth surgery, close in order of magnitude to expected fidelities of ∼63 % and ∼57 %, respectively.
- Gate resources: The complete circuit includes local 1-qubit gates, local N-qubit gates, and N-qubit MS gates.
- Smooth-surgery measurements: The smooth-surgery circuit reports encoded-state stabilizer and logical-state fidelity measurements before merging, during merging, and after splitting.
A.VI. Ancilla Readout
Ancilla readout maps merging-stabilizer information onto measured ions, but the experiment proceeds only for selected outcomes because in-sequence fluorescence heats the ion chain and leaves qubits outside the computational subspace.
- Ancilla measurement: Merging-stabilizer information is mapped onto ancillas A1 and A2, which are measured simultaneously after decoupling the data qubits.
- Readout constraints: In-sequence fluorescence detection reveals only the number of bright ions and can heat the ion chain, causing qubits to leave the computational subspace.
- Post-selection: The circuit can continue only when both ancillas are found in the dark state |1⟩, so the experiment uses the outcome m = m′ = 1.
- Outcome verification: Testing all four ancilla-outcome combinations verifies the expected sign changes in the merging stabilizers.
- Logical outcome: After splitting, the implemented operation I+(−1)m+m′XA produces |φ+⟩ or |φ−⟩ depending on the ancilla outcomes.
A.VII. Post-selected stabilizer measurements
Post-selected stabilizer measurements reduce the usable data because only selected ancilla outcomes are retained, with survival probabilities depending on the surgery boundary and measurement stage.
- Merging: Rough merging theoretically retains 25 % of measurements because two ancilla outcomes are measured and only one outcome is used.
- Merging: Smooth merging theoretically retains 50 % of measurements because only one ancilla is used.
- Splitting: Splitting also has a theoretical survival probability of 50 % when stabilizer information is mapped onto one ancilla.
- Experimental effects: Spectroscopic decoupling imperfections can lower survival probabilities while increasing fidelity by detecting certain decoupling errors.
- Future improvement: Re-cooling and state-preparation techniques could eliminate the survival-probability reduction.
A.VIII. Error Detection
The utilized surface code is a distance-(2, 2) error detection code whose available stabilizer measurements detect some single-qubit errors but not all multi-qubit errors.
- The code uses four data qubits and has distance (2, 2), allowing theoretical detection of arbitrary single-qubit errors.
- Detecting all single-qubit errors requires measuring all three code stabilizers of one logical qubit with additional ancilla qubits.
- In the Z basis, stabilizers S1 = Z1Z2 and S2 = Z3Z4 detect single-qubit errors on any of the four data qubits.
- In the X basis, S3 = X1X2X3X4 detects single-qubit errors, while Y-basis checks detect errors only on data qubits 3 and 4.
- The code generally cannot detect two-qubit or multi-qubit errors, so measurements with erroneous stabilizer values are discarded.
A.IX. Additional Measurements
Additional measurements examine lattice-surgery Bell-state generation across rough and smooth boundaries for varied inputs. The experiment implements I + XA_L, generating three of four logical Bell states, with fidelity differences attributed in one case to suspected calibration problems.
- A.IX. Additional Measurements: Lattice surgery merges and splits logical qubits, corresponding to the operation I ± XA_L.
- A.IX. Additional Measurements: The experiment implements I + XA_L along the rough boundary and therefore generates three out of four logical Bell states.
- A.IX. Additional Measurements: Bell-state fidelity is estimated from three common stabilizer expectation values, while post-selected fidelity and survival probability are also reported.
- A.IX. Additional Measurements: The additional measurements cover varied input states along rough and smooth boundaries, with results summarized in Fig. A.7 and Table III.
- A.IX. Additional Measurements: The |−A_L⟩ input produces significantly lower Bell-state fidelity than the other inputs, which the authors suspect results from bad calibration.
- A.IX. Additional Measurements: Stabilizer magnitudes differ because stabilizers involve different numbers of physical qubits and physical qubits experience unequal exposure to error-prone gates.