Entangling logical qubits with lattice surgery

Surface code

• The surface code is a prominent QEC code with error thresholds of up to 1%.The code has a simple description in the stabilizer formalism.Source: Surface code · p. 1
• The minimal code encodes one logical qubit on a 2 × 2 bicolorable lattice, with face colors assigning X- or Z-type stabilizers.Logical X and Z operators anticommute with each other and commute with all stabilizers.Source: Surface code · p. 1
• ⟨|Si|⟩= 0.603(3) for the split codes, which are recovered in a logical Bell state |φ+⟩.The split preserves the eigenstate of XA L by measuring one boundary stabilizer and reusing ancilla A1.Source: Figure 1 · p. 2, Surface code · p. 2
• The excerpt reports an average stabilizer value of ⟨|Si|⟩=0.669(8) and mentions logical state fidelities, but truncates before providing a complete fidelity value.Source: Surface code · p. 2
• ⟨|Si|⟩= 0.868(4) for the two initially separate encoded surface codes.The reported error is calculated from individual stabilizer errors.Source: Figure 1 · p. 2, Surface code · p. 2
• A sign flip in a code stabilizer provides an error syndrome; for example, syndrome −1 for SA 1 detects an X1 or X2 error.The passage states that the surface code can in theory be scaled by scaling the lattice.Source: Surface code · p. 2

• The entangling operation merges two surface codes to measure a joint logical observable, then splits them while preserving the measured eigenstate.The merge-and-split procedure implements MXX, which can entangle logical qubits and teleport information.Source: Lattice surgery · p. 3
• The generated logical Bell state had fidelity 58.0(1.6)% raw and 75.3(1.6)% after post-selection; the raw fidelity exceeded the 50% separability limit by 5 sigma.The reported fidelity format gives uncorrected and post-selected values, respectively.Source: Lattice surgery · pp. 3–4
• Lattice surgery teleported logical information by measuring MXX, then measuring the original logical qubit in the Z basis and applying a conditional logical Pauli correction.The experiment demonstrated teleportation for input states |0A_L⟩, |1A_L⟩, and |+A_L⟩.Source: Lattice surgery · p. 4, Figure 3 · p. 5
• The experiment demonstrated both rough and smooth lattice surgery between logical qubits encoded in 4-qubit surface codes, using a 10-qubit ion-trap processor.The paper describes lattice surgery as a fault-tolerant interface between quantum memories and processors.Source: Lattice surgery · p. 4

Theoretical background

• Appendix A.I covers the stabilizer formalism, while Appendices A.II and A.III provide further details on the surface code and lattice surgery.Source: APPENDIX: SUPPLEMENTAL INFORMATION · p. 7

Experimental realization

• Appendix A.IV describes the experimental realization of smooth lattice surgery, and Appendix A.V gives details of the experimental circuits.Source: APPENDIX: SUPPLEMENTAL INFORMATION · p. 7

Measurements and error detection

• Appendices A.VI–A.VIII address ancilla readout, ancilla-measurement survival probabilities, and error detection; Appendix A.IX discusses measurements estimating logical Bell-state fidelities.Source: APPENDIX: SUPPLEMENTAL INFORMATION · p. 7

A.I. Stabilizer Quantum Error Correction

• The three-qubit repetition code encodes |0L⟩ as |000⟩ and |1L⟩ as |111⟩, corrects one X error by majority vote, but fails when two X errors occur.For example, |100⟩ could result from X1|0L⟩ or X2X3|1L⟩; the code assumes the single-qubit error is more likely.Source: A.I. Stabilizer Quantum Error Correction · p. 7
• Measuring the commuting stabilizers Z1Z2 and Z2Z3 yields syndromes that identify errors, enabling active correction without disturbing the logical information.An X1 error flips the first stabilizer outcome; syndrome analysis and majority vote distinguish it from the corresponding two-qubit error X2X3.Source: A.I. Stabilizer Quantum Error Correction · pp. 7–8
• Logical operators commute with all stabilizers but are not stabilizers; their minimum weight determines code distance and the code’s error-correction and detection limits.For this code, the minimum logical Z weight is 1, so d = 1, while the minimum logical X weight is 3; it corrects one X error and detects up to two.Source: A.I. Stabilizer Quantum Error Correction · p. 8

A.II. Surface Code

• Face stabilizers on a bicolored square lattice commute and leave one encoded qubit; boundary-connecting X and Z strings define its logical operators, with distance d=3.The lattice has n−1 independent stabilizers, and the shortest logical string crosses three vertices.Source: A.II. Surface Code · p. 8
• Continuous stabilizer measurements detect errors through syndrome sign changes; assuming fewer than (d−1)/2 errors, correction restores the +1 stabilizers without a logical error.Source: A.II. Surface Code · p. 8
• Lattice surgery implements logical gates and state teleportation through joint logical measurements, including an ancilla-assisted CNOT and smooth-surgery teleportation.Source: Figure A.2 · p. 9

A.III. Lattice Surgery

• LS projects two stabilizer codes onto a joint logical-Pauli eigenstate through a joint measurement, implemented by merging and then splitting the codes.Source: A.III. Lattice Surgery · p. 9
• A measurement-based scheme uses joint Pauli measurements to implement logical CNOT, Hadamard, and code teleportation; the discussion also allows Z-logical projections beyond smooth and rough surgery.A projection onto a logical Z eigenstate can be used to generate a logical Hadamard.Source: A.III. Lattice Surgery · p. 9
• For two 2 × 2 surface codes, merging creates an asymmetric surface-code lattice that encodes one logical qubit, with the measured logical operator included as a stabilizer.The measurement outcome is determined by the product of merging-stabilizer outcomes.Source: A.III. Lattice Surgery · p. 9
• Merging stabilizers are measured d times so measurement errors can be identified by comparing outcomes across time.A faulty measurement can produce a syndrome inconsistent with its expected value.Source: A.III. Lattice Surgery · p. 9
• Splitting measures the separated codes’ stabilizers along aligned boundaries, discards the merging stabilizers, and restores the two codes while preserving the measured eigenstate.The text notes that quantum error correction is required at the end for full fault tolerance.Source: A.III. Lattice Surgery · p. 9

Results Smooth Lattice Surgery

• Smooth lattice surgery rotates both codes by 90 degrees and merges along the upper or lower boundaries to measure a joint logical operator.This projects onto a joint eigenstate of Z^A_L Z^B_L.Source: A.IV. Results Smooth Lattice Surgery · pp. 9–10, Figure A.3 · p. 10
• For two 2 × 2 surface codes, measuring the merging stabilizer produces a 4 × 2 asymmetric surface code that can be split by discarding that stabilizer.Source: A.IV. Results Smooth Lattice Surgery · p. 9
• 63.9(2.8)|78.0(2.7)% Bell-state fidelity was obtained for |φ+_L⟩.The measured data are presented in Fig. A.4.Source: A.IV. Results Smooth Lattice Surgery · p. 9

A.V. Experimental Circuits

• The circuits use one 4-qubit MS gate to prepare |0L⟩, two 2-qubit MS gates for |+L⟩, and 6 3-qubit rough-boundary or 4 5-qubit smooth-boundary MS gates for stabilizer measurements.Local operations combine multi-qubit X or Y rotations with single-qubit Z rotations. |0L⟩ and |1L⟩ use the same number of gates on different qubits.Source: A.V. Experimental Circuits · p. 10
• Measured Bell-state fidelities were 58(2) % (rough) and 64(3) % (smooth), compared with expected fidelities of ∼63 % (rough) and ∼57 % (smooth).The authors suggest that the largest deviation may arise from error protection in a decoherence-free subspace, which was not included in the estimates.Source: A.V. Experimental Circuits · p. 10
• The rough-boundary circuit encodes logical qubits in |0L⟩, then merges and splits them using local rotations and multi-qubit MS gates.Decoupling and recoupling operations move qubits into or out of the computational subspace; the sequence includes an ancilla measurement after merging and a measurement of all qubits at its end.Source: A.V. Experimental Circuits · p. 10, Figure A.5 · p. 11

A.VI. Ancilla Readout

• Merging maps specific stabilizer information onto ancillas A1 and A2, which are measured simultaneously in a projective readout.Illuminating the ion string with 397 nm light and collecting fluorescence performs the ancilla-only measurement after the data qubits are decoupled from the computational subspace.Source: A.VI. Ancilla Readout · pp. 11–12
• Across the tested outcomes, the merged state remained unchanged, and splitting generated |φ+⟩ or |φ−⟩ according to m and m′.The splitting operation depended on the parity of m and m′: I + (−1)^(m+m′)X_A^L.Source: Figure A.6 · p. 12, A.VI. Ancilla Readout · p. 13
• Because PMT readout identifies the number of bright ions but not their identities and scatters light that heats the chain, the algorithm proceeded only when both ancillas were dark.Without in-sequence cooling and state preparation, the scattered 397 nm light left qubits partly outside the computational subspace, preventing high-fidelity gates after detection.Source: A.VI. Ancilla Readout · p. 12
• Inverting ancilla states before measurement enabled tests of all four outcome combinations; the measured stabilizer SM7 changed sign as expected.The experiment’s algorithm data used only m = m′ = 1, so the other outcomes were investigated by inverting the individual ancilla states before measurement.Source: A.VI. Ancilla Readout · p. 13

A.VII. Post-selected stabilizer measurements

• Table II summarizes survival probabilities by measurement basis and input state after merging and splitting, with 9000 measurements per basis and input state.The caption defines SPM i after merging, SPS i after splitting, and ⟨SPi⟩ as their average; it supplies no measured values here.Source: Table II · p. 13
• Theoretical survival probability is 25% for rough-boundary merging, 50% for smooth-boundary merging, and 50% when splitting the logical qubit.Rough-boundary merging retains one of four ancilla outcomes; smooth-boundary merging uses one ancilla, and splitting maps stabilizer information onto one ancilla.Source: A.VII. Post-selected stabilizer… · p. 13
• Imperfect spectroscopic decoupling can lower survival probability while increasing fidelity by detecting some decoupling errors.Source: A.VII. Post-selected stabilizer… · p. 13
• Introducing re-cooling and state-preparation techniques into the apparatus could eliminate the survival-probability reduction.Source: A.VII. Post-selected stabilizer… · p. 13

A.VIII. Error Detection

• Post-selection increased fidelities by 28(3) % with average survival probabilities (SPs) of 64(4) %.Source: A.VIII. Error Detection · p. 13
• Z-basis checks S1 and S2 detect any single-qubit data error, X-basis check S3 detects any, and Y-basis check S2 detects errors only on qubits 3 and 4.The code has four data qubits and distance (2, 2).Source: A.VIII. Error Detection · p. 13
• Detecting arbitrary single-qubit errors requires all three stabilizers and additional ancillas; the implemented checks generally cannot detect two-qubit or multi-qubit errors.The work focuses on lattice-surgery processing capabilities and leaves full error detection to future investigations.Source: A.VIII. Error Detection · p. 13
• Undetected multi-qubit errors from MS gates leave SPs below unity; Y-basis checks yield higher SPY, while the five-qubit smooth-merge MS gate introduces more multi-qubit errors than the two three-qubit rough-merge gates.The merge difference is reflected in the X-basis survival probabilities.Source: A.VIII. Error Detection · p. 13

• The implemented smooth I + XA_L operation generated three of the four logical Bell states.The full merge-and-split procedure corresponds to I ± XA_L, but the experiment implemented only I + XA_L.Source: A.IX. Additional Measurements · p. 13
• Table III reports individual code-stabilizer values, mean absolute stabilizer value, Bell-state fidelities before and after post-selection, survival probabilities, and single-shot counts per basis.The Bell-state fidelity is estimated from measurements of the three common stabilizers.Source: A.IX. Additional Measurements · pp. 14–15, Table III · p. 15
• For rough-boundary lattice surgery, the |−_A L⟩ input produced significantly lower Bell-state fidelity than other inputs, possibly because of bad calibration.Results for the additional input-state experiments are shown in Fig. A.7 and Table III.Source: Figure A.7 · p. 14, A.IX. Additional Measurements · p. 15, Table III · p. 15