Entangled logical qubits on a superconducting quantum processor

The paper in detail

ibm_phoenix is a 120-qubit square lattice with a new unconditional fast reset. Calibration maps show a few poor qubits and couplers inside the distance-5 footprint. Ancilla re-purposing removes them: the stabilizer that a bad ancilla would measure is inferred from two weight-2 gauge checks measured by neighbours in alternate rounds, so syndrome extraction takes two rounds and some detectors span both. For each of the ten distance-3 patches inside the footprint, all four single-ancilla removals were tried, and every patch improved with exactly one of them.

Logical error per round is fitted over 50 rounds rather than the first few, because leakage on data qubits makes early rounds look better. The fit model is \(2S-1=Ap^m\), with error per round \((1-p)/2\). Belief-matching decoding is informed by a detector error model built from averaged circuit eigenvalue sampling and rescaled with daily calibrations. Logical \(Y\) memories show an error per round close to the sum of the \(X\) and \(Z\) rates, as predicted for independent flips.

Compressed lattice surgery merges adjacent codes without a buffer region. Either the two codes share the seam ancilla in alternate rounds, or one code takes it and the other measures its seam stabilizer with a re-purposed neighbour; the merged patch is \(3\times6\) instead of \(3\times7\). A logical \(XX\) measurement between two \(|0\rangle\) codes prepares a Bell state with fidelity above 93%, and the same circuit teleports cardinal states. With three codes, A–B Bell preparation followed by B–C surgery teleports states and gates (identity, or \(S^\dagger\) by choosing B’s readout basis) and transfers a Bell pair to non-neighbouring patches. That transfer has fidelity between 85.4% and 91.4%.

Stability experiments on a \(4\times4\) patch compare no reset, reset of ancillas only, and a walking circuit that swaps data and ancilla roles so that every qubit is reset every two rounds. Without reset the error stops falling as rounds accumulate, a signature of leakage. Ancilla reset helps but still levels off, while the walking circuit keeps improving: 8 failures in 50 million runs at 170 rounds. Simultaneously running two logical memories shows crosstalk with a lift of about 2 in conditional failure probability.

Transferable methods

1. Bounding Bell-state fidelity from two stabilizer expectations

   Measuring \(\langle YY\rangle\) on logical qubits is expensive and noisy here, so the fidelity with \(|\Phi^+\rangle\) is bracketed using only \(\langle XX\rangle\) and \(\langle ZZ\rangle\). Positivity of the Bell-basis populations limits the unknown \(\langle YY\rangle\).

   Editorial derivation: write \(x=\langle XX\rangle\), \(z=\langle ZZ\rangle\), \(y=\langle YY\rangle\). The Bell populations are \(\tfrac14(1+x-y+z)\) for \(\Phi^+\), \(\tfrac14(1-x+y+z)\) for \(\Phi^-\), \(\tfrac14(1+x+y-z)\) for \(\Psi^+\) and \(\tfrac14(1-x-y-z)\) for \(\Psi^-\). Nonnegativity gives \(|x-z|-1\le y\le1-|x+z|\), so \[\frac{x+z}2\le F_{\Phi^+}\le\frac{1+\min(x,z)}2\quad(x+z\ge0).\]

   Editorial check: the reported Bell-pair transfer bounds 85.4% and 91.4% correspond to \(\langle XX\rangle,\langle ZZ\rangle\approx0.828\) and 0.880. For 20,000 random two-qubit states, the bracket contained the true fidelity every time.

   How to reuse it

   Whenever one Pauli correlator of a two-qubit (or logical) state is costly to measure, report this bracket rather than assuming depolarizing noise. The bracket closes when the two measured correlators are equal, and it extends to GHZ-type states with stabilizer subsets.

2. Ancilla re-purposing with gauge checks

   When an ancilla or its couplers are bad, its weight-4 stabilizer can still be obtained as the product of two weight-2 checks measured by neighbouring ancillas. These anticommute with some neighbouring stabilizers, so measurement alternates between rounds and detectors are formed from products that commute.

   Round 1 measures the ordinary stabilizers except the missing one; the stabilizers that will anticommute with the new weight-2 checks become gauge operators. Round 2 measures the two weight-2 checks with re-purposed ancillas. Their product equals the missing weight-4 stabilizer and commutes with the products of the gauge operators, so detectors span two rounds (6- and 8-body).

   The fit used for every memory: \(2S-1=Ap^m\), single-round fidelity \((p+1)/2\), error per round \((1-p)/2\). Editorial check: independent \(X\) and \(Z\) flips predict a \(Y\) error per round of \(p_x+p_z-2p_xp_z=1.712\%\) from the measured 0.872% and 0.855%, against a measured 1.690%.

   How to reuse it

   On any planar code with fabrication defects, try removing each candidate ancilla in turn and measuring logical error per round before trusting decoder reweighting. The method preserves distance when defects are isolated, but each removal lengthens detectors and slows syndrome extraction, so it helps only when the excluded component is bad enough.

Source: Section II.1–II.4; Appendix A (noise model, crosstalk); Appendix B (ancilla re-purposing); Appendix C (fitting, Y basis); Appendices D–E (compressed lattice surgery, three-qubit protocols). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. It shows that defect-aware syndrome circuits are necessary on current large chips, since informing the decoder alone did not get below 1% per round, and that hardware reset removes the leakage floor seen in earlier IBM experiments. These are commercial-device demonstrations of the primitives needed for logical computation.

Limitations. Codes are distance 3 and 5 only, and the entangling results post-select on clean code-space preparation (yields about 40% for two qubits and 1.2% for Bell-pair transfer). Bell fidelities are bounds from \(\langle XX\rangle\) and \(\langle ZZ\rangle\), without \(\langle YY\rangle\). T states are prepared non-fault-tolerantly, and gates are checked by state tomography, not process tomography. Crosstalk roughly doubles correlated failures of neighbouring logical qubits, and only Clifford logic is demonstrated.

Description based on the full text of the linked arXiv version, read 2026-10-26; equations follow the paper’s notation.