Modular fault-tolerant quantum computing on a non-CSS code

AI summary

Overview

Research area: Experimental quantum error correction and fault-tolerant quantum computing, specifically modular logical operations on a non-CSS stabilizer code implemented on a trapped-ion processor.

Technical level: Advanced. The paper assumes familiarity with stabilizer codes, transversal gates, flagged syndrome extraction, magic-state injection and quantum process tomography.

Scope (one sentence): The paper demonstrates, on a 16-qubit trapped-ion device, the full set of logical primitives needed for modular universal fault-tolerant computing with the non-CSS [[5,1,3]] code, including logical state teleportation, magic-state preparation and injection, and SPAM-removed logical process characterization.

What This Paper Is About

Fault-tolerant quantum computing usually relies on CSS codes such as the [[7,1,3]] code, whose structure makes logical measurements and entangling operations comparatively easy. Non-CSS codes can encode the same distance more compactly — the [[5,1,3]] code needs only 5 physical qubits per logical qubit — but they lack transversal CNOT gates and independent X/Z error correction, so the logical toolbox had not been demonstrated experimentally. This paper builds and demonstrates that missing toolbox on a trapped-ion quantum computer, and uses it to teleport logical states between two separate code blocks with real-time classical feed-forward.

Key Contributions

1. First experimental implementation of modular universal logical quantum computation with a non-CSS code. All logical operations required for modular fault-tolerant universal computing with the [[5,1,3]] code are implemented on a trapped-ion quantum computer, and logical state teleportation between two non-CSS code blocks is demonstrated without physically relocating or directly coupling the encoded registers.

2. A complete fault-tolerant logical toolbox for the [[5,1,3]] code. This includes logical Pauli-state preparation, QEC with real-time feedback, fault-tolerant logical measurement, a new circuit for fault-tolerant magic-state preparation (verified by inserting all possible single faults), a pieceably fault-tolerant logical CY_L entangling gate built from a round-robin CZ_L construction, and magic-state injection realizing the non-Clifford gate T^{XZ}_L = e^{-iπY_L/8}.

3. Quantification of logical spectator error. The first measurement, in the 5-qubit code, of how an idling logical qubit degrades while fault-tolerant circuits run on a distinct logical register; one FT QEC cycle on the neighbouring qubit increases the spectator's state infidelity by roughly 20%.

4. Logical Pauli quantum process tomography with reduced sampling. A scheme that separates state-preparation and readout errors from the error of the logical operation itself, exploiting the fact that syndrome extraction projects noise onto logical Pauli channels — reducing the parameter count from 4^5 − 1 = 1023 physical parameters to 4 parameters for a single-qubit logical operation.

Main Findings

• Logical SPAM baseline: Fault-tolerant preparation plus fault-tolerant measurement of a single logical qubit over six cardinal states gives an average infidelity of 5.24^{+0.16}{-0.14}% with an 80% acceptance rate; the numerically simulated error model predicts 6.16^{+0.08}{-0.08}% at 77% acceptance.

• Quantum memory: Inserting one round of fault-tolerant QEC between preparation and measurement raises the average infidelity to 26.8^{+0.3}{-0.3}% at 75% acceptance (simulation: 21.9^{+0.3}{-0.3}% at 77%). The authors attribute the limitation mainly to two-qubit gate errors.

• Spectator error: Performing one FT QEC cycle on one logical qubit increases the state infidelity of an idling second logical qubit by roughly 20%, explained by dephasing of idling qubits and mid-circuit-measurement-induced errors; all gates run sequentially, so parallel execution could substantially reduce idling times.

• Teleportation-building-block SPAM: Using error detection rather than error correction for preparation and measurement yields an average infidelity of 0.35^{+0.07}{-0.05}% at 46% acceptance (simulation: 0.20^{+0.03}{-0.02}% at 39%) — lower infidelity but lower acceptance than the correction-based baseline.

• Two-logical-qubit error detection: After preparing two logical qubits and applying one round of quantum error detection to each, followed by a fault-tolerant measurement of one, the average logical infidelity is 2.9^{+0.8}{-0.5}% at 0.85% acceptance (simulation: 3.3^{+0.7}{-0.5}% at 1.15%).

• Logical state teleportation: Averaged over three input states, teleportation achieves a logical infidelity of 21^{+7}{-5}% at 0.035% acceptance (simulation: 13^{+4}{-3}% at 0.044%). The infidelity is mainly limited by mid-circuit measurement errors. Pauli corrections are applied by real-time feedback.

• Magic-state preparation: The prepared state |H_XZ⟩_L = cos(π/8)|0⟩L + sin(π/8)|1⟩L reaches a logical state infidelity of 4^{+3}{-2}% (simulation: 3.4^{+1.0}{-1.0}%) at an acceptance rate of 2.3% experimentally versus 7.3% in simulation.

• Pieceably fault-tolerant entangling gate: The logical CY_L gate, built from three layers of CZ gates with an intermediate error-detection block after the second layer plus transversal S_L†H_L and H_LS_L, produces maximally entangled logical states with an average infidelity of 10^{+3}{-2}% at 0.36% acceptance (simulation: 6.6^{+1.0}{-1.0}% at 0.67%).

• Magic-state injection: The full injection protocol applied to |+⟩L gives a logical state infidelity of 4^{+16}{-3}% at 0.023% acceptance, with uncertainties from parametric bootstrapping over 1000 repetitions. Logical state tomography of T^{XZ}_L|+⟩L gives a state fidelity of 96^{+3}{-16}%, and the process matrix determined by the logical tomography gives a process fidelity of 85 ± 8%.

• Code structure matters: The [[5,1,3]] code measures XZZX-type stabilizers (for example S₁ = X₁Z₂Z₃X₄I₅) and requires a dedicated non-transversal fault-tolerant logical measurement: three distinct representations of the logical operator are mapped onto flagged auxiliary qubits, followed by unencoding and destructive readout of the data qubits, with a single flag qubit rather than two.

• Not reported in the supplied content: The entanglement-fidelity-to-average-fidelity conversion formula (Eq. 10 and following) is truncated, and the per-experiment accepted shot counts appear only in App. Tab. 6, which is not included.

Methodology in Plain English

The team encoded one logical qubit into five physical qubits using the [[5,1,3]] code, where each physical qubit is a single trapped ⁴⁰Ca⁺ ion in a 16-ion chain. Gates come from laser-driven single-qubit rotations and Mølmer–Sørensen two-qubit interactions between any chosen ion pair, with Z rotations applied virtually; the device supports mid-circuit measurement and outcome-conditional operations inside a single circuit execution.

To catch errors without spreading them, they use "flagged" circuits: an extra auxiliary qubit is coupled so that dangerous errors propagating from the ancilla are heralded. To keep hardware requirements down, one flag qubit is shared across a pair of stabilizers under a specific gate ordering. Error handling then follows lookup tables — a standard table when no flag is raised and a flag-based table otherwise.

Fault-tolerant logical measurement is harder than in CSS codes, so three separate weight-3 representations of the logical operator are each mapped onto flagged ancillas; if the three agree and no flag is raised the value is accepted, otherwise the outcome is corrected using the extracted syndrome. Logical teleportation is built from two logical measurements — first X_L^(S)X_L^(T), then Z_L^(S) — where each logical operator is measured at least twice, with intermediate QEC rounds, and classically controlled Pauli corrections are applied to the target in real time. For the deeper protocols the team postselects on trivial syndromes and unraised flags to keep the decision tree tractable.

Non-Clifford capability comes from preparing a verified magic state and injecting it via a pieceably fault-tolerant entangling gate: a non-transversal operation is broken into pieces that are each fault-tolerant up to correctable errors, with error detection inserted between pieces. Finally, to remove SPAM contamination from the reported operation quality, they perform tomography at the logical level rather than over all five physical qubits, assuming a Pauli-noise model so that the process matrix is diagonal and the noise map can be inverted out of the estimated transfer matrix.

Why This Matters

Impact on research. Every prior modular or lattice-surgery-style logical demonstration used topological or other CSS codes, where code structure simplifies logical measurement and entangling operations. This work shows that the same architectural primitives — teleportation between code blocks via auxiliary registers, verified magic states, pieceable entangling gates — can be carried out on a non-CSS code with a fraction of the physical qubits per logical qubit, and it supplies the concrete circuits, lookup tables and fault-tolerance verification that the [[5,1,3]] code previously lacked.

Real-world applications (as pathways implied by the paper):

• Modular quantum processors in which separate small qubit modules are linked only through auxiliary registers or shuttling, rather than requiring all-to-all physical connectivity.
• Compact logical memories for devices with tight qubit budgets, since the [[5,1,3]] code uses 5 physical qubits where the smallest error-correcting CSS code, [[7,1,3]], uses 7 for the same distance.
• Distributed or networked quantum computing, where logical teleportation moves quantum information between zones without physically moving encoded registers.
• Benchmarking and validation tooling for quantum hardware, via the SPAM-free logical process tomography demonstrated here.

Industry relevance. The work targets the practical engineering question of how to scale trapped-ion processors built by academic groups and companies: modularity trades long-range connectivity requirements for classical feed-forward and shared entanglement, and real-time mid-circuit feedback with conditional corrections — demonstrated here — is a prerequisite for any such architecture. The reported error budgets (two-qubit gate errors limiting the memory, mid-circuit measurement errors limiting teleportation and the entangling gate) give hardware developers concrete targets.

Future Directions

• Reduce mid-circuit measurement and idling errors. Teleportation, the logical CY_L gate and magic-state injection are all dominated by mid-circuit measurement errors and dephasing of idling qubits; the authors note that all gates are executed sequentially and that parallel execution could substantially reduce idling times.
• Replace postselection with full real-time decision logic. The deep protocols rely on postselection — acceptance rates fall to 0.035% for teleportation and 0.023% for magic-state injection — so moving the complete lookup-table decision tree into real-time classical control is the obvious next step.
• Scale modularity beyond two code blocks. Teleportation is demonstrated between two [[5,1,3]] blocks; extending to networks of modules connected through auxiliary registers, and combining with quantum LDPC codes for higher encoding rates, remains open.
• Make logical tomography practical at the full-encoding level. The authors present a model-independent scheme to characterize the entire encoding circuit in App. H but state that its overhead prevents application to the current [[5,1,3]] experiments, leaving a resource-efficient SPAM-resolved tomographic protocol as an open problem.

Target Audience

Experimentalists and theorists working on quantum error correction and fault-tolerant architectures — particularly groups building trapped-ion, superconducting or neutral-atom processors who need concrete logical gate constructions for non-CSS codes. It is also valuable for architects designing modular or distributed quantum computers, and for readers who already know stabilizer formalism and want a detailed, quantitative experimental reference for the [[5,1,3]] code's complete logical gate set and its error budget.

Authors’ abstract

Modularization promises to break down the design and implementation complexity of large scale quantum processors into smaller manageable subtasks. In this approach, quantum channels, realized for instance through physical rerouting of qubits or quantum teleportation, connect multiple modules. Each of those modules hosts a subset of qubits, e.g. multiple logical qubits, and provides quantum operations on them. In this work, we implement for the first time all logical operations required for modular fault-tolerant universal quantum computing with a non-Calderbank-Shor-Steane (CSS) code, the perfect $[[5, 1, 3]]$ code, on a trapped-ion quantum computer. This code is the smallest quantum error-correcting (QEC) code capable of correcting any single-qubit error, making it a compact alternative to larger CSS codes. We demonstrate logical state teleportation and a full suite of fault-tolerant operations required for universal logical control, including logical state preparation, QEC with real-time feedback, logical measurements, magic-state preparation, logical entangling operations, and magic-state injection. Moreover, we characterize the logical spectator error picked up by idling logical qubits during quantum operations on distinct qubit registers and demonstrate a logical Pauli quantum process tomography that minimizes required sampling resources for logical tomography.