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arxiv: 2412.01446 · v3 · submitted 2024-12-02 · 🪐 quant-ph

Magic State Injection on IBM Quantum Processors Above the Distillation Threshold

Younghun Kim , Martin Sevior , Muhammad Usman This is my paper

Pith reviewed 2026-05-23 07:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords magic state injectionsurface codequantum error correctionfault tolerancelogical magic statespost-selectionIBM quantum processors
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The pith

Logical magic states are prepared above the distillation threshold on IBM quantum processors via a rotated surface code.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a qubit-efficient embedding of the surface code into IBM hardware allows post-selected injection of logical magic states. Fidelities for the |H_L> and |T_L> states reach 0.8806 and 0.8665, exceeding the level needed for distillation into higher-fidelity resources. This matters because non-Clifford operations require such states for universal fault-tolerant computation. The reported thresholds for logical bit-flip and phase-flip errors are higher than those obtained with standard embeddings. The work also establishes a lower bound on fidelity for arbitrary injected logical states.

Core claim

By employing a rotated heavy-hexagonal surface code on IBM processors, the post-selection-based magic state injection protocol prepares logical magic states |H_L⟩ and |T_L⟩ with fidelities of 0.8806±0.0002 and 0.8665±0.0003, both above the distillation threshold, while achieving error thresholds of approximately 0.37% for bit-flip and 0.31% for phase-flip errors.

What carries the argument

The qubit-efficient rotated heavy-hexagonal surface code embedding combined with post-selection during magic state injection.

If this is right

  • Non-Clifford logical gates can be realized by distilling the prepared states into higher-fidelity resources.
  • The improved error thresholds indicate the rotated embedding tolerates more noise than traditional lattice embeddings.
  • Arbitrary single logical qubit states can be injected with fidelity no lower than 0.8356.
  • Fault-tolerant quantum computation becomes more feasible on devices with connectivity constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This embedding strategy may lower overall qubit overhead when scaling surface codes to larger distances.
  • Similar post-selection techniques could be adapted to test magic state distillation on the same hardware.
  • The reported thresholds suggest hardware-specific lattice choices can outperform generic embeddings in near-term devices.

Load-bearing premise

The post-selection criteria and noise model assumptions accurately reflect the underlying physical error processes without unaccounted bias from hardware-specific embedding or readout errors.

What would settle it

An experiment repeating the injection with relaxed post-selection or on hardware yielding logical fidelities below 0.85 would show the states fall short of the distillation threshold.

Figures

Figures reproduced from arXiv: 2412.01446 by Martin Sevior, Muhammad Usman, Younghun Kim.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a and b show the logical error rates for the two states, either in the X (|+L⟩) basis or in Z (|0L⟩) basis, corresponding to the probability of logical Z and X errors, respectively. The logical error rates are calculated under various physical error rates (p) ranging from 5 × 10−4 to 10−2 and code distances (d) ranging from 3 to 15. Although the number of qubits required for the rotated code scales with th… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavy-hexagonal surface code for IBM quantum processors (\texttt{ibm\_fez}) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of $\approx0.37\%$ and $\approx0.31\%$, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states $|H_L\rangle$ and $|T_L\rangle$ achieve fidelities of $0.8806\pm0.0002$ and $0.8665\pm0.0003$, respectively, which are both above the magic state distillation threshold. Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as $0.8356\pm0.0003$. Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports an experimental implementation of magic state injection for logical |H_L⟩ and |T_L⟩ states using a qubit-efficient rotated heavy-hexagonal surface code embedded on the IBM ibm_fez processor. It claims logical error thresholds of ≈0.37% (bit-flip) and ≈0.31% (phase-flip), post-selection fidelities of 0.8806±0.0002 and 0.8665±0.0003 (both above the distillation threshold), and a minimum fidelity of 0.8356±0.0003 for arbitrary injected logical states.

Significance. If the post-selection protocol is free of systematic bias, the work demonstrates a concrete advance in preparing high-fidelity logical magic states on connectivity-constrained hardware, with reported thresholds exceeding those from prior embeddings. The direct hardware measurements and explicit numerical claims constitute a falsifiable experimental result that could inform near-term fault-tolerance roadmaps.

major comments (2)
  1. [Results section on post-selected state preparation] The central fidelity claims (abstract and results section) rest on post-selection applied to the rotated heavy-hex injection circuit; the manuscript must explicitly define the post-selection criteria, show they are independent of readout errors and embedding connectivity, and provide an independent cross-check (e.g., full tomography or alternative decoder) to confirm the conditional fidelities are not inflated. Without this, the assertion that both values exceed the distillation threshold cannot be verified from the reported data alone.
  2. [Threshold extraction and comparison] § on threshold extraction: the reported logical error thresholds (≈0.37% and ≈0.31%) are extracted from physical error rates on ibm_fez; the fitting procedure, number of data points, and any assumptions about the noise model (including how the heavy-hex embedding affects syndrome extraction) must be detailed so that the improvement over traditional embeddings can be reproduced and assessed for robustness.
minor comments (2)
  1. [Abstract] The abstract states a minimum fidelity of 0.8356±0.0003 for arbitrary states but does not specify the sampling method over the Bloch sphere or the number of tested states; this detail belongs in the methods or supplementary material.
  2. [Figures and captions] Figure captions and circuit diagrams should explicitly label the post-selection windows and the rotated heavy-hex lattice mapping to allow readers to assess the embedding efficiency claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and will incorporate the necessary clarifications and details in the revised version to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Results section on post-selected state preparation] The central fidelity claims (abstract and results section) rest on post-selection applied to the rotated heavy-hex injection circuit; the manuscript must explicitly define the post-selection criteria, show they are independent of readout errors and embedding connectivity, and provide an independent cross-check (e.g., full tomography or alternative decoder) to confirm the conditional fidelities are not inflated. Without this, the assertion that both values exceed the distillation threshold cannot be verified from the reported data alone.

    Authors: We acknowledge that the post-selection criteria require explicit definition for full reproducibility and verification. In the revised manuscript, we will add a dedicated subsection detailing the post-selection protocol, including the specific criteria used in the rotated heavy-hexagonal surface code embedding. We will include analysis demonstrating that the criteria are independent of readout errors by comparing with and without post-selection under varying noise models, and address embedding connectivity effects through additional simulations. For an independent cross-check, we will provide results from an alternative decoder or partial tomography where feasible on the hardware. These additions will allow verification that the reported fidelities of 0.8806±0.0002 and 0.8665±0.0003 indeed exceed the distillation threshold without inflation. revision: yes

  2. Referee: [Threshold extraction and comparison] § on threshold extraction: the reported logical error thresholds (≈0.37% and ≈0.31%) are extracted from physical error rates on ibm_fez; the fitting procedure, number of data points, and any assumptions about the noise model (including how the heavy-hex embedding affects syndrome extraction) must be detailed so that the improvement over traditional embeddings can be reproduced and assessed for robustness.

    Authors: We agree that the threshold extraction procedure needs more detailed exposition to enable reproduction. In the revised manuscript, we will expand the section on threshold extraction to include: the exact fitting procedure (e.g., linear or polynomial fit to logical error rate vs physical error rate), the number of data points used from the ibm_fez processor, and explicit assumptions in the noise model. We will also discuss how the heavy-hex embedding influences syndrome extraction, including any modifications to the standard surface code decoding. This will clarify the robustness of the reported thresholds (≈0.37% bit-flip and ≈0.31% phase-flip) and the improvement over prior embeddings. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental measurements on hardware

full rationale

The paper reports direct experimental fidelities (0.8806±0.0002, 0.8665±0.0003) and error thresholds (≈0.37%, ≈0.31%) extracted from post-selected runs on ibm_fez hardware using a rotated heavy-hex surface code. These are measured quantities, not quantities derived from the paper's own equations or reduced to fitted inputs by construction. No self-citation chains, ansatzes, or uniqueness theorems are invoked as load-bearing steps for the central claims. The derivation chain consists of circuit execution and data analysis on physical hardware, which is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central experimental claims rest on standard quantum error correction assumptions and the validity of the surface-code error model on the specific hardware; no additional free parameters or invented entities are introduced beyond the experimental setup itself.

axioms (1)
  • domain assumption The surface code error model and post-selection protocol accurately capture the dominant noise processes on ibm_fez without significant unmodeled effects.
    Invoked to interpret measured thresholds and fidelities as logical quantities.

pith-pipeline@v0.9.0 · 5763 in / 1241 out tokens · 39628 ms · 2026-05-23T07:56:00.568720+00:00 · methodology

discussion (0)

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