This paper applies the Quantum Operations Model (QOM)¹ cost framework to the five mostconsequential developments in quantum computing from late 2025 through early 2026: theGoogle/Oratomic cryptographic resource estimate compression, Microsoft's contestedtopological qubit claims, IBM's qLDPC-based Starling architecture roadmap, the first deliveriesof error-corrected "level-2" quantum machines (systems implementing real-time quantum errorcorrection on physical hardware), and emerging continuous-variable (CV) error correctiontechniques that bypass Gottesman-Kitaev-Preskill (GKP) state preparation. We ask: how dothese recent advances alter the cost structure of fault-tolerant quantum computation as modeledby QOM?The answer is nuanced. Algorithmic improvements — particularly in Shor's algorithmcompilation and quantum LDPC error correction codes — have compressed the gap betweencurrent hardware and useful fault-tolerant computation more than any single hardware advance,though these algorithmic results carry their own assumptions that require scrutiny. The QECoverhead,² identified in prior work as the dominant cost factor in quantum computation, is beingrenegotiated through new codes but has not yet been reduced in practice on any shippinghardware. Meanwhile, the topological qubit program suffered a significant credibility correctionwhen replication studies published in Science found simpler explanations for previouslycelebrated experimental signatures. Updated QOM cost tables are provided for all four majorarchitectures under both legacy surface-code and projected qLDPC overhead assumptions, withsensitivity ranges across physical error rates from 10⁻³ to 10⁻⁴. The central thesis of the QuantumAudit series — that quantum computing's near-term value lies in capability rather than costreduction — remains intact, but the timeline to that capability shortened meaningfully this year.¹ The Quantum Operations Model (QOM) is an original cost-accounting framework developed bythe author across three prior papers in this series: "Time Is Money: Wall-Clock Economics ofFault-Tolerant Quantum Computation," "The Decoder Is Architecture: Classical Bottlenecks inFault-Tolerant Quantum Systems," and "Show Me the Numbers: Concrete Costs of QuantumOperations." All three are available on Zenodo and Academia.edu. The QOM source code isavailable at github.com/kase1111-hash.² Throughout this paper, "QEC overhead" refers to the combined physical-to-logical qubit ratioand the code-distance time multiplier required to achieve a target logical error rate — thequantity introduced in the first Quantum Audit paper as the "error correction tax."
This paper applies the Quantum Operations Model (QOM)¹ cost framework to the five mostconsequential developments in quantum computing from late 2025 through early 2026: theGoogle/Oratomic cryptographic resource estimate compression, Microsoft's contestedtopological qubit claims, IBM's qLDPC-based Starling architecture roadmap, the first deliveriesof error-corrected "level-2" quantum machines (systems implementing real-time quantum errorcorrection on physical hardware), and emerging continuous-variable (CV) error correctiontechniques that bypass Gottesman-Kitaev-Preskill (GKP) state preparation. We ask: how dothese recent advances alter the cost structure of fault-tolerant quantum computation as modeledby QOM?The answer is nuanced. Algorithmic improvements — particularly in Shor's algorithmcompilation and quantum LDPC error correction codes — have compressed the gap betweencurrent hardware and useful fault-tolerant computation more than any single hardware advance,though these algorithmic results carry their own assumptions that require scrutiny. The QECoverhead,² identified in prior work as the dominant cost factor in quantum computation, is beingrenegotiated through new codes but has not yet been reduced in practice on any shippinghardware. Meanwhile, the topological qubit program suffered a significant credibility correctionwhen replication studies published in Science found simpler explanations for previouslycelebrated experimental signatures. Updated QOM cost tables are provided for all four majorarchitectures under both legacy surface-code and projected qLDPC overhead assumptions, withsensitivity ranges across physical error rates from 10⁻³ to 10⁻⁴. The central thesis of the QuantumAudit series — that quantum computing's near-term value lies in capability rather than costreduction — remains intact, but the timeline to that capability shortened meaningfully this year.¹ The Quantum Operations Model (QOM) is an original cost-accounting framework developed bythe author across three prior papers in this series: "Time Is Money: Wall-Clock Economics ofFault-Tolerant Quantum Computation," "The Decoder Is Architecture: Classical Bottlenecks inFault-Tolerant Quantum Systems," and "Show Me the Numbers: Concrete Costs of QuantumOperations." All three are available on Zenodo and Academia.edu. The QOM source code isavailable at github.com/kase1111-hash.² Throughout this paper, "QEC overhead" refers to the combined physical-to-logical qubit ratioand the code-distance time multiplier required to achieve a target logical error rate — thequantity introduced in the first Quantum Audit paper as the "error correction tax."
This paper is the six-month follow-up to *The 2026 Quantum Audit* (April 2026), applying the Quantum Operations Model (QOM) cost framework to what changed in quantum computing between April and September 2026. The two events April named as the next audit's triggers — IBM's Kookaburra processor and the Magne neutral-atom machine — have not fired. Five other things did. Quantinuum ran a heterogeneous-code architecture (Helix) on hardware: a memory code, a compute code, and an interface between them on one machine, decoded offline. The industry adopted reliable logical operations as its unit — proposed as rQOPS and QLOPS in 2023–2025, named by Preskill's megaquop, specified by the US Department of Energy's "hard operations," and, in Sandia's QUOPS benchmark, measured for the first time on logical hardware. NVIDIA shipped a compiler that emits fault-tolerant resource estimates. DOE published a procurement specification for a quantum computer — Genesis Q, with up to $215 million in appropriation-dependent milestone pools for machines of at least 100 logical qubits performing an illustrative 10⁵ "hard" operations, evaluated in September 2028 — that is the same width and, over one run, the same total operation volume as the QOM reference computation, at a tighter error target. And Microsoft's topological program produced a second contested preprint. Two new instruments are introduced. A Von Neumann scorecard asks, for each of seven components a computer must have — memory, logic, control unit, I/O, bus, compiler, clock — whether any platform has demonstrated it below break-even, with dates. Every component has at least one public demonstration somewhere; on this report's criteria five of the seven rows have a Pass on some platform and two — the control unit, and sustained operation at the logical level — have none; no platform has more than three Passes. And a price per megaquop, from the identity C_megaquop = 10⁶ · w_sys / (U · Ω_L), where w_sys is the machine's rate in dollars per second, Ω_L its throughput in reliable logical operations per second, and U the fraction of machine time that produces the buyer's operations. On the only logical rate measured on commercially accessible hardware that we identified (Helios-1, 4.9 logical QUOPS per second, at a width of four and a completed circuit size of 40), a megaquop is a derived 57 hours — a lower bound, since the benchmark's accuracy threshold sits far above megaquop-grade reliability — and, against separately assembled cross-platform system-rate proxies, an estimated $141,000–408,000 at list prices for dedicated machine time or $61,000–184,000 at an operator's cost, with a second route from per-gate list prices giving $0.6–1.5 million; the modeled fault-tolerant machines of April's tables price it at three cents to about $550 and seconds to minutes. The distance between the two is two and a half to five orders of magnitude in price and three to four in time; the time gap is entirely the throughput term, and the system-rate term narrows the price gap rather than widening it. The April thesis holds: the QEC overhead is the dominant modeled cost, this audit identifies no independently replicated advantage over the best classical method on a workload with a classical solution in production use, and quantum computing's near-term value lies in capability rather than cost reduction. The verdict in one line: the parts exist, no platform yet satisfies the eight-row criterion for a computer, and the invoice can now be estimated per operation from a measured rate. Three errors in the April paper are corrected.
Quantum computing has reached a pivotal inflection point with the recent demonstration ofbelow-threshold quantum error correction and the emergence of logical qubits that outperformtheir physical counterparts. This review synthesizes the remarkable progress achieved between2024 and 2026 across multiple quantum hardware platforms, including superconducting circuits,trapped ions, and neutral atom arrays. We examine the theoretical foundations of quantumerror correction codes, with particular emphasis on surface codes and the rapidly advancingquantum low-density parity-check (qLDPC) codes. Key experimental milestones are analyzed,including Google’s Willow chip demonstration of exponential error suppression, IBM’s modularfault-tolerant roadmap culminating in the Starling system, IonQ’s achievement of 48 logicalqubits with a 2:1 encoding ratio, and QuEra’s record-efficiency neutral atom error correctionapproaching the Teraquop regime. The review further explores critical enabling technologiessuch as real-time decoding architectures, magic state distillation protocols, and fault-tolerant gateimplementations. We discuss the remaining challenges in scaling to hundreds of logical qubits,achieving practical logical error rates of 10−10 to 10−15, and realizing end-to-end fault-tolerantalgorithms. By synthesizing these developments, this article provides a comprehensive assessmentof the field’s trajectory toward practical fault-tolerant quantum computing and identifies the keymilestones that will define the next phase of progress.
Quantum error correction (QEC) represents the cornerstone technology for realizing fault-tolerant quantum computing, addressing the fundamental challenge of quantum state decoherence in noisy intermediate-scale quantum (NISQ) devices. This comprehensive review examines recent advances in QEC implementations from 2020-2024, spanning theoretical foundations to practical hardware demonstrations. We analyze the evolution from pioneering stabilizer codes to modern topological approaches, with particular emphasis on surface codes and low-density parity-check (LDPC) codes that have demonstrated below-threshold error correction in superconducting and trapped-ion systems. The paper provides a comparative analysis of classical and machine learning-based decoder algorithms, evaluating their performance through complexity analysis and threshold comparisons in real-time error correction scenarios. Through examination of breakthrough experiments by Google, IBM, Quantinuum, and emerging platforms, we assess the current state of fault-tolerant quantum computing and identify critical engineering challenges including correlated noise models, cryogenic control systems, and scalable logical gate implementations. Our quantitative analysis reveals that while significant progress has been made toward practical fault-tolerance thresholds, achieving the resource efficiency and reliability required for large-scale quantum computation remains challenging. We provide detailed resource estimates for key applications including Shor's algorithm and optimization problems, along with analysis of industry roadmaps toward thousand-logical-qubit systems. This review serves as a comprehensive guide for researchers and engineers working toward the next generation of fault-tolerant quantum computers, with particular focus on information-theoretic perspectives relevant to quantum information entropy and syndrome processing.
Quantum computation holds the promise of solving certain complex problems exponentially faster than classical computers. However, the high prevalent noise in current quantum devices impedes the accurate execution of even basic algorithms. This can be remedied by protecting quantum information with a quantum error-correcting code, where the logical information of an algorithmic qubit is spread across multiple physical qubits. Individual quantum errors are then located and corrected by the fault-tolerant measurement of multi-qubit stabilizer operators (parity checks). Unfortunately, error correction and fault tolerance both impose large demands on the qubit overhead: hundreds to thousands of physical qubits per logical qubit.
We reduce the spacetime cost of fault tolerance by redesigning key building blocks of an error-corrected quantum computer. First, we develop a combinatorial proof with flag fault tolerance that exponentially reduces the extra qubits needed to measure a stabilizer of any size, while tolerating one fault. We leverage these proofs to then design state preparation circuits for the Steane and Golay codes with 100% yield. Next, we improve error correction on a planar layout by showing that a distance-four code encoding six logical qubits protects information as well as the distance-five surface code, using one-tenth as many physical qubits. Finally, we optimize the time overhead of logical gates in surface code quantum computers by protecting measurement results with a classical code, cutting computation time by a factor of two to six. Our hardware-agnostic optimizations of fault tolerance overheads thus suggest new routes to advance the timeline of error-free quantum computing.
Quantum error correction (QEC) represents the cornerstone technology for realizing fault-tolerant quantum computing, addressing the fundamental challenge of quantum state decoherence in noisy intermediate-scale quantum (NISQ) devices. This comprehensive review examines recent advances in QEC implementations from 2020-2024, spanning theoretical foundations to practical hardware demonstrations. We analyze the evolution from pioneering stabilizer codes to modern topological approaches, with particular emphasis on surface codes and low-density parity-check (LDPC) codes that have demonstrated below-threshold error correction in superconducting and trapped-ion systems. The paper provides a comparative analysis of classical and machine learning-based decoder algorithms, evaluating their performance through complexity analysis and threshold comparisons in real-time error correction scenarios. Through examination of breakthrough experiments by Google, IBM, Quantinuum, and emerging platforms, we assess the current state of fault-tolerant quantum computing and identify critical engineering challenges including correlated noise models, cryogenic control systems, and scalable logical gate implementations. Our quantitative analysis reveals that while significant progress has been made toward practical fault-tolerance thresholds, achieving the resource efficiency and reliability required for large-scale quantum computation remains challenging. We provide detailed resource estimates for key applications including Shor's algorithm and optimization problems, along with analysis of industry roadmaps toward thousand-logical-qubit systems. This review serves as a comprehensive guide for researchers and engineers working toward the next generation of fault-tolerant quantum computers, with particular focus on information-theoretic perspectives relevant to quantum information entropy and syndrome processing.
Quantum error correction (QEC) has transitioned from a theoretical promise to an experimental reality, with recent demonstrations of below-threshold logical qubits marking a turning point in the quest for fault-tolerant quantum computing. Over the past five years, breakthroughs in surface code implementations, low-latency decoders, and improved physical qubit fidelities have brought us to the brink of scalable logical qubit systems. Yet the road to practical quantum advantage remains steep: correlated noise, cryogenic control complexity, and the resource overhead of magic state distillation still pose major barriers. Here, we synthesize recent progress across superconductng, trapped-ion, and neutral-atom platforms, assess the state-of the-art in topological and LDPC codes, and highlight engineering challenges that must be overcome to achieve systems with thousands of logical qubits. We argue that the 2025-2035 decade will be decisive for quantum computing, with success hinging on co-design of hardware, software, and algorithms. This perspective outlines a roadmap for the next generation of fault tolerant systems and discusses the societal and economic implications of entering the era of practical quantum advantage.