Chapter 4 of Quantum from the Ground Up explains that a logical qubit is built from many noisy physical qubits. Chapter 13 cites Craig Gidney's 2025 estimate of roughly one million physical qubits to break RSA-2048. Both chapters depend on two numbers: how reliable a logical qubit is, and how many physical qubits it consumes. Two results published within a week of each other address one number each.
The Vocabulary
A physical qubit is the actual device: an ion, a superconducting circuit, or a neutral atom. It makes errors constantly. Error correction spreads one unit of quantum information across many physical qubits, and the group acts as one logical qubit. Repeated measurements called syndrome checks reveal where errors occurred without reading the stored information. Software called a decoder decides how to fix them. I covered one example in a post on IonQ decoding errors in real time on a single CPU.
Code distance is the smallest number of physical errors that can corrupt a logical qubit without detection. A larger distance needs more physical qubits. When the physical qubits are good enough, each increase in distance lowers the logical error rate. Logical fidelity is the fraction of logical operations that finish without an error. Color codes and surface codes are two families of error correcting codes. They use different qubit layouts and have different qubit costs.
Quantinuum: Quality
On October 19, Quantinuum reported 99.9995% logical fidelity on its Helios computer. That equals five errors per million operations, or about one error per 200,000. The company says the error rate is ten times lower than its own September record. It also says the rate is 100 times lower than the best published result on any other hardware type, which came from Google on a superconducting system.
The experiment used a color code. The team raised the code distance in steps, and each step cut the remaining logical errors by a factor of 12. That is the below-threshold behavior described above, and the factor matters as much as the headline fidelity. Two more steps at that rate divide the error rate by 144. A Sandia researcher who was not involved in the work said a logical qubit nearly 100 times better than its physical components could, when scaled up, reach useful reliability with fewer than one million physical qubits. The full study is linked from Quantinuum's announcement.
The September record used a different code, called Helix, on the same hardware. Helix needed 10 physical qubits per logical qubit. The color code needed up to 37 and delivered the lower error rate. Quantinuum presents this as evidence that its architecture can swap codes to trade accuracy against qubit count. Its August result built a universal gate set from 28 qubits, which I covered in an earlier post on that 28-qubit gate set. The cost of the magic states behind such gate sets is the subject of a post on why quantum chemistry waits on the T gate.
Google: Cost
On October 15, researchers at Google Quantum AI, Google DeepMind, and MIT posted a preprint on reducing the physical qubit bill. The standard surface code lays qubits on a flat grid. At the distances in the paper it needs more than 1,000 physical qubits per logical qubit. Higher-rate codes pack many logical qubits into one block, but on a chip they need long-range connections that cross each other.
The Google team tiles a hyperbolic surface with small flat modules of 72 or 116 qubits, joined only at their edges. Each module uses ordinary nearest-neighbor gates. Circuit-level simulations show a 10x reduction in physical qubits versus the surface code at a logical error rate of one in ten billion. One simulated hyperbolic surface code spread 146 logical qubits across 120 modules, 13,920 physical qubits in all, which works out to about 95 physical qubits per logical qubit.
The headline figure of more than 30x comes from extrapolation. The authors projected color codes too large to simulate or decode, which reach 36 physical qubits per logical qubit at distances of 20 to 24. The paper is a preprint that has not been peer reviewed.
Reading the Two Together
One result is measured on hardware and the other is simulated. Quantinuum lists all-to-all qubit connectivity as part of its architecture, so its codes need no special layout. Google restricts its design to mostly nearest-neighbor gates inside modules, which suits solid-state chips. IBM's work on linking two quantum cryostats toward fault tolerance is another modular route on superconducting hardware. The diagram below puts the two new results on one page.
Quantinuum measured a 12x drop in logical errors for each increase in code distance (left). Google simulated far fewer physical qubits per logical qubit than the surface code needs (right). Blue bars come from hardware. Orange bars come from simulation or projection. Compare orders of magnitude only, because the codes and counting conventions differ.
Quantinuum's 10 and 37 physical qubits per logical qubit sit in the same range as Google's projected 36. The sources do not state a common counting basis, so compare orders of magnitude only. Neither result runs a useful algorithm. Both benchmark error suppression and qubit cost, and the two numbers they move are the ones that decide how large a fault tolerant machine has to be.
The arithmetic is direct at the scale of the DOE's funding target of 100 logical qubits. At more than 1,000 physical qubits per logical qubit, the surface code reference implies over 100,000 physical qubits. At 36 per logical qubit, the same target implies about 3,600.
What This Changes in the Book
Chapter 4 covers error correction basics. It should add the logical qubit figures above as the current benchmark: 99.9995% logical fidelity and a 12x error reduction per distance step.
Chapter 6 lists trapped-ion figures for Quantinuum's H2 at 56 qubits and IonQ's 99.99% two-qubit gate fidelity. Those are physical layer numbers. Quantinuum's result is a logical layer number on newer Helios hardware, and the chapter needs that distinction along with the update from H2 to Helios.
Chapter 12 covers NVIDIA's Ising Decoder. Quantinuum's architecture integrates tailored AI decoders, so decoding belongs in the story of this result. Chapter 16 covers Google's superconducting and neutral atom platforms. The modular hyperbolic design targets the superconducting side, where long-range wiring is the constraint.
Chapter 13 carries the Gidney estimate of about one million physical qubits for RSA-2048. Neither result runs Shor's algorithm at scale, so neither changes the Q-Day range in the chapter by itself. Both lower the overhead per logical qubit, which is one of the inputs to that estimate, and the chapter should say so.
This post will be added to the next edition of the book. The current edition is described in the second edition announcement, and the download is on the Quantum from the Ground Up Drive.
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