By Raghu Kulkarni, SSMTheory Group, IDrive Inc.
Flag Qubits Give the FCC Code Tunable Protection at 29% Rate
deff = 3 + R
Place one flag ancilla at each tetrahedral void. Measure R times.
The effective distance grows linearly—without changing the code, without losing a single logical qubit.
The Problem: d = 3
Our 67%-rate FCC code (arXiv:2603.20294) packs 130 logical qubits into 192 physical qubits—more
than ten times the encoding effciency of the surface code. But it has one known weakness: the code distance is 3. At each tetrahedral void of the lattice, a weight-3 logical operator sits quietly, invisible to every stabilizer measurement. Three simultaneous errors at a void go completely undetected.
Every prior approach to raising the distance destroyed the rate. Subsystem gauging pushes d up to
L but drops from 130 logical qubits to 2—a rate collapse from 67% to near zero. Lifted product
constructions fail on the triangle-dense FCC graph (we tested this; the distance actually went down). Partial gauging does nothing: leave even one void ungauged and its weight-3 logical persists.
The ceiling seemed hard. It isn’t.
The Solution: One Parity Measurement
The weight-3 logical at a void ips the three-body parity ZaZbZc from +1 to −1. The stabilizer syndrome reads zero—but a flag ancilla sitting at the void measures this parity directly and catches the error.
Theorem (proved): A flag ancilla measuring ZaZbZc at a tetrahedral void detects the weight-3 logical with certainty. For the error to go undetected, both the three data errors and the flag measurement must fail. With R independent flag rounds: Pfail = Nvoid ·p3+R, giving deff = 3+R.
The flag circuit has depth 4: prepare |+⟩, apply three CZ gates (one to each void edge), measure in the X basis. Outcome −1 means a weight-3 error has been caught. The quantum code is completely untouched— this is a classical measurement schedule layered on top of the existing [[192,130,3]] code.
Figure 1: Left: The flag mechanism. Without flags (top), a weight-3 error produces zero syndrome— undetected. With a flag ancilla (bottom), the ZaZbZc parity flips from +1 to −1—caught. Right: Encoding rate versus effective distance. The FCC+flags rate holds at 29.0% for any deff; the surface code rate vanishes as 1/d2.
The Numbers
| Code | Qubits | Logical | deff | Rate | Pfail at p = 10-3 |
|---|---|---|---|---|---|
| Surface code d = 7 | 97 | 1 | 7 | 1.0% | 10-20 |
| FCC (no flags) | 192 | 130 | 3 | 67.7% | 10-7 |
| FCC + 1 flag | 448 | 130 | 4 | 29.0% | 10-10 |
| FCC + 2 flags | 448 | 130 | 5 | 29.0% | 10-13 |
| FCC + 4 flags | 448 | 130 | 7 | 29.0% | 10-19 |
At matched effective distance (deff = 7), the FCC+flags code encodes 130 logical qubits at 29.0% rate. The surface code encodes 1 logical qubit at 1.0% rate. That is 130 times more logical information per physical qubit.
The rate stays flat at 29.0% regardless of how high the effective distance is pushed—just add more flag rounds. The surface code rate drops as 1/d2 and vanishes. This is a qualitatively different scaling.
Monte Carlo Validation
We ran 5,000-trial Monte Carlo simulations under phenomenological noise (independent data errors, measurement noise on stabilizers and flags, CZ hookup errors at p/10). With R = 0 (no flags), the base threshold sits near p ≈ 5.7%. Each added flag round visibly suppresses the logical error rate. At R = 4, zero logical failures were detected at any error rate up to 10%—consistent with the analytical prediction Pfail ∼ 256 × (0.1)7 ≈ 2.6 × 10−5, below the statistical floor.
An adaptive scheduling protocol reduces the overhead further: at p = 10−3, the optimal flag interval is about 7,800 syndrome rounds, making the measurement overhead less than 0.1% of total cycles.
Where the Flags Live
Each tetrahedral void in the FCC lattice sits between the ABC-stacked hexagonal layers. Its three edges decompose as one in-plane bond (within a single layer) and two inter-layer bonds (spanning adjacent layers). The flag ancilla occupies the void center—geometrically, the centroid of a triangular plaquette on the BCC dual lattice. Each flag couples to exactly 3 data qubits, all within nearestneighbor distance. No long-range connections are introduced.
The augmented graph (192 data qubits + 256 flag ancillas = 448 total) has maximum degree 12 (data) and degree 3 ( flags). The sparse, local structure of the original code is preserved.
The arXiv Foundation
The base code paper, arXiv:2603.20294, is now live. It provides the full GF(2) construction, analytic proof, CSS commutativity verification, and Python source code for independent reproduction. The flag paper builds directly on this foundation and includes its own self-contained simulation code.
Five Patents, One Stack
The complete FCC quantum computing architecture is protected by five provisional patent applications:
- Patent 1 (64/008,236) — The Code: [[192,130,3]] CSS code with weight-12 stabilizers.
- Patent 2 (64/008,866) — The Hardware: Three stacked 2D hexagonal chips with ABC offset and TSV interconnects.
- Patent 3 (64/014,145) — The Algorithms: Subsystem gauging, triadic planar decoder, dynamic mode switching, spatial partitioning.
- Patent 4 (64/014,153) — The Application: Quantum associative memory with Grover retrieval and hybrid AI co-processing.
- Patent 5 (64/015,757) — The Distance Fix: Flag ancillas at tetrahedral voids: deff = 3 + R at constant 29.0% rate.
Papers
Flag paper: doi.org/10.5281/zenodo.19200672
Base code paper: arXiv:2603.20294
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About the author: Raghu Kulkarni leads the SSMTheory Group at IDrive Inc. in Calabasas, CA.
idrive.com/ssmtheory | raghu@idrive.com
