Topological Communication Cost

How much nonlocal classical information is needed to prepare topological order quickly? This project determines the optimal scaling between local circuit depth and communicated classical bits for physical toric-code ground-state preparation. A separate noisy-report theorem determines the preparation depth when public classical communication is free but every local measurement report has an independent unflagged error.

The system has $N=2L^2$ data qubits on an $L\times L$ torus. It starts without entanglement between lattice cells. Local quantum channels, measurements, private records, resets and arbitrary finite local ancillas are permitted. Local propagation consumes depth $D$; classical information sent outside those local layers consumes bits $C$. The output is the corrected quantum state itself, with unconditional ground-space weight $F=\mathrm{Tr}(P_{\mathrm{TC}}\rho)$.

For every fixed target overlap $F\ge f>0$ and fixed interaction range,

$$(C+1)(D+1)^2=\Omega(N).$$

The same scaling holds for expected communication $\overline C$ in almost surely terminating protocols. An explicit exact preparation matches the resource frontier: for every integer block scale $1\le\ell\le L$,

$$D\le12\ell+2,\qquad C\le2\!\left(\left\lfloor L/\ell\right\rfloor^2-1\right) \le\frac{N}{\ell^2}.$$

At the wholly local endpoint, $C=0$ and $D\le2L+8$. Both the collection of block parities and the returned correction instructions are included in the communication cost.

Local processing budget	Communication needed at fixed overlap	Attained preparation
Constant depth	Order $N$ bits	Constant-size blocks
Growing depth $D=o(L)$	Order $N/(D+1)^2$ bits, up to depth rescaling by constants	Blocks with side proportional to $D$
Depth of order $L$	Zero nonlocal bits are possible	A full-lattice local syndrome sweep

Sister result: independent report noise

With Readout Entanglement Limits, the local braiding obstruction and physical syndrome sweep combine with retained outside outcome registers and protected-pointer readout. The complete two-dimensional account proves, at fixed $p\in(0,1/2)$ and fixed target physical overlap $F_0\in(0,1)$,

$$T_{\mathrm{opt}}(L,p;F_0) =\Theta_{p,F_0}\!\bigl((\log L)^{1/3}\bigr).$$

The converse blinds buffered cores of a local two-loop stabilizer test, retaining all outside outcome registers before conditioning. The matching construction physically compresses true syndrome defects to block roots, copies their orthogonal quantum pointers, and applies the actual decoded corrections. It uses three memory qubits per cell, nearest-neighbor gates and at most one report per cell per readout layer. Global processing of already public reports is free; gates and storage are ideal. This model differs from the charged-bit model above, and no uniform detector-quality/accuracy crossover is claimed.

The new two-dimensional proof is maintained here; the sister's companion account gives its independent definitions and consequences for the one-dimensional readout program. Inherited braiding, blind-strip and pointer-readout results retain their original attribution.

Finite converse and its mechanism

For local gate range $r\ge1$, define

$$m=\left\lfloor\frac{L}{32(rD+1)}\right\rfloor^2, \qquad \lambda=\log_2(17/16).$$

Every protocol with a hard cap of $C$ nonlocal bits obeys

$$F\le\min\{1,\,2^{C-\lambda m}\}.$$

Thus constant-depth preparation with $o(N)$ bits has exponentially small global ground-space weight. For prefix-free variable-length physical transcripts,

$$\overline C\ge \max\{0,\,F\lambda m+F\log_2F\},$$

with $0\log_2 0=0$.

The proof places a braiding obstruction in each of $m$ disjoint causal patches. A local trace-preserving circuit has ground-space weight at most $(16/17)^m$, even when the logical state is entangled with discarded environments. Fixing a communicated transcript produces a local trace-preserving completion that dominates that branch's subnormalized output. Counting transcript leaves, or bounding their entropy, then prices the communication.

The topological obstruction and spatial exponent build on Li, Lee and Yoshida. Standard measurement-assisted toric preparation supplies the ingredients of the upper construction. The source comparison identifies these antecedents and distinguishes the charged-bit model from free feedforward, measurement-region counts and other communication tasks.

Read the science

Start with the scientific overview and the precise protocol model.

Question	Authoritative account
What operations, schedules and messages are allowed?	Model and output contract
Why does braiding obstruct a product reference on one patch?	Local braiding lemma and explicit geometry
Why do arbitrary local channels and discarded ancillas obey the exponential overlap bound?	Channel-overlap theorem
How do physical transcript bits become a lower bound?	Hard-cap and expected-cost converses
How are root parities compressed and corrections physically executed?	Exact preparation protocol
What does the frontier imply, and what changes under an ideal bandwidth limit?	Consequences and scope
What is the depth cost of independent unflagged measurement-report errors?	Buffered blind cores, physical syndrome compression and protected readout
Which ingredients are established prior work?	Related work and bibliography
Which finite calculations can be reproduced?	Verification guide

The LLM reading guide maps claims to their complete proofs and records the distinctions needed when explaining the result.

Learn the project

The single educational anchor is John Preskill, Quantum Information, Chapter 7: Quantum Error Correction, using the author's March 2026 draft. Its stabilizer, toric-code and string-operator sections provide the physical starting point.

The Preskill bridge identifies the relevant sections, translates conventions, and develops the additional connections to local channels, communicated transcripts and the physically corrected output. It separates the chapter's foundations from the primary research ingredients and the derivations supplied here.

The 30-milestone map follows the project's background and research narrative from local objects to the complete resource frontier. Each milestone records a central question, understanding target, earlier dependencies, conceptual connection and precise source pointers. It is a common route for self-directed study; specialists can use the direct proof route above.

Reproduce the finite checks

From the repository root, use Python 3.10 or later:

python3 tools/verify_geometry.py
python3 tools/verify_construction.py
python3 tools/verify_compression.py

These programs use only the standard library. The checks cover the explicit Pauli/string geometry, packing inequalities, physical correction boundaries, causal schedules, delivered messages, stabilizer ranks and exact small-state amplitudes. The compression audit additionally checks the pointer/actual-defect invariant, uniform-even root distributions, copied pointer values, fixed memory/report density, fully counted local schedules and exact physical overlap after decoder errors. The saved reports describe their scope. Universal converses and arbitrary-size correctness follow from the written arguments; finite enumeration does not establish them.

Scope

The lower bound grants each charged bit global broadcast, while the construction pays ordinary point-to-point delivered bits. It applies to the full toric ground space and hence also constrains preparation of a specified pure ground state. The construction prepares one fixed pure logical sector exactly.

The target is global physical ground-space overlap. An unapplied remotely described Pauli frame, small energy density, a noisy topological phase and a free global success flag define different tasks. Both models assume exact finite-depth causal cones and initially unentangled cells. Constants are conservative and not optimized. The charged-communication preparation-time law requires the additional ideal-switch assumptions stated in the consequences. The report-noise theorem instead assumes free global classical feedforward and includes the actual correction gates in its operation depth; it is not a laboratory-latency or gate-noise threshold theorem.

Purpose and contact

This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.