Quantum Routing Bandwidth

How much does reusable initialization increase the amount of unknown quantum information that can pass through a one-qubit interface? Two internally controllable modules exchange complete registers through a shared bus. Occupied qubits can assist a selected contact without losing their data; independently polarized helpers provide a further improvement even when their preparation is imperfect.

Each module has $n\ge1$ payload qubits and $s\ge0$ helpers. Both module–bus interfaces are bounded: every pairwise contact has operator norm at most $J$, and contacts may operate simultaneously. Internal module operations and bus-only rotations have zero charged duration. There are no measurements, resets, or extra initialized qubits.

For independent known preparations $\rho_p=(1-p)|0\rangle\langle0|+p|1\rangle\langle1|$, with $0\le p\le1/2$, the optimal fixed-error interaction time is

$$T_{\epsilon,p}(n,s) =\Theta_\epsilon\!\left( \frac{n}{J\sqrt{(n+s)\{1+s[1-h_2(p)]\}}} \right).$$

For every fixed $0\lt\epsilon\le0.1$, the constants are uniform in $n,s,p$. The error is a complete-operation half diamond distance, including arbitrary reference-entangled payload/bus inputs and the fixed helpers' inaccessible preparation references. The construction returns the physical helper density matrix exactly; remaining joint correlations and bus errors are included in the accuracy guarantee.

Helper promise on the same hardware	Accuracy	Optimal interaction time
Pure helpers, returned clean	Exact	$\Theta(n/[J\sqrt{(n+s)(s+1)}])$
Arbitrary unknown helpers, all information preserved	Exact	$\Theta(n/[J\sqrt{n+s}])$
Independent known mixed helpers	Fixed complete-operation error	The entropy-deficit law above

The exact tasks restore an arbitrary bus and all reference correlations. A full-rank mixed preparation has the unknown-helper scale at zero error; the fixed-error law is not uniform as the error tends to zero. At $s=0$, the routing time is $\Theta(\sqrt n/J)$.

Read the science

Start with the scientific account, then follow the information-flow bound, exact helper laws and contact construction, and imperfect initialization. The physical interpretation joins their mechanisms and resource consequences. Specialists can use the theorem and proof map directly.

Question	Authoritative account
Why can arbitrary internal processing not eliminate the bottleneck?	Conditional entropy and weighted interfaces
How do occupied qubits assist without sacrificing their information?	Coherent block construction and exact laws
What survives imperfect initialization, including helper references?	Matched mixed-helper law
Is there a finite advantage against the entire unknown-helper class?	Exact certificates, error composition, and resource budgets
What changes when internal gates have finite duration?	Control models and the bounded-strength compiler
Which ingredients are established prior work?	Related work and bibliography

The converse combines a conditional entropy-rate estimate with a conserved physical entropy deficit. The construction concentrates preparation weight reversibly, uses all remaining qubits as occupied spectators, and decompresses after routing. One common high-probability sector controls preparation error for the entire operation; new control errors have a separate composition bound.

Learn the project

The single educational anchor is John Watrous, The Theory of Quantum Information (2018). The Watrous bridge identifies relevant sections, the author's errata, and the notation changes needed for this project. Three worked examples connect references, coherent contact amplification, and preparation concentration to the proofs.

The 30-milestone map retains 14 background/preliminary and 16 narrative/innovation targets. Each milestone has a central question, an understanding target, earlier dependencies, a conceptual connection, and scientific sources. The learning entrypoint provides the route; the LLM guide maps study and research questions to authoritative files.

Reproduce the checks

From the repository root, using Python 3.13:

python -m pip install -r verification/requirements.txt
python check_repository.py
python verification/tools/test_structure.py
python verification/run_all.py

The verification guide describes ten deterministic scientific suites, structural regression tests, and the separate actual-GitHub rendering check. Executed scientific outputs and source hashes are in verification/results. Finite numerical identities support the implementations; the written proofs and cited theorems establish universal and asymptotic claims.

Scope and physical resources

The matching results charge contact time with free internal controls, under a per-contact rather than fixed total-norm budget. Initialization and internal compression are resources, not cost-free device operations. The displayed bounded-strength compiler has linear total duration at fixed internal/contact strength ratio; that is not a universal finite-speed impossibility theorem.

Synchronized block translations substantially shorten the exact contact list. With fixed nonzero polarization and $s=\Theta(n)$, the route uses $O(n)$ exact contact intervals, while global compression is still assigned zero duration. A fixed quadratic contact budget retains the initialization enhancement; a fixed sum of contact strengths removes its parametric growth. A correlated-helper counterexample shows why entropy deficit alone cannot determine achievable bandwidth for arbitrary helper states. Preparation robustness does not imply robustness to ongoing decoherence or noisy pulses.

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.

License

MIT.