Error-Transparent Locality

Exact resource laws for logical Hamiltonians that keep working through physical faults at unknown times.

A correctable fault during a gate can change the logical evolution rate until recovery. Exact error transparency removes this timing dependence: the physical Hamiltonian implements the same logical generator in the code and every protected error image. This repository proves general interaction-order and native coupling bounds, gives a sharp four-body witness, and constructs resource-optimal protected controls and entanglers on fixed concatenated five-qubit codes.

Sister projects: Protected Gates and Syndrome Readout

Sister repository	Central question
Error-Transparent Locality (this repository)	Which native interaction order and coupling budget let a logical gate tolerate faults at unknown times?
Syndrome Reversibility Distance	When does the original syndrome measurement preserve logical information after a known product unitary?

The shared gate/readout theorem connects the two requirements on a fixed code. The gate and readout models retain separate assumptions.

General resource bounds

Let a fixed code have detection distance $d$ and encoding $V$. A Hermitian, code-preserving Hamiltonian is transparent through Pauli-error weight $t$ when

$$HEV=EV(h_L+c_E I) \qquad\text{for every tensor Pauli }E\text{ with }\mathrm{wt}(E)\le t.$$

The lower bounds allow consistent scalar sector shifts $c_E$; the explicit constructions satisfy the strict identity with every $c_E=0$. This is a full physical-space equality, including preservation of each error image. With ideal final recovery, it preserves the intended logical evolution independently of the fault time.

Write $\ell$ for the largest weight of a nonzero native Pauli term, $J$ for the sum of absolute nonidentity bare Hermitian Pauli coefficients, and $\omega$ for half the spectral width of $h_L$. For nontrivial logical action, the general bounds are

$$\boxed{\ell\ge d+t,\qquad J\ge\left[\sum_{k=0}^{t}2^k\binom{d+k-1}{k}\right]\omega \quad\text{when }\ell=d+t.}$$

Correction of all errors through weight $t$ requires $d\ge2t+1$, hence $\ell\ge3t+1$. For one protected error and any $\ell>d$, the exact scalar moment bound is $J/\omega\ge(\ell+d)/(\ell-d)$. These statements allow degenerate codes and arbitrary logical dimension. See the model, weight-moment proof, and coupling theorem.

At $d=3,t=1$, a nine-qubit CSS code attains $\ell=4$ and $J/\omega=7$ with nine commuting terms for one logical $Z$ axis. The established canonical five-qubit construction uses $\ell=5$, $J/\omega=4$, and sixteen terms per axis. These are optima at their respective interaction orders under the declared native budget. The four-body result establishes neither universal four-body control nor minimal block size. Attainment of the unrestricted order $\ell=d+t$ for $t>1$ remains open here.

Optimal controls with reversible syndrome readout

For the concatenated five-qubit codes $Q_m=[[5^m,1,3^m]]$, put $n=5^m$ and $d=3^m$. Their stabilizers have even weight and their nontrivial logical Pauli representatives have odd weight. The resulting available-weight obstruction strengthens the gate bound to $\ell\ge d+2t$ whenever $2t+1\le d$.

Every logical Pauli axis attains this interaction order for $t=1,2,3$, with the following exact native coupling optima for this code family at that order:

Protected Pauli-error weight	Depth	Minimum interaction order	Optimal axis budget $J_t(d)=J/\omega$	Account
$t=1$	$m\ge1$	$d+2$	$d+1$	Single-error controls
$t=2$	$m\ge2$	$d+4$	$(4d^2+21d+8)/8$	Two-error optima
$t=3$	$m\ge2$	$d+6$	$(4d^3+51d^2+170d+24)/24$	Three-error optima

The higher-error coupling proofs resolve interaction weights inside code blocks and apply to arbitrary competing Hamiltonians, including consistent scalar sector shifts. A strict truncated tensor lifting carries the zero- through three-error hierarchy to every larger depth. In the fixed canonical one-qubit Pauli frame, the three-error result also gives the exact all-direction cost $J_{\min}=J_3(d)(|\Omega_X|+|\Omega_Y|+|\Omega_Z|)$ at $\ell=d+6$.

Native term count is a separate resource. The single-error sparse recursion uses at most $4n$ terms per axis with $O(\log n)$ coefficient bits; a separate support bound is $3n+1$. The two-error sparse synthesis supplies a 69,125-term optimal gate on 125 qubits. The scientific overview compares the finite witnesses, coefficient sizes, and exact coverage certificates.

These codes also have syndrome reversibility distance $r_S=\infty$: every known product unitary is recoverable after full ideal measurement of the original stabilizer syndrome, with recovery informed of that unitary. For odd-distance stabilizer codes with $d\ge3$, $t\ge1$ and $d\ge2t+1$, attaining the unrestricted gate order $\ell=d+t$ forces $r_S=d+1$. Thus $r_S>d+1$ requires $\ell\ge d+t+1$. The coexistence of these guarantees does not establish a joint noisy threshold.

Protected entanglers on fixed product codes

On $M$ fixed $Q_m$ blocks with $m\ge2$, prescribe a pure nonidentity logical Pauli string $P$ of logical weight $p$. Its minimum physical representative weight is $D=pd$, while the product code's ordinary distance remains $d$. For $h_L=\Omega P+cI$, $\Omega\ne0$, exact transparency through three physical errors has the sharp resource law

$$\boxed{\ell_{\min}=D+6,\qquad J_{\min}(\ell=D+6)=|\Omega|J_3(D).}$$

The protected-string construction and lower bound keep the code and original syndrome measurement fixed, with $r_S=\infty$. The first $ZZ$ entangler acts on a $[[50,2,9]]$ product code at $\ell=24$, $J/\omega=1789$. Together with protected local Pauli controls, it generates universal continuous encoded control in this model. The pure-string law does not assign a globally optimal sum of costs to an arbitrary mixed logical Hamiltonian at one common interaction-order cap.

The first entangler's compact dense specification contains 41,032,000,000 native terms. This proves an exact resource optimum in interaction order and coupling budget; practical sparse synthesis and physical implementation require further work.

Reading and learning routes

Reader's aim	Start here
Follow the complete scientific argument	Scientific overview and proof map
Check definitions and general bounds	Model → locality → coupling cost
Inspect explicit controls and entanglers	Construction catalog and protected logical strings
Understand fault-sensitive control schedules	Control robustness and physical interpretation
Study from the shared educational anchor	Learning guide, Preskill bridge, and 30 milestones
Check attribution and nearest precedents	Related work and bibliography

The shared educational anchor is John Preskill, Quantum Information, Chapter 7, “Quantum Error Correction”, in the author-hosted draft updated March 2026. This project's bridge selects relevant sections and translates notation and Pauli phases; its twelve background and eighteen narrative milestones lead through protected entanglers and universal encoded control. The sister project's Preskill bridge follows the readout argument with its own 30-milestone map.

Reproduce the certificates

Only Python 3.10 or later and its standard library are required:

python verification/run_all.py

The verification guide describes fourteen exact checks and their committed JSON records. They cover full error-image identities, code and stabilizer properties, coefficient normalization, moment primal/dual certificates, sparse constructions, recursive response coverage, entanglers, and control phases. No floating-point tolerance is used. Finite certificates support the stated instances; the general and arbitrary-depth conclusions use the analytical proofs.

Physical scope

Interaction order counts qubits in one Pauli term. Native coupling budget counts absolute coefficients in a fixed basis. Neither specifies geometric range, term count, coefficient description size, or complete hardware cost.

Transparency must hold throughout a pulse. Even commuting terms can lose protection when applied sequentially: a correct fault-free final unitary can coexist with an incorrect logical evolution after an intermediate fault. The control analysis derives the exact phase and a bounded repeated-cycle approximation.

The model uses a fixed code and ideal final recovery. Multiple faults are protected while every accumulated error stays in the declared weight-limited set. State preparation, noisy syndrome extraction and recovery, efficient decoding, and synthesis of native many-body interactions require their own accounting.

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

The repository retains the MIT license.