Exact and Fault-Tolerant Compilation of Hopf Differential Frames

Exact state preparation normally specifies one initialized input:

$$|0^n\rangle|0^m\rangle \longmapsto |\psi\rangle|0^m\rangle.$$

The original inverse-frame Hopf decoder uses a prescribed unitary completion: the state in its first column and coordinate-frame directions in designated other columns. Its inverse resolves a common objective response.

Synthesis task	Required action
Exact state preparation	fix $U\lvert 0^n\rangle$
Hopf differential-frame compilation	fix $W\lvert x\rangle$ for every system basis state $\lvert x\rangle$ and restore workspace according to its input promise

This repository asks whether the prescribed Hopf completion retains the state-preparation size–depth frontier, and what finite precision costs. The publication scope fixes the full-frame claims.

The LLM reading guide maps proofs and assumptions. Resume research from the workspace checkpoint.

Two resource models

Both models compile the complete frame; different circuits attain their optimal resource bounds.

Model	Resource question	Scope of the matching theorem
Arbitrary one-qubit gates and CNOTs	exact size, CNOT count, and depth versus clean workspace	real and phase-dressed complex magnitude frames, every clean budget
Clifford+T	T-count versus accuracy and clean/dirty workspace	real and phase-dressed complex magnitude frames, every precision, under the sufficient clean reservation below

Exact logical theorem

Let

$$N=2^n$$

and let $m\geq0$ be the number of clean ancillary qubits. In the exact all-to-all logical model with arbitrary one-qubit gates and CNOTs,

$$\boxed{ S_{\mathbb R}(n,m) =S_{\mathbb C,\mathrm{mag}}(n,m) =\Theta(N) }$$

and

$$\boxed{ D_{\mathbb R}(n,m) =D_{\mathbb C,\mathrm{mag}}(n,m) =\Theta\left(n+\frac{N}{n+m}\right). }$$

The upper bounds hold for every parameter tuple; lower bounds hold in the worst case over the Hopf-frame family, uniformly in the clean-workspace budget.

The worst-case CNOT count alone is $\Theta(N)$ for $n\ge2$, with free one-qubit gates and arbitrary clean workspace; it is zero for $n=1$. See the separate lower-bound proof.

The real result concerns the complete Hopf differential frame. The complex result concerns the phase-dressed magnitude frame

$$W_{\mathbb C,\mathrm{mag}} =D_{\mathrm{ph}}W_{\mathbb R}.$$

The leaf-phase derivatives form a separate direct measurement stream; they are not additional columns of the same $N$-dimensional unitary.

The construction restores at most $m$ clean ancillas exactly, reaching the optimal state-preparation frontier at every clean-workspace budget.

Fault-tolerant theorem

Let $a$ be the number of initialized clean qubits and $b$ the number of borrowed qubits in an arbitrary unknown state. Write

$$q=n+a+b,\qquad L=\max\{6,\lceil\log_2(1/\eta)\rceil\},\qquad h=1+\lceil\log_2(L+n+2)\rceil.$$

For $0\lt \eta\leq1/64$ and a sufficiently large fixed constant $C$, both the real frame and phase-dressed complex magnitude frame have matching worst-case T-count

$$\boxed{ T^\star(n,a,b,\eta) =\Theta\left(\sqrt{NL}+L+\frac{NL}{n+a+b}\right), \qquad a\geq C(n+h). }$$

The upper construction uses $O(NL)$ Clifford gates. Every supplied precision source is prepared and charged. Borrowed qubits are restored jointly with any external reference; complete-input approximation includes clean-work leakage. There are no measurements, resets, or free supplied catalysts in this model.

At $L=N$, sufficient $a=\Theta(n)$ and $b=\Theta(N)$ give $T^\star=\Theta(N)$. With one clean qubit and $b\ge L+n+7$, the one-clean compiler gives $T=O(N+L\ell_*(n))$, $G=O(NL)$ for real frames, where $\ell_*(n)=1+\log_2^*(n+2)$. The iterated logarithm counts base-two logs to at most one. At $L=N$, the open endpoint is $\Omega(N)\le T^\star\le O(N\ell_*(n))$. Smaller allocations retain the borrowed-workspace bound. At $b\ge L+n+7$, zero-clean real-frame synthesis gives $T=O(N+nL)$, $G=O(NL)$; the grouped improvement still uses one clean qubit.

Composing with a literal diagonal of separately supplied certified phases extends the one-clean bound to phase-dressed complex magnitude frames at $b\ge L+n+8$. With $b\ge2(L+n+8)$, it gives $O(\sqrt{NL}+L\ell_*(n)+NL/b)$ T gates, still $G=O(NL)$. Leaf-phase derivatives retain a separate QBP stream.

Hybrid lookup uses two clean flags and $b\ge17(L+n+7)$. Fixed L gives $T=O(\sqrt N+N/b)$ and

$$D_T=O(N/b^2+n\log(n+2)).$$

For $L=\Theta(n)$, sufficient $b=\Theta(n)$ gives optimal worst-case $T=\Theta(N)$ and $D_T=\Theta(N/n)$ in one complete real-frame circuit.

At fixed accuracy, grouped programs and chunked queries give, with two clean flags and sufficient $b=\Theta(\sqrt N)$, one complete real-frame circuit with

$$T=O(\sqrt N),\qquad G=O(N),\qquad D_T=O\!\left(n\log\log(n+2)\right).$$

T-count is optimal in worst-case order. Depth optimality and the high-precision endpoint remain open; the variable-precision theorem is unchanged.

Beyond Hopf frames, literal diagonals and general one-target U(2) multiplexors attain $\Theta(\sqrt{NL}+L+NL/b)$ with one clean qubit, respectively at $b\ge2(L+n+7)$ and $b\ge2(L+n+9)$. For the multiplexor, $n$ counts address qubits and $N$ counts its blocks. These matched one-stage frontiers do not close the multi-layer frame gap.

Two update classes attain $T=O(N+L)$, $G=O(NL)$ at $b\ge L+n+7$, requiring literal agreement outside changed nodes with a native baseline whose determinant-one local words have length $O(n-d+1)$ at depth d.

• Antichain changes use zero clean qubits; no changed node is an ancestor of another.
• Sparse nested changes use one clean qubit. If S is the ancestor closure of the changed nodes, a sufficient condition is

$$n\ge 2\lceil\log_2(|S|+1)\rceil+32.$$

The sparse result includes a full root-to-leaf path for every n. Both preserve the complete frame and QBP interface. Generic rounding supplies neither promise.

Exact construction at a glance

The Hopf frame is a product of addressed tree layers. At depth $d$, the prefix selects one rotation angle and the complete lower suffix supplies a shared all-zero predicate. Three schedules exploit this structure.

Workspace	Schedule	Mechanism
$m=0$	borrowed-suffix echo	one original suffix data qubit carries the predicate temporarily and is restored exactly
$1\leq m\lt 4n$	direct flagged UCG	one reusable clean flag stores the suffix-zero predicate
larger $m$	routed parallel subframes	a tree cut turns the tail into a direct sum; the suffix is routed coherently and the subtree frames run in parallel

Read the repository in three passes

Time	Route	Purpose
5 minutes	this page	problem, theorem, and construction map
30–40 minutes	complete technical narrative	shared contract, the two resource models, and the QBP consequence
full audit	exact theorem, T-count theorem, one-clean theorem, and verification map	proofs, register schedules, evidence limits, and source dependencies

The documentation map includes the QBP consequence, related work, and Hopf geometry.

Why the completion matters

The complete frame satisfies

$$W_{\mathbb R}|0^n\rangle=|\psi\rangle, \qquad W_{\mathbb R}|\lambda(j)\rangle=|e_j\rangle.$$

At a regular coordinate,

$$\partial_{\theta_j}|\psi\rangle =a_j|e_j\rangle, \qquad g_{j,j}=a_j^2,$$

where $a_j$ is the oriented amplitude entering the corresponding tree node. On the canonical Hopf domains, $a_j\geq0$ and equals the principal metric square root. At zero metric weight the raw derivative vanishes, while the marker column remains the chart-selected orthogonal continuation determined by the complete parameter tuple.

A state-preparation-equivalent completion may move these marker columns. The repository contains an exact two-qubit example in which the state is unchanged but the decoded gradient changes from

$$(2,0,0) \quad\longmapsto\quad (0,\sqrt2,0).$$

For this fixed decoder, the relevant compiler contract is

$$\widetilde W \bigl(|\varphi\rangle|0^m\rangle\bigr) =(W|\varphi\rangle)|0^m\rangle$$

for every system input $\lvert\varphi\rangle$, not only the forward preparation input. At regular points, universal preservation of the fixed decoder's means also forces this contract up to common phase; see the necessity theorem.

Relation to the all-workspace state-preparation framework

Four imported exact tools supply the construction:

Imported result	Role here
optimal QSP frontier	benchmark and matching comparison
ancilla-free multi-controlled X	zero-suffix predicates and toggles
all-workspace UCG synthesis	prefix-selected rotations, subtree frames, and the phase diagonal
coherent CNOT copy–uncopy	control fanout for the decoder and router

The Hopf-specific contributions are the complete-operator factorization and three workspace schedules, which adapt these primitives while preserving marker columns.

The normative compiler citation is:

P. Yuan and S. Zhang, “Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits,” Quantum 7, 956 (2023).

Consequence for quantum backpropagation

Frame-safe compilation preserves the complete inverse-frame measurement distribution. Under the primary finite-shot target of simultaneous absolute accuracy for the raw Hopf-coordinate gradient, the global magnitude stream uses

$$O(\log n)=O(\log\log M)$$

independent executions at fixed accuracy and confidence, where $M=\Theta(2^n)$ is the number of coordinates.

This is a matched-program statement: scalar and gradient programs use the same forward preparation family and controlled observable, while the gradient program adds one inverse frame of the same asymptotic logical depth as optimal state preparation. Classical materialization of an $M$-entry output is not included in that quantum-depth ratio.

At finite precision, the approximation bridge controls the bias of the same fixed-parameter, bounded-score estimator. It uses the actual compiled circuit and its actual adjoint. It does not differentiate a discontinuous family of compiled words. Frame-synthesis lower bounds alone do not establish optimal gradient-query or training complexity. The same proof now covers the complete complex gradient, reflection-sum observables, rounded classical weights, and correlated dirty-bank reuse between executions with fresh declared clean inputs.

The separate state-based QBP theorem changes the decoder and avoids compiling the fine complete frame. For real and phase-dressed complex Hopf states, observable coefficient norm $\Lambda$, and error $\epsilon$, put $K=\max\{6,\lceil\log_2(80\Lambda/\epsilon)\rceil\}$ and $P=\max\{n,K\}$. Two initialized compiler flags and $b\ge P+n+7$ dirty wires give $T=O(N+P)$, $G=O(NP)$ per execution apart from the observable; the protocol branch is separate. The actual coarse inverse and classical correction retain the shot scaling. Depth schedules and bounded-input costs are explicit. Bounded residual QBP implements both streams. General software remains optional; the complete-frame endpoint is open. No end-to-end speedup is claimed.

Verification boundary

Checks cover complete logical identities, explicit decoder/router gates, arbitrary-entangled inputs, exact resource ledgers, and gradient decoders.

UCG and multi-controlled-X decompositions are imported. Finite checks expose indexing, phase, order, cleanup, and resource errors; the asymptotic claims rest on proofs.

The fault-tolerant evidence adds exact finite source/kernel identities and rational resource checks. It does not yet provide a general elementary emitter for the complete asymptotic shared-source compiler. See the focused reproduction guide.

Device connectivity, physical noise thresholds, the full T-depth frontier, arbitrary non-Hopf charts, and application-independent observable costs remain outside the claims. Clifford work, T work, quantum executions, and classical output costs are reported separately.

Reproduce the checks

python -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
python -m pip install -r requirements.txt
python scripts/reviewer_walkthrough.py
python validate.py
python scripts/verify_fault_tolerant.py
python scripts/unified_resource_ledger.py --n 12
python scripts/strict_zero_echo_ledger.py --n 12

Relationship to the Hopf repositories

Repository	Role
Hopf-ansatz	coordinate chart, inverse map, metric, tangent preparation, and optimization interface
Hopf-QBP	global, direct-phase, and checkpoint gradient records, including the earlier Möttönen-style robustness result
This repository	complete-frame contracts, exact and fault-tolerant compilation, and their scoped QBP consequences

Status and license

Evidence and sources delimit the claims. The repository uses the MIT license.