Quantum Conditional Learning

How many quantum examples are needed to supply a conditional state, and how does the cost change when the prediction must first be described classically?

An unknown pure $N$-qubit stabilizer supplies independent training copies. A fixed public measurement reads $n=N-m$ commuting Pauli observables on a fresh verifier copy. The learner receives the classical outcome and must supply the conditional state on the remaining $m$ qubits. Both output classes may keep their training quantum data until the same outcome arrives.

Output channel	Training-copy result
Direct quantum processing	$n+\lceil\log_2(1/\delta)\rceil+3$ copies suffice, with exact output on a heralded event of probability at least $1-\delta$
Arbitrary classical density description followed by preparation	$N-o(N)$ copies are necessary for any constant fidelity when $m$ is a positive fraction of $N$

The quantum converse is

$$F_Q^{\mathrm{av}}(N,m,k;W) \le\min\{1,\;2^{-m}+2^{k-(N-m)}\}.$$

It matches the leading measured-size term of the constructive upper bound. The classical-output lower bound allows arbitrary collective measurements, unlimited classical record length and precision, arbitrary mixed predictions, and measurements chosen after the outcome is supplied. Full stabilizer identification attains its leading rate.

For $m/N\to\beta\in(0,1)$, the leading copy rates per source qubit are $1-\beta$ for quantum output and $1$ for classical descriptions. Taking $n=\lceil\sqrt N\rceil$ gives $\sqrt N+O(\log(1/\delta))$ versus $N-o(N)$ training copies. A typical specified measurement branch is rare, but different observed branches can be combined coherently to supply it.

Read the science

Start with the operational model and exact theorems. The construction selects an affine combination of observed outcomes and applies a published Clifford-covariant quantum convolution. The two converse arguments quantify what can be predicted before enough constraints or a sufficient classical description can be recovered.

Scientific question	Authoritative account
What access, timing, outputs and fidelity are compared?	Model and theorem statements
How can an unknown conditional branch be supplied efficiently?	Synthesis algorithm and success bound
Why does quantum output need the measured-size copy cost?	Quantum converse and full-prior reduction
Why does every classical density description retain the full rate?	Classical-output converse
What do the copy savings mean physically?	Resources and interpretation
How do the output classes differ with one training copy?	Exact one-copy comparison
Which methods are established, and what is the proposed contribution?	Related work and bibliography
What has been checked computationally?	Verification

The construction uses Bu–Gu–Jaffe's convolution; an alternative uses Gross–Nezami–Walter's copy symmetries. The converse develops a conditional-output application of Hayashi–Lu's finite quotient and completion framework. The proposed contribution is the matched conditional-learning copy separation. These imported methods and the scope of the comparison are part of the scientific account.

Learn the project

The single educational anchor is John Watrous, The Theory of Quantum Information (Cambridge University Press, 2018). The Watrous bridge gives a targeted reading map, translates the book's notation and conventions, and connects its channel and state-discrimination framework to conditional stabilizer learning.

The 30-milestone map follows the project's scientific dependencies. Its 14 background/preliminaries milestones join quantum states, channels, measurements, stabilizer algebra and statistical reasoning into the shared experiment. Its 16 narrative/innovation milestones develop the exact one-copy comparison, synthesis mechanism, two converses and resulting resource frontier. Each milestone includes a central question, an understanding target, earlier dependencies, a conceptual connection and source pointers.

Use the bridge alongside the milestones for self-directed study. In particular, its convention dictionary distinguishes Watrous's fidelity from the squared-overlap score used here, and fixes the Choi normalization. The bridge also identifies the stabilizer, convolution and counting ingredients supplied by the project and its primary research references. The LLM reading guide provides story, study, theorem and reproduction routes.

Reproduce the finite checks

The verification guide specifies dependencies and expected outputs. From the repository root, run:

python3 verification/run_all.py

The checks cover finite-field moments, quotient/frame incidence and unsigned report-subspace containment, exact one-copy channel witnesses, and conditional synthesis on complete small stabilizer ensembles. They test concrete implementations and finite cases. The general copy laws are established by the written proofs.

Physical scope

The source promise is an ideal pure stabilizer state and the public measurements are exact. The quantum protocol retains $km$ output qubits, stores $kn$ outcome bits, and consumes training systems to return one prediction. Its final mixing uses at most $2m(k-1)$ CNOTs, in addition to state preparations and public-measurement standardization. It assumes no QRAM or inverse source circuit.

The theoretical score is conditional-state fidelity. Comparing one unknown prediction with one independently prepared unknown target through a SWAP test has acceptance $(1+F)/2$; the verifier preparation and any repetitions are separate resources. The finite bounds determine leading copy costs, without asserting optimal critical-window constants or robustness to arbitrary physical noise.

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.