Quantum Precision Limits

How can coherent access to a quantum circuit determine a multiparameter Holevo precision bound without reconstructing its large physical state?

This project studies Bures tomography from a state-preparation unitary and uses it to compute certified Holevo bounds for low-rank sensing models. The central tomography guarantee permits an arbitrarily large purification workspace. The application learns a small register containing the state-and-derivative Gram matrix, then certifies a classical optimization over that model.

Result	Guarantee	Essential conditions
Coherent Bures tomography	$\widetilde\Theta(dr/\varepsilon_B)$ worst-case preparation/inverse queries	A rank-$r$, $d$-dimensional reduced state; no occupied-eigenvalue floor; arbitrary supplied purification
Certified Holevo computation	$\widetilde O(1/\varepsilon)$ precision dependence at fixed model size and conditioning	Explicit coherent factor/derivative access; occupied-state and Fisher lower bounds; no tangent-Gram eigenvalue floor
State-weight obstruction	$\Omega(\varepsilon^{-3/2})$ queries on a rank-three family without an occupied-state floor	Opaque global preparation oracles; bounded fixed generators and uniformly conditioned Fisher information

The matching tomography lower bound is a worst-case oracle statement for sufficiently small constant-bounded accuracy and $d\ge2$.

The tomography upper bound combines published coherent Frobenius tomography with adaptive spectral peeling. Its lower bound follows from published trace-norm tomography. The arbitrary-purifier Bures guarantee and its certified precision application are developed here; the related-work account identifies the inherited ingredients and closest competing methods.

Why Bures accuracy matters

A local model of rank $r$ with $p$ parameters has factors $V=[\Psi,\Phi_1,\ldots,\Phi_p]$. Its Holevo value depends only on $K=V^\dagger V$, a matrix of order $m=r(p+1)$. A controlled preparation encodes $K^{\mathsf T}$ into a label register of at most $m+1$ dimensions. The physical sensor becomes its purification.

Near a singular tangent Gram, entrywise accuracy gives only square-root control of the factors. Learning directly in Bures distance avoids that generic accuracy loss. The algorithm returns a classical confidence interval for the local Holevo value. Constructing an attaining measurement is a separate task.

Read the science

Start with the question, model, and exact scope.

To inspect	Read
Optimal Bures query scaling, conditioning cost, and finite gates	Coherent Bures tomography
Small tangent-support model and quantitative perturbation control	Compression and stability
Physical Gram fitting and checkable classical value certificates	Certified optimization
Input circuits, normalization, confidence budget, and full resources	Circuit access and algorithm
Circuit examples, matched baselines, and approximate-low-rank bounds	Applications and limits
Why Fisher conditioning alone is insufficient	Rare-branch obstruction
Primary sources and novelty boundaries	Related work and bibliography

For specialists, the proof route is tomography → compression/stability → certified optimization → circuit access. The applications explain the physical meaning and comparisons. The LLM reading guide maps questions to their authoritative files.

Study the project

The single educational anchor is Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă's Multi-parameter estimation beyond Quantum Fisher Information (2020). It explains the statistical meaning of simultaneous precision and the Holevo bound.

Start with the source-to-project bridge, which selects the relevant sections, translates notation and assumptions, and works through how the estimation problem becomes a certified quantum computation. The 30-milestone map organizes the full project into background/preliminaries and narrative/innovation, with explicit understanding targets, dependencies, and links to the proofs. The quantum acquisition and certification arguments are developed in this repository and retain their primary research dependencies.

A checkable circuit example

A four-qubit parity-preserving circuit prepares a rank-two probe with occupied weights $3/4$ and $1/4$. It senses two local Pauli quadratures. Exact rational primal and dual witnesses give

$$2.250762525\le C_H(I_2)\le2.250762539,$$

while the SLD benchmark lies between $2.233360231$ and $2.233360232$. The Fisher matrix is certified to satisfy $J\succeq(4/5)I_2$. Thus the example has a nonzero incompatibility penalty despite good conditioning. A numerical comparison also agrees after compressing the physical model from 16 dimensions to 6. The full circuit and certificate are reproducible.

Reproduce the finite checks

With the dependencies in verification/requirements.txt, run:

python3 verification/run_all.py

The suite checks spectral-peeling geometry, exact physical-fit and Holevo certificates, and the four-qubit example. Its verification guide distinguishes exact rational assertions from floating-point comparisons and explains the committed expected outputs. These checks do not simulate the coherent tomography circuit or establish hardware runtime advantage.

Resource and physical boundaries

Preparation circuits, inverses, controls, derivative normalization, small-register gates, and classical optimization are charged. No QRAM or free loading of exponentially many amplitudes is assumed. The Holevo application requires a full-column-rank factor with an explicitly small environment, even though the general Bures learner allows an arbitrarily large purifier.

Classically contractible circuits and direct overlap methods remain strong baselines. The exact closed-form hardness witness has its own simpler quantum algorithm. The general result establishes a query guarantee for the stated input model; it does not establish a practical hardware crossover.

Approximate low rank is controlled only with the stated common-encoding and bounded-generator assumptions. A small discarded state weight can change the precision benchmark at square-root order.

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 [email protected].