# Techniques: Verifiable blind observable estimation

## What the paper contributes (one paragraph)

The paper closes a gap in cryptographic verification of delegated quantum computing: existing efficient verification protocols only cover decision problems where correctness is amplified by classical majority voting, leaving out observable (expectation-value) estimation, the statistical task behind most near-term quantum advantage claims. The authors first define the Secure Delegated Observable Estimation (SDOE) ideal resource, a formal Abstract Cryptography object that returns an estimate of an observable's expectation value within a chosen bias or aborts, while keeping the observable and result blind. They then give the Verifiable Blind Observable Estimation (VBOE) protocol, which constructs SDOE with composable security and a security error negligible in the number of rounds, using sequential sampling and a client-side classical empirical average rather than costly quantum averaging. Crucially this requires zero extra qubit overhead, making rigorous, composable verification of observable estimation feasible on near-term hardware.

## Techniques offered

- **Verifiable blind observable estimation (delegated expectation-value verification)** - an interactive protocol letting a near-classical client delegate observable estimation to an untrusted noisy server and verify the empirical average via interleaved trap-based test rounds plus client-side classical averaging, with no extra qubit overhead.
  - guarantee: composable security in the Abstract Cryptography framework, with distinguishing (soundness) error negligible in the number of computation rounds for a fixed test-to-computation ratio
  - quote: "We then present the Verifiable Blind Observable Estimation (VBOE) protocol, which efficiently constructs this resource."
- **Secure Delegated Observable Estimation (SDOE) ideal resource** - a formal cryptographic resource that abstracts trustworthy, blind expectation-value estimation, returning an estimate within a specified bias or aborting, and serving as the composable security target other (possibly hybrid) protocols can build on.
  - guarantee: ideal-resource definition with perfect blindness and bounded estimation bias, enabling composable security guarantees without reproving security in a wider context
  - quote: "To resolve this, we introduce the Secure Delegated Observable Estimation (SDOE) ideal resource, the first formal cryptographic framework for trustworthy expectation-value estimation within Abstract Cryptography."

## Where it could apply

- Target primitive(s): VQE, QAOA, and other expectation-value or observable-estimation routines (including sampling-based estimation and Trotterised-simulation observables); out-of-time-order correlators
- Target application group(s): platforms_tooling, quantum_ml, chemistry_materials, optimization
- Code family / hardware assumptions: MBQC measurement patterns on a fixed graph with a given flow and chromatic number K; client with single-qubit preparation and a quantum link to the server; no fault tolerance and no extra qubit overhead required; demonstrated substrates for related protocols include photonic and trapped-ion platforms

## Caveats

The protocol assumes the client can prepare and send single qubits over a quantum channel to the server (it is not a fully classical-client scheme), and the analysis is developed for a binary observable on a single-qubit output, with extension to bounded non-binary observables argued but not fully proved. Security is stated up to a bias epsilon and an abort option, so a malicious server can always force an abort and can perturb the result within epsilon. Secure integration with quantum error mitigation is raised as an open question, not delivered.