Efficient certification of intractable quantum states with few Pauli measurements

TL;DR

This paper presents an efficient, Pauli-measurement-based protocol for certifying a broad class of quantum states essential for fault-tolerant quantum computing, enabling practical verification with minimal experimental requirements.

Contribution

It introduces the first efficient Pauli-only certification protocol for Magic-State Injection states, bridging a gap in quantum state verification methods.

Findings

Requires only single-qubit Pauli measurements and classical post-processing.

Efficient in both i.i.d. and adversarial settings.

Enables practical verification of universal quantum computation.

Abstract

Verification of quantum computations is crucial as experiments advance toward fault-tolerant quantum computing. Yet, no efficient protocol exists for certifying states generated in the Magic-State Injection model -- the foundation of several fault-tolerant quantum computing architectures. Here, we introduce an efficient protocol for certifying Clifford-enhanced Product States, a large class of quantum states obtained by applying an arbitrary Clifford circuit to a product of single-qubit, possibly magic, states. Our protocol only requires single-qubit Pauli measurements together with efficient classical post-processing, and has efficient sample complexity in both the independent (i.i.d.) and adversarial (non-i.i.d.) settings. This fills a key gap between Pauli-based certification schemes for stabilizer or (hyper)graph states and general protocols demanding non-Pauli measurements or classically intractable information about the target state. Our work provides the first efficient, Pauli-only certification protocol for the Magic-State Injection model, leading to practical verification of universal quantum computation under minimal experimental assumptions.