Efficient certification of intractable quantum states with few Pauli measurements

TL;DR

The paper addresses the challenge of certifying quantum states produced under Magic-State Injection, a cornerstone of fault-tolerant quantum computing. It introduces Clifford-enhanced Product States (CPS) and a Pauli-only certification protocol with efficient sample complexity in both i.i.d. and adversarial settings, enabling practical verification of universal quantum computations under minimal experimental assumptions. The approach combines back-propagation of Clifford operations, Direct Fidelity Estimation, and a robust fidelity witness to bound fidelity from simple Pauli measurements. This work provides a practical, scalable route to verify MSI-based quantum computations and bridges gaps between stabilizer-based certification and more general, non-Pauli verification methods.

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.

Figures (5)

• Figure 1: Relationship between Clifford-enhanced Product States ($\mathcal{C}$PS), Stabilizer States (STAB), Graph States (GS), and Hypergraph States (HGS). $\mathcal{C}$PS reduces to STAB if the product state contains only stabilizer states ; and it furthermore reduces to GS if in addition the Clifford circuit is only made of $CZ$. A HGS is a GS if the underlying hypergraph is a graph.
• Figure 2: Magic-State Injection model. The blue dashed box captures the fact that teh state prior to measurements has the structure of the $\mathcal{C}\text{PS}$ class introduced previously. The green box consists of a layer of adaptive single-qubit Pauli measurements to drive the computation.
• Figure 3: Measurement back-propagation: measuring observable $P$ on $C^\dagger \rho C$ yields the same outcome distribution as measuring $C P C^\dagger$ on $\rho$. By definition of the Clifford group, $CPC^\dagger$ is also a Pauli observable if $C\in\mathcal{C}_n$.
• Figure 4: Visual description of our protocol to certify $\mathcal{C}\text{PS}$. With the notations of Protocol \ref{['protocol:certification']}, here for copy $j$ for which we sampled $i, P$, we have $g(x) = \mathrm{sgn}(\chi_{\psi_{i_j}}(P_j)) \,x_j$. Also, the estimate for the witness $\bar{W}$ is computed from the estimator $\bar{X}$ using $\bar{W} = 1-n+m\times \bar{X}$. The samples in red boxes are used for certification, while the blue box is a sample that is untouched, left to be used for the computation.
• Figure 5: Illustration of the protocol to run against an adversarial prover. The verifier receives different $n-$qubit subsystems of a global system $\rho^{1\ldots N}\in(\mathcal{H}^{\otimes n} )^{\otimes N}$ The verifier chooses a random partition, discards $K$ systems, applies the certification protocol on $N-K-1$ chosen samples, and leaves one for further computation.

Theorems & Definitions (8)

• Lemma 1: Hoeffding's inequality
• Theorem 1: Efficient certification of $\mathcal{C}$PS
• proof
• Lemma 2: State certification in the non-i.i.d. setting FKMO24learning
• Theorem 2: Efficient verification of $\mathcal{C}$PS
• proof
• Theorem 3: Efficient verification of MSI-based quantum computation
• proof