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(2020) Optimal Verification of Stabilizer States PHYSICAL REVIEW RESEARCH 2, 043323 (2020) Optimal verification of stabilizer states Ninnat Dangniam,1,2,* Yun-Guang Han,1,2 and Huangjun Zhu 1,2,3,4,† 1Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China 2State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 3Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China (Received 25 July 2020; accepted 30 October 2020; published 4 December 2020) Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, Greenberger-Horne-Zeilinger (GHZ) states, and antisymmetric basis states. In this work, we investigate systematically the optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the number of qubits and the specific stabilizer state. Then we propose a simple algorithm for constructing optimal protocols based on Pauli measurements. Our calculations suggest that optimal protocols based on Pauli measurements can saturate the above bound for all entangled stabilizer states, and this claim is verified explicitly for states up to seven qubits. Similar results are derived when each party can choose only two measurement settings, say X and Z. Furthermore, by virtue of the chromatic number, we provide an upper bound for the minimum number of settings required to verify any graph state, which is expected to be tight. For experimentalists, optimal protocols and protocols with the minimum number of settings are explicitly provided for all equivalent classes of stabilizer states up to seven qubits. For theorists, general results on stabilizer states (including graph states in particular) and related structures derived here may be of independent interest beyond quantum state verification. DOI: 10.1103/PhysRevResearch.2.043323 I. INTRODUCTION state σ is “close” to the target ρ with some confidence. More precisely, the verification scheme accepts a density operator Engineered quantum systems have the potential to effi- σ that is close to the target state with the worst-case fidelity ciently perform tasks that are believed to be exponentially 1 − and confidence 1 − δ. In other words, the probabil- difficult for classical computers such as simulating quantum ity of accepting a “wrong” state σ with |σ | 1 − systems and solving certain computational problems. With the is at most δ. For the convenience of practical applications, potential comes the challenge of verifying that the quantum usually the verification protocols are constructed using lo- devices give the correct results. The standard approach of cal operations and classical communication (LOCC). Such quantum tomography accomplishes this task by fully char- verification protocols have been gaining traction in the quan- acterizing the unknown quantum system, but with the cost tum certification community [7–19] because they are easy exponential in the system size. However, rarely do we need to implement and potentially require only a small number to completely characterize the quantum system as we often of copies of the state. However, sample-optimal protocols have a good idea of how our devices work, and we may only under LOCC have been found only for bipartite maximally need to know if the state produced or the operation performed entangled states [2,3,12], two-qubit pure states [13], n-partite is close to what we expect. The research effort to address Greenberger-Horne-Zeilinger (GHZ) states [17], and most re- these questions have grown into a mature subfield of quantum cently antisymmetric basis states [18]. certification [1]. Maximally entangled states and GHZ states are subsumed Statistical verification of a target quantum state ρ = under the ubiquitous class of stabilizer states, which can be || [2–6] is an approach for certifying that an unknown highly entangled yet efficiently simulatable [20,21] and ef- ficiently learnable [22] under Pauli measurements. Another notable example of stabilizer states are graph states [23,24], *[email protected] which have simple graphical representations that transform †[email protected] nicely under local Clifford unitary transformations. They find applications in secret sharing [25], error correcting codes Published by the American Physical Society under the terms of the [26,27], and cluster states in particular are resource states Creative Commons Attribution 4.0 International license. Further for universal measurement-based quantum computing [28]. distribution of this work must maintain attribution to the author(s) Stabilizer states and graph states can be defined for multiqudit and the published article’s title, journal citation, and DOI. systems with any local dimension; nevertheless, multiqubit 2643-1564/2020/2(4)/043323(36) 043323-1 Published by the American Physical Society DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020) stabilizer states are the most prominent because most quantum suggests that the maximum spectral gap achievable by X information processing tasks build on multiqubit systems. In and Z measurements is 1/2. For the ring cluster state, we this paper, we only consider qubit stabilizer states and graph prove this result rigorously by constructing an explicit optimal states unless stated otherwise, but we believe that many results verification protocol. We also prove that three settings based presented here can be generalized to the qudit setting as long on Pauli measurements (or X and Z measurements) are both as the local dimension is a prime. Efficient verification of necessary and sufficient for verifying the odd ring cluster state stabilizer states have many applications, including but not with at least five qubits. limited to blind quantum computing [29,30] and quantum gate In the course of study, we generalize the concepts of verification [31–33]. canonical test projectors, admissible Pauli measurements, and While one might expect that the determination of an opti- admissible test projectors, first introduced in Ref. [17]for mal verification strategy to be difficult in general, one could GHZ states, to arbitrary stabilizer states. We also clarify their hope for the answer for stabilizer states in view of their basic properties, which are of interest to quantum state ver- relatively simple structure. Given an n-qubit stabilizer state, ification in general. Meanwhile, we introduce several graph Pallister, Montanaro, and Linden [4] showed that the optimal invariants that are tied to the verification of graph states and strategy when restricted to the measurements of nontrivial clarify their connections with the chromatic number. In ad- stabilizers (to be introduced below) is to measure all 2n − 1 dition to their significance to the current study, these results of them with equal probabilities, which yields the optimal provide additional insights on stabilizer states and graph states constant scaling of the number of samples, themselves and are expected to find applications in various other related problems. n − δ−1 δ−1 ≈ 2 1 ln ≈ 2ln . The rest of this paper is organized as follows. First, we N − (1) 2n 1 present a brief introduction to quantum state verification in One could choose to measure only n stabilizer generators at Sec. II and preliminary results on the stabilizer formalism the expense of now a linear scaling [4]: in Sec. III. In Sec. IV, we study canonical test projec- tors and admissible test projectors for stabilizer states and n ln δ−1 graph states and clarify their properties. In Sec. V, we de- N ≈ . (2) rive an upper bound for the spectral gap of verification operators based on separable measurements. Moreover, we This trade-off is not inevitable in general. Given a graph state propose a simple algorithm for constructing optimal verifica- associated with the graph G, by virtue of graph coloring, tion protocols based on Pauli measurements and provide an χ Ref. [7] proposed an efficient protocol which requires (G) explicit optimal protocol for each connected graph state up to χ −1 δ−1 measurement settings and (G) ln samples. Here seven qubits. In Sec. VI, we discuss optimal verification of χ the chromatic number (G)ofG is the smallest number of graph states based on X and Z measurements. In Sec. VII, colors required so that no two adjacent vertices share the same we consider verification of graph states with the minimum color. With this protocol, one can verify two-colorable graph number of settings. Section VIII summarizes this paper. To states, such as one- or two-dimensional cluster states, with streamline the presentation, the proofs of several technical −1 δ−1 2 ln tests. results are relegated to the Appendices, which also contain In this paper, we study systematically optimal verification Tables I and II. of stabilizer states using Pauli measurements. We prove that the spectral gap of any verification operator of an entangled stabilizer state based on separable measurements is upper II. STATISTICAL VERIFICATION / bounded by 2 3. To verify the stabilizer state within infidelity A. The basic framework and significance level δ, therefore, the number of tests re- quired is bounded from below by Let us formally introduce the framework of statistical ver- ification of quantum states. Suppose we want to prepare the 1 3lnδ−1 target state ρ =||, but actually obtain the sequence N = ln δ ≈ . (3) σ ,...,σ ln[1 − 2/3] 2 of states 1 N in N runs. Our task is to determine whether all of these states are sufficiently close to the target Moreover, we propose a simple algorithm for constructing state on average (with respect to the fidelity, say). Following optimal verification protocols of stabilizer states and graph Refs. [4–6], we perform a local measurement with binary out- states based on nonadaptive Pauli measurements.
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