(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 efficiently 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 projectors 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 required is bounded from below by Let us formally introduce the framework of statistical verification 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. An optimal comes {E j, 1 − E j}, labeled as “pass” and “fail” respectively, protocol for each equivalent class of graph states with respect on each state σk for k = 1,...,N with some probability p j. to local Clifford transformations (LC) and graph isomorphism Each operator E j is called a test operator. Here we demand is presented in Table I in the Appendix, and our code is avail- that the target state ρ can pass the test with certainty, which able in Ref. [34]. These results suggest that for any entangled means E jρ = ρ. The sequence of states passes the verification stabilizer state the bound in (3) can be saturated by protocols procedure iff every outcome is “pass.” built on Pauli measurements. The efficiency of the above verification procedure is deter- In addition, we study the problem of optimal verification mined by the verification operator based on X and Z measurements and the problem of verification with the minimum number of measurement settings. m = , The two problems are of interest in certain scenarios in which p jE j (4) the accessible measurement settings are restricted. Our study j=1

043323-2 OPTIMAL VERIFICATION OF STABILIZER STATES PHYSICAL REVIEW RESEARCH 2, 043323 (2020) where m is the total number of measurement settings. If the fi- which implies that delity |σk| is upper bounded by 1 − , then the maximal σ ν() min pk 1/m. (9) average probability that k can pass each test is [4,6] k max tr(σ ) = 1 − [1 − β()] = 1 − ν(). (5) Here the second inequality is saturated iff p = 1/m for all |σ |1− k k. Here, β() is the second largest eigenvalue of the verification operator , and ν():= 1 − β()isthespectral gap from C. Verification of a tensor product the maximum eigenvalue. | Now and throughout the paper, we assume that the states Suppose the target state is a tensor product of the form σ ,σ ,...,σ |= J | | 1 2 N received in different runs are independent of j=1 j , where J 2 and each tensor factor j each other (more general situations can be dealt with using could be a multipartite state and either separable or entangled. the recipe proposed in Refs. [5,6]). Then these states can pass It is instructive to clarify the relation between verification N tests with probability at most operators of | and that of each tensor factor. Given a verification operator for |,thereduced verifi- N N | N cation operator of for the tensor factor j is defined as tr(σ j ) [1 − ν() j] [1 − ν()¯] , (6) j=1 j=1 := || , (10) j j j = / = −|σ | where ¯ j j N with j 1 j is the average where | := | . Note that | =| ,so infidelity [5,6]. If N tests are passed, then we can ensure the j j = j j j j j j is indeed a verification operator for | .If is separable, condition< ¯ with significance level δ = [1 − ν()]N .To j then each is also separable. Reduced test operators can be verify these states within infidelity and significance level δ, j defined in a similar way. the number of tests required is [4–6] Proposition 2. Suppose is a verification operator for |= J | = , ,..., 1 1 1 j=1 j , and j for j 1 2 J are reduced N(,δ,) = ln δ ln . (7) ln[1 − ν()] ν() δ verification operators of . Then the second largest eigenvalue and spectral gap of satisfy the following inequalities: If there is no restriction on the measurements, the optimal performance is achieved by performing the projective mea- β() max β( j ),ν() min ν( j ). (11) jJ jJ surement onto the target state || itself, which yields 1 1 − ν() = 1 and N =ln δ/ln(1 − ) ln δ 1/ as the ul- Proof. timate efficiency limit allowed by quantum theory.1 β( j ) = max j| j| j | j : j | j =0 B. Minimal sets of test operators = max ( j|⊗ j|)(| j⊗| j) { }m | | : | =0 A set of test operators E j j=1 for is minimal if any j j j { }m | proper subset of E j j=1 cannot verify reliably because max || =β(), (12) the common pass eigenspace of operators in the subset has | :| =0 dimension larger than one. A minimal set of test operators has which implies (11). the following properties. m Conversely, suppose are verification operators for Proposition 1. Suppose = p E is a verification j j=1 j j | with spectral gap ν( )for j = 1, 2,...,J.Let = operator based on a minimal set of m test operators. Then j j J | ν() 1/m. If the inequality is saturated then p = 1/m for j=1 j; then is a verification operator for , and j j all j. are reduced verification operators of by the definition in Proof. By assumption, for each k ∈{1, 2,...,m}, there (10). Straightforward calculation shows that the spectral gap of reads exists a pure state |k that is orthogonal to the target state | and belongs to the pass eigenspace of E j for all j = k, ν() = min ν( j ), (13) that is, E j|k=|k. Therefore 1 jJ β() k||k p jk|E j|k= p j which saturates the upper bound in (11). In addition, if each j=k j=k j can be realized by LOCC (Pauli measurements), then so can . On the other hand, the number of distinct test opera- = 1 − p ∀k, (8) k tors (measurement settings) required to realize (naively as suggested by the definition) increases exponentially with the number J of tensor factors. It is of practical interest to reduce this number. 1A related certification framework by Kalev and Kyrillidis [35]for Suppose can be realized by the set of test operators stabilizer states is, in a sense, opposite to ours. In our framework, j { ( j)}m j = m j ( j) ( j) ( j) m j we are given the worst case fidelity and are asked to find an optimal Ek k=1, that is, j k=1 pk Ek , where (pk )k=1 is a | measurement, whereas in their work we are given a (stabilizer) mea- probability vector. In addition, j is the unique common − ( j) = surement and are asked to bound the worst case fidelity 1 to the eigenstate of Ek with eigenvalue 1. Let m max j m j; then desired stabilizer state within some radius r (their “”). | can be reliably verified by the following set of test

043323-3 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020) operators the symplectic form J T 0 −1n = ( j), = , ,..., . [μ, ν]:= μ Jν, J = . (19) Ek : Ek k 1 2 m (14) 1n 0 j=1 Let μx (μz) be the vector composed of the first (last) n ( j) = < μ μ = μx μz Here we choose Ek 1 for m j k m, so that each test Ek elements of ; then ( ; ), where the semicolon de- can be realized by virtue of the original test for each tensor notes the vertical concatenation. In addition, we have [μ, ν] = factor or the trivial test (with the test operator equal to the μz · νx + μx · νz. (Addition and subtraction are the same in identity). To verify this claim, first note that Ek|=| for arithmetic modulo 2.) The Weyl representation [36] of each = , ,..., | μ ∈ 2n k 1 2 m, so each Ek is a test operator for . Suppose vector Z2 yields a Pauli operator | is a common eigenstate of all Ek with eigenvalue 1, that μx·μz μx μz | | = ρ = | | g(μ) = i X Z , (20) is, Ek 1. Let j tr j¯( ), where tr j¯ denotes the partial trace over all tensor factors except for the jth factor. x μx μx z μ = 1 ⊗···⊗ n μ where X X1 Xn and similarly for Z . Each Then we have P k μ = , , , Pauli operator in n is equal to i g( )fork 0 1 2 3 and ρ ( j) = | ( j) ⊗ | | | = μ ∈ 2n = = 1 tr jEk Ek 1 Ek 1 (15) Z2 . For example, we have X g((1;0)), Z g((0;1)), and Y = g((1; 1)). By the following identity for all j, k. This equation implies that tr(ρ E ( j)) = 1, so each j k μ ν = [μ,ν] μ + ν = 2[μ,ν] ν μ , ρ ( j) g( )g( ) i g( ) i g( )g( ) (21) j is supported in the eigenspace of Ek with eigenvalue 1 for all k.Itfollowsthatρ j =| j j| and | |=||,so g(μ) and g(ν) commute iff [μ, ν] = 0. | 2n is the unique common eigenstate of all Ek with eigenvalue The symplectic complement of a subspace W in Z2 is 1 and it can be reliably verified by the test operators in (14). defined as m Let (qk ) = be any probability vector with qk > 0for k 1 W ⊥ = μ ∈ Z2n | [μ, ν] = 0, ∀ν ∈ W . (22) all k (sufficient number of settings are measured) and = 2 m | ⊥ k=1 qkEk; then is a verification operator for with The subspace W is isotropic if W ⊂ W , in which case ν > ( ) 0 according to the above discussion. In addition, the [μ, ν] = 0 for all μ, ν ∈ W . Hence all Pauli operators associ- | reduced verification operator of for tensor factor j reads ated with vectors in W commute with each other. The maximal m dimension of any isotropic subspace is n, and such a maximal = || = ( j). = ⊥ j j j qkEk (16) isotropic subspace satisfies the equality W W and is called k=1 Lagrangian. Each isotropic subspace W of dimension k is determined by a 2n × k basis matrix over Z2n whose columns According to proposition 2,wehave 2 form a basis of W . Conversely, a 2n × k matrix M is a basis m m matrix for an isotropic subspace if the following condition ν ν = ν ( j) ν ( j) , ( ) ( j ) qkEk max qkEk holds: (q )k k=1 k k=1 T (17) M JM = 0k×k. (23) where the maximization is taken over all probability vectors Two Lagrangian subspaces W and W of Z2n are complemen- with m components. The right-hand side coincides with the 2 tary if their intersection is trivial (consists of the zero vector maximum spectral gap achievable by any verification operator only), in which case Span(W ∪ W ) = Z2n. of | that is based on the set of test operators {E ( j)}m j .Note 2 j k k=1 The Clifford group is the normalizer of the Pauli group P . q = m < k m n that k 0for j when the maximum spectral gap Up to phase factors, it is generated by phase gates, Hadamard ( j) = < is attained given that Ek 1 for m j k m. gates for individual qubits and controlled-not gates for all pairs of qubits. Its quotient over the Pauli group is isomorphic III. STABILIZER FORMALISM to the symplectic group with respect to the symplectic form in (19). A. Pauli group H = 2 ⊗n Let (C ) be the Hilbert space of n qubits. The B. Stabilizer codes Pauli group for one qubit is generated by the following three S P S matrices: A subgroup of n is a stabilizer group if is commutative and does not contain −1. Since S cannot contain − = 01, = 0 i , = 10. a Pauli operator with phases ±i (otherwise −1 ∈ S), every X Y Z − (18) 10 i0 0 1 element except the identity has order 2. Thus S is isomorphic k k The n-fold tensor products of Pauli matrices and the identity to an elementary abelian group Z2 of order 2 , where k n {1, X,Y, Z}⊗n form an orthogonal basis for the space B(H) is the cardinality of a minimal set of generators. Suppose of linear operators on H. Together with the phase factors that the stabilizer group S is generated by the k generators S , S ,...,S ; then the elements of S can be labeled by {±1, ±i}, these operators generate the Pauli group Pn, which 1 2 k n+1 k has order 4 . Two elements of the Pauli group either com- vectors in Z2 as follows: mute or anticommute. k y = y j , ∈ k . Up to phase factors, n-qubit Pauli operators can be labeled S S j y Z2 (24) 2n by vectors in the binary symplectic space Z2 endowed with j=1

043323-4 OPTIMAL VERIFICATION OF STABILIZER STATES PHYSICAL REVIEW RESEARCH 2, 043323 (2020)

The stabilizer code HS of S is the common eigenspace of between S and S reads eigenvalue 1 of all Pauli operators in S, which has dimen- n−k 1 sion 2 . Alternatively, it is also defined as the common S S = SS tr( ) |S|·|S | tr( ) eigenspace of eigenvalue 1 of the k generators S1, S2,...,Sk. S∈S,S ∈S The projector onto the code space reads n = 2 |S ∩ S |−| −S ∩ S | |S|·|S | ( ( ) ) k + 1 1 S j = = . n S S (25) 2 |S¯∩S | S ∩ S = S¯ ∩ S , |S| 2 |S|·|S | S∈S j=1 = (29) 0 otherwise. Conversely, S happens to be the group of all Pauli operators in S ∩ S S¯ ∩ S S ∩ P that stabilize the stabilizer code HS . So there is a one-to- Note that is a subgroup of of index 2 if n S = S¯ ∩ S one correspondence between stabilizer groups and stabilizer . Equation (29) implies the following equation codes. To later establish the relation between stabilizer groups 2n|S¯∩S | S ∩ S = S¯ ∩ S , |S|·|S | w w and isotropic subspaces, we introduce the signed stabilizer tr( S,w S ,w ) = (30) 0 otherwise. group of the stabilizer code HS to be the union k for all w ∈ Z given that S¯ = S¯ and S¯ ∩ S = S¯ ∩ S .In S¯ := S ∪ (−S), (26) 2 w w addition, we have where −S :={−S|S ∈ S}. 2n S , ,..., tr( S, S ) = tr( S ) = (31) Given the stabilizer group with generators S1 S2 Sk, w |S | for each w ∈ Zk , define S as the group generated by w 2 w − w j = , ,..., S ( 1) S j for j 1 2 k; then w is also a stabilizer thanks to the equality S, = 1. So the number of H S w w group. The stabilizer code S of w is the common ∈ n = w vectors w Z2 at which tr( S,w S ) 0 is equal to , ,..., − w j = eigenspace of S1 S2 Sk with eigenvalue ( 1) for j |S|/|S¯ ∩ S |. 1, 2,...,k and is also denoted by HS, . The projector onto w Lemma 1. Suppose S j, T j are stabilizer groups on H j for the stabilizer code reads j = 1, 2,...,J;letS = S1 × S2 ×···×SJ and T = T1 × n T ×···×T H ⊗ H ⊗···⊗H + − w j 2 J be stabilizer groups on 1 2 J 1 ( 1) S j y y ¯ S,w = = χw(S )S , (27) and let T be the signed stabilizer group associated with T . = 2 ∈ n j 1 y Z2 Then S ∩ T¯ = L × L ×···×L , where 1 2 J (32) y w·y where L = S ∩ T¯ with T¯ being the signed stabilizer groups χw(S ) = χw(y) = (−1) (28) j j j j associated with T¯j. S k can be understood as a character on or Z2 [37]. Note that Proof. To simplify the notation, here we prove (32)inthe H ∈ k = all stabilizer codes S,w for w Z2 share the same signed case J 2; the general case can be proved in a similar way. stabilizer group, that is, S¯w = S¯. Any S ∈ S has the form S = S1 ⊗ S2 with S1 ∈ S1 and S2 ∈ According to the Weyl representation in (20), each ele- S2. If in addition S ∈ T¯ , then S1 ∈ T¯1 and S2 ∈ T¯2. Therefore ment in the stabilizer group S is equal to g(μ)or−g(μ) S1 ∈ L1 and S2 ∈ L2, so that S ∈ L1 × L2, which implies that μ ∈ 2n S S ∩ T¯ ⊆ L × L for Z2 . In this way, is associated with an isotropic 1 2. ⊂ 2n ∈ L × L = ⊗ subspace W Z2 of dimension k, and there is a one-to-one Conversely, any S 1 2 has the form S S1 S2 correspondence between elements in S and vectors in W . with S1 ∈ S1 ∩ T¯1 and S2 ∈ S2 ∩ T¯2, which implies that S ∈ S ¯ Suppose the k generators S1, S2,...,Sk of S correspond to and S ∈ T¯ . Therefore L1 × L2 ⊆ S ∩ T , which confirms (32) μ ,μ ,...,μ the k symplectic vectors 1 2 k, which form a basis in view of the opposite inclusion relation derived above. y μ = in W . Then S corresponds to the vector j y j j My for ∈ k = μ ,μ ,...,μ each y Z2, where M : ( 1 2 k ) is a basis matrix C. Stabilizer states for W . Note that all the stabilizer groups S for w ∈ Zk are w 2 When the stabilizer group S is maximal, that is, |S|=2n, associated with the same isotropic subspace according to the the stabilizer code HS has dimension 1 and is represented by above correspondence, and this correspondence extends to |S S¯ a normalized state called a stabilizer state and denoted by . the signed stabilizer group . Conversely, given an isotropic |S S k Note that is uniquely determined by up to an overall subspace W of dimension k,2 stabilizer groups can be con- |S k phase factor. According to (25), the projector onto reads structed as follows. Let {μ j} = be any basis for W ; for each j 1 n k + vector a in Z , a stabilizer group can be constructed from 1 1 S j 2 |SS|= S = S = , (33) a j n the k generators (−1) g(μ j )for j = 1, 2,...,k. All these 2 2 S∈S j=1 stabilizer groups extend to a common signed stabilizer group. In this way, there is a one-to-one correspondence between where S1, S2,...,Sn are a set of generators of S. For each ∈ n S − w j signed stabilizer groups and isotropic subspaces. w Z2, define w as the group generated by ( 1) S j for

Suppose S and S are two n-qubit stabilizer groups of j = 1, 2,...,n; then Sw is also a maximal stabilizer group. k k orders 2 and 2 , respectively; let S and S be the projec- In addition, the associated stabilizer state |Sw is the com- w j tors onto the corresponding stabilizer codes. Then the overlap mon eigenstate of S1, S2,...,Sn with eigenvalue (−1) for

043323-5 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020) y n y j n S S j = 1, 2,...,n.LetS = = S for y ∈ Z as in (24) with Lemma 2. Suppose and are two maximal stabilizer j 1 j 2 k = n; then the projector onto |Sw reads groups with basis matrix M and M , respectively. Then T |S¯ ∩ S |=|ker(MT JM )|=2n−rank(M JM ),

n n−rank(MT M +MT M ) + − w j = 2 z x x z , (39) 1 ( 1) S j w·y y |S S |= S, = = (−1) S , w w w 2 j=1 y∈Zn 2 −n T 2 max |Sw|S | = 2 | ker(M JM )| ∈ n w (34) w Z2 which reduces to (33) when w = 00 ···0. The set of stabilizer − MT JM − MT M +MT M = 2 rank( ) = 2 rank( z x x z ), states |S for w ∈ Zn forms an orthonormal basis in H, w 2 (40) known as a stabilizer basis. Stabilizer bases are in one-to- 2n one correspondence with Lagrangian subspaces in Z2 . Based where Mx (Mz) denotes the submatrix of M composed of the on this observation, one can determine the total number of ﬁrst (last) n rows, and Mx (Mz) is deﬁned in a similar way. n-qubit stabilizer states, with the result [36] D. Graph states

n Before introducing graph states, it is helpful to briefly = , 2n (2 j + 1) 2n(n+3)/2, (35) review basic concepts related to graphs. A graph G (V E ) is defined by a vertex set V and an edge set E in which each j=1 element of E is a two-element subset of V . The graph G is nonempty if it has at least one edge, that is, E is nonempty. Without loss of generality, the vertex set can be chosen to be which is exponential in the number n of qubits. V ={1, 2,...,n}. Two distinct vertices i, j ∈ V are adjacent Note that all stabilizer states in a stabilizer basis share if {i, j}∈E. The adjacency relation is characterized by the the same signed stabilizer group as defined in (26), that is, × n adjacency matrix A, which is an n n symmetric matrix with S¯w = S¯ for all w ∈ Z . In this way, there is a one-to-one cor- 2 A , = 1ifi and j are adjacent and A , = 0 otherwise. The respondence between signed stabilizer groups and stabilizer i j i j neighbor of a vertex j is composed of all vertices that are bases (and Lagrangian subspaces). adjacent to j; a vertex is isolated if it has no neighbor. A Suppose S and S are two n-qubit maximal stabilizer subset B of V is an independent set if every two vertices in B groups. Then the fidelity between |S and |S reads are not adjacent. The independence number α(G)ofG is the maximum cardinality of independent sets of G. A coloring of G is an assignment of colors to the vertices such that every −n 2 |S¯ ∩ S | S ∩ S = S¯ ∩ S , |S|S |2 = (36) two adjacent vertices receive different colors. The chromatic 0 otherwise, number χ(G)ofG is the minimum number of colors required to color G. The graph G is two colorable if G can be colored using two distinct colors, that is, √χ(G) 2. according to (29), where S¯ = S ∪ (−S) is the signed stabi- Denote by |+ = (|0+|1)/ 2 the eigenstate of X with lizer group of |S. In addition, eigenvalue 1. To each graph G = (V, E ) with n vertices, a graph state |G of n-qubits (corresponding to the n vertices) can be constructed from the product state |+⊗n by applying −n|S¯ ∩ S | S ∩ S = S¯ ∩ S , the controlled-Z gate 2 2 w w |S |S | = (37) w w 0 otherwise CZ =|00|⊗1 +|11|⊗Z (41) to every pair of qubits that are adjacent. In other words, , ∈ n | = |+⊗n , for all w w Z2 according to (30). The number of vec- G CZi, j (42) − ∈ n |S |S |2 = n|S¯ ∩ S | { , }∈ tors w Z2 that satisfy the equality w 2 i j E is equal to 2n/|S¯ ∩ S |. Two maximal stabilizer groups S where CZ , denotes the CZ gate acting on the adjacent qubits and S are complementary if |S¯ ∩ S |=1 or, equivalently, i j i and j. Note that all these CZ gates commute with each other, |S ∩ S¯ |=1, which is the case iff the Lagrangian subspaces so their order in the product does not matter. To give some associated with S and S , respectively, are complementary. In examples, a path or linear graph yields a linear cluster state; a addition, S and S are complementary iff the stabilizer bases ring or cycle yields a ring cluster state; a square lattice yields a associated with S and S are mutually unbiased [38], that is, two-dimensional cluster state. A star graph or complete graph yields a GHZ state up to a local Clifford transformation. Alternatively, the graph state |G is uniquely defined (up to |S |S |2 = −n ∀ , ∈ n. w w 2 w w Z2 (38) a phase factor) as the common eigenstate with eigenvalue 1 of the n stabilizer generators = ⊗ , The following lemma is useful to computing the fidelities S j Xj Zk (43) between stabilizer states; see Appendix A for a proof. ( j,k)∈E

043323-6 OPTIMAL VERIFICATION OF STABILIZER STATES PHYSICAL REVIEW RESEARCH 2, 043323 (2020) which generate the stabilizer group SG of |G. The elements S n = of G can be labeled by vectors in Z2 as in (24) with k n, that is, n y = y j , ∈ n. S S j y Z2 (44) j=1 ∈ 2n | For each w Z2 ,let Gw be the common eigenstate of S j with eigenvalue (−1)w j for j = 1, 2,...,n. Then (34) reduces to FIG. 1. A star graph (a) and a complete graph (b) with the same n number of vertices (depicted with minimum colorings) are equivalent + − w j 1 ( 1) S j w·y y under LC. Their associated graph states are both LC-equivalent to the |GwGw|= = (−1) S . (45) = 2 ∈ n GHZ state. j 1 y Z2 The stabilizer states |G for w ∈ Zn form a stabilizer basis, w 2 under LC with respect to any given vertex. This fact explains known as a graph basis. why the corresponding graph states are equivalent under LC. Let 1 be the n × n identity matrix over Z .Thecanonical 2 Both graph states are equivalent to GHZ states, as illustrated basis matrix of the graph state |G is defined as the vertical in Fig. 1. concatenation of 1 and the adjacency matrix A and is denoted by A˜ = (1; A) [in contrast with the horizontal concatenation denoted by (1, A)]. The symplectic vector associated with the IV. TEST PROJECTORS FOR STABILIZER STATES y stabilizer operator S in (44) can be expressed as A˜y,sothe A. Canonical test projectors for stabilizer states isotropic subspace V associated with the graph state |G is G Pauli measurements are the simplest measurements that given by can be applied to verifying stabilizer states. Each Pauli mea- = ˜ | ∈ n . VG Ay y Z2 (46) surement for a single qubit can be specified by a symplectic 2 vector in Z2; to be specific, Pauli X, Y , and Z measurements Thanks to the special form of the canonical basis matrices correspond to the vectors (1; 0), (0; 1), and (1; 1), respec- of graph states, lemma 2 can be simplified as follows. tively, while the trivial measurement corresponds to the vector Lemma 3. Suppose G is an n-vertex graph with adjacency S (0; 0). Each Pauli measurement on n qubits can be specified matrix A and is a maximal stabilizer group of n qubits with μ = μx μz 2n by a symplectic vector ( ; )inZ2 , which means basis matrix M. Then μxμz μx μz the Pauli operator measured for qubit j is i j j X j Z j for − T + T j j |S ∩ S¯|= T + T = n rank(Mz Mx A), = , ,..., G ker Mz Mx A 2 (47) j 1 2 n. The weight of the Pauli measurement is the number of qubit j such that (μx,μz ) = (0, 0). The Pauli mea- − j j | |S |2 = n T + T max Gw w 2 ker Mz Mx A surement is complete if the weight is equal to n, so that the w ∈Zn 2 Pauli measurement for every qubit is nontrivial. In that case, − MT +MT A μ = 2 rank( z x ), (48) the corresponding symplectic vector is also called complete. 2n The set of complete symplectic vectors in Z2 is denoted by where Mx (Mz) denotes the submatrix of M composed of the 2n (Z2 )C. first (last) n rows. Let Tμ be the stabilizer group generated by the n local μx μz Lemma 4. Suppose G and G are two n-vertex graphs with μxμz j j Pauli operators i j j X Z associated with the Pauli mea- adjacency matrices A and A , respectively. Then j j k surement; then |Tμ|=2 , where k is the weight of the Pauli n−rank(A+A ) |SG ∩ S¯G |=|ker(A + A )|=2 , (49) measurement. Let T¯μ = Tμ ∪ (−Tμ) be the signed stabilizer group. Each outcome of the Pauli measurement corresponds 2 −n − rank(A+A ) max |Gw|Gw | = 2 | ker(A + A )|=2 . (50) to a common eigenspace of the stabilizer group Tμ. It can w ∈Zn 2 ∈ n be specified by a vector v Z2, which corresponds to the μxμz μx μz It is known that every stabilizer state is equivalent to a common eigenspace of i j j X j Z j with eigenvalue (−1)v j graph state under some local Clifford transformation (LC) for j = 1, 2,...,n. The projector onto the eigenspace reads

[26,39,40]. In addition, every Calderbank-Shor-Steane (CSS) μxμz μx μz n + − v j j j j j state is equivalent to a graph state of a two-colorable graph and 1 ( 1) i Xj Z j μ, = . (51) vice versa [41]. Recall that a stabilizer state is a CSS state if its v = 2 stabilizer group can be expressed as the product of two groups j 1 one of which is generated by X operators for certain qubits, Note that v j must be 0 if the Pauli measurement for qubit j while the other is generated by Z operators. Two graph states is trivial; otherwise, μ,v = 0. So those vectors v of interest | | n G and G are equivalent under LC iff the corresponding to us belong to a subspace of Z2 of dimension k. Neverthe- graphs G and G are equivalent under local complementation, less, vectors outside this subspace do not affect the following hence both transformations are abbreviated as LC. An LC with analysis. respect to a vertex j turns adjacent (nonadjacent) vertices in Suppose we want to verify the n-qubit stabilizer state |S the neighborhood of j into nonadjacent (adjacent) vertices. associated with the stabilizer group S. Then any test operator For example, a complete graph is turned into a star graph P based on the Pauli measurement μ is a linear combination

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∈ n |S of μ,v for v Z2. To guarantee that the target state then the test projector Pμ is determined by aμ given that can always pass the test, P must satisfy P μ,v whenever Sw|Pμ|Sw=1 when w = 0. In view of this fact, the vector S| μ,v|S > 0. The canonical test projector based on the aμ is referred to as the test vector associated with the test Pauli measurement μ is defined as projector Pμ or the Pauli measurement μ (with respect to a given stabilizer basis). Here the index w can also be replaced Pμ = μ,v. (52) j−1 by the natural number j 2 w j if necessary. S| μ, |S>0 v To efficiently compute the diagonal elements Sw|Pμ|Sw In the special case of GHZ states, the concept of canonical of the test projector Pμ and determine the test vector aμ,we test projectors was introduced in Ref. [17], while here our need to introduce additional tools. Let U1 (respectively, U2) definition applies to all stabilizer states. By construction we be the index set of qubits for which the Pauli measurement μ have Pμ E for any other test operator E based on the same is nontrivial (respectively, trivial), that is, Pauli measurement, so it is natural to choose canonical test U = |μx = μz = , projectors if we want to construct an optimal verification 1 j j 1or j 1 (56) protocol. U = |μx = μz = . To clarify the properties of canonical test projectors, we 2 j j 0 and j 0 (57) need to introduce an additional concept. The stabilizer group Let MS be the basis matrix of S associated with the generators Lμ = S ∩ T¯μ is called a local subgroup of S associated with x z S1, S2,...,Sn; denote by MS and MS the first n rows and last the Pauli measurement μ. Given any two stabilizer operators n rows, respectively. Define S, S in Lμ, the tensor factors of S, S for any given qubit com- Lμ = μz x + μx z ,μ∈ 2n, mute with each other. Therefore is locally commutative, MS,μ : diag( )MS diag( )MS Z2 (58) and hence the name. The significance of the local subgroup Lμ x z is tied to the fact that its stabilizer operators can be measured M˜ S,μ := (MS,μ(U1); MS (U2 ); MS (U2 )), (59) simultaneously by the Pauli measurement μ. 2n U Lemma 5. For any μ ∈ Z , the canonical test projector Pμ where MS,μ( 1) is the matrix composed of the rows of MS,μ 2 U x U z U is identical to the stabilizer code projector associated with Lμ, indexed by the set 1 and similarly for MS ( 2 ) and MS ( 2 ). ˜ that is, Note that MS,μ reduces to MS,μ when the Pauli measurement μ is complete. In this case, MS,μ is nothing but the symplectic 1 T Pμ = S. (53) inner product Mμ JMS between the basis matrix Mμ associ- |Lμ| μ S∈Lμ ated with the Pauli measurement and the basis matrix MS associated with the stabilizer state |S. (Recall that addition S| |S Proof of lemma 5. According to (30), μ,v is equal and subtraction are the same in arithmetic modulo 2.) |T¯ ∩ S|/|T |=|L |/|T | to either 0 or μ μ μ μ . Moreover, by (31), For an m × n matrix M defined over Z , denote by rank(M) |T |/|L | n 2 it is nonzero for exactly μ μ vectors v in Z2. When the rank of M and ker(M) the kernel of M: S| μ,v|S is nonzero, the support of μ,v is contained in H L = ∈ n = . the stabilizer code μ associated with μ. So the support of ker(M) y Z2 My 0 (60) Pμ lies in Hμ. In addition, we have Denote by rspan(M) the row span of M: n n |Tμ| 2 2 tr(Pμ) = = , (54) = ∈ m . |Lμ| |Tμ| |Lμ| rspan(M) vM v Z2 (61) so the support of Pμ has the same dimension as Hμ.Itfollows Theorem 1. The local subgroup Lμ, canonical test pro- that Pμ must be the projector onto Hμ, which implies (53). jector Pμ, and test vector aμ,w associated with any Pauli If the local subgroup Lμ is trivial, that is, |Lμ|=1, then measurement μ are determined by M˜ S,μ as follows: the canonical test projector Pμ is equal to the identity, so L ={ y| ∈ ˜ }, the corresponding Pauli measurement is useless to verify the μ S y ker(MS,μ) (62) stabilizer state |S. The following corollary is an immediate 1 y consequence of lemma 5. Pμ = S , (63) | ker(M˜ S,μ)| Corollary 1. Every canonical test projector of the stabi- y∈ker(M˜ S,μ ) |S lizer state is diagonal in the stabilizer basis associated 1 w ∈ rspan(M˜ S,μ), S aμ, =S |Pμ|S = (64) with the stabilizer group ; the diagonal elements are equal w w w 0 otherwise. to either 0 or 1, and the rank of the test projector is a power of 2. All canonical test projectors of a given stabilizer state Theorem 1 is proved in Appendix B. As an implication, commute with each other. the order of the local subgroup Lμ and the rank of the test To determine the diagonal elements of Pμ in the stabilizer projector Pμ read basis, we need to specify a concrete basis. To this end, we can n−rank(M˜ S,μ ) |Lμ|=|ker(M˜ S,μ)|=2 , (65) choose any minimal set of generators for S,sayS1, S2,...,Sn. For each w ∈ Zn,let|S be the common eigenstate of S 2 w j rank(M˜ S,μ ) w j tr(Pμ) =|rspan(M˜ S,μ)|=2 . (66) with eigenvalues (−1) for j = 1, 2,...,n. Then {|Sw}w∈Zn n 2 forms a stabilizer basis (cf. Sec. III C). Define aμ as the (2 − The test projector Pμ is trivial iff M˜ S,μ has rank n (full rank). 1) × 1 column vector composed of the entries Let r = rank(M˜ S,μ) and let w1, w2,...,wr be a basis in =S | |S , ∈ n, = , ˜ ˜ aμ,w : w Pμ w w Z2 w 0 (55) rspan(MS,μ). Then rspan(MS,μ) coincides with the span of

043323-8 OPTIMAL VERIFICATION OF STABILIZER STATES PHYSICAL REVIEW RESEARCH 2, 043323 (2020) these basis vectors, that is, and the corresponding graph basis associated with the graph = , r G (V E ) as defined in Sec. III D.LetA be the adjacency r ˜ = rspan(M˜ S,μ) = a w |(a ) ∈ Z ; (67) matrix of G; then A (1; A) is the canonical basis matrix for j j j j 2 | j=1 G . Define = μz + μx ,μ∈ 2n. this observation is helpful to computing the diagonal elements Aμ diag( ) diag( )A Z2 (75) of Pμ in the stabilizer basis. Equation (64) is equivalent to the Then Aμ can be regarded as the symplectic inner product following equation: between the basis matrix Mμ associated with the Pauli mea- μ | 1 w · y = 0 ∀y ∈ ker(M˜ S,μ), surement and the basis matrix (1; A) for the graph state G . G |Pμ|G = (68) w w 0 otherwise, The rank, kernel, and row span of Aμ are denoted by rank(Aμ), ker(Aμ), and rspan(Aμ), respectively. since w ∈ rspan(M˜ S,μ)iffw · y = 0 for all y ∈ ker(M˜ S,μ). Theorem 2. The local subgroup Lμ, canonical test projec- , ,..., Let y1 y2 yn−r be a basis in ker(M˜ S,μ), which has di- tor Pμ, and test vector aμ,w associated with a complete Pauli mension n − r; then w ∈ rspan(M˜ S,μ)iffw · y j = 0for j = measurement μ are determined by Aμ as follows: 1, 2,...,n − r. L ={ y| ∈ }, To determine the minimum rank of canonical test pro- μ S y ker(Aμ) (76) jectors, it suffices to consider complete Pauli measurements. 1 y According to (66), we have Pμ = S , (77) | ker(Aμ)| κ(S) y∈ker(Aμ ) min tr(Pμ) = min tr(Pμ) = 2 , (69) μ∈ 2n μ∈ 2n ∈ , Z2 (Z2 )C 1 w rspan(Aμ) aμ, =G |Pμ|G = (78) w w w 0 otherwise. where Theorem 2 is a special case of theorem 1 tailored to the κ(S):= min rank(MS,μ), (70) μ∈ 2n (Z2 )C verification of a graph state based on a complete Pauli measurement; a simplified proof is presented in Appendix B.Asa ˜ = μ ∈ 2n κ(S) given that MS,μ MS,μ for (Z2 )C. Note that 2 is corollary, we have also the minimum rank of all test operators of |S based on n−rank(Aμ ) Pauli measurements since the canonical test projector attains |Lμ|=|ker(Aμ)|=2 , (79) the minimum rank for a given Pauli measurement. Denote by =| |= rank(Aμ ). Prod the set of pure product states and by ProdP the set of pure tr(Pμ) rspan(Aμ) 2 (80) product states of which each factor is an eigenstate of a Pauli The test projector Pμ is trivial iff Aμ has full rank. operator. Define = ,..., Let r rank(Aμ) and let w1w2 wr be a basis 2 ={ r | ∈ r } (S) = (|S):= max |ϕ|S| , (71) in rspan(Aμ). Then rspan(Aμ) j=1 a jw j (a j ) j Z2 . |ϕ∈Prod Equation (78) is equivalent to the following equation: S = |S = |ϕ|S|2. P( ) P( ): max (72) · = ∀ ∈ , |ϕ∈Prod 1 w y 0 y ker(Aμ) P G |Pμ|G = (81) w w 0 otherwise, The following lemma relates the geometric measure of entanglement (S) of any stabilizer state |S [42]toκ(S). given that w ∈ rspan(Aμ)iffw · y = 0 for all y ∈ ker(Aμ). Lemma 6. Let y1, y2,...,yn−r be a basis in ker(Aμ), which has dimension n − r; then w ∈ rspan(Aμ)iffw · y = 0 for all j = S S = −κ(S). j ( ) P( ) 2 (73) 1, 2,...,n − r. These observations are helpful to computing Proof. The inequality in (73) follows from the definitions Gw|Pμ|Gw efficiently. of (S) and P(S)in(71) and (72). To prove the equality Similar to (69), the minimum rank of test projectors Pμ for −κ(G) | P(S) = 2 , note that each state in ProdP is a stabilizer G is given by state, and the state projector has the form in (51) with μ ∈ κ(G) min tr Pμ = 2 , (82) 2n μ∈ 2n (Z2 )C. Therefore (Z2 )C −n P(S) = max G| μ,v∈Zn |G= max 2 |Lμ| where μ∈ 2n , 2 μ∈ 2n (Z2 )C v (Z2 )C − −κ S κ(G):= min rank(Aμ) = κ(SG). (83) = rank(MS,μ ) = ( ), μ∈ 2n max 2 2 (74) (Z2 )C μ∈ 2n (Z2 )C Define where the second equality follows from (37), and the third 2 equality follows from (65) given that M˜ S,μ = MS,μ for μ ∈ (G) = (|G):= max |ϕ|G| = (SG), (84) 2n |ϕ∈Prod (Z2 )C. 2 P(G) = P(|G):= max |ϕ|G| = P(SG). (85) |ϕ∈ProdP B. Canonical test projectors for graph states Then lemma 6 reduces to For graph states, the discussions in the previous section Lemma 7. can be simplified. Here we only consider complete Pauli mea- −κ(G) surements. Suppose |G and {|G } ∈ n are the graph state (G) (G) = 2 . (86) w w Z2 P

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The following lemma sets an upper bound for κ(G), which gives an admissible test projector PXZX, whereas measuring Y in turn yields an upper bound for the minimum rank of test or Z results in the same inadmissible test projector that has projectors Pμ. higher rank than PXZX. Further calculation shows that there Lemma 8. κ(G) n − α(G), where α(G) is the indepen- are ﬁve admissible Pauli measurements in total, namely, XZX, dence number of G. ZXZ, ZYY, YYZ, YXY (cf. Table I). Lemma 8 and (86) imply the following inequalities: Different Pauli measurements may give rise to the same α − canonical test projector. When more than one measurement (G) (G) 2 (G) n, (87) P settings share the same canonical test projector, the projector which was originally derived in Ref. [43]. can be realized by an incomplete Pauli measurement that these Proof of lemma 8. Let B be an independent set of the graph settings have in common. Therefore this canonical test projec- G = (V, E ) with cardinality α(G). Consider the Pauli mea- tor cannot be admissible by by proposition 3. This observation surement in which X measurements are performed on all yields the following lemma. qubits in B and Z measurements are performed on all qubits Lemma 9. No two admissible Pauli measurements lead to in the complement B = V \ B.Letμ be the corresponding the same canonical test projector. μx = ∈ μz = symplectic vector, then j 1iff j B, while j 1iff Lemma 9 shows that there is a one-to-one correspon- j ∈ B. Since B is an independent set, rspan(diag(μx )A) ⊆ dence between admissible Pauli measurements and admissible z z rspan(diag(μ )). Consequently, rank Aμ = rank(diag(μ )) = canonical test projectors. |B|, and we have Corollary 2. A canonical test projector based on a Pauli measurement is admissible iff it cannot be realized by an κ(G) rank(Aμ) =|B|=n − α(G). (88) incomplete Pauli measurement. Proof. If the test projector can be realized by an incomplete Pauli measurement, then it is not admissible by proposition 3. If the test projector is realized by a complete C. Admissible test projectors Pauli measurement μ, but is not admissible, then we can ﬁnd

Suppose we want to verify the stabilizer state |S based a complete Pauli measurement μ , such that Pμ Pμ and on Pauli measurements. Let E be a test operator based on tr(Pμ ) < tr(Pμ). Note that Pμ can also be realized by the in- the Pauli measurement speciﬁed by the symplectic vector complete Pauli measurement that the two Pauli measurements μ ∈ 2n μ μ Z2 . The test operator E is not admissible if there exists and have in common. This observation completes the another test operator E based on a Pauli measurement such proof of corollary 2. that E E and tr(E ) < tr(E ). Let Pμ be the canonical test The following lemma is a useful tool for determining projector associated with the Pauli measurement μ, then E admissible test projectors of stabilizer states; it is a direct Pμ, so an admissible test operator is necessarily a canonical consequence of theorem 1. test projector. The test vector aμ is (respectively, not) admis- Lemma 10. Suppose Pμ and Pμ are the canonical test sible if the test projector Pμ is (respectively, not) admissible. projectors for the stabilizer state |S based on Pauli measure-

A Pauli measurement is (respectively, not) admissible if the ments μ and μ , respectively. Then the following statements corresponding canonical test projector is (respectively, not) are equivalent: admissible. Previously, the concepts of admissible measure- (1) Pμ Pμ; ments and admissible test projectors were considered only for (2) aμ ◦ aμ1 =aμ 1; GHZ states [17]. (3) rspan(M˜ S,μ ) rspan(M˜ S,μ); Proposition 3. Any admissible Pauli measurement is (4) ker(M˜ S,μ ) ker(M˜ S,μ); complete. (5) (M˜ S,μ; M˜ S,μ ) and M˜ S,μ have the same rank. Proof. Following the proof for GHZ states in Ref. [17], we Here, aμ and aμ are the test vectors associated with Pμ and prove the contrapositive, that an incomplete Pauli measure- Pμ , respectively; aμ ◦ aμ denotes the elementwise product of ment is inadmissible. Without loss of generality, consider a aμ and aμ; and aμ 1 denotes the 1-norm of aμ , that is, − aμ 1 = ∈ n, = aμ ,w. Pauli measurement of weight n 1onthen-qubit target sta- w Z2 w 0 bilizer state |S.Aftern − 1 single-qubit Pauli measurements, In the case of graph states, lemma 10 can be simplified as the reduced states of the remaining party for all possible follows, assuming that the Pauli measurements are complete. outcomes are eigenstates of a Pauli operator. So we can always Lemma 11. Suppose Pμ and Pμ are the canonical test find an extra Pauli measurement on the remaining qubit that projectors for the graph state |G based on the complete makes the canonical test projector smaller. Pauli measurements μ and μ , respectively. Let A be the As an example, consider the three-qubit linear cluster state adjacency matrix of G. Then the following statements are defined by the three stabilizer generators S1 = XZ1, S2 = equivalent: ZXZ, and S3 = 1ZX. The state can be written as (1) Pμ Pμ; (2) aμ ◦ aμ =aμ ; |+ |0 |+ + |− |1|− 1 1 |G= √ , (89) (3) rspan(Aμ ) rspan(Aμ); 2 (4) ker(Aμ ) ker(Aμ); where |± is an eigenstate of X with the eigenvalue ±1. If we (5) (Aμ; Aμ ) and Aμ have the same rank. perform X and Z measurements on the first and second qubit, The total number of admissible test projectors for |G is respectively, then the reduced state of the third qubit is left in denoted by η(G). Note that η(G) is also the total number one of the X eigenstates. Thus measuring X on the last qubit of admissible Pauli measurements by lemma 9.Thevalueof

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η(G) for every equivalent class of connected graph states up probabilities that X, Y , and Z measurements are performed to seven qubits is shown in Table I. on the jth qubit, respectively. Then β X , Y , Z 1, ( ) max max p j p j p j (90) j∈V V. OPTIMAL VERIFICATION OF GRAPH STATES NI 3 2 ν() ν(G) , (91) In this section, we study optimal verification of entangled 3 (possibly nonconnected) graph states based on Pauli measure- ν = / ments. (Generalization to stabilizer states is straightforward.) where VNI is the set of nonisolated vertices. If ( ) 2 3or β = / X = Y = Z = / ∈ First, we show that the spectral gap of any verification pro- ( ) 1 3, then p j p j p j 1 3 for all j VNI. tocol based on separable measurements cannot surpass 2/3. Proof. Equation (91) follows from (90). To prove (90), We also derive a necessary condition on Pauli measurements we can assume that the verification protocol employs only to attain the maximum spectral gap. Then we propose a sim- canonical test projectors. By assumption G is nonempty, so ∈ | ple algorithm for constructing optimal verification protocols VNI is also nonempty. For any j VNI, the reduced state of G X based on Pauli measurements. Using this algorithm, we con- for qubit j is completely mixed [23,24]. Let Pj be the product struct an optimal verification protocol, attaining the maximum of all canonical test projectors based on Pauli measurements X spectral gap of 2/3, for every equivalent class of connected with X measurement on qubit j, then Pj is also a test projector | graph states up to seven qubits. We believe that the maximum for G (not necessarily associated with a Pauli measurement) spectral gap of 2/3 can be attained for all graph states associ- since all canonical test projectors commute with each other X ated with nonempty graphs. Recall that a nonempty graph is a according to corollary 1. In addition, Pj has rank at least 2 | graph that has at least one edge (cf. Sec. III D). since, otherwise, the reduced state of G on qubit j would be a pure state, in contradiction with the assumption that j is a Y Z nonisolated vertex. Define Pj and Pj in a similar way. Then A. Efficiency limit of separable and Pauli measurements we have | = X X + Y Y + Z Z ∀ , Given a graph state G associated with the graph G p j Pj p j Pj p j Pj j (92) (V, E ), denote by ν(G) the maximum spectral gap of verification operators that are based on nonadaptive Pauli β X , Y , Z 1 ∀ , ( ) max p j p j p j j (93) measurements, and by νsep(G) the maximum spectral gap 3 attainable by separable measurements. A verification operator which confirms (90) and implies (91). The last statement in ν = ν based on Pauli measurements is optimal if ( ) (G). theorem 3 is an immediate consequence of (90). Note that ν(G) = ν (G) = 1ifG is empty, in which case sep If G is nonempty and has J connected components |G is a tensor product of eigenstates of Pauli matrices. So J G , G ,...,GJ . Then |G= = |G j and we have it remains to consider graph states associated with nonempty 1 2 j 1 graphs. ν(G) = min ν(G j ) (94) Lemma 12. Suppose G = (V, E ) is a (possibly noncon- j ν ν / nected) nonempty graph. Then (G) sep(G) 2 3. according to proposition 9 in Appendix D. In addition, if J Proof. The inequality ν(G) νsep(G) follows from the | j are optimal verification operators for G j , then j=1 j fact that Pauli measurements are separable measurements. is an optimal verification operator for |G. As a corollary, ν / To prove the inequality sep(G) 2 3, note that the graph ν(G) = 2/3ifν(G ) = 2/3 for each nonempty component | j state G is maximally entangled with respect to at least one G j, note that ν(G j ) = 1ifG j is empty. To construct optimal bipartition into a single qubit and the other n − 1 qubits since verification protocols, therefore, we can focus on graph states G is nonempty [23,24]. A separable measurement for |G is of connected graphs. necessarily separable with respect to the bipartition. Now note that the spectral gap for a Bell state or two-qubit maximally B. Algorithm for constructing optimal verification protocols entangled state cannot be increased by increasing the local dimension of one of the subsystems. Therefore the spectral To construct an optimal verification protocol based on gap of any verification operator for |G that is based on Pauli measurements, it suffices to consider canonical test separable measurements cannot be larger than the maximum projectors associated with admissible Pauli measurements. spectral gap 2/3 achievable for a Bell state [3,4,12,13], that is, Suppose we perform the Pauli measurement μ with proba- ν() 2/3. This result implies the inequality νsep(G) 2/3 bility pμ, then the resulting verification operator reads and confirms lemma 12. While lemma 12 establishes the bound for a vast class of = pμPμ, (95) measurements, it provides no information as to what kind of μ measurements can saturate the bound. Specializing to Pauli where Pμ is the canonical test projector associated with the measurements, we prove a simple necessary requirement for Pauli measurement μ. According to corollary 1, all canonical constructing an optimal verification protocol: X, Y , and Z test projectors are diagonal in the graph basis, so the operator should be measured with an equal probability for each qubit. is also diagonal in the graph basis. Let Theorem 3. Suppose G = (V, E ) is a (possibly nonconnected) nonempty graph and is a verification operator of λ = pμaμ. (96) | X Y Z G based on Pauli measurements. Let p j , p j , p j be the μ

043323-11 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020) where aμ is the test vector associated with the projector Pμ and C. Special examples μ the Pauli measurement . Then all eigenvalues of except To illustrate the algorithm presented in Sec. III, we first λ the one associated with the target state are contained in .In consider graph states associated with star graphs (Nos. 3, 5, particular, we have 9, and 20 in Ref. [23]; omitted in Table I). These states can be turned into GHZ states by applying the Hadamard gate on β() =λ∞ = max pμaμ,w ∈ n, = w Z2 w 0 each noncentral qubit. All information about the canonical test projector asso- = max pμ, (97) μ = μx μz ∈ 2n w∈Zn,w=0 ciated with the Pauli measurement ( ; ) Z2 is 2 μ, rspan(Aμ ) w encoded in the matrix Aμ defined in (75). For the star graph with n vertices we have where λ∞ denotes the maximum of the elements in the ⎛ ⎞ λ 01··· 1 vector , and the last equality follows from theorem 2. Max- ⎜ ⎟ imizing the spectral gap ν() = 1 − β() is equivalent to ⎜ ⎟ ⎜1 ⎟ minimizing β(). To this end we need to only consider ad- A = ⎜. ⎟, ⎝. ⎠ missible test vectors (cf. Sec. IV C). 0 Define the test matrix A as the matrix composed of all 1 ⎛ ⎞ admissible test vectors aμ as column vectors, and let p be μz μx μx ··· μx 1 1 1 1 the column vector composed of the probabilities pμ. Then ⎜μx μz ⎟ λ = Ap and β() =Ap∞. The minimization of β() can ⎜ 2 2 0 ⎟ = ⎜μx μz ⎟. be cast as a constrained optimization problem Aμ ⎜ 3 3 ⎟ (100) ⎝ . .. ⎠ . 0 . minimize Ap∞ μx μz p∈Rm n n According to (80), the rank of the canonical test projector subject to pμ 0, (98) associated with μ is the cardinality of the subspace spanned pμ = 1. by the rows of Aμ. Analysis shows that the smallest subspace μz = μx = = μ is obtained when 1 1, j 1for j 1, and every other component is zero; that is, when the Pauli measurement is After choosing a proper order of the test vectors the minimiza- ZX ···X. This Pauli measurement effectively measures the tion can be expressed as a linear programming: n − 1 stabilizer generators associated with the n − 1 noncentral qubits. The resulting rank-2 test projector reads minimize y ⊗n−1 ⊗n−1 P0 =|00|⊗(|+ +|) +|11|⊗(|− −|) . subject to (Ap) j y, p j 0 ∀ j, (99) (101) p j = 1. It is the only admissible Pauli measurement that measures j Z on the central qubit and, thus, must be included in any optimal verification protocol with probability 1/3 according Putting all these together, we have a recipe for finding an to theorem 3. optimal verification protocol as shown in algorithm 2. Our By virtue of the connection with the GHZ state and the PYTHON module that implements the algorithm using the con- result derived in Ref. [17], the other admissible Pauli mea- vex optimization package CVXPY [44] and open source solvers surements can be described compactly as follows. We perform are available in Ref. [34]. either X or Y measurement on the central qubit, while either Y or Z measurement on each of the other qubits. Let Y be the set Algorithm 1. Finding an optimal verification protocol for graph of parties that perform Y measurements assuming that |Y| is states. even, and Y be the set of parties that perform a Z measurement on a noncentral qubit or an X measurement on the central Input: qubit. Let |Y|=2t with t = 0, 1,...,n/2 and A Adjacency matrix for an n-qubit graph state |G t Output: SY = (−1) Yj O j , (102) M The set of Pauli measurements (represented by j∈Y j ∈Y 2n vectors in (Z2 )C) employed in the optimal verification protocols where O1 = X1 and O j = Z j for all j = 1. The Pauli mea- {pμ}μ∈M Probabilities for individual measurement settings surement effectively measures the product of the stabilizer S as in (95) 1 ν of the central qubit together with all the stabilizers of all non- ( ) Optimal spectral gap Y 1: Determine all nontrivial test vectors. central qubits in the set (cf. Fig. 2). Thus the corresponding 2n canonical test projector 1. For each μ ∈ (Z ) , compute the matrix Aμ in (75). 2 C + 2. Discard Aμ that has full rank by checking the determinant. 1 SY PY = (103) 3. Compute the test vector aμ by virtue of theorem 2. 2 2: Determine all admissible test vectors and has rank 2n−1, and there are 2n−1 such canonical test projec- A construct the test matrix . tors. The unique optimal verification strategy is constructed 3: Solve the linear programming (99) to obtain β()and by performing the test P0 with probability 1/3 and all other compute ν() = 1 − β(). n−2 admissible tests PY with probability 1/(3 × 2 ) each. The

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up to seven qubits [23]. It turns out the maximum spectral gap can always attain the upper bound 2/3 for separable measurements presented in lemma 12. Therefore we believe that the upper bound 2/3 for the spectral gap can be saturated by Pauli measurements for all graph states. More details can be found in Table I. Note that optimal verification strategies FIG. 2. Graphical representation of the optimal verification pro- are in general not unique, and it is possible that the numbers tocol for the star graph. Each term in the sum represents a Pauli of measurement settings in the optimal strategies presented in measurement setting. Measurement settings in the parentheses are Table I can be reduced. According to this table, linear cluster chosen with the probability 1/6 each. states require fewer measurement settings to construct optimal protocols compared with most other graph states: only 5, 6, 6, resulting verification operator reads 8, and 12 settings are required for 3, 4, 5, 6, and 7 qubits, respectively. Curiously, graph state No. 42 requires only six 1 1 1 = P + PY = (2|GG|+1). (104) measurement settings to construct an optimal verification pro- 0 n−2 3 Y 2 3 tocol, which is fewer than the number 7 of qubits. This is the only connected graph state that has this property as far as we Compared with other n-qubit graph states with n 7, it turns know. out the graph state associated with the star graph requires the most number of measurement settings to construct an optimal verification protocol. VI. OPTIMAL VERIFICATION OF GRAPH STATES In general, optimal verification strategies based on Pauli WITH X AND Z MEASUREMENTS measurements are not unique. For example, let us consider According to Ref. [7], all graph states can be verified the five-qubit ring cluster state (No. 8 in Table I). Our al- using only X and Z measurements. Here we present an upper gorithm shows that this graph state has 21 admissible Pauli bound for the spectral gap of any verification protocol based measurements. Two distinct optimal verification strategies can on X and Z measurements and show that this bound can be be constructed from six admissible Pauli measurements each saturated for all connected graph states up to seven qubits as presented below (left and right columns); each setting is (which again implies that the bound can be saturated for measured with probability 1/6, (the measurement strategy in nonconnected graph states built from these graphs according the left column is depicted in Fig. 3) which is consistent with to proposition 9 in Appendix D). In addition, we construct an theorem 3. optimal verification protocol for every ring cluster state.

XZZXZ ZXZZX A. Verification protocols based on X and Z measurements ZZXZX ZXZXZ Given a graph state |G associated with the graph G = XXYYY XYYYX (V, E ), in analogy to ν(G) defined in Sec. VA, denote by ZYYZX XZYYZ νXZ(G) the maximum spectral gap of verification operators YXXYY YYXXY that are based on X and Z measurements. YYZXZ YZXZY Corollary 3. Suppose G = (V, E ) is a (possibly nonconnected) nonempty graph, and is a verification operator of | Note that the two strategies have no measurement setting the graph state G based on X and Z measurements. Then ν ν / in common. All 12 canonical test projectors associated with the spectral gap satisfies ( ) XZ(G) 1 2. To saturate these Pauli measurements have rank 8 (some of other nine the upper bound both X and Z should be measured with / admissible test projectors have higher ranks). probability 1 2 for each qubit associated with a nonisolated vertex. This result is a simple corollary of theorem 3. Moreover, D. Generic case the bound ν() 1/2 applies whenever each party can per- By means of the algorithm presented in Sec. I,wehave form only two types of Pauli measurements. found an optimal verification protocol based on Pauli mea- To construct an optimal protocol for the graph state |G surements for each equivalent class of connected graph states based on X and Z measurements, it suffices to consider canon-

FIG. 3. Graphical representation of an optimal veriﬁcation protocol for the ﬁve-qubit ring cluster state. Each term in the sum represents a Pauli measurement setting, and each setting is chosen with probability 1/6.

043323-13 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020) ical test projectors based on complete X and Z measurements. Optimal verification protocols based on X and Z measure- Each subset B of V determines a complete Pauli measurements can be found using a similar algorithm as presented ment by performing X measurements on qubits in B and Z in Sec. I with minor modification. It turns out the bound measurements on qubits in the complement B = V \ B;the ν() 1/2 in corollary 3 can be saturated for all entangled corresponding local subgroup and canonical test projector are graph states up to seven qubits (see Table II and Appendix E). denoted by LB and PB, respectively. Conversely, each com- Proposition 4. For any entangled CSS state or entangled plete Pauli measurement based on X and Z measurements is graph state associated with a two-colorable graph, the max- determined by such a subset in V . In view of this fact, com- imum spectral gap achievable by X and Z measurements plete Pauli measurements based on X and Z measurements is 1/2. and the corresponding canonical test projectors can be labeled Proof. First, consider an entangled graph state. According by subsets in V .Letμ be the symplectic vector associated to corollary 3, the spectral gap of any verification operator μx = / with the Pauli measurement determined by B, then j 1iff based on X and Z measurements is upper bounded by 1 2. ∈ μz = ∈ ={ |μx = } If the graph is two-colorable, then the bound 1/2 can be j B, while j 1iff j B. Conversely, B j j 1 . Based on this observation, the local subgroup LB and the test saturated by a coloring protocol proposed in Ref. [7], so the projector PB can be determined by virtue of theorem 2, where maximum spectral gap achievable by X and Z measurements the subscript μ characterizing the Pauli measurement can be is 1/2. replaced by B to manifest the role of the set B; in particular, Next, consider an entangled CSS state. According to theo- Aμ can be expressed as AB. By the form of AB (75) and the rem 3, the spectral gap of any verification operator based on above discussion, we have X and Z measurements is also upper bounded by 1/2. Mean- while, the bound can be saturated by the protocol composed of | |= −| |, rank(AB ) B n B (105) the two measurement settings XX ···X and ZZ ···Z chosen and the inequality is saturated iff B is an independent set (cf. with an equal probability. This result is consistent with the the proof of lemma 8). In conjunction with the corollaries (79) fact that any CSS state is equivalent to a graph state associated and (80) of theorem 2, this equation implies the following. with a two-colorable graph and vice versa [41]. Corollary 4. Let B be a subset of the vertex set V of the graph G. Then B. Admissible test projectors based on X and Z measurements |B| |LB| 2 , (106) In contrast to the definitions in Sec. IV C, there are two sensible definitions of admissible measurements and test pro- n−|B| tr(PB ) 2 . (107) jectors based on X and Z measurements. Such a test projector The inequality (106) is saturated iff B is an independent set; (and corresponding measurement) is admissible if there is the same holds for the inequality (107). no smaller test projectors based on Pauli measurements. The When B is an independent set, L is generated by the test projector (and corresponding measurement) is weakly B admissible if there is no smaller test projectors based on X stabilizer operators S j for all j ∈ B, that is, and Z measurements. Let ηXZ be the number of admissible L ={ | ∈ } η B S j j B ; (108) test projectors based on X and Z measurements and XZ the number of weakly admissible test projectors. Obviously, an accordingly, the canonical test projector reads admissible test projector based on X and Z measurements is 1 + S j weakly admissible, so we have η η . This inequality P = . (109) XZ XZ B 2 is usually strict since a weakly admissible measurement is j∈B η η not necessarily admissible. In fact, as XZ and XZ are not In addition, the test vector in (78) reduces to invariant under LC, so the equality η = η is too strong to XZ XZ expect. 1 supp(w) ∈ B, aB,w =Gw|PB|Gw= (110) As an example, let U be an n-qubit Clifford unitary opera- 0 otherwise, n tor with n 3 that interchanges Y and Z for every qubit, and where consider the state |=Un |G, where |G is the star-graph state as discussed in Sec. VC. Here, the measurement X ···X supp(w) ={j|w j = 1}. (111) yields a rank-4 test projector Pμ that is weakly admissible. Test projectors based on independent sets play a key role in However, this projector cannot be admissible according to the constructing the cover and coloring protocols in [7]. Com- discussion in Sec. VC (cf. Ref. [17]). Specifically, let Pμ be pared with other test projectors, these test projectors are easy the unique rank-2 admissible test projector based on the mea- to visualize, and it is easier to compute the spectral gap of surement YX ···X (which would have been ZX ···X before verification operators based on such test projectors. Neverthe- the unitary operator Un is applied), then we have Pμ Pμ and less, more general test projectors are helpful to enhance the tr(Pμ ) < tr(Pμ), so Pμ is not admissible. spectral gap. For graph states, it turns out the two notions of ad- The following lemma follows from (43), (108), and (109). missibility coincide. The following proposition is proved in Lemma 13. Suppose B and B are two subsets of the vertex Appendix C. set of the graph G = (V, E ), and B is an independent set. If Proposition 5. Given any graph state |G, a test projector

PB PB or equivalently, LB ⊇ LB, then B ⊆ B . If both B based on X and Z measurements is admissible iff it is weakly

⊆ η = η and B are independent sets, then PB PB iff B B . admissible; XZ(G) XZ(G).

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η The values of XZ(G) for graph states up to seven qubits are = 1 |{ | ∈ }| + max ( j supp(w) B j aB + ,w ) shown in Table II. These results suggest that the number of ad- + ∈ n, = n 1 n 1 w Z2 w 0 missible settings based on X and Z measurements grows only 1 linearly with the number of qubits, although the total number = . (116) of admissible measurement settings grows exponentially (cf. 2 Table I). Here the last inequality follows from (115) and the fact that The following lemma clarifies all admissible test projectors |{ j| supp(w) ∈ B j}| (n + 1)/2, and the inequality is satu- that are based on independent sets; see Appendix C for a rated iff w has weight 1. proof. This lemma provides further insight on the cover and coloring protocols proposed in [7]. VII. VERIFICATION WITH MINIMUM Lemma 14. Suppose B is an independent set of the graph NUMBER OF SETTINGS G = (V, E ). Then the test projector PB given in (109) is admissible iff B is a maximal independent set. Besides the total number of samples, the number of distinct measurement settings is another figure of merit of practical C. Optimal verification of ring cluster states interest if it is difficult to switch measurement settings. If it A ring cluster state is the graph state associated with a ring is easy to switch measurement settings, then this figure of graph, i.e., a cycle. When the number n of vertices is odd, merit is not so important to practical applications compared the ring graph is also one of the simplest graphs that are not with the total number of samples. Nevertheless, it is of in- 2-colorable. Here we show that the upper bound 1/2forthe trinsic theoretical interest to determine the minimum number spectral gap in corollary 3 can be saturated for all ring cluster of measurement settings required. As we shall see shortly, this states by constructing an optimal verification protocol using number is tied to several graph invariants, which are of interest X and Z measurements only. beyond the focus of this paper. Let |G be the n-qubit ring cluster state associated with the Since every stabilizer state is equivalent to a graph state ring graph G = (V, E ), where V ={1, 2,...,n}. When n is under local Clifford transformations, we can focus on graph even, according to [7], an optimal protocol for verifying |G states. All statements proven in this section that are not spe- can be constructed using two measurement settings associated cific to a particular graph apply to connected graphs as well with the two independent sets B ={1, 3,...,n − 3, n − 1} as nonconnected ones. The relation between verification of 1 a nonconnected graph state and verification of its connected and B2 ={2, 4,...,n − 2, n}, respectively. To construct an optimal protocol when n is odd, we first components is clarified in Appendix D. introduce n + 1 subsets of V defined as follows, A. Local cover number B j ={j, j + 2, j,..., j + n − 3}, j = 1, 2,...,n, (112) Suppose G = (V, E ) is a graph with adjacency matrix A. | Bn+1 = V ={1, 2,...,n}; (113) Let G be the graph state associated with the graph G and the stabilizer group SG.Toverify|G based on Pauli mea- note that the first n subsets are independent sets of G. Each surements, we need to construct a verification operator set B j for j = 1, 2,...,n + 1 defines a canonical test by per- with positive spectral gap, that is, ν() > 0 or, equivalently, forming X measurements on qubits in B j and Z measurements β() < 1. The following lemma clarifies the necessary re- on qubits in the complement V \ B j as described in Sec. VI A. quirements. Now an optimal protocol can be constructed by performing M ⊆ 2n Lemma 15. Suppose (Z2 )C is a set of complete n + P / n + | the 1tests B j with probability 1 ( 1) each; the result- Pauli measurements employed in the verification of G . ing verification operator reads Suppose = μ∈M pμPμ is a verification operator of |G, n+1 where Pμ are canonical test projectors and pμ > 0 for all 1 = P . (114) μ ∈ M . Then the following statements are equivalent: n + 1 B j j=1 (1) ν() > 0; ∪ L =S To corroborate our claim, note that the test projectors P (2) μ∈M μ G; B j ∪ = n for j = 1, 2,...,n are determined by (109) and all have rank (3) Span( μ∈M ker(Aμ)) Z2; ∩ = 2(n+1)/2; the corresponding test vectors are determined by (4) μ∈M rspan(Aμ) 0; ◦ = (5) μ∈M aμ 0. (110). To determine the test vector aBn+1 associated with the ◦ ◦ = = Here the expression μ∈M aμ = 0 means μ∈M aμ,w = test projector PB + , note that AB + A, so that aB + ,w 1iff n 1 n 1 n 1 ◦ | | = ∈ n = w ∈ rspan(A) according to theorem 2. In addition, A has rank μ∈M Gw Pμ Gw 0 for all w Z2 with w 0. n − 1, and ker(A) is spanned by {1, 1,...,1}, which implies Proof. The equivalence of statements 1 and 2 follows from that the fact that Pμ is the stabilizer code projector associated with L 1 w has even weight, the local subgroup μ according to lemma 5. The equivalence a , =G |P |G = (115) Bn+1 w w Bn+1 w 0 otherwise. of statements 2 and 3 follows from (76) in theorem 2.The equivalence of statements 3 and 4 is a simple fact in lin- Therefore ear algebra. The equivalence of statements 1 and 5 follows n+1 from (97). 1 β = local cover number χ G ( ) max aB j ,w The ˜ ( ) is defined as the minimum n + 1 w∈Zn,w=0 2 j=1 number of Pauli measurement settings required to verify |G.

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It is equal to the minimum cardinality of M with M ⊂ B. Connection to the chromatic number 2n (Z2 )C that satisfies one of the statements in lemma 15 and Here we discuss the connection between the local cover can be expressed as follows: numbersχ ˜ (G), χ˜XZ(G) and the chromatic number χ(G). Note χ G G χ = {|M ||∪ L =S } that ˜ ( ) is invariant under LC of the graph (corresponding ˜ (G) min μ∈M μ G (117) to LC of the graph state |G), but this is not the case for χ˜ (G) and χ(G). To remedy this defect, defineχ ˜ (G)asthe = |M || ∪ = n XZ 2 min Span( μ∈M ker(Aμ)) Z2 (118) minimum number of settings required to verify |G when each = {|M || ∩ = } party can perform only two different Pauli measurements. min μ∈M rspan(Aμ) 0 (119) Note that each party needs to perform at least two different ◦ Pauli measurements to verify any graph state of a connected = min{|M || μ∈M aμ = 0}. (120) graph with two or more vertices [6,7]. Define χLC(G)asthe minimum chromatic number of any graph that is equivalent to Here the terminology is inspired by (117) according to which G under LC, that is, χ˜ (G) is equal to the minimum number of local stabilizer groups required to generate the stabilizer group of |G.The χLC(G) = min χ(G ), (124) χ LC above equations can be applied to computing ˜ (G), although G G such algorithms are not efficient. In general, we cannot expect LC to find an efficient algorithm in view of the connection (via where the symbol means equivalence under LC. χ proposition 6 below) between ˜ (G) and the chromatic number Proposition 6. χ˜ (G) χ˜ (G) χ (G) χ(G) for any χ 2 LC (G), which is NP-hard to compute [45]. The local cover graph G. χ number ˜XZ(G) can be defined and computed in a similar way, Proof. Here the first and third inequalities follow from except that only X and Z measurements are considered. the definitions. To prove the second inequality, let G be a If one of the statements in Lemma 15 holds, then we graph that is equivalent to G under LC and satisfies χ(G ) = n n have Span(∪μ∈M ker(Aμ)) = Z , which implies that 2 2 χLC(G). According to Ref. [7], |G can be verified by a | | μ∈M ker(Aμ) , so that coloring protocol composed of χ(G ) distinct settings based on X and Z measurements. Therefore n dim(ker(Aμ)) |M | max dim(ker(Aμ)) μ∈M μ∈M χ˜2(G) = χ˜2(G ) χ˜XZ(G ) χ(G ) = χLC(G), (125)

|M | max [n − rank(Aμ)] =|M |[n − κ(G)], (121) which confirms the second inequality. μ∈ n (Z2 )C Conjecture 1. χ˜ (G) = χ˜2(G) = χLC(G) for any graph G. where the last equality follows from the definition of κ(G)in We have verified Conjecture 1 for all connected graphs up (83). As a corollary, we have to seven vertices (graph states up to seven qubits). Actually we have n χ˜ (G) . (122) χ = χ = χ = χ = χ n − κ(G) ˜ (G) ˜2(G) ˜XZ(G) (G) LC(G) (126)

If one of the statements in lemma 15 holds, and pμ = 1/m for all the graphs shown in Table II and all nonconnected for all μ ∈ M , where m =|M |, then graphs built from these graphs thanks to proposition 9 in Appendix D. [This result does not mean that (126) holds for 1 m − 1 all connected graphs up to seven vertices sinceχ ˜XZ(G) and β() = max aμ,w (123) ∈ n, = χ(G) are not invariant under LC; see Appendix E for more m w Z2 w 0 m detail.] Therefore, for all such graphs, the maximum spectral according to (97). Here the inequality follows from the fact gap of verification operators based on the minimum number = μ ∈ M ∈ n /χ that aμ,w 0 for at least one for each w Z2 of settings is 1 (G) according to theorem 4. Incidentally, all with w = 0. Therefore ν() 1/m.Ifm =|M |=χ˜ (G), protocols with the minimum number of settings in Table II then ν() 1/m according to proposition 1 given that the are chosen to be coloring protocols. In addition, Table II number of measurement settings cannot be reduced. These also contains the fractional chromatic number χ ∗(G) for all observations imply the following theorem, which clarifies the the graphs listed. The inverse fractional chromatic number efficiency limit of verification protocols based on the mini- 1/χ ∗(G) is the maximum spectral gap achievable by the cover mum number of Pauli measurement settings. protocol proposed in [7]. Theorem 4. The maximum spectral gap of verification op- Proposition 7. χ˜ (G) = 1iffG is an empty graph (with no erators of |G based onχ ˜ (G) distinct Pauli measurements is edges). 1/χ˜ (G). Proof. If G is empty, then |G is a product state of the form Theorem 4 follows from proposition 1 and the commuta- |+⊗n, which can be verified by performing X measurements tivity of canonical test projectors, so it applies to all graph on all qubits, soχ ˜ (G) = 1. Conversely, if |G can be veri- states, irrespective whether the graph is connected or not. fied by a single setting based on a Pauli measurement, then According to the coloring protocol proposed in Ref. [7], by |G must be a tensor product of eigenstates of local Pauli virtue of χ(G) settings based on X and Z measurements, we operators, so G must be an empty graph. Alternatively, propo- can achieve spectral gap 1/χ(G). Theorem 4 may be seen as sition 7 follows from lemma 3 in [12] and proposition 3 in a generalization of this result. Ref. [6].

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As an implication of propositions 6 and 7,wehave˜χ(G) = For a graph state associated with a nonempty graph G,the 2 for any nonempty two-colorable graph. The following theo- two conjectures amount to the following equalities, respec- rem provides a partial converse and confirms conjecture 1 in a tively, special case of practical interest. See Appendix F for a proof. χ = χ = ν = 2 , Theorem 5. ˜ (G) 2iff LC(G) 2. A stabilizer state (G) 3 (130) can be verified by two settings based on Pauli measurements ν = 1 , iff it is equivalent to a CSS state or, equivalently, a graph state XZ(G) 2 (131) of a two-colorable graph. ν ν The following proposition determines the local cover num- where (G) [respectively, XZ(G)] denotes the maximum | bers of odd ring graphs, which are typical examples of graphs spectral gap of verification operators for G that are based on that are not two colorable. Pauli measurements (respectively, X and Z measurements); Proposition 8. Suppose G is an odd ring graph with at least cf. Secs. VA and VI A. Conjecture 2 holds for GHZ states five vertices; then according to Ref. [17]. When the local dimension is an odd prime p instead of 2, we believe that the maximum spectral gap of verification operators based on Pauli measurements is χ˜ (G) = χ˜ (G) = χ˜ (G) = χ (G) = χ(G) = 3. (127) 2 XZ LC p/(p + 1), which holds for GHZ states according to Ref. [17]. χ χ = Conjecture 3 holds for graph states associated with two- To prove proposition 8, note that ˜ (G) (G) 3forthe colorable graphs and ring graphs according to Sec. VI. odd ring graph thanks to proposition 6. Conversely,χ ˜ (G) 3 κ In addition, we studied the problem of verifying graph according to the bound (122) with the value of (G) supplied states with the minimum number of settings. For any given by the following lemma, which is proved in Appendix G. |G graph state , it turns out that the minimum number of Lemma 16. Suppose G is a ring graph with n 4 vertices. settings requiredχ ˜ (G) is upper bounded by the chromatic Then number χLC(G) minimized over LC equivalent graphs. In addition, the upper bound still applies even if each party −(n+1)/2 P(G) = 2 , (128) can perform only two types of Pauli measurements, so we haveχ ˜ (G) χ˜2(G) χLC(G) (cf. proposition 6). Actually, κ(G) =(n + 1)/2. (129) the two inequalities are saturated for all two-colorable graphs, all graphs up to seven vertices, and ring (or cycle) graphs (cf. Sec. VII and Table II). These facts lead to the following VIII. SUMMARY AND OPEN PROBLEMS conjecture originally stated in Sec. VII. χ = χ = χ We have investigated systematically optimal verification of Conjecture 1. ˜ (G) ˜2(G) LC(G) for any graph G. stabilizer states (including graph states in particular) using In the future, it would be desirable to prove or disprove the Pauli measurements. We proved that the spectral gap of any above conjectures. In either case, we may gain further insights verification operator of any entangled stabilizer state based on quantum state verification and stabilizer states themselves. on separable measurements (including Pauli measurements) The number of admissible Pauli measurements (or X and Z is bounded from above by 2/3. Moreover, we introduced the measurements) and its scaling behavior with the number of concepts of canonical test projectors, admissible Pauli mea- qubits are also of interest from the theoretical perspective. In surements, and admissible test projectors and clarified their practice, it is desirable to find more efficient approaches for properties. By virtue of these concepts, we proposed a sim- constructing optimal verification protocols and protocols with ple algorithm for constructing optimal verification protocols the minimum number of measurement settings. Furthermore, based on (nonadaptive) Pauli measurements. Although this our study leads to the following question, which is of inter- algorithm is not efficient for large systems, it enables us to est beyond the immediate focus of this work: What are the χ χ χ construct an optimal protocol for any stabilizer state up to ten generic and maximum values of ˜ (G), ˜2(G), and LC(G), qubits without difficulty. In particular, our calculation shows respectively, for graphs of n vertices. that the bound 2/3 for the spectral gap can be saturated for all entangled stabilizer states up to seven qubits. It is quite ACKNOWLEDGMENTS surprising that the maximum spectral gap seems to be independent of the specific stabilizer state, although different We thank Zihao Li for stimulating discussion. This work stabilizer states may have very different entanglement struc- is supported by the National Natural Science Foundation of tures. In the case of graph states, we also prove that the upper China (Grant No. 11875110) and Shanghai Municipal Science bound for the spectral gap is reduced to 1/2 if only X and and Technology Major Project (Grant No. 2019SHZDZX01). Z measurements are accessible. Again, this bound can be saturated for all entangled graph states up to seven qubits. APPENDIX A: PROOF OF LEMMA 2 These results naturally lead to the following conjectures. Conjecture 2. For any entangled stabilizer state, the max- Proof of lemma 2. Let W and W be the isotropic sub- = imum spectral gap of verification operators based on Pauli spaces associated with S and S , respectively. Then W { | ∈ n} measurements is 2/3. M y y Z2 and Conjecture 3. For any graph state of a nonempty graph, the ∩ = ⊥ ∩ = | ∈ n, T = . maximum spectral gap of verification operators based on X W W W W M y y Z2 M JM y 0 and Z measurements is 1/2. (A1)

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Therefore 1 = (−1)w·y |L | |S¯ ∩ S |=| ∩ | μ n W W y∈Z ,Sy∈Lμ 2 n T T = M y | y ∈ Z , M JM y = 0 =|ker(M JM )| 1 · 2 = (−1)w y − T |Lμ| n rank(M JM ) ∈ n, ∈ = 2 , (A2) y Z2 MS y Uμ 1 which conﬁrms the ﬁrst two equalities in (39). The last equal- = (−1)w·y ity in (39) follows from the following equality: |Lμ| ∈ n, ˜ = y Z2 MS,μy 0 T = T + T . M JM Mz Mx Mx Mz (A3) 1 = (−1)w·y, (B6) Equation (40) follows from (37) and (39). |Lμ| y∈ker(M˜ S,μ )

APPENDIX B: PROOFS OF THEOREM 1 AND THEOREM 2 where the fourth equality follows from (B3). The summation in (B6) is nonzero iff w · y = 0 for all y ∈ ker(M˜ S,μ). This is Proof of theorem 1. Let Vμ be the isotropic subspace the case iff w ∈ rspan(M˜ S,μ), in which case we have associated with Tμ. Then the column span of Mμ = x z w·y (diag(μ ); diag(μ )), where the semicolon denotes the verti- (−1) =|ker(M˜ S,μ)|=|Uμ|=|Lμ|, (B7) Vμ Mμ cal concatenation, coincides with .( is a basis matrix y∈ker(M˜ S,μ ) for Vμ when the Pauli measurement is complete). Let Nμ be the n × n diagonal matrix over Z2 such that (Nμ) jj = 1iff which implies (64). μ j ∈ U2. Then by construction, the column span of the block Proof of theorem 2. Let V be the Lagrangian subspace as- x z matrix sociated with Tμ and Mμ := (diag(μ ); diag(μ )); then Mμ is Vμ V μx a basis matrix for .Let G be the Lagrangian subspace as- ⊥ = diag( ) Nμ 0 |G V ={A˜y|y ∈ Mμ μz (B1) sociated with the graph state ; then we have G diag( )0Nμ n} ˜ = Z2 , where A (1; A) is the canonical basis matrix for VG.Let ⊥ Uμ = Vμ ∩ V ; then Uμ is the isotropic subspace associated coincides with Vμ , the symplectic complement of Vμ.(Note G ⊥ with the local subgroup Lμ. In addition, that the first n columns of Mμ coincide with Mμ.) Therefore = ∩ = ⊥ ∩ = ⊥ ⊥ = ⊥ T , Uμ Vμ VG Vμ VG Vμ (Vμ ) ker((Mμ ) J) (B2) n T = A˜y|y ∈ Z , Mμ JA˜y = 0 where J is the symplectic form (19). 2 n Let VS be the Lagrangian subspace associated with the = A˜y|y ∈ Z , Aμy = 0 n 2 stabilizer group S. Then VS ={MS y|y ∈ Z }, where M is 2 S ={˜ | ∈ }= ˜ , the basis matrix of S. Uμ = Vμ ∩ VS is the isotropic subspace Ay y ker(Aμ) A ker(Aμ) (B8) L associated with the local subgroup μ. In addition, we have where J is defined in (19), and the fourth equality follows T ⊥ T from the fact that Mμ JA˜ = Aμ. As an implication of (B8), we Uμ = Vμ ∩ VS = ker((Mμ ) J) ∩ VS have | ker(Aμ)|=|Uμ|=|Lμ| and = | ∈ n, ⊥ T = MS y y Z2 (Mμ ) JMS y 0 y Lμ ={S |y ∈ ker(Aμ)}, (B9) n = MS y|y ∈ Z , M˜ S,μy = 0 2 which confirms (76). Equation (77) follows from (76) and ={MS y|y ∈ ker(M˜ S,μ)}=MS ker(M˜ S,μ). (B3) lemma 5. Furthermore, lemma 5 and (45) imply that To derive the fourth equality, note that ⎛ ⎞ 1 w·y y diag(μz )Mx + diag(μx )Mz G |Pμ|G = (−1) tr(S S ) S S w w n|L | ⊥ 2 μ T ⎝ z ⎠ S ∈Lμ y∈Zn (Mμ ) JMS = NμMS 2 x μ N MS 1 w·y = (−1) |Lμ| = U z x , ∈ n, y∈L MS,μ( 1); NμMS ; NμMS (B4) y Z2 S μ ˜ z x 1 · which reduces to MS,μ after interchanging NμMS and NμMS = (−1)w y and deleting rows of zeros. As an implication of (B3), we have |Lμ| y∈Zn,A˜y∈Uμ | ker(M˜ S,μ)|=|Uμ|=|Lμ| and 2 y 1 w·y Lμ ={S |y ∈ ker(M˜ S,μ)}, (B5) = (−1) |Lμ| y∈Zn,Aμy=0 which confirms (62). Equation (63) follows from (62) and 2 lemma 5. 1 · = (−1)w y, (B10) Furthermore, lemma 5 and (34) imply that |L | μ ∈ y ker(Aμ ) 1 w·y y S |Pμ|S = (−1) tr(S S ) where the fourth equality follows from (B8). The summation w w n|L | 2 μ ∈Lμ ∈ n · = ∈ S y Z2 in (B10) is nonzero iff w y 0 for all y ker(Aμ). This is

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| = the case iff w belongs to the row span of Aμ, in which case we Lemma 17. Every canonical test projector P for G J | = J have j=1 G j has a tensor-product form P j=1 Pj, where | w·y Pj is a canonical test projector for G j , and vice versa. The (−1) =|ker(Aμ)|=|Uμ|=|Lμ|, (B11) test projector P is admissible iff each tensor factor Pj is y∈ker(Aμ ) admissible. Proof. The stabilizer group S of |G is a direct prod- which implies (78). uct of the form S = S1 × S2 ×···×SJ , where S j for j = 1, 2,...,J are the stabilizer groups of |G j, respectively. APPENDIX C: PROOFS OF PROPOSITION 5 Suppose the test projector P is associated with the Pauli AND LEMMA 14 measurement specified by the symplectic vector μ;letTμ and T¯μ be the stabilizer group and signed stabilizer group Proof of proposition 5. Note that an admissible test projec- associated with the Pauli measurement μ. Then Tμ has the tor based on X and Z measurements is automatically weakly T = T × T ×···×T T = , ,..., admissible. To prove proposition 5 we need to prove that form μ 1 2 J , where j for j 1 2 J are stabilizer groups associated with certain Pauli measure- if a test projector Pμ based on X and Z measurements is XZ | L = S ∩ T¯ inadmissible, then it is not weakly admissible either; in other ments on G j , respectively. Let μ μ be the local subgroup associated with the Pauli measurement μ. Thanks to words, there always exists a smaller test projector PξXZ lemma 1, Lμ has the form Lμ = L × L ×···×L , where Pμ with tr(Pξ ) < tr(Pμ ) that is also based on X and 1 2 J XZ XZ XZ L = S ∩ T¯ | Z measurements. According to corollary 2, the inadmissible j j j are local subgroups of G j . According to (53), we have measurement μXZ can be replaced by an incomplete measure- μ < μ J J ment XZ on k n qubits. After the measurement XZ,the 1 1 reduced state on the remaining n − k qubits is a stabilizer state P = S = S = P , |L | |L | j j (D1) | − μ j of the form ULC G , where G is a graph of n k vertices, S∈Lμ j=1 S j ∈L j j=1 and ULC is an outcome-dependent local Clifford unitary [23]. Crucially, when μ consists of X and Z measurements, the where XZ unitary operator ULC can only map X to Z and vice versa up 1 P = S to an overall phase factor. Therefore we can obtain a smaller j |L | j (D2) j ∈L test projector by performing suitable X and Z measurements S j j on the remaining n − k qubits, which implies that Pμ is not XZ are canonical test projectors for |G . weakly admissible and confirms the proposition. j Conversely, suppose Pj are canonical test projectors for Proof of lemma 14. In one direction, suppose that B is | L = G j that are associated with the local subgroups j for j not maximal and let B be a larger independent set containing J 1, 2,...,J. Then P = = Pj is a canonical test projector B. Then we have PB PB and tr(PB ) < tr(PB ) according to j 1 for |G that is associated with the local subgroup L1 × L2 × (109), so PB is not admissible. ···×LJ . This observation confirms the first statement in In the other direction, suppose that B is maximal. If PB is not admissible, then it is not weakly admissible either by lemma 17. proposition 5. So there exists a subset B of V such that Next, let us prove the second statement in lemma 17. Suppose the test projector P for |G with 1 k J is not P P and tr(P ) < tr(P ), that is, k k B B B B admissible, then one can find a canonical test projector P k L ⊃ L ={ | ∈ }, for G such that P P and tr P < tr P .LetP = P for all B B S j j B (C1) k k k k k j j = = J < j k and P j=1 Pj; then P P and tr P tr P,soP is where LB and LB are the local subgroups associated with the not admissible. Pauli measurements determined by B and B , respectively, and Conversely, if P is not admissible, then one can find a the equality follows from (108). Equation (C1) implies that canonical test projector P for G such that P P and tr P < B ⊂ B by lemma 13. Since B is a maximal independent set tr P. In addition, P can be expressed as a tensor product by assumption, there must exist a vertex j ∈ B that is adja- = J | P j=1 Pj, where Pj is a canonical test projector for G j , cent to some vertex k ∈ B \ B, which implies that S ∈/ L , j B which satisfies P P . Moreover, tr P < tr P for at least one in contradiction with (C1). This contradiction completes the j j j j component j; in other words, at least one of the tensor factors proof of lemma 14. Pj is not admissible. This observation completes the proof of lemma 17. APPENDIX D: VERIFICATION OF GRAPH STATES Proposition 9. Suppose the graph G is a disjoint union of OF NONCONNECTED GRAPHS (possibly empty) subgraphs G j for j = 1, 2,...,J. Then Let G be a disjoint union of (possibly empty) connected ν(G) = min ν(G j ), (D3) { }J | = J | j graphs G j j=1. Then G j=1 G j is a graph state which is not genuinely multipartite entangled [23,24]. Here we clar- νXZ(G) = min νXZ(G j ), (D4) ify the relations between optimal verification of |G based on j |G Pauli measurements and that of j . It is worth pointing out χ˜ (G) = max χ˜ (G j ), (D5) that optimal protocols (with respect to the spectral gap or the j number of measurement settings) can always be constructed χ˜2(G) = max χ˜2(G j ), (D6) from canonical test projectors; cf. Sec. IV B. j

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χ˜XZ(G) = max χ˜XZ(G j ), (D7) χ˜2(G) max χ˜2(G j ), (D14) j j η(G) = η(G j ), (D8) χ˜XZ(G) max χ˜XZ(G j ). (D15) j j Conversely, suppose |G j can be verified by the set of ηXZ(G) = ηXZ(G j ). (D9) { ( j)}m j = | j canonical test projectors Pk k=1.Letm max j m j; then G can be verified by the following canonical test projectors Here, ν(G) (respectively, νXZ(G)) denotes the maximum spectral gap of verification operators of |G that are based on J = ( j), = , ,..., , Pauli measurements (respectively, X and Z measurements); cf. Pk : Pk k 1 2 m (D16) Secs. VAand VI A. Equation (D3) still applies if we consider j=1 ν the maximum spectral gap sep(G) achievable by separable ( j) = < wherePk 1 if m j k m;cf.(14) in Sec. II C.Let measurements, which follows from almost the same reason- = / ν / χ k Pk m, then ( ) 1 m since these canonical test ing as the one presented below. ˜ (G) denotes the minimum ( j) number of Pauli measurement settings required to verify |G, projectors commute with each other. If each Pk is based on XZ measurements or two measurement settings for each party, whileχ ˜XZ(G) andχ ˜2(G) denote the minimum numbers of settings based on XZ measurements and two measurement then each Pk has the same property. These observations imply settings for each party, respectively (cf. Sec. VII). Incidentally, that the minimum number of measurement settings based on the χ˜ (G) max χ˜ (G j ), (D17) coloring protocol [7] is equal to the chromatic number χ(G), j which satisfies χ˜2(G) max χ˜2(G j ), (D18) j χ(G) = max χ(G j ). (D10) j χ˜XZ(G) max χ˜XZ(G j ), (D19) So proposition 9 may be seen as a generalization of this j equation. Lastly, η(G) and ηXZ(G) denote the number of which confirm (D5)–(D7) in view of the opposite inequalities admissible test projectors based on Pauli measurements and derived above. those based on X and Z measurements, respectively (cf. Finally, (D8) and (D9) follow from lemma 17 (see also Secs. IV C and VI B). proposition 5). Proof of proposition 9. Suppose is an optimal verifica- | ν = ν tion operator for G with ( ) (G) that can be realized by APPENDIX E: GENERAL CONNECTED GRAPHS canonical test projectors. Let j be the reduced verification UP TO SEVEN VERTICES operator of for |G j. Then j can also be realized by canonical test projectors according to lemma 17. Therefore When restricted to X and Z measurements, many results on the verification of graph states are not invariant under LC. Therefore it is of interest to consider those connected graphs ν(G) = ν() min ν( j ) min ν(G j ), (D11) j j up to seven vertices that are not necessarily listed in Table I. Here we briefly discuss optimal verification protocols (with where the second inequality follows from proposition 2. respect to the spectral gap and the number of measurement Conversely, suppose for j = 1, 2 ...,J are optimal j settings) of graph states associated with these graphs. There verification operators of G that are based on Pauli measure- j are 996 such (nonisomorphic) graphs [46]. Our calculation ν = ν = J ments, so that ( j ) (G j ). Let j=1 j; then is shows that the maximum spectral gap achievable by X and | a verification operator of G that is based on Pauli measure- Z measurements is 1/2 for all these graph states. Since every ments. Therefore such graph G is equivalent under LC to a graph in Table I, χ ν(G) ν() = min ν( j ) = min ν(G j ). (D12) LC(G) is either 2 or 3. By proposition 6 and the results 1 jJ j presented in Table II,wehave˜χ(G) = χ˜2(G) = χ˜LC(G)for Equations (D11) and (D12) together imply (D3). all these graphs. In contrast,χ ˜XZ(G) can take on any value Equation (D4) follows from a similar reasoning. fromχ ˜2(G)uptoχ(G), so (126) in Sec. VII B does not hold To prove (D5)–(D7), suppose |G can be verified in general. A graph G for which the inequalitiesχ ˜2(G) , ,..., = χ G χ G by m canonical test projectors P1 P2 Pm;let ˜XZ( ) ( ) are strict is shown in Fig. 4. ( j) / | = | = | | Proposition 10. For the complete graph of n vertices, k Pk m.Let G j j = j G j , P G j Pk G j , and k χ˜ (G) = χ(G) = n. = P( j)/m; then P( j) for k = 1, 2,...,m are canonical XZ j k k k Proof. The equality χ(G) = nis immediate from the fact test projectors for |G according to lemma 17. Moreover, j that G is a complete graph of n vertices. According to the |G can be verified by these canonical test projectors since j coloring protocol proposed in [7], any graph state |G can ν( ) ν() 1/m. If each P is based on XZ measure- j k be verified by χ(G) settings based on X and Z measurements or two measurement settings for each party, then each ments, which implies thatχ ˜ (G) χ(G) = n. To complete P( j) has the same property. These observations imply that XZ k the proof, it remains to prove thatχ ˜XZ(G) n.

χ˜ (G) max χ˜ (G j ), (D13) It is straightforward to verify that the canonical test pro- j jector based on Y ⊗n has rank 2. Moreover, all canonical test

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to [41]. In other words, G is equivalent to a two-colorable graph under LC, which implies that χLC(G) = 2 given that G is nonempty. If G is not connected, then each connected component of G is equivalent to a two-colorable graph under LC according to the above reasoning, and the same conclusion holds for G. Therefore we still have χLC(G) = 2. The second statement in theorem 5 follows from the first statement and the fact that every stabilizer state is equiva- FIG. 4. (a) An example of a graph G for which the inequalities lent to a graph state under a local Clifford transformation χ χ χ ˜2(G) ˜XZ(G) (G) are strict. This six-vertex graph is No. 94 [26,39,40]. in the graph database [46], withχ ˜2(G) = χLC(G) = 2, χ˜XZ(G) = 4, and χ(G) = 5. The corresponding graph state can be verified by two measurement settings, say YYYYYZ and YZXYYY [˜χ2(G) = 2], or four settings based on X and Z measurements, say ZZZZXX, ZZZXZX, XXXZZX,andZZXZZZ [˜χXZ(G) = 4]. The graph G is APPENDIX G: PROOF OF LEMMA 16 equivalent to the two-colorable graph in plot (b) (corresponding to Proof of lemma 16. Equation (129) follows from (128) and No. 12 in Table I) under LC with respect to the vertices No. 1, 2, 6, and 3 in succession. lemma 7, so it suffices to prove (128). When n is even, (128) is proved in [43]. When n is odd, lemmas 7 and 8 imply that

n−α(G) −(n+1)/2 −(n+1)/2 P(G) 2 = 2 = 2 , (G1) projectors based on X and Z measurements have ranks either n−1 n 2 or 2 [cf. (80)], so the corresponding local subgroups given that α(G) = (n − 1)/2. So it remains to prove the op- are either trivial or have order 2, given that all canonical −(n+1)/2 posite inequality P(G) 2 . test projectors of the standard GHZ state based on X and |ϕ=|ϕ ⊗|ϕ ⊗···⊗|ϕ n−1 n Suppose that 1 2 n is a tensor Y measurements have ranks either 2 or 2 according to product of eigenstates of Pauli X,Y , and Z such that [17] (cf. Sec. VC). So at least n settings based on X and Z |ϕ|G|2 = (G). Then measurements are required to verify |G in view of lemma 15, P that is,χ ˜XZ(G) n, which completes the proof.

=|ϕ| |2 = 1 |ϕ | |2 1 | , P(G) G 2 2 ( ) (G2)

where√ the kets |ϕ =|ϕ1⊗|ϕ2⊗···⊗|ϕn−1 and | = APPENDIX F: PROOF OF THEOREM 5 2ϕn|G denote (n − 1)-qubit stabilizer states. If |ϕn is an | − Proof of theorem 5. If χ (G) = 2, then G is nonempty, so eigenstate of Z, then is an (n 1)-qubit linear cluster LC |ϕ χ˜ (G) = 2 according to propositions 6 and 7. state by the general set of rules presented in [23]. If n is an | Conversely, ifχ ˜ (G) = 2, then G is nonempty according to eigenstate of Y , by contrast, then is equivalent to a ring | = −(n−1)/2 proposition 7. In addition, |G can be verified by two settings cluster state. In both cases, we have ( ) 2 [43], |ϕ| |2 −(n+1)/2 based on Pauli measurements. First, suppose G is connected, which implies that G 2 . The same inequality |ϕ then |G is genuinely multipartite entangled, so the Pauli oper- holds if at least one of the tensor factors j is an eigenstate ators measured for each qubit associated with the two settings of Z or Y . |ϕ must be different according to theorem 3 (cf. proposition 3 in It remains to consider the case in which every j is an |ϕ Ref. [6]). By a suitable local Clifford transformation U,the eigenstate of X, so that belongs to the graph basis asso- state U|G can be verified by two measurement settings in ciated with the empty graph. According to lemma 4, then we which one setting is based on X measurements only, while the have other setting is based on Z measurements only. Up to a sign factor, each generator of the local subgroup associated with the first (second) setting is a product of some X (Z) operators |ϕ|G|2 2− rank(A) = 2−(n−1) 2−(n+1)/2, (G3) for individual qubits. Therefore the stabilizer group of U|G can be generated by a set of generators each of which is a product of local X operators only or a product of local Z operators only. It follows that U|G is a CSS state, so |G is where A is the adjacency matrix of G. It follows that P(G) equivalent to a graph state of a two-colorable graph according 2−(n+1)/2, which implies (128)inviewof(G1).

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APPENDIX H: TABLE OF OPTIMAL VERIFICATION PROTOCOLS

TABLE I. Optimal verification protocols for connected graph states up to seven qubits. There are 45 equivalent classes with respect to LC and graph isomorphism and here the labeling follows from Refs. [23]. Graph states associated with an edge and star graphs (Nos. 3, 5, 9, and 20) are omitted since optimal protocols for these states have a simple description as discussed in Sec. VC (cf. Fig. 2). For each graph, the optimal protocol is specified by Pauli measurement settings shown in the fifth column together with the corresponding probabilities shown in the sixth column. The spectral gap ν() of the verification operator is 2/3, which attains the upper bound presented in theorem 3.For completeness, the table also shows a minimum coloring of each graph, the number #() of measurement settings in the optimal protocol and the ranks of canonical test projectors. In addition, η(G) denotes the total number of admissible test projectors (that is, the number of admissible Pauli measurements) for the graph state |G.

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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TABLE I. (Continued.)

Number of measurements #(Ω)

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APPENDIX I: TABLE OF PROTOCOLS BASED ON X AND Z MEASUREMENTS

TABLE II. Optimal veriﬁcation protocols based on X and Z measurements and protocols with the minimum number of settings for the same graph states as shown in Table I (cf. [23]). For each graph, the optimal protocol can achieve the spectral gap ν() = 1/2, which attains the upper bound in corollary 3. The minimum number of settingsχ ˜ (G) is equal to the chromatic number χ(G) shown in the fourth column (cf. proposition 6); the corresponding protocol achieves the spectral gap 1/χ(G) (cf. theorem 4). All protocols with the minimum number of settings shown in the table are coloring protocols [7] determined by the colorings shown in Table I. Each protocol is speciﬁed by Pauli measurement settings shown in the seventh column: all settings are measured with the same probability unless noted otherwise (for graphs No. 41 and No. 42). When χ(G) = 2, only one protocol for |G is shown because the protocol can achieve the maximum spectral gap ν() = 1/2 and meanwhile requires the minimum number of settings. For completeness, the table also shows the qubit number n in the second column, the rank of each canonical test projector in the eighth column, and the minimum number of generators of the local subgroup in the ninth column (cf. Sec. IV B). In addition, ηXZ(G) denotes the total number of admissible test projectors for the graph state |G that are based on X and Z measurements (cf. Sec. VI B). χ ∗(G) denotes the fractional chromatic number of G [45], whose inverse is the maximum spectral gap ∗ achievable by the cover protocol proposed in [7]. Note that ηXZ(G), χ(G), and χ (G) are not LC-invariant.

Number of ∗ No. n ηXZ(G) χ(G) χ (G) ν() Setting Rank generators XZX 2 2 23 2 2 2 1/2 ZXZ 4 1 ZXX 2 3 34 2 2 2 1/2 XZZZ 8 1 ZXZX 4 2 44 3 2 2 1/2 XZXZ 4 2 ZXXXX 2 4 55 2 2 2 1/2 XZZZZ 16 1 XZXZX 4 3 65 3 2 2 1/2 ZXZXZ 8 2 XZXZX 4 3 75 4 2 2 1/2 ZXZXZ 8 2 XZXZZ 8 2 1/3 ZXZXZ 8 2 ZZZZX 16 1 ZZXZX 8 2 85 6 3 5/2 ZXZZX 8 2 ZXZXZ 8 2 1/2 XZZXZ 8 2 XZXZZ 8 2 XXXXX 16 1 ZXXXXX 2 5 96 2 2 2 1/2 XZZZZZ 32 1 XZXZXX 4 4 10 6 3 2 2 1/2 ZXZXZZ 16 2 ZXZXZX 8 3 11 6 3 2 2 1/2 XZXZXZ 8 3 XZXZXX 4 4 12 6 4 2 2 1/2 ZXZXZZ 16 2 ZXZXZX 8 3 13 6 5 2 2 1/2 XZXZXZ 8 3 ZXZXZX 8 3 14 6 5 2 2 1/2 XZXZXZ 8 3 XZXZXX 4 4 15 6 5 2 2 1/2 ZXZXZZ 16 2 XZZZXX 8 3 1/3 ZXZXZZ 16 2 ZZXZZZ 32 1

043323-31 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020)

TABLE II. (Continued.)

Number of ∗ No. n ηXZ(G) χ(G) χ (G) ν() Setting Rank generators 16 6 5 3 3 ZZXXXZ 8 3 ZXZXZX 8 3 1/2 XZZZXX 8 3 XXXZZZ 32 1 ZXZZXZ 16 2 1/3 XZXZZX 8 3 ZZZXZZ 32 1 ZXZXZX 8 3 17 6 6 3 5/2 XZZXZX 8 3 XZXZZX 8 3 1/2 ZZXZXZ 16 2 ZXZZXZ 16 2 XXXXXZ 32 1 18 6 6 2 2 1/2 ZXZXZX 8 3 XZXZXZ 8 3 XZZZZX 16 2 1/3 ZXZXZZ 16 2 ZZXZXZ 16 2 ZZXZXZ 16 2 19 6 12 3 3 ZZXXZZ 16 2 ZXZZZX 16 2 1/2 XZZZZX 16 2 XXZXXZ 16 2 XXXXXX 16 2 ZXXXXXX 2 6 20 7 2 2 2 1/2 XZZZZZZ 64 1 XXXXZXZ 4 5 21 7 3 2 2 1/2 ZZZZXZX 32 2 XXXZZXZ 8 4 22 7 3 2 2 1/2 ZZZXXZX 16 3 XXXXZXZ 4 5 23 7 4 2 2 1/2 ZZZZXZX 32 2 XXXXZXZ 4 5 24 7 4 2 2 1/2 ZZZZXZX 32 2 XZXXZXZ 8 4 25 7 5 2 2 1/2 ZXZZXZX 16 3 XXZXZXZ 8 4 26 7 5 2 2 1/2 ZZXZXZX 16 3 XZXZXZX 8 4 27 7 5 2 2 1/2 ZXZXZXZ 16 3 ZXZXXZX 8 4 28 7 6 2 2 1/2 XZXZZXZ 16 3 XZXZXZX 8 4 29 7 8 2 2 1/2 ZXZXZXZ 16 3 XZXZXZX 8 4 30 7 7 2 2 1/2 ZXZXZXZ 16 3 ZXZXXXX 4 5 31 7 5 2 2 1/2 XZXZZZZ 32 2

043323-32 OPTIMAL VERIFICATION OF STABILIZER STATES PHYSICAL REVIEW RESEARCH 2, 043323 (2020)

TABLE II. (Continued.)

Number of ∗ No. n ηXZ(G) χ(G) χ (G) ν() Setting Rank generators ZXZZXZX 16 3 1/3 XZXZZXZ 16 3 ZZZXZZZ 64 1 32 7 5 3 3 XXZXZXZ 8 4 XXXZXZZ 8 4 1/2 ZZXXZZX 16 3 ZZZZXXX 64 1 ZXZZXZZ 32 2 1/3 XZXXZXZ 8 4 ZZZZZZX 64 1 XXZZXZX 8 4 33 7 6 3 5/2 XXZXZZX 8 4 XXZXZXZ 8 4 1/2 ZZXZZXZ 32 2 ZZXZXZZ 32 2 ZZXXXXX 64 1 XZXZXZX 8 4 34 7 6 2 2 1/2 ZXZXZXZ 16 3 ZXZXZZZ 32 2 1/3 XZXZXZX 8 4 ZZZZZXZ 64 1 XXZZXZX 8 4 35 7 6 3 5/2 XXZXZZX 8 4 ZXZXZXZ 16 3 1/2 XZXZXZZ 16 3 ZZXZZXZ 32 2 ZZXXXXX 64 1 XZXZZXX 8 4 1/3 ZXZXZZZ 32 2 ZZZZXZZ 64 1 36 7 7 3 3 XZXZZXX 8 4 ZXZZXZX 16 3 1/2 ZXZXZXZ 16 3 XZXXXZZ 32 2 ZXZXZZX 16 3 1/3 XZXZZXZ 16 3 ZZZZXZZ 64 1 XZXZXZX 8 4 37 7 7 3 5/2 ZXZZXZX 16 3 ZXZXZZX 16 3 1/2 XZZXZXZ 16 3 XZXZZXZ 16 3 ZXXXXXZ 64 1 XZXZXZX 8 4 38 7 7 2 2 1/2 ZXZXZXZ 16 3 ZXZZXZX 16 3 1/3 XZXZZXZ 16 3 ZZZXZZZ 64 1

043323-33 DANGNIAM, HAN, AND ZHU PHYSICAL REVIEW RESEARCH 2, 043323 (2020)

TABLE II. (Continued.)

Number of ∗ No. n ηXZ(G) χ(G) χ (G) ν() Setting Rank generators ZZXZXZX 16 3 39 7 9 3 5/2 ZXZZXZX 16 3 ZXZXZXZ 16 3 1/2 XZZXZXZ 16 3 XZXZZXZ 16 3 XXXXXZX 32 2 XZXZXZZ 16 3 1/3 ZXZXZXZ 16 3 ZZZZZZX 64 1 ZZXZXZX 16 3 ZXZZXZX 16 3 40 7 8 3 7/3 ZXZXZZX 16 3 ZXZXZXZ 16 3 1/2 XZZXZXZ 16 3 XZXZZXZ 16 3 XZXZXZZ 16 3 XXXXXXX 64 1 XZXZZZX 16 3 1/3 ZXZZXZZ 32 2 ZZZXZXZ 32 2 ZZXZXZX 16 3 ZXZZXZX 16 3 41 7 10 3 3 ZXZXZXZ (1/4) 16 3 1/2 XZZXZZX 16 3 XZXZZZX 16 3 XZXZXXZ 32 2 XXXXXXZ 32 2 ZXZZXZZ 32 2 1/3 XZXZZXZ 16 3 ZZZXZZX 32 2 ZZXZXZX (1/4) 16 3 42 7 10 3 5/2 ZXZXZZX 16 3 XZZXZXZ (1/4) 16 3 1/2 XXZZXZZ 16 3 ZXXXXXZ 64 1 XXXZZXX 64 1 ZXZXZXX 8 4 43 7 9 2 2 1/2 XZXZXZZ 16 3 ZXZZZXZ 32 2 1/3 XZXZXZZ 16 3 ZZZXZZX 32 2 44 7 11 3 3 ZXZXZXZ 16 3 XXZZXZZ 16 3 1/2 ZZXZZZX 32 2 XZXXXXX 32 2 ZXZZZXZ 32 2 1/3 XZZXZZX 16 3 ZZXZXZZ 32 2 45 7 12 3 3 XZZZXZX 16 3 XZXXZXX 16 3 1/2 ZXZZZXZ 32 2 ZXXXXZZ 32 2

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