Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory
PPrepared for submission to JHEP
BRX-TH-6672
Hypergraph States in
SU( N ) , N odd prime, Chern-Simons Theory
Howard J. Schnitzer
Martin Fisher School of Physics, Brandeis University, Waltham MA 02453
E-mail: [email protected]
Abstract:
Graph states and hypergraph states can be constructed from products ofbasic operations that appear in SU( N ) . The level-rank dual of a theorem of Salton,Swingle, and Walter implies that these operations can be prepared topologically in the n -torus Hilbert space of Chern-Simons theory for N (cid:54) = 5 mod 4.For SU( N ) , N = 5 mod 4, only stabilizer states can be prepared on the n -torusHilbert space, which restricts the construction to graph states. a r X i v : . [ h e p - t h ] F e b ontents SU( d ) , d odd prime 2 d ) Pauli group 22.2 SU( d ) Clifford operators, d odd 3 The generalized Pauli group and Clifford operators [1–3] are obtained from productsof basic operations of SU( N ) Chern-Simons theory for N odd prime. Graph andhypergraph states [4–8] are described as products of such operations, which allow forexplicit representations of graph and hypergraph states. For N odd prime, N (cid:54) = 5mod 4, these operations can be obtained topologically in the n -torus Hilbert spaceof Chern-Simons theory as a result of the level rank dual [9] of a theorem of Salton,Swingle, and Walter [10]. Thus graph and hypergraph states of SU( N ) , N (cid:54) = 5 mod 4,are topological on the n -torus Hilbert space of Chern-Simons theory.For SU( N ) , N = 5 mod 4, only graph states are topological, as in this case onlystabilizer states are obtained from the n -torus Hilbert space. Hypergraph states forSU( N ) , N = 5 mod 4, are therefore not topological in the above sense.In Section 2 the construction of the generalized Pauli group and Clifford operationsfor SU( N ) is reviewed. Section 3 presents the construction of graph and hypergraphstates in terms of basic operations of SU( N ) . Section 4 is a discussion of related issues.– 1 – SU( d ) , d odd prime We first review the generalized Pauli group and Clifford operations for SU( d ) , d oddprime, following [1–3]. SU( d ) Pauli group
Representations of SU( d ) describing qudits are given by a single column Young tableau,with zero, one, . . . , ( d −
1) boxes. The fusion tensor of the theory is N abc ; a + b = c mod d (2.1)so that N | a (cid:105) | b (cid:105) = | a (cid:105) | a + b, mod d (cid:105) . (2.2)The modular transformation matrix S ab satisfies | a (cid:105) = d − (cid:88) b =0 S ab | b (cid:105) , a = 0 . . . d − . (2.3)For ω a primitive d th root of unity ω = exp (cid:18) πid (cid:19) (2.4)so that S ∗ = 1 √ d d − (cid:88) a =0 d − (cid:88) b =0 ω ab | a (cid:105) (cid:104) b | (2.5)which is the d -dimensional generalization of the Hadamard gate.The Pauli operator Z is given by Z ac = d − (cid:88) b =0 S ab (cid:16) S † b +1 ,a (cid:17) δ ac (2.6)so that Z = d − (cid:88) a =0 ω a | a (cid:105) (cid:104) a | . (2.7)The Pauli operator X is obtained from the fusion matrix, since N a, b | a (cid:105) = | a + 1 , mod d (cid:105) (2.8)which is identical to X | a (cid:105) = | a + 1 , mod d (cid:105) (2.9)– 2 –he single qudit Pauli group is the collection of operators ω r X a Z b ; a, b, r ∈ Z d . (2.10)Thus the one-qudit Pauli group is constructed from basic operations of SU( d ) , d odd.The n -qudit Pauli group is obtained from products of operators of the one-quditPauli group. That is X a Z b = X a Z b ⊗ X a Z b ⊗ · · · ⊗ X a n Z b n (2.11)The operator X a Z b , along with all scalar multiples thereof, (cid:8) ω c X a Z b | c ∈ Z d (cid:9) (2.12)defines the n -qudit Pauli group. SU( d ) Clifford operators, d odd The necessary gates for the single-qudit Clifford operators are [1–3] i) the QFT gate,Eq. (2.5), and ii) the phase gate P | j (cid:105) = ω j ( j − / | j (cid:105) . (2.13)The multi-qudit Clifford operators are obtained from the generalizations of (2.5) and(2.8) as well as the SUM gate, C SUM | i (cid:105) | j (cid:105) = N | i (cid:105) | j (cid:105) = | i (cid:105) | i + j, mod d (cid:105) (2.14) There are many equivalent constructions of graph states [4–8]. We follow arxiv:1612.06418for a definition of qudit graph states. The multigraph is G = ( V, E ), with vertices V and edges E , where an edge has multiplicity m e ∈ Z d . To G associate a state | G (cid:105) suchthat to each vertex i ∈ V , there is a local state | + (cid:105) = | p (cid:105) = 1 √ d d − (cid:88) q =0 | q (cid:105) (3.1)– 3 –ecall that the Hadamard gate generalizes to (2.5), so that S ∗ | (cid:105) = 1 √ d d − (cid:88) q =0 | q (cid:105) = | + (cid:105) = | p (cid:105) . (3.2)To each edge e = { i, j } apply the unitary Z m e e = d − (cid:88) q i =0 | q i (cid:105) (cid:104) q i | ⊗ (cid:0) Z m e j (cid:1) q i (3.3)to the state | + (cid:105) V = (cid:79) i ∈ V | + (cid:105) i (3.4)The graph state is | G (cid:105) = (cid:89) e ∈ E Z m e e | + (cid:105) V (3.5)= (cid:89) e ∈ E Z m e e (cid:79) i ∈ V | + (cid:105) i (3.6)The level-rank dual [9] of Theorem 1 of Salton, Swingle, and Walter [10] for d odd primeimplies that the graph state | G (cid:105) can be constructed from topological operations on the n -torus Hilbert space of Chern-Simons SU( d ) by means of the operations detailed inSection 2. Every stabilizer state is LC equivalent to a graph state, while the Cliffordgroup enables conversion between different multigraphs [4–8]. We again follow arxiv:1612.06418 for the construction of qudit multi-hypergraph states.Given a multi-hypergraph H = ( V, E ), associate a quantum state | H (cid:105) , with m e ∈ Z d the multiplicity of the hyperedge e . To each vertex i ∈ V , associate a local state | + (cid:105) = 1 √ d d − (cid:88) q =0 | q (cid:105) = S ∗ | (cid:105) (3.7)To each hyperedge e ∈ E , with multiplicity m e , apply the controlled unitary Z m e e tothe state | + (cid:105) V = (cid:79) i ∈ V | + (cid:105) i (3.8)– 4 –he hypergraph state is | H (cid:105) = (cid:89) e ∈ E Z m e e | + (cid:105) V (3.9)The elementary hypergraph state is | H (cid:105) = d − (cid:88) q =0 | q (cid:105) (cid:104) q | ⊗ (cid:16) Z m e e \{ } (cid:17) q | + (cid:105) V (3.10)For d prime, all n -elementary hypergraph states are equivalent under SLOCC.Hypergraph and graph states admit a representation in terms of Boolean functions, | H (cid:105) = d − (cid:88) q =0 ω f ( q ) | q (cid:105) (3.11)with f : Z nd → Z d , where f ( x ) = (cid:88) i ,...,i k ∈ V { i ,...,i k }∈ E x i · · · x i k (3.12)For graph states, f ( x ) is quadratic, i.e. f ( x ) = (cid:88) i ,i ∈ V { i ,i }∈ E x i x i (3.13)while for f ( x ) cubic or higher, | H (cid:105) is a hypergraph state. Therefore, for quadratic f ( x ),one has a representation of stabilizer states, up to LC equivalence. For f ( x ) cubic orhigher, | H (cid:105) represents hypergraph states which contain “magic” states. Examples ofmagic states are the CCZ state and Toffoli states, constructed from appropriate gates[3, 11–18]. Thus CCZ | x x x (cid:105) = ω x x x | x x x (cid:105) (3.14)with | CCZ (cid:105) = CCZ (cid:12)(cid:12) + ⊗ (cid:11) (3.15)as an example of a magic hypergraph state. Similarly | Toff (cid:105) = Toff (cid:12)(cid:12) + ⊗ (cid:11) (3.16)where the Toffoli gate can be expressed in terms of the fusion matrix (2.2). Explicitly,Toff | i, j, k (cid:105) = N ij,k ( ij + k ) = | i, j, ij + k, mod d (cid:105) (3.17)– 5 –here the Young tableau for ( ij ) has i + j vertical boxes, mod d .For SU( d ) , d odd prime, d (cid:54) = 5 mod 4, both graph and hypergraph states canbe obtained from operators which can be constructed from products of topologicaloperations on the n -torus Hilbert space [10].For SU( d ) , d = 5 mod 4, the topological argument does not apply, since in thiscase Theorem 1 of Salton et al [10] implies that only stabilizer states can be con-structed on the n -torus Hilbert space. Thus, graph states can be so constructed butnot hypergraph states with cubic or higher functions (3.12). It was shown above that graph states and hypergraph states for SU( d ) , d odd, canbe constructed from the basic operations N abc , S ab , Z , and X of SU( d ) Chern-Simonstheory. For d odd prime, d (cid:54) = 5 mod 4, these operations can be constructed topolog-ically on the n -torus Hilbert space of Chern-Simons theory. A subset of hypergraphstates are “magic.” For d (cid:54) = 5 mod 4, they are topological in the above sense.Fliss [19] has studied knot and link states of SU(2) d Chern-Simons theory, and hasshown that knot and link states are generically magical. However for U(1) d , magic isabsent for all knot and link states. Since U(1) d is level-rank dual to SU( d ) , the knotand link states for this theory also have zero magic [19–21],[22].There is a great deal of recent interest in magic states [6, 23–25]. One featurethat deserves further study is to understand which magic states are topological. Forexample, universal topological computing is possible for SU(2) [26] and SU(3) [27]Chern-Simons theory. Implicitly this implies that magic states are present in thesetheories, presumably due to the braiding operations. It would be interesting to makethis explicit. Acknowledgments
We thank Zi-Wen Liu for clarifying the difference between hypergraph / stabilizer cones[28–31], and hypergraph / stabilizer states, and for emphasizing that universal quantumcomputation models for SU(2) and SU(3) have gates / actions which are magical.We are grateful to Greg Bentsen and Isaac Cohen for their assistance in the prepa-ration of the paper. – 6 – eferences [1] H.J. Schnitzer, Clifford group and stabilizer states from Chern-Simons theory ,[ arXiv:1903.06789 ].[2] H.J. Schnitzer, SU( N ) Chern-Simons theory, the Clifford group, and entropy cone ,[ arXiv:2008.02406 ].[3] H.J. Schnitzer, Clifford operators in
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