We introduce a dichromatic calculus (RG) for qutrit systems. We show that the
decomposition of the qutrit Hadamard gate is non-unique and not derivable from
the dichromatic calculus. As an application of the dichromatic calculus, we
depict a quantum algorithm with a single qutrit. Since it is not easy to
decompose an arbitrary d by d unitary matrix into Z and X phase gates when d >
2, the proof of the universality of qudit ZX calculus for quantum mechanics is
far from trivial. We construct a counterexample to Ranchin's universality
proof, and give another proof by Lie theory that the qudit ZX calculus contains
all single qudit unitary transformations, which implies that qudit ZX calculus,
with qutrit dichromatic calculus as a special case, is universal for quantum
mechanics.Comment: In Proceedings QPL 2014, arXiv:1412.810

Bian, Xiaoning

Wang, Quanlong

Electronic Proceedings in Theoretical Computer Science

English

arXiv

We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics

Quanlong Wang

Xiaoning Bian

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B. Coecke, I. Hasuo & P. Panangaden (Eds.):Quantum Physics and Logic 2014 (QPL 2014).EPTCS 172, 2014, pp. 92–101, doi:10.4204/EPTCS.172.7c© Q. Wang & X. BianThis work is licensed under theCreative Commons Attribution License.Qutrit Dichromatic Calculus and Its UniversalityQuanlong Wang Xiaoning BianSchool of Mathematics and Systems ScienceBeihang UniversityBeijing, Chinaqlwang@buaa.edu.cn bianxiaoning@smss.buaa.edu.cnWe introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition ofthe qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As anapplication of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since itis not easy to decompose an arbitraryd×d unitary matrix into Z and X phase gates whend > 2, theproof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We constructa counterexample to Ranchin’s universality proof, and giveanother proof by Lie theory that the quditZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus,with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.1 IntroductionIn [3], Coecke and Duncan developed dichromatic ZX-calculus for qubit systems. To extend the graph-ical calculus to higher dimensions, Ranchin considered qudit (d-dimensional quantum system) ZX-calculus [8]. At almost the same time, the authors of this paper investigated the theory and applicationof qutrit ZX-calculus [1]. Unlike in [8] and [1], we introduce two new rules P1 and P2 in this paper. Thenecessity of these two rules is demonstrated by depicting indichromatic calculus the simplest quantumspeed-up algorithm with a single qutrit [6].In the qubit case, Duncan and Perdrix [5] proved that the Euler decomposition is not derivable fromZX calculus. In this paper, we prove similarly that the decomp sition of the qutrit Hadamard gate isnon-unique and not derivable from a dichromatic qutrit ZX-calculus.For any d-dimensional quantum system (d ≥ 2), universality is a very important problem for ZXcalculus. This means that the qudit ZX calculus can express any quantum state and gate. To the best ofour knowledge, it is not easy to decompose an arbitraryd × d unitary matrix into Z and X phase gateswhend > 2. Thus the proof of the universality of qudit ZX calculus forquantum mechanics is far fromtrivial. Due to Brylinski [2], to prove the universality of qudit ZX calculus, it suffices to prove that thequdit ZX calculus contains all single qudit unitary transformations. Such a proof given in [8] is based onthe fact [7] that the d-dimensional phase gatesZd andXd are sufficient to simulate all single qudit unitarytransforms. For our understanding, only part of the whole family of Zd phase gates can be represented byΛX phase gates (i.e., X phase gates) in [8]. Actually, we have a counterexample that someZd phase gatescannot be realized byΛX only. Thus another proof that the qudit ZX calculus containsall single quditunitary transformations is requested. We solve this problem by the method of Lie algebra. Therefore thequdit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.2 Red and Green GraphsWe fix some notations here. LetFdHilb be the symmetric monoidal†−category(SM†−category) offinite-dimensional complex Hilbert spaces and linear maps between them. LetFdHilbp be The SM†−categoryQ. Wang & X. Bian 93of finite-dimensional complex Hilbert spaces and linear maps modulo the relationf ≡ g if ∃z ∈ C,z , 0 :f = zg. FdHilbQ is defined as the full subcategory ofFdHilbp generated by the objects{Q⊗ · · ·⊗Q︸       ︷︷       ︸n| n ≥0, whereQ := C3. This is essentially the category of qutrits.2.1 RG categoryWe define a categoryRG where the objects aren-fold monoidal products of an object∗, denoted∗n(n ≥0). In RG, a morphism from∗m to ∗n is a finite undirected open graph fromwires ton wires, builtfromδZ = δ†Z = ǫZ = ǫ†Z = PZ(α,β) =αβ H = HδX = δ†X = ǫX = ǫ†X = PX(α,β) =αβ H† = H†whereα,β ∈ [0,2π). For convenience, we denote the frequently used angles2π3 and4π3 by 1 and 2respectively. The generator H is called a Hadamard gate. Additionally, the identity morphism on∗ isrepresented as the straight wire. Composition is connecting up the edges, while tensor is simply puttingtwo diagrams side by side. We also mention here that we ignoreconnected components of a graph whichare connected to neither input nor output. This is in order tonot have to deal with scalars.RG morphisms are also subject to the equations depicted below.1. Equations in Figure1.2. All equations hold under flip of graphs, negation of angles, and exchange ofH andH†.3. All equations hold under flip of colours (except for rulesK2 andH2).The equations below can be derived from the rules ofRG given above. They are very useful whendemonstrating some more complex equalities in describing quantum protocols [1] and algorithms[6].= (1)= =αββα(2)= = = (3)D=D= (4)94 Qutrit Dichromatic Calculus and Its Universality... = α+ηβ+ θηθ............αβ......(S 1) := 00 = (S 2)= (B1) = (B2)21=1221 =121221 (K1)12αβ=12=21αβ21β-α-α-βα-β=12=21αβ12αβ21β-α-α-βα-β(K2)H†H†=H=H(H1)H... ......αβH† H†...= αβH(H2)D =:= (P1)DDαβ=D= =DDDβαD(P2)Figure 1: RG rulesIt is worth noting that there are some remarkable differences between qutrit rules and qubit rules.First, in qubit case we have = , while in qutrit case we have = . Second, the dualizerof the two observables Z and X is an even permutation, i.e., thidentical permutation. And there is onlyone odd permutationπ in qubit case such thatπ=π -α=π=παπ=πππWhile in qutrit case, the dualizer of Z and X is an odd permutation which satisfies ruleP2. Third, inqubit case the K2 rule still holds when flipping the colours, while it doesn’t hold under flip of colours inqutrit case.Now RG is a symmetric monoidal category, which can further be made into a†− SMC by having†act on the generators as follows:Q. Wang & X. Bian 95†=( )†=†=†=αβ†= -α-βH†= H††=( )†=†=†=αβ†= -α-βH††= Hwhere functoriality of (·)† is guaranteed by Rule 2.2.2 RG interpretationHere, we give an interpretation for these graphs by describing a monoidal functor [·]RG : RG→FdHilbQ,mapping the morphisms as follows (expressed in Dirac notation):RG= |+〉[ ]RG= 〈+|RG= |00〉 〈0|+ |11〉 〈1|+ |22〉 〈2|RG= |0〉 〈00|+ |1〉 〈11|+ |2〉 〈22|αβRG= |0〉 〈0|+ eiα |1〉 〈1|+ eiβ |2〉 〈2|RG= |0〉[ ]RG= 〈0|RG= |++〉 〈+|+ |ωω〉 〈ω|+ |ω̄ω̄〉 〈ω̄|RG= |+〉 〈++|+ |ω〉 〈ωω|+ |ω̄〉 〈ω̄ω̄|αβRG= |+〉 〈+|+ eiα |ω〉 〈ω|+ eiβ |ω̄〉 〈ω̄|HRG= |+〉 〈0|+ |ω〉 〈1|+ |ω̄〉 〈2|H†RG= |0〉 〈+|+ |1〉 〈ω|+ |2〉 〈ω̄|whereω = e23πi, ω̄ = e43πi, and|+〉 = |0〉+ |1〉+ |2〉|ω〉 = |0〉+ω |1〉+ ω̄ |2〉|ω̄〉 = |0〉+ ω̄ |1〉+ω |2〉Proposition 2.1 [·]RG is a symmetric monoidal †−functor.Proof: This involves checking for each rulef = g in RG that [f ]RG = [g]RG, that[·]RG respects the sym-metric monoidal structure on the generators, and for each generator f , we have [f ]†RG = [ f†]RG. 96 Qutrit Dichromatic Calculus and Its Universality3 Decomposition of the Hadamard GateIt can be directly checked that inFdHilbQ, we have[H]RG = [PX(4π3,4π3)]RG ◦ [PZ(4π3,4π3)]RG ◦ [PX(4π3,4π3)]RGWe call the following graph an Euler decomposition of the Hadamard gate:22H =2222Proposition 3.1 The Euler decomposition is not unique:H = 222222⇒ 2222=22HProof:22=11=H22H222222H=1122222222=11In the qubit case, Duncan and Perdrix [5] proved that the Euler decomposition is not derivable from ZXcalculus. Similarly, we haveProposition 3.2 The Euler decomposition is not derivable from RG.Proof: We define an alternative interpretation functor [·]0 : RG→ FdHilbQ exactly as [·]RG with thefollowing change:[PX(α,β)]0 = [PX(0,0)]RG [PZ(α,β)]0 = [PZ(0,0)]RGThis functor preserves all the rules introduced in Figure 1,so its image is indeed a valid model of thetheory. However we have the following inequality[H]0 , [PX(4π3,4π3)]0 ◦ [PZ(4π3,4π3)]0 ◦ [PX(4π3,4π3)]0hence the Euler decomposition is not derivable fromRG.Q. Wang & X. Bian 974 Quantum Algorithm with a Single QutritRecently, Gedik[6] introduces a simple algorithm using only a single qutrit to determine the parity ofpermutations of a set of three objects. As in the case of Deutsch’s algorithm, a speed-up relative tocorresponding classical algorithms is obtained.Consider the six permutations of the set{0,1,2}. Each permutation can be treated as a functionf (x)defined on the setx ∈ {0,1,2}. Then the task is to determine its parity. The problem could be solved byevaluatingf (x) for two different values of x.The function f has a domain and range of three values. These three values corr pond to the threestates of a qutrit|m〉 wherem = 0,1,2. The unitaryU f corresponding to the functionf is a simpletransposition of orthonormal states|m〉. Applying U f to the eigenstate|ω〉 of theX observable we obtain{U f |ω〉 = |ω〉 (up to a phase) if is an even permutation;U f |ω〉 = |ω̄〉 (up to a phase) if is an odd permutation.Thus, a single evaluation of the function is enough to determine its parity.The above algorithm can be depicted by the dichromatic calculus as follows:f (0) (1 2)(0 1)(1 2)(0 2) (1 2) (0 1) (0 2)U f 00 12 21 DD1221DU f |w〉 00122112= = =1212 1212=211212 21=12=DDDParity Even Odd5 The Qudit ZX Calculus Is UniversalIt is important to prove that the qudit ZX calculus is universal for quantum mechanics for anyd. Sinceit is not easy to decompose an arbitraryd × d unitary matrix into Z and X phase gates (i.e.,ΛZ andΛX gates) whend > 2, the proof of university is far from trivial. Due to Brylinski [2], to prove theuniversality of qudit ZX calculus for quantum mechanics, itsuffices to prove that the qudit ZX calculuscontains all single qudit unitary transformations. Such a proof given in [8] is based on the fact [7] thatthe d-dimensional phase gatesZd,Xd are sufficient to simulate all single qudit unitary transforms, whereZd(b0,b1...,bd−1) : b0 |0〉+b1 |1〉+ ...+bd−1 |d−1〉 7→ |d−1〉(thed complex coefficients,b0,b1...,bd−1 are normalized to unity)Xd(φ) :{|d−1〉 7→ eiφ |d−1〉|p〉 7→ |p〉 for p , d−1It was checked in [8] that eachXd can be encoded to a phase gateΛZ of the qudit ZX calculus, whereΛZ(α1,α2, ...,αd−1) :=1eiα1. . .eiαd−198 Qutrit Dichromatic Calculus and Its UniversalityMeanwhile, someZd phase gates were shown to be realized byΛX phase gate in the qudit ZXcalculus, whereΛX(α1,α2, ...,αd−1) :=1dc0 cd−1 cd−2 ... c2 c1c1 c0 cd−1 ... c3 c2c2 c1 c0 ... c4 c3... ... ... ... ... ...cd−1 cd−2 cd−3 ... c1 c0ck = 1+∑d−1l=1 ηrk(l)eiαl , rk permutes the entries 1 (there is onerk for each k).However, not everyZd phase gate can be represented byΛX(α1,α2, ...,αd−1). In fact, to realize anyZd(b0,b1...,bd−1) in this way, we need to findα1,α2, ...,αd−1 such thatcd−1b0+ cd−2b1+ ...c1bd−2+ c0bd−1 = d (1)ckb0+ ck−1b1+ ...c0bk + cd−1bk+1+ ...ck+1bd−1 = 0, ∀k , d−1 (2)Since∑d−1k=0 ck = d, summing up all the equations in (1) and (2), we have∑d−1k=0 bk = 1. Clearly, notevery unit complex vector (b0,b1...,bd−1) satisfies∑d−1k=0 bk = 1 or∑d−1k=0 bk = eiα up to a global phase.For example, (b0,b1...,bd−1) = (0, 1√2,1√2,0, ...,0),d > 2, is such a counterexample.The above argument means that we need to find another proof that the qudit ZX calculus contains allsingle qudit unitary transformations. Next we solve this problem using the theory of Lie algebra.LetH =eiα0. . .eiαd−1∣∣∣∣∣∣∣∣∣∣α0, ...,αd−1 ∈ R, V =1√dd−1∑j,k=0ω jk | j〉 〈k| ,ω = ei2πd ,H′ = VHV−1Proposition 5.1 Both H and H′ are closed connected subgroups of the compact Lie group of unitariesG = U(d).Proof: Let S 1 = {eiα|α ∈ R}. Clearly, the circleS 1 is closed and connected. SinceH  S 1× · · ·×S 1,H is also a closed connected group. It is obvious thatH′ is topologically isomorphic toH. ThusH′ is aclosed connected group. We need two lemmas as follows.Lemma 5.2 [2] Let G be a compact Lie group. If H1, ...,Hk are closed connected subgroups and theygenerate a dense group of G, then in fact they generate G.Lemma 5.3 [2] Let h = Lie H, h′ = Lie H′, g = Lie U(d). If h and h′ generate g as a Lie algebra, and Hand H′ are closed connected groups, then H and H′ generate a dense subgroup of U(d).We choose the following matrices [4] as a basis of the Lie algebrag:σ( jk)x (0≤ j < k ≤ d−1),σ( jk)y (0≤ j < k ≤ d−1),σ( jk)z ( j = 0,1≤ k ≤ d−1), iIdQ. Wang & X. Bian 99whereσ( jk)x = i | j〉 〈k|+ i |k〉 〈 j| ,σ( jk)y = | j〉 〈k| − |k〉 〈 j| ,σ( jk)z = i | j〉 〈 j| − i |k〉 〈k|.It is easily checked thath =d−1∑j=0iα j | j〉 〈 j|∣∣∣∣∣∣∣∣α j ∈ R= span{σ(0 j)z (1≤ j ≤ d−1), iId}h′ =d−1∑j=0iα jV | j〉 〈 j|V−1∣∣∣∣∣∣∣∣α j ∈ R= span{Vσ(0 j)z V−1(1≤ j ≤ d−1), iId}Theorem 5.4 Letm be the Lie subalgebra of g generated by h and h′. Then all the σ( jk)x (0≤ j< k ≤ d−1)and σ( jk)y (0≤ j < k ≤ d−1) are included in m.Proof: ∀t ∈ {1, ...,d−1}, Vσ(0t)z V−1 ∈m.Vσ(0t)z V−1=1√dd−1∑j,k=0ω jk | j〉 〈k| (i |0〉 〈0| − i |t〉 〈t|)1√dd−1∑j1,k1=0ω̄ j1k1 |k1〉 〈 j1|=idd−1∑j, j1=0(1−ω( j− j1)t | j〉 〈 j1|)Thus∑d−1t=1 Vσ(0t)z V−1 ∈m. By direct calculation,χ :=d−1∑t=1Vσ(0t)z V−1 =d−1∑t=1idd−1∑j, j1=0(1−ω( j− j1)t | j〉 〈 j1|=(d−1)∑j, j1=0, j, j1i | j〉 〈 j1| =∑0≤ j<k≤d−1σ( jk)x ∈mWe give the Lie products betweenσx, σy andσz as follows.[σ(0t)x ,σ(0t)z ] = 2σ(0t)y[σ(0k)x ,σ(0t)z ] = σ(0k)y ,k , t[σ(tk)x ,σ(0t)z ] = −σ(tk)y ,0< t , k[σ( jt)x ,σ(0t)z ] = σ( jk)y ,0< j , t(1)[σ(0k)y ,σ(0k)z ] = −2σ(0k)x[σ(0k)y ,σ(0t)z ] = −σ(kt)x ,k , t[σ( jk)y ,σ(0 j)z ] = −σ( jk)x ,0< j , k[σ( jk)y ,σ(0k)z ] = σ( jk)x ,0< j , k(2)From the Lie products listed above, we have[χ,σ(0t)z ] =d−1∑k=1σ(0k)y +d−1∑k=0σ(kt)y ∈m,∀t ∈ {1, ...,d−1}.100 Qutrit Dichromatic Calculus and Its Universality∀u ∈ {1, ...,d−1}, [d−1∑k=1σ(0k)y +d−1∑k=0σ(kt)y ,σ(0u)z ] =−σ0ux −∑d−1k=0,k,uσ(ku)x , t , u−2σ(0u)x −2∑d−1k=0,k,uσ(ku)x , t = uTherefore,σ(0u)x +∑d−1k=0,k,uσ(ku)x ∈m,∀u ∈ {1, ...,d−1}.Furthermore, foru, v ∈ {1, ...,d−1},[σ(0u)x +d−1∑k=0,k,uσ(ku)x ,σ(0v)z ] =2σ(0u)y −σ(uv)y ∈m, v , u4σ(0u)y +∑d−1k=1,k,uσ(ku)y ∈m, v = uThusd−1∑v=1,v,u(2σ(0u)y −σ(vu)y)= 2(d−2)σ(0u)y −d−1∑v=1,v,uσ(vu)y ∈m2(d−2)σ(0u)y −d−1∑v=1,v,uσ(vu)y+4σ(0u)y +d−1∑k=1,k,uσ(ku)y = 2dσ(0u)y ∈mi.e.,σ(0u)y ∈m,∀u ∈ {1, ...,d−1}.Immediately, we getσ(vu)y ∈m,∀u , v,u, v ∈ {1, ...,d−1}.Up to now, all theσ( jk)y (0≤ j < k ≤ d−1) are included inm. Still from the Lie products listed in (2),we know that all theσ( jk)x (0≤ j < k ≤ d−1) are included inm. Theorem (5.4) means thath andh′ generate the Lie algebrag. It follows from proposition(5.1), lemma(5.2) and lemma (5.3) thatH andH′ generateU(d), which means qudit ZX Calculus contains all singlequdit unitary transformations. Therefore the qudit ZX calculus is universal for quantum mechanics.6 Conclusion and Future WorkIn this paper, we introduce a dichromatic calculus (RG) for qutrit systems. We show that the decompo-sition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As anapplication of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Furthermore,for any d, we prove that the qudit ZX calculus contains all single qudit nitary transformations. It fol-lows that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantummechanics.There are many issues requiring further exploration. Here we just list a few of them as follows. First,does there exist a formula in which each unitary is decomposed into X and Z phase gates? Second, is thedichromatic ZX calculus complete for qutrit stabilizer quantum mechanics? Finally, is the dichromaticZX calculus incomplete for qutrit quantum mechanics?References[1] Xiaoning Bian & Quanlong Wang (2014):Graphical calculus for qutrit systems. accepted for publication inthe IEICE Transactions.Q. Wang & X. Bian 101[2] Jean-Luc Brylinski & Ranee Brylinski (2002):Universal quantum gates. Mathematics of Quantum Compu-tation79, doi:10.1201/9781420035377.pt2.[3] Bob Coecke & Ross Duncan (2011):Interacting quantum observables: categorical algebra and diagrammat-ics. New Journal of Physics13(4), p. 043016, doi:10.1088/1367-2630/13/4/043016.[4] Yao-Min Di & Hai-Rui Wei (2011): Elementary gates for ternary quantum logic circuit. arXiv preprintarXiv:1105.5485.[5] Ross Duncan & Simon Perdrix (2013):Pivoting makes the ZX-calculus complete for real stabilizers. arXivpreprint arXiv:1307.7048.[6] Z. Gedik (2014):Computational Speed-up with a Single Qutrit. arXiv:1403.5861.[7] Ashok Muthukrishnan & CR Stroud Jr (2000):Multivalued logic gates for quantum computation. PhysicalReview A62(5), p. 052309, doi:10.1103/PhysRevA.62.052309.[8] André Ranchin (2014):Depicting qudit quantum mechanics and mutually unbiased qudit theories. arXivpreprint arXiv:1404.1288.

Qutrit Dichromatic Calculus and Its Universality

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