Open N-Qubit System As a Quantum Computer with Four-Valued Logic

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Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia

E-mail: [email protected]

Preprint of Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow SINP MSU 2001-31/671 20 August 2001

Abstract In this paper we generalize the usual model of quantum computer to a model in which the state is an operator of density matrix and the gates are general superoperators (quantum operations), not necessarily unitary. A mixed state (operator of density matrix) of n two- level quantum system (open or closed n-qubit system) is considered as an element of 4n- dimensional operator Hilbert space (Liouville space). It allows to use quantum computer (circuit) model with 4-valued logic. The gates of this model are general superoperators which act on n-ququats state. Ququat is quantum state in a 4-dimensional (operator) Hilbert space. Unitary two-valued logic gates and quantum operations for n-qubit open system are considered as four-valued logic gates acting on n-ququats. We discuss properties of quantum 4-valued logic gates. In the paper we study universality for quantum four-valued logic gates. PACS 3.67.Lx; 03.65-w; 3.65.Bz Keywords: Quantum computation, open quantum systems, ququats, many-valued logic

1 IIntroduction II Quantum stateand qubit Usual models for quantum computer use closed n-qubit sys- II.1 Pure states tems and deal only with unitary gates on pure states. In these models it is difficult or impossible to deal formally with mea- A quantum system in a pure state is described by unit vector surements, dissipation, decoherence and noise. It turns out, in a Hilbert space . In the Dirac notation a pure state is denoted by Ψ >. TheH Hilbert space is a linear space with an that the restriction to pure states and unitary gates is unneces- | H sary [1]. inner product. The inner product for Ψ1 >, Ψ2 > is denoted by < Ψ Ψ >. A quantum bit| or qubit,| the fundamen-∈H One can describe an open system starting from a closed 1| 2 system if the open system is a part of the closed system. How- tal concept of quantum computations, is a two-state quantum system. The two basis states labeled 0 > and 1 >,areor- ever, situations can arise where it is difficult or impossible to | | find a closed system comprising the given open system. This thogonal unit vectors, i.e. would render theory of dissipative and open systems a fun- = δkl, damental generalization of quantum mechanics [2]. Under- | standing dynamics of open systems is important for studying 2 where k, l 0, 1 . The Hilbert space of qubit is 2 = C . quantum noise processes [3, 4, 5], quantum error correction The quantum∈{ system} which corresponds to a quantumH com- [6, 8, 9, 11, 12], decoherence effects [13, 14, 15, 16, 17, 18, puter (quantum circuits) consists of n quantum two-state par- 19, 20, 21] in quantum computations and to perform simula- ticles. The Hilbert space (n) of such a system is a ten- tions of open quantum systems [22, 23, 24, 25, 26, 27]. H sor product of n Hilbert spaces 2 of one two-state particle: In this paper we generalize the usual model of quantum (n) H (n) n = 2 2 ... 2. The space is a N =2 computer to a model in which the state is a density matrix HdimensionalH ⊗H complex⊗ linear⊗H space. Let usH choose a basis for operator and the gates are general superoperators (quantum (n) which is consists of the N =2n orthonormal states operations), not necessarily unitary. Pure state of n two-level H n k>, where k is in binary representation. The state k>is a closed quantum systems is an element of 2 -dimensional Hilbert| (n) | tensor product of the states ki > in : space and it allows to realize quantum computer model with | H 2-valued logic. The gates of this computer model are unitary k>= k1 > k2 > ... kn >= k1k2...kn >, operators act on a such state. In general case, mixed state | | ⊗| ⊗ ⊗| | (operator of density matrix) of n two-level quantum systems where ki 0, 1 and i =1, 2, ..., n. This basis is usually is an element of 4n-dimensional operator Hilbert space (Li- called computational∈{ } basis which has 2n elements. A pure ouville space). It allows to use quantum computer model with state Ψ(t) > (n) is generally a superposition of the basis 4-valued logic. The gates of this model are general superop- states| ∈H erators (quantum operations) which act on general n-ququats N 1 state. Ququat [55] is quantum state in a 4-dimensional (op- − Ψ(t) >= ak(t) k>, (1) erator) Hilbert space. Unitary gates and quantum operations | | Xk=0 for quantum two-valued logic computer can be considered as four-valued logic gates of new model. In the paper we con- n N 1 2 with N =2 and − ak(t) =1. The inner product sider universality for general quantum 4-valued logic gates k=0 | | between Ψ > and PΨ0 > is denoted by < Ψ Ψ0 > and acting on ququats. | | | In Sections 2, 3 and 5, the physical and mathematical N 1 − background (pure and mixed states, Liouville space and su- < Ψ Ψ0 >= a∗a0k . | k peroperators, evolution equations for closed and open quan- kX=0 tum) are considered. In Section 4, we introduce generalized computational basis and generalized computational states for II.2 Mixed states 4n-dimensional operator Hilbert space (Liouville space). In the Section 6, we study some properties of general four-valued In general, a quantum system is not in a pure state. Open logic gates. Unitary gates and quantum operations of two- quantum systems are not really isolated from the rest of the valued logic computer are considered as four-valued logic universe, so it does not have a well defined pure state. Lan- gates. In Section 7, we introduce a four-valued classical logic dau and von Neumann introduced a mixed state and a density matrix into quantum theory. A density matrix is a Hermitian formalism. In Section 8, we realize classical 4-valued logic (n) gates by quantum gates. In Section 9, we consider a universal (ρ† = ρ), positive (ρ>0) operator on with unit trace (Trρ =1). Pure states can be characterizedH by idempotent set of quantum 4-valued logic gates. In Section 10, quantum 2 four-valued logic gates of order (n,m) as a map from density condition ρ = ρ. A pure state (1) is represented by the operator ρ = Ψ >< Ψ . matrix operator on n-ququats to density matrix operator on | | m-ququats are discussed. One can represent an arbitrary density matrix operator ρ(t) for n-qubits in terms of tensor products of Pauli matrices σµ: 1 ρ(t)= Pµ ...µ (t)σµ ... σµ . (2) 2n 1 n 1 ⊗ ⊗ n µ1X...µn

2 where each µi 0, 1, 2, 3 and i =1, ..., n.Hereσµ are called Liouville space attached to or the associated Hilbert Pauli matrices ∈{ } space, or Hilbert-Schmidt space [28]-[54].H Let k> be an orthonormal basis of (n): 01 0 i {| } H σ1 = ,σ2 = − , 10 i 0 N 1 − = δkk0 , k>= k> l>and | 6 | | ⊗| N 1 1 − k, l)= k1,l1) k2,l2) ... kn,ln) , (5) ρ(t)= σµPµ(t), | | ⊗| ⊗ ⊗| 2n µX=0 where ki,li 0, 1 , i =1, ..., n and ∈{ } n where σµ = σµ ... σµ , µ =(µ1...µn) and N =4 . 1 n ki,li) kj,lj)= ki > kj >, < li

2 2 2 (k, l A)=Tr( l>= Akl. P1 (t)+P2 (t)+P3 (t)=1. | | | | | An operator ρ(t) of density matrix for n-qubits can be con- The mixed state can be identiﬁed with close ball (n) sideredasanelement ρ(t)) of the space .From(6)we P 2(t)+P 2(t)+P 2(t) 1. get | H 1 2 3 ≤ N 1 N 1 Not all linear combinations of quantum states ρj(t) are − − states. The operator ρ(t)) = k, l)(k, l ρ(t)) , (7) | | | kX=0 Xl=0 ρ(t)= λ ρ (t) j j where N =2n and Xj N 1 − is a state iff j λj =1. (k, k ρ(t)) = 1. | P Xk=0

III Liouville space and superoperators III.2 Superoperators For the concept of Liouville space and superoperators see Operators, which act on , are called superoperators and we [28]-[54]. denote them in general byH the hat. For an arbitrary superoperator Λˆ on , which is deﬁned III.1 Liouville space by H The space of linear operators acting on a N =2n-dimensional Λˆ A)= Λ(A)), Hilbert space (n) is a N 2 =4n-dimensional complex linear | | (n) H (n) we have space . We denote an element A of by a ket-vector H H (n) N 1 N 1 A). The inner product of two elements A) and B) of − − | | | H (k, l Λˆ A)= (k, l Λˆ k0,l0)(k0,l0 A)= is deﬁned as | | | | | kX0 =0 lX0=0 (A B)=Tr(A†B) . (3) | N 1 N 1 − − The norm A = (A A) is the Hilbert-Schmidt norm of = Λ A , k k | klk0 l0 k0l0 operator A. A new Hilbertp space with scalar product (3) is kX=0 lX=0 H 0 0

3 n where N =2 . σ0 = I = 0 >< 0 + 1 >< 1 = 0, 0) + 1, 1). Let A be a linear operator in Hilbert space. Then the su- | | | | | | Let us use the formulas peroperators LÂ and RÂ will be defined by 1 1 LÂ B)= AB) , RÂ B)= BA). 0, 0) = ( σ0)+ σ3)) , 1, 1) = ( σ0) σ3)), | | | | | 2 | | | 2 | −| In the basis k, l) we have | 1 1 0, 1) = ( σ )+i σ )) , 1, 0) = ( σ ) i σ )). N 1 N 1 1 2 1 2 − − | 2 | | | 2 | − | (k, l LÂ B)= (k, l LÂ k0,l0)(k0,l0 B)= | | | | | It allows to rewrite operator basis kX0=0 lX0=0

k, l)= k ,l ) k ,l ) ... kn,ln) | | 1 1 ⊗| 2 2 ⊗ ⊗| N 1 N 1 − − by complete basis operators = (LÂ)klk l , 0 0 | | kX0=0 lX0=0 σµ)= σµ σµ ... σµ ), | | 1 ⊗ 2 ⊗ ⊗ n Note that where µi =2ki + li,i.e. µi 0, 1, 2, 3 and i =1, ..., n. ∈{ } (k, l AB)== The basis σµ) is orthogonal | | | | n (σµ σµ )=2 δµµ | 0 0 N 1 N 1 − − and complete operator basis = < k0 B l0 >< l0 l>. | | | | | kX0=0 lX0=0 N 1 1 − σµ)(σµ = I.ˆ Finally, we obtain 2n | | Xµ ˆ (LA)klk0 l0 =< l0 l>= Akk0 δll0 . (n) | | | For an arbitrary element A) of we have Pauli represen- | H The superoperator Pˆ = A)(B is defined by tation by | | N 1 Pˆ C)= A)(B C)= A)Tr(B†C). 1 − | | | | A)= σµ)(σµ A) | 2n | | µX=0 A superoperator ˆ† is called the adjoint superoperator for ˆ if E E with the complex coefficients ˆ A B A ˆ B ( †( ) )=( ( )) (σµ A)=Tr(σµA). E | |E | for all A) and B) from . For example, if ˆ = LÂRˆB ,then We can rewrite formulas (2) using the complete operator basis ˆ ˆ| ˆ | ˆ ˆH ˆ ˆ E ˆ ˆ ˆ (n) † = LA RB .If = LA,then † = LA .If = LARA , E † † E E † E † σµ) in Liouville space : then ˆ† = Lˆ RÂ. | H E A† A superoperator ˆ is called unital if ˆ I)= I). N 1 E E| | 1 − ρ(t)) = σµ)(σµ ρ(t)), | 2n | | µX=0 IV Generalized computational basis and where σµ = σµ ... σµ , µ =(µ ...µn) and (σµ ρ(t)) = 1 ⊗ ⊗ n 1 | ququats Pµ(t). The density matrix operator ρ(t) is a self-adjoint operator Let us introduce generalized computational basis and general- with unit trace. It follows that ized computational states for 4n-dimensional operator Hilbert space (Liouville space). P ∗(t)=Pµ(t) ,P(t)=(σ ρ(t)) = 1. µ 0 0| IV.1 Pauli representation In general case, N 1 Pauli matrices (II.2) can be considered as a basis in operator 1 − P 2(t)=(ρ(t) ρ(t)) = Trρ2(t) 1. space. Let us write the Pauli matrices (II.2) in the form 2n µ | ≤ µX=0

σ1 = 0 >< 1 + 1 >< 0 = 0, 1) + 1, 0), Note that Schwarz inequality | | | | | | (A B) 2 (A A)(B B) σ2 = i 0 >< 1 + i 1 >< 0 = i( 0, 1) 1, 0)), | | | ≤ | | − | | | | − | −| leads to

2 σ3 = 0 >< 0 1 >< 1 = 0, 0) 1, 1), (I ρ(t)) (I I)(ρ(t) ρ(t)), | |−| | | −| | | | ≤ | |

4 N 1 − where k, l 1, 2, 3 . 1= Trρ(t) 2 2n(ρ(t) ρ(t)) = P 2(t), ∈{ } | | ≤ | µ The usual computational basis k> is not a basis of µX=0 general state ρ(t) which has a time{| dependence.} In general i.e. case, a pure state evolves to mixed state. Pure state of n two-level closed quantum systems is an N 1 1 − element of 2n-dimensional functional Hilbert space (n).It Tr(ρ2(t)) 1 or 1 P 2(t) 2n. H √ n ≤ ≤ ≤ µ ≤ leads to quantum computer model with 2-valued logic. In 2 µX=0 general case, the mixed state ρ(t) of n two-level (open or n An arbitrary general one-qubit state ρ(t) can be repre- closed) quantum system is an element of 4 -dimensional op- (n) sented in Liouville space 2 as erator Hilbert space (Liouville space). It leads to 4- H valued logic model forH quantum computer. 3 1 The state of the quantum computation at any point time is ρ(t)) = σµ)Pµ(t), | 2 | a superposition of basis elements µX=0 N 1 2 2 2 − where Pµ =(σµ ρ), P0 =1and P (t)+P (t)+P (t) 1. | 1 2 3 ≤ ρ(t)) = µ)ρµ(t), Note that the basis σµ) is orthogonal, but is not orthonornal. | | | µX=0

where ρµ(t) are real numbers (functions) satisfying normal- IV.2 Generalized computational basis n ized condition ρ0(t)=1/√2 ,i.e. Let us define the orthonormal basis µ) of Liouville space (n) | √2n(0 ρ(t)) = Tr(ρ(t)) = 1. . In general case, the state ρ(t) of the n-qubits system is | H (n) (n) an element of Hilbert space . The basis for consists Any state ρ(t)) for basis element 0...0) has P =1in all H H 0...0 of the N 2 =22n =4n orthonormal basis elements denoted cases. | | by µ). | (n) Definition A basis of Liouville space is defined by H IV.3 Generalized computational states 1 1 Generalized computational basis elements µ) are not quan- µ)= µ1...µn)= σµ)= σµ1 ... σµn ), | | | √2n | √2n | ⊗ ⊗ tum states for µ =0. It follows from normalized condition (0 ρ(t)) = 1/√26. The general quantum state in Pauli repre- where each µi 0, 1, 2, 3 and | ∈{ } sentation (2) has the form N 1 − N 1 ˆ 1 − (µ µ0)=δµµ0 , µ)(µ = I, | | | ρ(t)) = n σµ)Pµ(t), µX=0 | 2 | µX=0 is called a generalized computational basis. where P (t)=1in all cases. Let us define simple computa- µ 0 Here is 4-valued representation of tional quantum states. n 1 Definition A quantum states in Liouville space defined by µ = µ14 − + ... + µn 14+µn . − 1 ρ t Example. In general case, one-qubit state ( ) of open quan- µ]= n σ0)+ σµ)(1 δµ0) tum system is | 2 | | − or 1 ρ)= 0) + 1)ρ1 + 2)ρ2 + 3)ρ3, 1 | | √2 | | | µ]= 0) + µ)(1 δµ0) . | √2n | | − where four orthonormal basis elements are is called generalized computational states. 1 1 1 Note that all states µ],whereµ =0, are pure states, since 0) = σ0)= I) , 1) = σ1), | 6 | √2| √2| | √2| [µ µ]=1.Thestate 0] is maximally mixed state. The states | | (n) µ] are elements of Liouville space . | H 1 1 Quantum state in a 4-dimensional Hilbert space is usually 2) = σ2) , 3) = σ3). | √2| | √2| called ququat or qu-quart[55] or qudit [56, 57, 58, 59, 60] with D =4. Usually ququat is considered as 4-level quantum Example. Two-qubit state ρ(t) is an element of 16-dimensional system. We consider ququat as general quantum state in a 4- Hilbert space with the orthonormal basis dimensional operator Hilbert space. Definition A quantum state in 4-dimensional operator Hilbert 1 1 00) = I I) , 0k)= I σ ), (1) 2 2 k space (Liouville space) associated with single qubit of | | ⊗ | | ⊗ (1) H n = 2 is called single ququat. A quantum state in 4 - H H (n) 1 1 dimensional Liouville space associated with n-qubits k0) = σk I) , kl)= σk σl), H | 2| ⊗ | 2| ⊗ system is called n-ququats.

5 Example. For the single ququat the states µ] are VEvolution equations and quantum op- | 1 1 erations 0] = σ0) , k]= σ0)+ σk) , | 2| | 2| | or In this section I review the description of open quantum sys- 1 1 tems dynamics in terms of evolution equations and quantum 0] = 0) , k]= 0) + k) . operations. | √2| | √2| | It is convenient to use matrices for quantum states. In matrix representation the single ququat computational basis V.1 Evolution equation for pure state of closed µ) and computational states µ] can be represented by systems | | Let H be the Hamilton operator, then in the Schroedinger 1 0 0 0 picture the equation of motion for the pure state Ψ(t) > of 0 1 0 0 | 0) =   , 1) =   , 2) =   , 3) =   .closed system is given by the Schroedinger equation | 0 | 0 | 1 | 0         d  0   0   0   1  Ψ(t) >= iH Ψ(t) >. (8)         dt In this representation single qubit generalized computa- | − | tional states µ] is represented by The change in the state Ψ(t) > of a closed quantum system | between two fixed times| t and t is described by a unitary 1 1 0 operator U(t, t ) which depends on those times 1  0  1  1  0 0] = , 1] = , | √ 0 | √ 0 2   2   Ψ(t) >= U(t, t0) Ψ(t0) >.  0   0  | |     If the Hamilton operator H has no time dependence, then the 1 1 unitary operator U(t, t ) has the form 1 0 1 0 0 2] =   , 3] =   . | √ 1 | √ 0 2   2   U(t, t0)=exp i(t t0)H .  0   1  {− − }     N 1 In general case, the unitary operator U(t, t0) is defined by A general single ququat quantum state ρ)= − µ)ρµ is | µ=0 | represented P d U(t, t )= iHU(t, t ) ,U(t ,t )=I. ρ0 dt 0 − 0 0 0  ρ1  ρ)= , (n) | ρ2 A pure state Ψ > of closed n-qubits system is gener-   | ∈H  ρ3  ally a superposition of the orthonormal basis states k>   | 2 2 2 where ρ0 =1/√2 and ρ + ρ + ρ √2. 1 2 3 Ψ(t) >= ak(t) k>. We can use the other matrix representation≤ for the states | | Xk ρ] which has no the coefficient 1/√2n. In this representa- |tion single qubit generalized computational states µ] is rep- Let the Hamilton operator H on the space (n) can be written resented by | in the form H 1 1 1 1 H = Hlm l>

6 2 2 where the Liouville superoperator Λˆ is given by The matrix Ckl is a positive matrix (n 1) (n 1) and { } 2 − × − I,Fk k =1, ..., n 1 is an operator basis for the space ˆ ˆ ˆ { | − } Λ= i(LH RH ). of bounded operators on n. The matrix Ckl is called a − − H { } positive matrix if all elements Ckl are real (C∗ = Ckl)and A change of pure and mixed states of closed (Hamiltonian) kl positive Ckl > 0. For the proofs of (13), we refer to [61]. quantum system is the unitary evolution. The final state ρ(t) For a given Λˆ, operator H is uniquely determined by the of the system is related to the initial state ρ(t = t0)=ρ by condition Tr(H)=0,andthematrix Ckl is uniquely de- unitary transformation U = U(t, t0): { } termined by the choice of the Fk. The conditions Tr(H)=0 and Tr(Fk)=0provide a canonical separation of the super- ρ ρ(t)= t(ρ)=UρU† , (10) → U operator Λˆ into a Hamiltonian plus dissipative part. If the condition of completely positivity is replaced by the where UU† = I. The superoperator ˆ is written in the form U weaker requirement of simple positivity, the generator for a n- ˆ = LÛ RÛ : ρ(t)) = ˆt ρ). U | U | level system can be again be written in the form (13), where the matrix Ckl is a matrix of positive defined Hermitian V.3 Evolution equation for mixed state of open form [67, 63],{ i.e.}

system n2 1 − A classiﬁcation of norm continuous (or, equivalently, with Cklzkzl∗ > 0, bounded generators) dynamical semigroup [66] of the Bahach klX=1 space of trace-class operators on , has been given by Lind- for all zk C. The matrix Ckl of Hermitian form is Her- blad ([62]). The general form ofH the generator Λˆ of such a ∈ { } mitian matrix (Ckl∗ = Clk). It is known [85] that Hermitian semigroup is the following form is positive if and only if

Λˆρ = i[H, ρ]+Φ(ρ) Φ(I) ρ, (11) C11 C12 ... C1k − − ◦  C21 C22 ... C2k  det > 0, where ......    Ck Ck ... Ckk  1  1 2  Φ(B)= Vj BV † ,AB = (AB + BA). j ◦ 2 2 Xj for all k =1, 2, ..., n 1. This condition is equivalent to condition of positivity for− matrix eigenvalues. Here H is a bounded self-adjoint Hamilton operator, Vj is a Let us consider a two-level quantum system (qubit) [61, sequence of bounded operators , Φ(I) is a bounded operator.{ } ?, 63, 64, 65] for a positive trace-preserving semigroup. Let The evolution equation has the form Fµ ,whereµ 0, 1, 2, 3 , be a complete orhonormal set {of self-adjoint} matrices:∈{ } d ρ(t)= i[H, ρ(t)] + Vjρ(t)Vj† ρ(t) (Vj Vj†) . 1 1 1 1 dt − − ◦ F0 = I, F1 = σ1,F2 = σ2,F3 = σ3. Xj √2 2 √2 √2 (12) Let Hamilton operator H and state ρ(t) have the form For the proofs of (11) and (12), we refer to [62, 66]. Using 3 equation (12) evolution equation for the mixes state ρ(t)) can 1 H = Hkσk ,ρ(t)= (P I + Pk(t)σk), be written by | 2 0 Xk=1 d where P =1in all cases. Using the relations ρ(t)) = Λˆ ρ(t)), 0 dt| | 3 3 σ σ = Iδ + i ε σ , [σ ,σ ]=2i ε σ , where the Liouville superoperator Λˆ is given by k l kl klm m k l klm m mX=1 mX=1 1 Λ=ˆ i(Lˆ Rˆ )+ 2Lˆ Rˆ Lˆ Lˆ Rˆ Rˆ and. εklmεijm = δkiδlj δkjδli, for (13) we obtain the equa- H H Vj V † Vj V † V † Vj − − − 2 j − j − j tions: Xj d 3 1 1 In the case of a n-level system (dim = n), evolution equa- Pk(t)= 2Hmεkml + (Ckl + Clk) Cδkl Pl(t) H dt 8 − 4 − tion (12) can be given in the form [61]: Xl=1

n2 1 d − 1 ρ(t)= i[H, ρ(t)] + Ckl Fkρ(t)F † ρ(t) (FkF †) , εijk(ImCij )P0, dt − l − ◦ l −4 k,lX=1 3 (13) where C = Cmm and k, l 1, 2, 3 . We can rewrite m=1 ∈{ } this equationP in the form where d 3 Pµ(t)= µν Pν (t) , (14) H† = H, Tr(H)=0,Tr(Fk)=0,Tr(Fk†Fl)=δkl, dt L Xν=0

7 where µ, ν 0, 1, 2, 3 and the matrix µν is By definition, Tr( (ρ)) is the probability that the process ∈{ } L represented by occurs,E when ρ is the initial state. The prob- 0000ability never exceedE 1. The quantum operation is trace-  B1 C(22) C(33) C(12) 2H3 C(13) +2H2  E − − − , decreasing, i.e. Tr( (ρ)) 1 for all density matrix operators B2 C +2H3 C C C 2H1  (12) − (11) − (33) (23) −  ρ. This condition canE be expressed≤ as an operator inequality  B3 C(13) 2H2 C(23) +2H1 C(11) C(22)   − − −  for Aj. The operators Aj must satisfy where m 1 1 A†Aj I. Bk = εijk(ImCij ) ,C(kl) = (Ckl + Clk). j ≤ −4 8 Xj=1 If the matrix Ckl and Hamilton operator H are not time- The normalized post-dynamics system state is defined by (15). dependent, then equation (14) has a solution The map (15) is nonlinear trace-preserving map. If the linear quantum operation is trace-preserving Tr( (ρ)) = 1,then 3 E E Pµ(t)= µν (t, t )Pν (t ), E 0 0 m Xν=0 Aj†Aj = I. j where the matrix µν is X=1 E 10 0 0 Notice that a trace-preserving quantum operation (ρ)=AρA† E T R R R must be a unitary transformation (A†A = AA† = I). =  1 11 12 13  . E T2 R21 R22 R23 The example of nonunitary dynamics is associated with    T3 R31 R32 R33  the measurement of quantum system. The system being mea-   sured is no longer a closed system, since it is interacting with The matrices T and R of the matrix µν are defined by the measuring device. The usual way [2] to describe a mea- E surement (von Neumann measurement) is a set of projectors ∞ n 1 τA 1 τ − n 1 P onto the pure state space of the system such that T =(e I)(τA)− B = A − , k − n! nX=0 PkPl = δklPk ,Pk† = Pk , Pk = I. Xk ∞ τ n R = eτA = An, n! The unnormalized state of the system after the measurement nX=0 is given by where τ = t t0 and elements of the matrix A are − k(ρ)=PkρPk. 1 1 E Akl =2Hmεkml + (Ckl + Clk) Cδkl. 8 − 4 The probability of this measurement result is given by

If Ckl is a real matrix, then all Tk =0,wherek =1, 2, 3. p(k)=Tr( k(ρ)). E The normalization condition, p k for all density V.4 Quantum operation k ( )=1 matrix operators, is equivalentP to the completeness condition Unitary evolution (10) is not the most general type of state k Pk = I. If the state of the system before the measure- change possible for quantum systems. A most general state Pment was ρ, than the normalized state of the system after the change of a quantum system is a positive map which is measurement is called a quantum operation or superoperator. For theE concept 1 ρ0 = p− (k) k(ρ). of quantum operations see [68, 69, 70, 71, 4]. In the formal- E ism of quantum operations the final (output) state ρ0 is related to the initial (output) state ρ by a map VI Quantum four-valued logic gates (ρ) ρ ρ0 = E . (15) → Tr( (ρ)) In this section we consider some properties of four-valued E logic gates. We connect quantum four-valued logic gates with The trace in the denominator is induced in order to preserve unitary two-valued logic gates and quantum operations by the the trace condition, Tr(ρ0)=1. In general case, this denom- generalized computational basis. inator leads to the map is nonlinear, where the map is a E linear positive map. VI.1 Generalized quantum gates The quantum operation usually considered as a completely positive map [66]. TheE most general form for com- Quantum operations can be considered as generalized quan- pletely positive quantum operation is tum gates act on general (mixed) states. Let us define a quan- E m tum 4-valued logic gates. Definition Quantum four-valued logic gate is a superopera- (ρ)= AjρA†. E j ˆ (n) Xj=1 tor on Liouville space which maps a density matrix E H

8 operator ρ) of n-ququats to a density matrix operator ρ0) of In general case, the linear trace-decreasing superoperator is n-ququats.| | not a quantum four-valued logic gate, since it can be not trace- A generalized quantum gate is a superoperator ˆ which preserving. The generalized quantum gate can be deﬁned as maps density matrix operator ρ) to density matrixE operator nonlinear trace-preserving gate ˆ by | N ρ0).Ifρ is operator of density matrix, then ˆ(ρ) should also | E ˆ be a density matrix operator. Any density matrix operator is 1 (ρ) ˆ ρ)=(ˆ ρ)I ˆ ρ)− or ˆ (ρ)= E , self-adjoint (ρ†(t)=ρ(t)), positive (ρ(t) > 0) operator with N| E| |E| N Tr( ˆ(ρ)) unit trace (Trρ(t)=1). Therefore we have some require- E ments for superoperator ˆ. where ˆ is a linear completely positive trace-decreasing su- E The requirements for a superoperator ˆ to be a general- peroperator.E E ized quantum gate are as follows: In the generalized computational basis the gate ˆ can be E 1. The superoperator ˆ is real superoperator, i.e. ˆ(A) † = represented by E E † N 1 N 1 ˆ(A†) for all A or ˆ(ρ) = ˆ(ρ). Real superoperator ˆ 1 − − E E E E ˆ = µν σµ)(σν . E 2n E | | maps self-adjoint operator ρ into self-adjoint operator ˆ(ρ): µX=0 νX=0 ( ˆ(ρ)) = ˆ(ρ). E † n 2.1.E The gateE ˆ is a positive superoperator, i.e. ˆ maps posi- where N =4 , µ and ν are 4-valued representation of E ˆ 2 E tive operators to positive operators: (A ) > 0 for all A =0 N 1 E 6 µ = µ14 − + ... + µN 14+µN , or ˆ(ρ) 0. − 2.2.E We≥ have to assume the superoperator ˆ to be not merely E positive but completely positive. The superoperator ˆ is com- N 1 ν = ν14 − + ... + νN 14+νN , pletely positive map of Liouville space, i.e. the positivityE is − (n) remained if we extend the Liouville space by adding ˆ H ˆ(m) σµ = σµ1 ... σµm , more qubits. That is, the superoperator I must be ⊗ ⊗ positive, where Iˆ(m) is the identity superoperatorE⊗ on some (m) µi,νi 0, 1, 2, 3 and µν are elements of some matrix. Liouville space . ∈{ } E 3. The superoperatorH ˆ is trace-preserving map, i.e. E VI.2 General quantum operation as four-valued

(I ˆ ρ)=(ˆ†(I) ρ)=1 or ˆ†(I)=I. logic gates |E| E | E ˆ Proposition 1 In the generalized computational basis µ) any 3.1. The superoperator is a convex linear maponthesetof | density matrix operators,E i.e. linear two-valued logic quantum operation can be represented as a quantum four-valued logic gate ˆEdeﬁned by E ˆ( λsρs)= λs ˆ(ρs), E E N 1 N 1 Xs Xs − − ˆ = µν µ)(ν , E E | | µX=0 Xν=0 where all λs are 0 <λs < 1 and s λs =1. Note that any convex linear map of density matrix operators can be P where uniquely extended to a linear map on Hermitian operators. Any linear completely positive superoperator can be repre- 1 ˆ µν = n Tr σµ (σν ) , sented by E 2 E m and σµ = σµ1 ... σµn . ˆ Lˆ Rˆ . ⊗ ⊗ = Aj A† E j Xj Proof. The state ρ(t) in the generalized computational basis µ) has the form If ˆ is trace-preserving superoperator, then | E N 1 − I ˆ ρ or ˆ I I. ρ(t)) = µ)ρµ(t) , ( )=1 †( )= | | |E| E µX=0

3.2. The restriction to linear gates is unnecessary. Let us con- n sider ˆ is a linear superoperator which is not trace-preserving. where N =4 and E This superoperator is not a quantum gate. Let (I ˆ ρ)= 1 ˆ |E| ρµ(t)=(µ ρ(t)) = Tr(σµρ(t)). Tr( (ρ)) is a probability that the process represented by the | √2n superoperatorE ˆ occurs. Since the probability is nonnegative and never exceedE 1, it follows that the superoperator ˆ is a The quantum operation deﬁne a four-valued logic gate by E trace-decreasing superoperator: E N 1 − 1 ˆ ˆ ρ(t)) = ˆt ρ)= t(ρ)) = t(σν )) ρν (t0). 0 (I ρ) 1 or †(I) I. | E | |E |E √ n ≤ |E| ≤ E ≤ νX=0 2

9 Then Proof.

N 1 m m − 1 1 1 (µ ρ(t)) = (σµ t(σν )) ρν (t0). µν = Tr σµAjσν A† = (A†σµ σν A†). | |E 2n E 2n j 2n j | j νX=0 Xj=1 Xj=1 Finally, we obtain m m N 1 − 1 1 µν∗ = (A†σµ σν A†)∗ = (σν A† A†σµ)= ρµ(t)= µν ρν (t0), E 2n j | j 2n j| j E Xj=1 Xj=1 νX=0 where m m 1 1 1 1 = (σ (σ )) = Tr σ (σ ) . = Tr Aj σν Aj†σµ = Tr σµAj σν Aj† = µν . µν n µ t ν n µ t ν 2n 2n E E 2 |E 2 E Xj=1 Xj=1 This formula deﬁnes a relation between general quantum op- n n Proposition 3 Any real matrix µν associated with linear eration and the real 4 4 matrix µν of four-valued logic E × E (trace-preserving) quantum four-valuedE logic gates (18) has gate ˆ. E ˆ Four-valued logic gates in the matrix representation are δ . n n E 0ν = 0ν represented by 4 4 matrices µν . The matrix µν of the E × E E gate ˆ is Proof. E ... a E00 E01 E0 1 1  10 11 ... 1a  = Tr σ (σ ) = Tr (σ ) = = E E E , 0ν n 0 ν n ν E ...... E 2 E 2 E    a0 a1 ... aa   E E E  m m where a = N 1=4n 1. 1 1 Tr A σ A† Tr A†A σ − − ˆ = n j ν j = n ( j j ) ν = In matrix representation the gate maps the state ρ)= 2 2 N 1 N E1 | Xj=1 Xj=1 − ν)ρν to the state ρ0)= − µ)ρ0 by ν=0 | | µ | µ P P N 1 − 1 ρ0 = µν ρν . (16) = Trσν = δ0ν . µ E 2n νX=0 The general linear n-ququats quantum gate has the form: n where ρ0 = ρ0 =1/√2 . It can be written in the form 10 0... 0 ρ0 00 01 ... 0a ρ0 E E E  T1 R11 R12 ... R1 N 1   ρ10   10 11 ... 1a   ρ1  − = E E E . = T2 R21 R22 ... R2 N 1 ...... E  −         ......   ρa0   a0 a1 ... aa   ρa       E E E     TN 1 RN 11 RN 12 ... RN 1 N 1   − − − − −  Since P = √2nρ and P = √2nρ , it follows that µ µ µ0 µ0 Completely positive condition leads to some inequalities [87, representation (16) for linear gate ˆ is equivalent to 88, 89] for matrix elements µν . E E N 1 In general case, linear quantum 4-value logic gate acts on − P 0 = µν Pν . (17) 0) by µ E | νX=0 N 1 It can be written in the form − ˆ 0) = 0) + Tk k). E| | | P 0 ... a P kX=1 0 E00 E01 E0 0  P 1   10 11 ... 1a   P1  = . For example, single ququat quantum gate acts by ...0 E...E ...... E ......        Pa0   a0 a1 ... aa   Pa  ˆ 0) = 0) + T 1) + T 2) + T 3).    E E E    E| | 1| 2| 3| where P =1. Note that if we use different matrix represen- 0 If all T ,k=1, ..., N 1 is equal to zero, then ˆ 0) = 0). tation of state ρ) or ρ] we can use identical matrices repre- k The linear quantum gates− with T =0conserve theE| maximally| sentation for gate| ˆ. | E mixed state 0] invariant. | Proposition 2 In the generalized computational basis µ) the Deﬁnition A quantum four-valued logic gate ˆ is called uni- | E matrix µν of general quantum four-valued logic gate tal gate or gate with T =0if maximally mixed state 0] is E invariant under the action of this gate: ˆ 0] = 0]. | m The output state of a linear quantumE| four-valued| logic ˆ Lˆ Rˆ (18) = Aj A† E j gate ˆ is 00...0] if and only if the input state is 00...0].If Xj=1 ˆ 00...E 0] =| 00...0],then ˆ is not unital gate. | E| 6 | E is real ∗ = µν . Eµν E

10 n Proposition 4 The matrix µν of linear trace-preserving n- where N =4 . The gate matrix (0,R) has the form E E ququats gate ˆ is an element of group TGL(4n 1, R) which 10 0... 0 is a semidirectE product of general linear group GL− (4n 1, R) n −  0 R11 R12 ... R1 N 1  and translation group T (4 1, R). − − (0,R)= 0 R21 R22 ... R2 N 1 . E  −   ......  Proof. This proposition follows from proposition 3. Any   n 0 R R ... R element (gate matrix µν ) of group TGL(4 1, R) can be  N 11 N 12 N 1 N 1  E −  − − − −  represented by In matrix representation the linear trace-preserving gates with T =0can be described by group GL(4n 1, R) which define 10 (T,R)= , −(n) E TR a set of all linear transformations of by (16) or (17). The group GL(4n 1, R) has (4n H1)2 independent one- n − n − where T is a column with 4 1 elements, 0 is a line with parameter subgroups GLkl(4 1, R) of one-parameter gates n − n n ˆ − 4 1 zero elements, and R is a real (4 1) (4 1) matrix (kl)(t) such that R −GL(4n 1, R).IfR is orthogonal−(4n× 1) − (4n 1) E matrix∈ (RT R−= I), then we have motion group− [94,× 95,− 96]. ˆ(kl)(t)= 0)(0 + t k)(l . E | | | | The group multiplication of elements (T,R) and (T ,R) 0 0 Generators are defined by is defined by E E d ˆ ˆ(kl) (T,R) (T 0,R0)= (T + RT 0,RR0). Hkl = (t) = k)(l . E E E dtE t=0 | | n In particular, we have where k, l =1, 2, ..., 4 1. The generators Hˆkl of the one- parameter subgroup GL − n , R are represented by n 1 kl(4 1 ) 4 (T,R)= (T,I) (0,R) , (T,R)= (0,R) (R− T,I). n − × E E E E E E 4 matrix Hkl with elements n n where I is unit (4 1) (4 1) matrix. (Hkl)µν = δµkδνl. Therefore any linear− × quantum− gate can be decompose on unital gate and translation gate. It allows to consider two The set of superoperators Hˆkl is a basis of Lie algebra { } types of linear trace-preserving gates: gl(4n 1, R) such that − 1) Translation gates ˆ(T ) defined by matrices (T,I). ˆ ˆ ˆ ˆ E ˆ(T =0) E [Hij , Hkl]=δjkHil δilHjk. 2) Unital quantum gates with the matrices (0,R). − Translation gate ˆ(T )Eis E E VI.3 Decomposition for linear quantum gates N 1 N 1 − − (T ) ˆ = µ)(µ + Tk k)(0 , Let us consider the n-ququats linear gate E | | | | µX=0 Xk=1 N 1 N 1 N 1 − − − ˆ = 0)(0 + Tµ µ)(0 + Rµν µ)(ν , (19) has the matrix E | | | | | | µX=1 µX=1 νX=1 100... 0 where N =4n. The gate matrix (T,R) is an element of Lie  T1 10... 0  group TGL(4n 1, R). The matrixE R is an element of Lie (T,I)= T2 01... 0 . E   n − R  ......  group GL(4 1, ).   Theorem 1. (Singular− Valued Decomposition for Matrix)  TN 1 00... 1  − T   Any real matrix R can be written in the form R = 1D 2 , n U U One-parameter subgroups T (4 1, R) of n-ququats transla- where 1 and 2 are real orthogonal (N 1) (N 1) − U U − × − tion gates consist of one-parameters 4n 1 gates matrices and D = diag(λ1, ..., λN 1) is diagonal (N 1) − − − × (N 1) matrix such that λ1 λ2 ... λN 1 0. N 1 − ≥ ≥ ≥ − ≥ − Proof. This theorem is proved in [90, 91, 92, 85]. ˆ(T,k)(t)= µ)(µ + t k)(0 , ˆ(T =0) E | | | | Let us consider the unital gates defined by (19), µ=0 E X where all Tµ =0. Theorem 2. (Singular Valued Decomposition for Gates) where t is a real parameter and k =1, 2, ..., 4n 1. Genera- Any unital linear gate ˆ defined by (19) with all T can tors of the gates are defined by − µ =0 be represented by E

d (T,k) Hˆk = ˆ (t) = k)(0 . ˆ = ˆ1 Dˆ ˆ2, dtE t=0 | | E U U The quantum n-ququats unital gate can be represented by where ˆ and ˆ are unital orthogonal quantum gates U1 U2 N 1 N 1 − − N 1 N 1 ˆ(T =0) − − = 0)(0 + Rkl k)(l , ˆ (i) E | | | | i = 0)(0 + µν µ)(ν , (20) kX=1 Xl=1 U | | U | | µX=1 νX=1

11 Dˆ is a unital diagonal quantum gate, such that Theorem 4. (Polar Decomposition for matrix) Any real (N 1) (N 1) matrix R can be written in the N 1 − × − − form R = S or R = S0 0,where and 0 are orthogonal Dˆ = 0)(0 + λµ µ)(µ , (21) U U U U | | | | (N 1) (N 1) matrices and S and S0 are symmetric µX=1 − × − T (N 1) (N 1) matrices such that S = √R R, S0 = −T × − where λµ 0. √RR . Proof. The≥ proof of this theorem can be easy realized in ma- Proof. This theorem is proved in [85]. trix representation by using theorem 1. Theorem 5. (Polar Decomposition for gates) In general case, we have the following theorem. Any linear four-valued logic gate (19) can be written in the (S) (S0) Theorem 3. (Singular Valued Decomposition for Gates) form ˆ = ˆ ˆ or ˆ = ˆ ˆ0,where E UE E E U Any linear quantum four-valued logic gate (19) can be repre- ˆ and ˆ0 are orthogonal gates (20). U U sented by ˆ(S0) and ˆ(S0) are symmetric gates (22). Proof.E TheE proof of this theorem can be easy realized in ma- (T ) ˆ = ˆ ˆ1 Dˆ ˆ2, trix representation by using Theorem 4. E E U U where VI.4 Unitary two-valued logic gates as orthog- ˆ1 and ˆ2 are unital orthogonal quantum gates (20). UDˆ is a unitalU diagonal quantum gate (21). onal four-valued logic gates ˆ(T ) is a translation quantum gate, such that Let us rewrite the representation (2) for the mixed state ρ(t)) E | N 1 N 1 using generalized computational basis in the form − − ˆ(T ) = 0)(0 + µ)(µ + Tµ µ)(0 . N 1 E | | | | | | − µ=1 µ=1 X X ρ(t)) = µ)ρµ(t) , | | µX=0 Proof. The proof of this theorem can be easy realized in matrix representation by using Proposition 4 and Theorem 1. where As a result we have that any trace-preserving gate can be 1 ρµ(t)=(µ ρ(t)) = Tr(σµρ(t)). realized by 3 types of gates: (1) unital orthogonal quantum | √2n gates ˆ with matrix SO(4n 1, R); (2) unital diagonal U U∈ − n n n quantum gate Dˆ with matrix D D(4 1, R); (3) nonunital Note that ρ0(t)=(0ρ(t)) = 1/√2 Trρ(t)=1/√2 for | translation gate ˆ(T ) with matrix∈ (T ) −T (4n 1, R). all cases. E E ∈ − Proposition 5 If the quantum operation has the form Proposition 6 In the generalized computational basis any uni- E tary two-valued logic gate U can be considered as a quantum m four-valued logic gate: (ρ)= AjρAj†, E N 1 N 1 Xj=1 − − ˆ = µν µ)(ν , (23) U U | | µX=0 νX=0 where A is a self-adjoint operator (Aj† = Aj), then quantum four-valued logic gate ˆ is described by symmetric matrix where is a real matrix such that E µν µν = νµ. U E E 1 µν = n Tr σν UσµU † . (24) Proof. If Aj† = Aj,then U 2

m Proof. Let us consider unitary two-valued logic gate U.Us- 1 ing equation (10), we get µν = Tr(σµAjσν Aj)= E √ n 2 Xj=1 ρ(t)) = ˆt ρ(t )). | U | 0

m Then 1 = Tr(σν Aj σµAj)= νµ. N 1 √2n E − Xj=1 (µ ρ(t)) = (µ ˆt ρ(t )) = (µ ˆt ν)(ν ρ(t )). | |U | 0 |U | | 0 νX=0 This gate is trace-preserving if µ0 = 0µ = δµ0. The symmetric n-ququat linearE (trace-preserving)E quan- Finally, we obtain tum gate has the form N 1 − ρµ(t)= µν (t, t )ρν (t ), N 1 N 1 U 0 0 (S) − − νX=0 ˆ = 0)(0 + Sµν µ)(ν . (22) E | | | | µX=1 νX=1 where where S = S and this gate is unital (T =0for all k). 1 µν νµ k µν (t, t )=(µ ˆt ν)= (σµ t(σν )) = U 0 |U | 2n |U

12 1 ˆ Tr σ U t, t σ U t, t . Proposition 9 If † is adjoint superoperator for linear trace- = n µ ( 0) ν †( 0) E 2 preserving gate ˆ, then matrices of the gates are connected E T by transposition † = : This formula defines a relation between unitary quantum two- E E valued logic gates U and the real 4n 4n matrix . × U ( †)µν = νµ. E E Proposition 7 Any four-valued logic gate associated with uni- Proof. Using tary 2-valued logic gate by (23,24) is unital gate, i.e. gate matrix defined by (24) has µ0 = 0µ = δµ0. m m U U U ˆ Lˆ Rˆ , ˆ Lˆ Rˆ , = Aj A† † = A† Aj E j E j Proof. Xj=1 Xj=1 1 1 1 we get µ = Tr σµUσ U † = Tr σµUU† = Trσν . U 0 2n 0 2n 2n m 1 Using Trσµ = δµ0 we get µ0 = δµ0. µν = Tr(σµAjσν A†), U E 2n j Let us denote the gate ˆ associated with unitary two- Xj=1 valued logic gate U by ˆ(U).U E m Proposition 8 If U is unitary two-valued logic gate, then in 1 ( †)µν = Tr(σµA†σν Aj)= the generalized computational basis a quantum four-valued E 2n j Xj=1 logic gate ˆ = ˆ(U) associated with U is represented by orthogonalU matrixE (U): E m (U) (U) T (U) T (U) 1 ( ) =( ) = I. (25) = Tr(σν AjσµA†)= νµ. E E E E 2n j E Xj=1 Proof. Let ˆ(U) is defined by E Obviously, if ˆ(U) ˆ(U †) ρ)= UρU†) , ρ)= U †ρU). N 1 N 1 E | | E | | − − ˆ = µν µ)(ν , If UU† = U †U = I,then E E | | µX=0 νX=0 ˆ(U) ˆ(U †) = ˆ(U †) ˆ(U) = I.ˆ E E E E then N 1 N 1 In the matrix representation we have − − ˆ† = νµ µ)(ν . N 1 N 1 E E | | − − µX=0 Xν=0 (U) (U †) (U †) (U) = = δµν , Eµα Eαν Eµα Eαν α=0 α=0 X X Proposition 10 If ˆ† ˆ = ˆˆ† = Iˆ,then ˆ is orthogonal gate, i.e. T = ETE= I. EE E i.e. (U †) (U) = (U) (U †) = I. Note that E E EE E E E E ˆ ˆ ˆ 1 1 Proof. If † = I,then (U †) (U) E E µν = n Tr σµU †σν U = n Tr σν UσµU † = νµ , E 2 2 E N 1 − (µ ˆ† α)(α ˆ ν)=(µ Iˆ ν), i.e. (U †) =( (U))T . Finally, we obtain (25). |E | |E| | | αX=0 InE matrix representationE orthogonal gates can be described by group SO(4n 1, R) which is a set of all linear transfor- i.e. − (n) N 1 N 1 2 mations of ρ0µ = ν=0− µν ρν such that µ=0− ρµ = N 1 H E n R − const and det[ µν ]=1P. The group SO(4 P 1, ) has ( †)µα αν = δµν . n n E1 − E E (4 1)(2 4 − 1) independent one-parameter subgroups αX=0 − n − SOkl(4 1, R) of one-parameter orthogonal gates which are − Using proposition 9 we have (kl) ˆ (α)= µ)(µ + cosα k)(k + l)(l + N 1 E | | | | | | − T µX=k,l ( ˆ )µα αν = δµν , 6 E E αX=0

+sinα l)(k k)(l . i.e. T = I. | |−| | E E If µν is real orthogonal matrix, then E This gate deﬁnes rotation in the ﬂat (k, l). Let us note that the n R N 1 generators of the one-parameter subgroup SOkl(4 1, ) − 2 − ( µν ) =1. are represented by antisymmetric (4n 1) (4n 1) matrix E − × − νX=0 Xkl with elements Therefore all elements of orthogonal gate matrix never ex- X δ δ δ δ . ( kl)µν = µk νl µl νk ceed 1, i.e. µν 1. − |E |≤

13 Note that n-qubit unitary two-valued logic gate U is an where α, θ and β are Euler angles. The correspondent 4 4- element of Lie group SU(2n). The dimension of this group matrix (α, θ, β) of four-valued logic gate has the form × is equal to dim SU(2n)=(2n)2 1=4n 1. The matrix of U (1) (2) (1) n-ququat orthogonal linear gate ˆ−= ˆ(U) −can be considered (α, θ, β)= (α) (θ) (β), U U U U as an element of Lie group SO(4Un 1)E. The dimension of this n − n n 1 where group is equal to dim SO(4 1) = (4 1)(2 4 − 1). For example, if n ,then− − · − =1 (1) 1 (α)= Tr σµU (α)σν U †(α) , Uµν 2 1 1 dim SU(21)=3,dimSO(41 1) = 3. −

If n =2,then (2) 1 µν (θ)= Tr σµU2(θ)σν U2†(θ) , U 2 dim SU(22)=15,dimSO(42 1) = 105. − Finally, we obtain Therefore not all orthogonal 4-valued logic gates for mixed and pure states are connected with unitary 2-valued logic gates 10 0 0  0cosα sin α 0  for pure states. (1)(α)= − , U 0sinα cos α 0    00 0 1 VI.5 Single ququat orthogonal gates   Let us consider single ququat 4-valued logic gate ˆ associ- 1000 U  0cosθ 0sinθ  ated with unitary single qubit 2-valued logic gate U. (2)(θ)= , U 0010    0 sin θ 0cosθ  Proposition 11 Any single-qubit unitary quantum two-valued  −  logic gate can be realized as the product of single ququat sim- where ple rotation gates ˆ(1)(α), ˆ(2)(θ) and ˆ(1)(β) defined by U U U 0 α<2π, 0 θ π, 0 β 2π. ≤ ≤ ≤ ≤ ≤ ˆ(1)(α)= 0)(0 + 3)(3 +cosα 1)(1 + 2)(2 + U | | | | | | | | Using U(α, θ +2π, β)= U(α, θ, β), we get that 2-valued logic gates U(α, θ, β) and−U(α, θ +2π, β) map into single 4-valued logic gate (α, θ, β). The back rotation 4-valued +sinα 2)(1 1)(2 , U | |−| | logic gate is defined by the matrix 1 − (α, θ, β)= (2π α, π θ, 2π β) . ˆ(2)(θ)= 0)(0 + 2)(2 +cosθ 1)(1 + 3)(3 + U U − − − U | | | | | | | | The simple rotation gates ˆ(1)(α), ˆ(2)(θ), ˆ(1)(β) are defined by matrices ˆ(1)(α),Uˆ(2)(θ) andU ˆ(1)(βU). U U U +sinθ 1)(3 3)(1 , Let us introduce simple reflection gates by | |−| | ˆ(1) = 0)(0 1)(1 + 2)(2 + 3)(3 , where α, θ and β are Euler angles. R | |−| | | | | |

Proof. Let us consider a general single qubit unitary gate ˆ(2) = 0)(0 + 1)(1 2)(2 + 3)(3 , [80]. Every unitary one-qubit gate U can be represented by R | | | |−| | | | 2 2-matrix × ˆ(3) = 0)(0 + 1)(1 + 2)(2 3)(3 . iασ /2 iθσ /2 iβσ /2 U(α, θ, β)=e− 3 e− 2 e− 3 = R | | | | | |−| | Proposition 12 Any single ququat linear gate ˆ deﬁned by i(α/2+β/2) i(α/2 β/2) E e− cos θ/2 e− − sin θ/2 orthogonal matrix : T = I can be realized by = i(α/2 β/2) − i(α/2+β/2) , E EE e − sin θ/2 e cos θ/2 simple rotation gates ˆ(1) and ˆ(2). i.e. • U U U(α, θ, β)=U (α)U (θ)U (β), inversion gate ˆ deﬁned by 1 2 1 • I where ˆ iα/2 = 0)(0 1)(1 2)(2 3)(3 . e− 0 I | |−| |−| |−| | U (α)= , 1 0 eiα/2 Proof. Using Proposition 11 and cos θ/2 sin θ/2 U (θ)= , (3) (1) (2) (2) (1) (1) (1) 2 sin θ/2cos− θ/2 ˆ = ˆ ˆ, ˆ = ˆ ˆ , ˆ = ˆ ˆ ˆ, R U I R U I R U U I iβ/2 e− 0 we get this proposition. U (β)= , 1 0 eiβ/2

14 Example 1. In the generalized computational basis the Pauli Proposition 13 A nonlinear four-valued logic gate ˆ for von N 1N matrices as two-valued logic gates are the four-valued logic Neumann measurement (27) of the state ρ = − α)ρα is α=0 | gates with diagonal 4 4 matrix. The gate I = σ0 is deﬁned by P × r 3 1 (σ0) ˆ (k) ˆ = µ)(µ = I,ˆ = µν µ)(ν , U | | N p(k)E | | µX=0 Xk=1

(σ0) where i.e. µν =(1/2)Tr(σµσν )=δµν . U For the unitary two-valued logic gates are equal to the N 1 − (k) 1 n (k) Pauli matrix σk,wherek 1, 2, 3 , we have quantum four- = Tr(σµPkσν Pk),p(k)=√2 ρα . ∈{ } Eµν 2n E0α valued logic gates αX=0 (28) 3 ˆ(σk) (σk) = µν µ)(ν , ˆ U U | | Proof. The trace-decreasing superoperator k is deﬁned by µ,νX=0 E

ρ) ρ0)= k ρ)= PkρPk). with the matrix | →| E | |

(σk) The superoperator ˆ has the form ˆk = LˆP RˆP .Then =2δµ δν +2δµkδνk δµν . (26) k k Uµν 0 0 − E E N 1 N 1 Example 2. In the generalized computational basis the uni- − − (k) ρ0 =(µ ρ0)=(µ ˆk ρ)= (µ ˆk ν)(ν ρ)= ρν , tary NOT gate (”negation”) of two-valued logic µ | |E | |E | | Eµν νX=0 Xν=0 01 X = 0 >< 1 + 1 >< 0 = σ = , where | | | | 1 10 (k) 1 =(µ ˆk ν)= Tr(σµPkσν Pk). is represented by quantum four-valued logic gate Eµν |E | 2n ˆ(X) = 0)(0 + 1)(1 2)(2 3)(3 , The probability that process represented by ˆk occurs is U | | | |−| |−| | E N 1 i.e. 4 4 matrix is − × n n (k) p(k)=Tr( ˆk(ρ)) = (I ˆ ρ)=√2 ρ0 = √2 ρα . E |E| 0 E0α 10 0 0 αX=0 X  01 0 0 ( ) = . U 00 10 If  −   00 0 1  N 1  −  − αρα =0, Example 3. The Hadamar two-valued logic gate E0 6 αX=0 1 H = (σ1 + σ3) then the matrix for nonlinear trace-preserving gate ˆ is √2 N N 1 − can be represented as a four-valued logic gate by n 1 µν = √2 ( αρα)− µν . N E0 E ˆ(H) = 0)(0 2)(2 + 3)(1 + 1)(3 , αX=0 E | |−| | | | | | Example. Let us consider single ququat projection operator with

(H) 1 = δµ δν δµ δν + δµ δν + δµ δν . P0 = 0 >< 0 = (σ0 + σ3). Eµν 0 0 − 2 2 3 1 1 3 | | 2 VI.6 Measurements as quantum 4-valued logic Using formula (28) we derive

gates (0) 1 µν = Tr σµ(σ0 + σ3)σν (σ0 + σ3) = It is known that superoperator ˆ of von Neumann measure- E 8 ment is deﬁned by E 1 r = δ δ + δ δ + δ δ + δ δ , 2 µ0 ν0 µ3 ν3 µ3 ν0 µ0 ν3 ˆ ρ)= PkρPk) , (27) E| | kX=1 i.e. 1/2001/2 where Pk k =1, .., r is a (not necessarily complete) se- (0)  0000 quence{ of orthogonal| projection} operators on (n). = . H E 0000 Let Pk are projectors onto the pure state k>which de-   |  1/2001/2  ﬁne usual computational basis k> ,i.e.Pk = k>

15 The linear trace-decreasing superoperator for von Neumann Note that measurement projector 0 >< 0 onto pure state 0 > is ( ) ( ) ( ) ( ) ( ) | | | ˆ M ˆ M = ˆ M , ( ˆ M )† = ˆ M . 1 E E E E E ˆ(0) = 0)(0) + 3)(3 + 0)(3 + 3)(0 . E 2 | | | | | | | Let ˆ be the restriction of ˆ to the subspace EM E M Example. For the projection operator m ˆ (ρ)= AjP ρP Aj† . (30) 1 EM M M P = 1 >< 1 = (σ σ ) Xj=1 1 | | 2 0 − 3 ˆ ˆ Using formula (28) we derive Notice that (ρ)= (ρ) if ρ lies wholly in .Note, that the adjointEM superoperatorE for trace-decreasingM quantum 1 operation is generally not a quantum operation, since it can be (1) δ δ δ δ δ δ δ δ . µν = µ0 ν0 + µ3 ν3 µ3 ν0 µ0 ν3 trace-increasing, but it is always a completely positive map. E 2 − − Equation (29) is equivalent to the requirement that super- ˆ(1) The linear superoperator for von Neumann measurement operator † ρ be a positive multiple of identity operation E ( ) projector onto pure state 1 > is on . ThisEM requirement can be formulated as theorem. | TheoremM 7. 1 ˆ(1) = 0)(0) + 3)(3 0)(3 3)(0 , A necessary and sufficient condition for reversibility of linear E 2| | |−| |−| | superoperator ˆ on the subspace is i.e. E M 1/200 1/2 ˆM ˆ† ˆˆM = γ ˆM. 0000− E E EE E (1) =   . E 0000    1/200 1/2  Proof. For the proofs we refer to [99].  −  The superoperators ˆ(0) and ˆ(1) are not trace-preserving. The probabilities that processesE E represented by superopera- VII Classical four-valued logic classi- tors ˆ(k) occurs are E cal gates 1 1 p(0) = (ρ0 + ρ3) ,p(1) = (ρ0 + ρ3). √2 √2 Let us consider some elements of classical four-valued logic. For the concept of many-valued logic see [72, 73, 74, 75, 76]. VI.7 Reversible quantum 4-valued logic gate VII.1 Elementary classical gates In the paper [97], Mabuchi and Zoller have shown how a measurement on a quantum system can be reversed under appro- A classical four-valued logic gate is called a function g(x1, ..., xn) priate conditions. In the papers [98, 99, 100, 12] was consid- if following conditions hold: ered necessary and sufficient conditions for general quantum all xi 0, 1, 2, 3 ,wherei =1, ..., n. operations to be reversible. • ∈{ } Let us consider quantum operation on a subspace of g(x1, ..., xn) 0, 1, 2, 3 . the total state space. E M • ∈{ } Theorem 6. A quantum operation It is known that the number of all classical logic gates n E with n-arguments x , ..., x is equal to 44 . The number of m 1 n classical logic gates g(x) with single argument is equal to (ρ)= Aj ρA† 1 E j 4 . Xj=1 4 = 256 is reversible on subspace if and only if there exists a pos- Single argument classical gates M itive matrix M such that x x 2x x x I0 I1 I2 I3 0 ∼3 0 ♦0 1 3 0 0 0 P Ak† Aj P = MjkP . (29) M M M 1 2 0 3 2 0 3 0 0 2 1 0 3 3 0 0 3 0 where P is a projector onto subspace . The trace of M M M 3 0 3 3 0 0 0 0 3 m 2 Mjj = µ Single argument classical gates Xj=1 x 0 1 2 3 g1 g2 g3 0 0 1 2 3 3 0 1 is the constant value of Tr( (ρ)) on . E M 1 0 1 2 3 0 1 1 Proof. This result was proved in [12, 98, 99]. 2 0 1 2 3 1 3 2 ( ) Let ˆ M is projection superoperator defined by E 3 0 1 2 3 2 2 3

ˆM(ρ)=P ρP . E M M

16 The number of classical logic gates g(x1,x2) with two- Note that the Luckasiewicz negation is satisfied arguments is equal to ( x)=x, (x1 x2)=( x1) ( x2). 2 ∼ ∼ ∼ ∧ ∼ ∨ ∼ 44 =416 = 42949677296. The shift x for x is not satisfied usual negation rules: Let us write some of these gates. x = x, x x = x x . 6 1 ∧ 2 6 1 ∨ 2 Two-arguments classical gates The analog of disjunction normal form of the n-arguments (x1,x2) V4 V4 4-valued logic gate is (0;0) ∧0 ∨0 1 ∼2 g(x , ..., xn)= Ik (x ) ... Ik g(k , .., kn). (0;1) 0 1 2 1 1 1 1 ∧ ∧ n ∧ 1 (0;2) 0 2 3 0 (k1,...,k_ n) (0;3) 0 3 0 3 (1;0) 0 1 2 1 VII.2 Universal classical gates (1;1) 1 1 2 1 (1;2) 1 2 3 0 Let us consider universal sets of universal classical gates of (1;3) 1 3 0 3 four-valued logic. Theorem 8. (2;0) 0 2 3 0 , , , ,I ,I ,I ,I ,x x ,x x (2;1) 1 2 3 0 The set 0 1 2 3 0 1 2 3 1 2 1 2 is universal. {x, x x ∧ ∨ } (2;2) 2 2 3 0 The set 1 2 is universal. {V x ∨,x } (2;3) 2 3 0 3 The gate 4( 1 2) is universal. (3;0) 0 3 0 3 Proof. This theorem is proved in [75]. Theorem 9. (3;1) 1 3 0 3 g x (3;2) 2 3 0 3 All logic single argument 4-valued gates ( ) can be gener- (3;3) 3 3 0 3 ated by functions: g (x)=x 1(mod4). Let us define some elementary classical 4-valued logic • 1 − gates by formulas. g (x): g (0) = 0, g (1) = 1, g (2) = 3, g (3) = 2. • 2 2 2 2 2 Luckasiewicz negation: x =3 x. g (x)=1if x =0and g (x)=x if x =0. • ∼ − • 3 3 6 Cyclic shift: x = x +1(mod4). Proof. This theorem was proved by Piccard in [78]. • Functions Ii(x),wherei =0, ..., 3, such that Ii(x)=3 • if x = i and Ii(x)=0if x = i. 6 VIII Quantum four-valued logic gates Generalized conjunction: x1 x2 = min(x1,x2). for classical gates • ∧ Generalized disjunction: x x = max(x ,x ). • 1 ∨ 2 1 2 VIII.1 Quantum gates for single argument clas- Generalized Sheffer function: sical gates • Let us consider linear trace-preserving quantum gates for clas- V4(x1,x2)=max(x1,x2)+1(mod4). sical gates , x, I ,I ,I ,I , 0, 1, 2, 3, g ,g ,g , , 2. ∼ 0 1 2 3 1 2 3 ♦ Commutative law, associative law and distributive law for Proposition 14 Any single argument classical gate g(ν) can the generalized conjunction and disjunction are satisfied: be realized as linear trace-preserving quantum four-valued Commutative law logic gate by • 3 x x = x x ,x x = x x 1 ∧ 2 2 ∧ 1 1 ∨ 2 2 ∨ 1 ˆ(g) = 0)(0 + g(k))(k + E | | | | kX=1 Associative law • 3 3 (x1 x2) x3 = x1 (x2 x3). ∨ ∨ ∨ ∨ +(1 δ ) g(0))(0 (1 δ ) µ)(ν . − 0g(0) | |− − µg(ν) | | µX=0 νX=0 (x1 x2) x3 = x1 (x2 x3). ∧ ∧ ∧ ∧ Proof. The proof is by direct calculation in

Distributive law (g) • ˆ α]= g(α)], E | | x (x x )=(x x ) (x x ). 1 ∨ 2 ∧ 3 1 ∨ 2 ∧ 1 ∨ 3 where ˆ(g) 1 ˆ(g) ˆ(g) x (x x )=(x x ) (x x ). α]= 0) + α) . 1 ∧ 2 ∨ 3 1 ∧ 2 ∨ 1 ∧ 3 E | √2E | E |

17 Examples. 7. The gate g2(x) is 1. Luckasiewicz negation gate is ˆ(g2) = 0)(0 + 1)(1 + 3)(2 + 2)(3 . E | | | | | | | | ˆ(LN) = 0)(0 + 1)(2 + 2)(1 + 3)(0 3)(3 . E | | | | | | | |−| | 1000 g  0100 100 0 ( 2) = . 001 0 E 0001 (LN) =   .   E 010 0  0010      100 1   −  8. The gate g3(x) can be realized by 2. The four-valued logic gate I can be realized by 0 ˆ(g3) = 0)(0 + 2)(2 + 3)(3 + 1)(0 1)(2 1)(3 . E | | | | | | | |−| |−| | 3 ˆ(I0) = 0)(0 + 3)(0 3)(k . 10 0 0 E | | | |− | | g  10 1 1  Xk=1 ( 3) = − − . E 00 1 0   10 0 0  00 0 1   I  00 0 0 ( 0) = . 9. The gate x is E 00 0 0 ♦    1 1 1 1  3 ( )  − − −  ˆ ♦ = 0)(0 + 3)(k . E | | | | 3. The gates Ik(x),wherek =1, 2, 3 is Xk=1 ˆ(Ik ) = 0)(0 + 3)(k . 1000 E | | | | ( )  0000 ♦ = . For example, E 0000    0111 1000   2x x I  0000 10. The gate = is ( 1) = , ∼♦ E 0000 ( )   ˆ ∼♦ = 0)(0 + 3)(3 .  0100 E | | | |   1000 4. The gate x can be realized by 0000 (2) =   . 3 E 0000   ˆ(x) = 0)(0 + 1)(0 + 2)(1 + 3)(2 1)(k .  0001 E | | | | | | | |− | |   Xk=1 Note that quantum gates ˆ(LN), ˆ(I0), ˆ(k), ˆ(g1) are not E E E E 1000 unital gates. x  1 1 1 1  ( ) = − − − . E 0100   VIII.2 Quantum gates for two-arguments clas-  0010   sical gates 5. The constant gates and k , , can be realized by 0 =1 2 3 Let us consider quantum gates for two-arguments classical gates. ˆ(0) = 0)(0 , ˆ(k) = 0)(0 + k)(0 . E | | E | | | | 1. The generalized conjunction x x = min(x ,x ) 1 ∧ 2 1 2 For example, and generalized disjunction x1 x2 = max(x1,x2) can be realized by two-ququat gate with∧T =0: 1000 1000 (1) =   . E 0000   x1 x1 x2  0000 ∨   CD 6. The gate g1(x) can be realized by x2 x1 x2. 3 ∧ ˆ(g1) = 0)(0 + 1)(2 + 2)(3 + 3)(0 3)(k . E | | | | | | | |− | | kX=1 Let us write the quantum gate which realizes the CD gate 1000 in the generalized computational basis by g  0010 ( 1) = . E 0001 N 1 N 1 3   − −  1 1 1 1  ˆ = µν)(µν + 0k) k0) (k0 +  − − −  E | | | −| | Xµ Xν Xk=1

18 3 1 1 = (k 0) + (k k) = =0, + 1k) k1) (k1 + 23) 32) (32 . √ n √ n | −| | | −| | 2 | | 2 6 Xk=2 i.e. Tk =0in the matrix µν . 2. The Sheffer function gate x1,x2] V4(x1,x2), From6 this propositionE we see that single argument classi- V (x ,x )] can be realized by two-ququat| →| gate with T =0∼: 4 1 2 6 cal gate g(x) can be realized by single ququat quantum gate 3 3 with T =0if and only if ˆ(SF) = 00)(00 + 12)(00 12)(µν + 21)(10 + 1. g(0) = 0,or E | | | |− | | | | µX=0 νX=1 2. g(µ)=const,i.e.g(0) = g(1) = g(2) = g(3) = k and k 1, 2, 3 . ∈{ } For example, classical gates x, I0, x, g1 and g3 can not be + 21)(11 + 30)(02 + 30)(20 + 30)(12 + 30)(21 + ∼ | | | | | | | | | | realized by single ququat unital quantum gates. Single argument classical logic gates such that g(0) =0 3 can not be realized by single ququat quantum gates ˆ with6 + 30)(22 + 03)(03 + 03)(13 + 03)(23 + 03)(3µ . T =0. This classical gates can be realized by two-qubitsE | | | | | | | | | | µX=0 unital quantum gates. Let us consider Luckasiewicz negation x =3 x.Ifx2 =0and x1,x2 0, 1, 2, 3 then we can ∼deﬁne quantum− Luckasiewicz6 negation∈{ gate by }

x1 V4(x1,x2)

SF x x 1 ∼ 1 x2 V4(x1,x2). ∼ LN2

x2 x2.

Note that this Sheffer function gate is not unital quantum gate and This gate realizes Luckasiewicz negation for x1: LN2 x1 ˆ(SF) | ⊗ = V4(x1,x2), V4(x1,x2))(x1,x2 . x x x iff x .Ifx , then the two- E 6 | ∼ | 2]= ( 1) 2] 2 =0 2 =0 ququat| gate∼ must⊗ be following VIII.3 Unital quantum gates for single argument classical gates It is interesting to consider a representation for classical gates x1 0 ˆ by linear unital quantum gates ( 0] = 0]). There is a restric- LN tion for representation single argumentE| classical| (4-valued logic) 2 gate by linear quantum four-valued logic gates with T =0 0 0. (all Tk =0). Any unital n-ququat quantum gate has the form

N 1 N 1 − − (T =0) ˆ = 0)(0 + Rµν µ)(ν , E | | | | Let us write the unital quantum four-valued logic gate µX=1 νX=1 which realizes Luckasiewicz negation in generalized compu- i.e. Tk =0for all k and the gate matrix is tational basis by

10 0... 0 ˆ(LN2) = 00)(00 + k3)(k0 + k2)(k1 + E | | | | | |  0 R11 R12 ... R1 N 1  k=1X,2,3 (T =0) − = 0 R21 R22 ... R2 N 1 . E  −   ......     0 RN 11 RN 12 ... RN 1 N 1  + k1)(k2 + k0)(k3 .  − − − −  | | | | Proposition 15 If the single argument classical (4-valued logic) By analogy to realization of Luckasiewicz negation we gate g(x) such that g(0) = k,wherek 1, 2, 3 and exists can derive quantum gates with T =0for classical gates m 1, 2, 3 : g(m)=l,wherel = k, then∈{ there} is no a rep- I (x), x, g and g . resentation∈{ of} this gate by some unital6 quantum four-valued 0 1 3 logic gates ˆ. E VIII.4 Unital quantum gates for two-arguments Proof. If ˆ 0] = k] and ˆ m]= l] ,wherek 1, 2, 3 , classical gates l = k,thenE| | E| | ∈{ } 6 By analogy with Proposition 15 we can proof the following. k =(k ˆ 0) = (k ˆ 0] = (k k]= E 0 |E| |E| |

19 Proposition 16 The classical n-arguments 4-valued logic gate exclusive-or (XOR) gate is universal in the sense that all uni- g(x1, ..., xn) can be realized as n-ququat unital quantum gate tary operations on arbitrary many qubits can be expressed as if and only if g(0, ..., 0) = 0,org(x1, ..., xn)=const. compositions of these gates. Recently in the paper [59] was considered universality for n-qudits quantum gates. The two arguments nonconstant classical gate g(x1,x2) The same is not true for the general quantum operations can be realized by two-ququat linear unital quantum gate ˆ if (superoperators) corresponding to the dynamics of open quan- E g(0, 0) = 0. tum systems. In the paper [24] single qubit open quantum Two arguments nonconstant classical gates such that g(0, 0) =system with Markovian dynamics was considered and the re- 6 0 can not be realized by two-ququat quantum gates ˆ with sources needed for universality of general quantum opera- E T =0. These classical gates can be realized by three-ququats tions was studied. An analysis of completely-positive trace- unital quantum gates. Let us consider Sheffer function V4(x1,x2preserving)= superoperators on single qubit density matrices max(x1,x2)+1(mod4).Ifx3 =0and x1,x2,x3 0, 1, 2, 3 ,was realized in papers [86, 87, 88]. 6 ∈{ } then we can deﬁne unital quantum Sheffer gate by Let us study universality for general quantum four-valued logic gates. A set of quantum four-valued logic gates is universal iff all quantum gates on arbitrary many ququats can be expressed as compositions of these gates. A set of quan- x 1 V4(x1,x2) tum four-valued logic gates is universal iff all unitary two- valued logic gates and general quantum operations can be x2 SF3 V4(x1,x2) ∼ represented by compositions of these gates. Single ququat x3 x3. gates cannot map two initially un-entangled ququats into an entangled state. Therefore the single ququat gates or set of single ququats gates are not universal gates. Quantum gates which are realization of classical gates cannot be universal

If x3 =0then we must have for unital quantum Sheffer by deﬁnition, since these gates evolve generalized computa- gate tional states to generalized computational states and never to the superposition of them. Let us consider linear completely positive trace-decreasing superoperator ˆ. This superoperator can be represented in the E x1 0 form

m x2 SF3 0 ˆ Lˆ Rˆ , (31) = Aj A† E j 0 0. Xj=1

where LÂ and RÂ are left and right multiplication superop- (n) erators on defined by LÂ B)= AB), RÂ B)= BA). Let us write the unital quantum gate which realizes Shef- The n-ququatsH linear gate |ˆ is completely| positive| | trace- fer function in generalized computational basis by preserving superoperator suchE that the gate matrix is an element of Lie group TGL(4n 1, R). In general case, the 3 ˆ − (SF3) n-ququats nonlinear gate is defined by completely posi- ˆ = 000)(µ1µ20 + N E | | tive trace-decreasing linear superoperator ˆ such that the gate µ1X,µ2=0 matrix is an element of Lie group GL(4n,ER). The condition of completely positivity leads to difficult inequalities for gate 3 matrix elements [89, 86, 87, 88]. In order to satisfy condition + V (µ ,µ ), V (µ ,µ ),µ )(µ µ µ . of completely positivity we use the representation (31). To | 4 1 2 ∼ 4 1 2 3 1 2 3| µ1X,µ2=0 find the universal set of completely positive (linear or nonlinear) gates ˆ we consider the universal set of the superopera- ˆ E ˆ tors LAj and R . The matrices of these superoperators are IX Universal set of quantum four-valued Aj† logic gates connected by complex conjugation. Obviously, the universal set of superoperators LÂ defines a universal set of completely positive superoperators ˆ of the quantum gates. The trace- The condition for performing arbitrary unitary operations to preserving condition forE linear superoperator (31) is equiva- realize a quantum computation by dynamics of a closed quan- lent to the requirement for gate matrix TGL(4n 1, R), tum system is well understood [80, 81, 82, 83]. Using a uni- i.e. = δ . The trace-decreasing conditionE∈ can be− satis- versal gate set, a quantum computer may realize the time se- 0µ 0µ fiedE by inequality of the following proposition. quence of operations corresponding to any unitary dynamics. Deutsch, Barenco and Ekert [81], DiVincenzo [82] and Lloyd Proposition 17 Ifthematrixelements µν of a superopera- [83] showed that almost any two-qubits quantum gate is uni- E versal. It is known [80, 81, 82, 83] that a set of quantum gates that consists of all one-qubit gates and the two-qubits

20 tor ˆ is satisﬁed the inequality (jA†) 1 R =(α Rˆ ν)= Tr σαRˆ σν = E αν A† n A† | | 2 N 1 − 2 ( µ) 1, (32) E0 ≤ 1 1 µX=0 = n Tr σασν A† = n Tr A†σασν . 2 2 then ˆ is a trace-decreasing superoperator. E The matrix elements can be rewritten in the form

Proof. Using Schwarz inequality (jA) 1 (jA†) 1 L = (σµσα A) ,R = (A σασν ). (34) µα 2n | αν 2n | N N N 1 2 1 1 − − 2 − 2 0µρµ ( 0µ) (ρν ) Lˆ E ≤ E Example. Let us consider the single ququat pseudo-gate A. µX=0 µX=0 νX=0 The elements of pseudo-gate matrix L(A) are deﬁned by and the property of density matrix (A) 1 Lµν = Tr(σµAσν ). N 1 2 2 − 2 Trρ =(ρ ρ)= (ρν ) 1, | ≤ Let us denote Xν=0 1 a = Tr(σ A). we have µ 2 µ N 1 N 1 − 2 − 2 2 2 Using Trˆ(ρ) = (0 ˆ ρ) = µρµ ( µ) . | E | | |E| | E0 ≤ E0 µX=0 µX=0 1 1 i L(A) = Tr(σ σ A)= δ TrA+ ε Tr(σ A), kl 2 l k 2 kl 2 lkm m Using (32), we get Trˆ(ρ) 1.Sinceˆ is completely positive (or positive)| superoperatorE |≤ ( ˆ(ρ) E 0), it follows where k, l, m =1, 2, 3,weget that E ≥ 3 3 LˆA = a µ)(µ + ak 0)(k + k)(0 + 0 Trˆ(ρ) 1, 0| | | | | | ≤ E ≤ µX=0 Xk=0 i.e. ˆ is trace-decreasing superoperator. LetE the superoperators Lˆ and Rˆ be called pseudo- A A† gates. These superoperators can be represented by +ia1 3)(2 2)(3 + ia2 1)(3 3)(1 + | |−| | | |−| | N 1 N 1 N 1 N 1 − − − − ˆ (A) ˆ (A†) LA = L µ)(ν , RA = R µ)(ν . µν | | † µν | | +ia3 2)(1 1)(2 . µX=0 νX=0 µX=0 νX=0 | |−| | Proposition 18 The matrix of the completely positive super- The pseudo-gate matrix is operator (31) can be represented by 3 (A) m N 1 Lµν = δµν TrA+ δµ0δνm + δµmδν0 Tr(σmA)+ − (jA) (jA†) mX=1 µν = L R . (33) E µα αν Xj=1 αX=0 3 Proof. Let us write the matrix µν by matrices of superoper- +i δ δ ε Tr(σ A), ˆ ˆ E µk νl lkm m ators LAj and RAj . mX=1 m i.e. µ ˆ ν µ Lˆ Rˆ ν µν =( )= ( Aj A† )= a0 a1 a2 a3 E |E| | j | Xj=1 A  a1 a0 ia3 ia2  L( ) = − . a2 ia3 a0 ia1  a ia ia −a  m N 1 m N 1  3 2 1 0  − −  −  (jA) (jA†) µ Lˆ α α Rˆ ν L R . (jA) = ( Aj )( A† )= µα αν | | | j | Let us consider properties of the matrix elements Lµα j=1 α=0 j=1 α=0 X X X X (jA†) and Rµα . Finally, we obtain (33), where Proposition 19 The matrices L(jA) and R(jA†) are complex 1 µα µα (jA) ˆ ˆ 4n 4n matrices and their elements are connected by complex Lµα =(µ LA α)= n Tr σµLAσα = | | 2 conjugation:×

(jA) (jA†) 1 1 (Lµα )∗ = Rµα . = n Tr σµAσα = n Tr σασµA , 2 2

21 Proof. Using complex conjugation of the matrix elements where hµν are complex coefﬁcients. (34), we get As a basis of Lie algebra gl(16, C) we can use 256 lin- early independent self-adjoint superoperators (jA) 1 1 (jA†) (L )∗ = (σ σ A)∗ = (A σ σ )=R . µα n µ α n µ α µα r 2 | 2 | Hαα = α)(α ,H = α)(β + β)(α , | | αβ | | | | We can write the gate matrix (33) in the form

m N 1 i − H = i α)(β β)(α . (jA) (jA) αβ µν = Lµα (Lαν )∗. − | |−| | E Xj=1 αX=0 where 0 α β 15. The matrices of these generators ≤ ≤ ≤ (jA) (jA†) is Hermitian 16 16 matrices. The matrix elements of 256 Proposition 20 The matrices Lµα and Rµα of the n-ququatsHermitian 16 16× matrices H , Hr and Hi are defined n C αα αβ αβ quantum gate (31) are the elements of Lie group GL(4 , ). by × Proof. The proof is trivial. (H ) = δ δ , (Hr ) = δ δ + δ δ , A two-ququats gate ˆ is called primitive [59] if ˆ maps αα µν µα να αβ µν µα νβ µβ να tensor product of singleE ququats to tensor product ofE single ququats, i.e. if ρ1) and ρ2) are ququats, then we can find i | | (Hαβ )µν = i(δµαδνβ δµβ δνα). ququats ρ0 ) and ρ0 ) such that − − | 1 | 2 For any Hermitian generators Hˆ exists one-parameter pseudo- ˆ ρ ρ )= ρ0 ρ0 ). E| 1 ⊗ 2 | 1 ⊗ 2 gates Lˆ(t) which can be represented in the form Lˆ(t)=exp itHˆ ˆ ˆ ˆ The superoperator ˆ is called imprimitive if ˆ is not primitive. such that L†(t)L(t)=I. It can be shownE that almost every pseudo-gateE that oper- Let us write main operations which allow to derive new ates on two or more ququats is universal pseudo-gate. pseudo-gates Lˆ from a set of pseudo-gates. 1) We introduce general SWAP (twist) pseudo-gateTˆ(SW).A Proposition 21 The set of all single ququat pseudo-gates and new pseudo-gate Lˆ(SW) defined by Lˆ(SW) = Tˆ(SW)LˆTˆ(SW) any imprimitive two-ququats pseudo-gate are universal set of is obtained directly from Lˆ by exchanging two ququats. (2) pseudo-gates. 2) Any superoperator Lˆ on generated by the commuta- ˆ ˆ H ˆ ˆ Proof. This proposition can be proved by analogy with [82, tor i[Hµν , Hαβ] can be obtained from Lµν (t)=exp itHµν ˆ ˆ 81, 59]. Let us consider some points of the proof. Expressed and Lαβ(t)=exp itHαβ because in group theory language, all n-ququats pseudo-gates are el- ˆ ˆ ements of the Lie group GL(4n, C). Two-ququats pseudo- exp t [Hµν , Hαβ]= gates Lˆ are elements of Lie group GL(16, C). The question of universality is the same as the question of what set of su- n ˆ ˆ ˆ ˆ peroperators Lˆ sufficient to generate GL(16, C). The group = lim Lαβ( tn)Lµν (tn)Lαβ(tn)Lµν ( tn) , n − − GL(16, C) has (16)2 = 256 independent one-parameter sub- →∞ (µν) ˆ ˆ groups GLµν (16, C) of one-parameter pseudo-gates Lˆ (t) where tn =1/√n. Thus we can use the commutator i[Hµν , Hαβ] such that Lˆ(µν)(t)=t µ)(ν . Infinitesimal generators of Lie to generate pseudo-gates. group GL(4n, C) are defined| | by 3) Every transformation Lˆ(a, b)=expiHˆ (a, b) of GL(16, C) generated by superoperator Hˆ (a, b)=aHˆµν + bHˆαβ,where d (µν) ˆ ˆ Hˆµν = Lˆ (t) , a and b is complex, can obtained from Lµν (t)=exp itHµν dt t=0 and Lˆαβ(t)=exp itHˆαβ by n ˆ where µ, ν =0, 1, ..., 4 1. The generators Hµν of the n − n a b one-parameter subgroup GLµν (4 , R) are superoperators of exp iHˆ (a, b) = lim Lˆµν ( )Lˆαβ( ) . n n n (n) →∞ the form Hˆµν = µ)(ν on which can be represented by n n | | H 4 4 matrix Hµν with elements For other details of the proof, see [82, 81, 59] and [79, 80, × 83]. (Hµν )αβ = δαµδβν.

The set of superoperators Hˆµν is a basis (Weyl basis [84]) of X Quantum four-valued logic gates of Lie algebra gl(16, R) such that order (n,m)

[Hˆµν , Hˆαβ]=δναHˆµβ δµβ Hˆνα, − In general case, a quantum gate is deﬁned to be the most gen- where µ, ν, α, β =0, 1, ..., 15. Any element Hˆ of the algebra eral quantum operation [1]: gl(16, C) can be represented by Deﬁnition Quantum gate Gˆ of order (n,m) is a positive (completely positive) linear (nonlinear) trace preserving map from 15 15 density matrix operator ρ) on n-ququats to density matrix Hˆ = hµν Hˆµν , | operator ρ0) on m-ququats. µX=0 Xν=0 |

22 In the generalized computational (operator) basis the gate where p = min N,M and λµ 0. Gˆ of order (n,m) can be represented by Let us rewrite formula Proof. The proof{ of this} theorem≥ can be easy realized in ma- by trix representation by using theorem 10. In general case, we have the following theorem. M N (n,m) 1 (n,m) Theorem 13. (Singular Valued Decomposition for Gates) Gˆ = G σµ)(σν . (35) √ n m µν | | Any linear gate Gˆ(n,m) of order (n, m) can be represented by 2 2 µX=0 νX=0 Gˆ(n,m) = Tˆ(m,m) ˆ(m,m) Dˆ (n,m) ˆ(n,n), where U U m 1 µ = µ14 − + ... + µm 14+µm, where − ˆ(m,m) is an orthogonal quantum gate of order (m, m). Uˆ(n,n) n 1 is an orthogonal quantum gate of order (n, n). ν = ν14 − + ... + νn 14+νn . Uˆ (n,m) − D is a diagonal quantum gate (37) of order (n, m). ˆ(m,m) For the gate matrices we use N =4n 1 and M =4m 1. T is a translation quantum gate of order (m, m): (n,m) − n m − The matrix Gµ,ν of linear gate is a real 4 4 -matrix p M × (m,m) 1 with Tˆ = 0)(0 + σµ)(σµ + Tµ σµ)(0 , √2n2m | | | | | | (n,m) µX=1 µX=0 G0,ν = δ0ν . where p = min N,M and λk 0. In general case, linear gates Gˆ(n,m) of order (n, m) have Proof. The proof{ of this} theorem≥ can be easy realized in ma- G(n,m) =0, i.e. this gate is not unital. i.e. trix representation by using theorem 10. µ0 6 Note that, any n-arguments classical gate g(ν1, ..., νn) can 10 0... 0 be realized as linear trace-preserving quantum four-valued  T1 R11 R12 ... R1N  logic gate Gˆ(n,1) of order (n, 1) by (n,m) G = T2 R21 R22 ... R2N .   (n,1)  ......  Gˆ = 0)(0...0 + g(ν , ..., νn))(ν , ..., νn +   | | | 1 1 |  TM RM1 RM2 ... RMN  ν ...νX=0...0   1 n6 Theorem 10. (Singular Valued Decomposition for Matrix) Any real N M matrix G can be written in the form +(1 δ0g(0,...,0)) g(0, ..., 0))(0...0 × − | |− T G = M DNM N , U U N 1 − where (1 δ ) (1 δ ) µ)(ν ...νn . − − 0g(0,...,0) − µg(ν1 ,...,νn) | 1 | M is an orthogonal M M matrix. µX=0 ν1X...νn U × N is an orthogonal N N matrix. U × In general case, this quantum gate is not unital gate. DNM is diagonal N M matrix such that × DNM = diag(λ , ..., λp) ,p= min N,M , 1 { } References where λ1 λ2 ... λp 0. [1] Aharonov D., Kitaev A. and Nisan N., ”Quantum cir- ≥ ≥ ≥ ≥ Proof. This theorem is proved in [90, 91, 92, 85]. cuits with mixed states.” in Proceedings of the Thirti- Let us consider the unital gates with T =0deﬁned by eth Annual ACM Symposium on Theory of Computa-

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