View metadata, citation and similar papers at Ocore.ac.ukpen n-Qubit System as a Quantum Computer brought to you by CORE with Four-Valued Logic provided by CERN Document Server 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 de- noted 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 de- noted 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- <kl>= 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 dimensionalH H ⊗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 oper- ator 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 − <kk0 >= δkk0 , k><k = I. 10 10 | | | σ = ,σ= I = . Xk=0 3 0 1 0 01 − Then k, l)= k><l) is an orthonormal basis of the Liou- The real expansion coefficients P (t) are given by | (||n) | µ1...µn ville space : H P t Tr σ ... σ ρ t . µ1...µn ( )= ( µ1 µn ( )) N 1 N 1 ⊗ ⊗ − − (k, l k0,l0)=δkk δll , k, l)(k, l = I,ˆ (4) Normalization (Trρ =1) requires that P0:::0(t)=1.Since | 0 0 | | Xk=0 Xl=0 the eigenvalues of the Pauli matrices are 1, the expansion ± n n coefficients satisfy Pµ1...µn (t) 1. Let us rewrite (2) in the where N =2 . This operator basis has 4 elements. Note | |≤ form: that k, l) = kl >= 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 <lj ). An arbitrary⊗ general⊗ one-qubit state ρ(t) can be repre- | ⊗| || ⊗| |⊗ | sented as (n) For an arbitrary element A) of we have 3 | H 1 N 1 N 1 ρ(t)= σµPµ(t), − − 2 A)= k, l)(k, l A) (6) µX=0 | | | kX=0 Xl=0 where P (t)=Tr(σ ρ(t)) and P (t)=1. The pure state µ µ 0 with can be identified with Bloch sphere 2 2 2 (k, l A)=Tr( l><kA)=<kA 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.