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Quantum theory of computation and relativistic physics

Alexander Yu. Vlasov∗ 197101, Mira Street 8, IRH St.–Petersburg, Russia Jan 1997 (Jul 1999)

quant-ph/9701027 Title: Theory of Computation and Relativistic Physics Authors: Alexander Yu. Vlasov (FCR/IRH, St.-Petersburg, Russia) Comments: 6 pages, LaTeX2e, 2 columns, 4 PostScript figures are included by epsfig.sty; based on poster in Proc. PhysComp ’96 Workshop (BU, Boston MA, 22-24 Nov 1996) pp. 332-333, a "no-go" result for bounded quantum networks; v3/4 -- typos corrected, minor changes. Report-no: RQC-VAY11/v4

In the e-print is discussed a few steps to introducing of "vocabulary" of relativistic physics in of information and computation (QTI&C). The behavior of a few simple quantum systems those are used as models in QTI&C is tested by usual relativistic tools (transformation properties of wave vectors, etc.). Massless and charged massive with 1/2 are considered. theory is also discussed briefly.

Abstract In the paper are described some steps for merger between relativistic quantum theory and theory of computation. The first step is consideration of transformation of state due to rotation of . The Lorentz transformation is considered after that. The some new properties of this transformation change usual model of qubit. The system of q2bit seems more fundamental relativistic model. It is shown also that such model as is really such q2bit system and for modelling of qubit is necessary to use massless like electron . The quantum field theory (QFT) is briefly discussed further. The wave vectors of interacted particle now described by some and it can produce some multiparticle (‘nonlinear’) effects.

PACS numbers: 03.30.+p, 03.65.-w, 03.70.+k, 89.70.+c Keywords: Quantum, Computation, Relativistic

∗E-Mails: [email protected], [email protected], [email protected] 1 Introduction with complex coefficients. The state of quantum system is described as a ray in complex and for two- The paper describes some approaches to relativistic quan- state system it can be considered as complex projective . The main purpose of the work tum theory of computation space CP C .Eachray(c0,c1) is presented by ∼ ∪{∞} is to consider essentially new properties of quantum com- ζ = c0/c1.The 0 corresponds to 0 and puters [1, 2, 3, 4, 5] due to relativistic phenomena rather the 1 to . There is correspondence|i [8] between the than some small corrections to nonrelativistic formulae. plane| iζ and∞ a sphere S due to stereographic projection ζ =(x iy)/(1 z) (see Fig. 1). Expressions for coordinate At first, in relativistic theory it is necessary to consider − − a qubit in different coordinate systems. In simplest case (x, y, z) on the unit sphere are: it may be 3D local rotations and SU(2) . 2Re ζ c c +c c = = 0 1 1 0 For consideration of temporal coordinate it is necessary x 2 ζ +1 c0c0 +c1c1 to use Lorentz transformations and 4D spinors. The more |2Im| ζ i(c c c c ) = = 0 1 1 0 correct approach include full Poincare group and quantum y 2 − − (2) − ζ +1 c0c0 + c1c1 field theory. |ζ|2 1 c c c c z = | | − = 0 0 − 1 1 ζ 2 +1 c c +c c | | 0 0 1 1 2 Qubit Due to the equation Eq. (1) we can consider (X, Y, Z) instead: A quantum two-state system is often called quantum bit X = c0c1 + c1c0 or ‘qubit’ [6, 7]. Let us consider a particle with spin 1/2as Y =i(c c c c ) (3) 0 1− 1 0 a model of the qubit. The quantum state of the system is Z=c0c0 c1c1 ψ = c 0 + c 1 ,wherec and c are complex numbers − 0 | i 1 | i 0 1 The 0 and 1 map to opposite poles of the sphere. and the norm of ψ is: | i | i 2.1 Spatial rotation of coordinate system 2 2 2 ψ ψ∗ψ = c0 + c1 =1,c0,c1 C (1) || || ≡ | | | | ∈ A transformation of the state due to a spatial rotation of coordinate system is described by unitary with determinant unity:

False ab a = d, c = b (c1 =0) ψ0 = ψ, − (4) ∞ cd ad bc= a 2 + b 2 =1 S   − | | | | This is the group of unitary 2 2 matrices, SU(2). It × z corresponds to principle, that transformation of the wave x vector is described by some representation of a group of y coordinate transformation. The group SU(2) is represen- CP tation of the group of spatial rotations SO(3) in a space Im of 2D complex vectors. c 2 0 Due to 2–1 isomorphism SU(2) and SO(3), any rota- True Re × c1 tion corresponds to unitary matrix up to sign. We can see simple correspondence between any 1–gate and “pas- sive” transformation, i.e. transition to other coordinate Figure 1: Riemann sphere for qubit system. A state of a qubit can be described as a superposition The equations Eq. (3) can be used for demonstration of two logical states of usual bit (False, True or 0, 1) of relation between SO(3) and SU(2). If we apply some

1 unitary transformation Eq. (4) U :(c ,c ) (c0 ,c0 )then Linear transformations with determinant unity of a 0 1 → 0 1 (X, Y, Z) (X0,Y0,Z0). Unitary matrices do not change qubit correspond to Lorentz transformation of the vector the norm→ Eq. (1) and length of the vector: (T,X,Y,Z): 2 2 2 2 2 2 X + Y + Z =(c0 + c1 ) (5) | | | | ψ = Aψ;detA=1 Angles between vectors also do not change. Unitary trans- 0 V0 =2Aψ(Aψ)∗ =2Aψψ∗A∗ = AVA ∗ formations of a state of the qubit correspond to rotations 2 2 2 2 (8) det V0 = T 0 X0 Y 0 Z0 = of the sphere (Fig. 1). Two matrices: U and U produce − − − − =detV=T2 X2 Y2 Z2 the same rotation due to Eq. (3). − − − The transformations of a state of n– due to spa- tial rotation can be described by unitary 22n matrices. t 3 The relativistic consideration of light cone a qubit ψ =1 || || 3.1 Lorentz transformation rotation (T,X,Y,Z) For Lorentz transformation of coordinate system there is similar isomorphism between the group SO(3, 1) and the group SL(2, C) of all complex 2 2 matrices with de- boost × terminant unity. The group SL(2, C) is isomorphic with x Lorentz group in the same way as the group SU(2) with group of 3D rotations [8]. The group SL(2, C)isarep- resentation of Lorentz group SO(3, 1) in a space of 2D complex vectors. On the other hand, we should not directly apply such y representation of relativistic group SL(2, C) to a qubit. Only the subgroup of unitary matrix saves the norm Figure 2: Null vector (T,X,Y,Z) Eq. (1). The expression Eq. (1) in relativistic theory is not invariant scalar, but temporal part of 4–vector. Sim- 1 ple relation between transformations of coordinate system Only if the matrix A is unitary, AVA ∗ = AVA − and and unitary matrices is broken here. V i.e. the norm Eq. (6) does not change. Otherwise Let us denote: Eq. (6) should be considered as the ‘T –component’ of a 2 4–vector. T = ψ ψ∗ψ = c c + c c (6) || || ≡ 0 0 1 1 We can write1 , using equations Eq. (3), Eq. (6) : The relation between SL(2, C) and Lorentz group T + ZXiY c c c c Eq. (8) is valid not only for null vectors. Any vector is a V − =2 0 0 0 1 ≡ X+iYTZ c1c0 c1c1 sum of two null vectors and  −   1 c0 A(V + U)A∗ = AVA ∗ + AUA∗. 2V= (c0 c1)=ψψ∗ (7) c1 The qubit is described by two-component complex vec-  2  2 2 2 det V = T X Y Z = tor or Weyl . It corresponds to =2c c 2c c− 2c−c 2c−c =0 0 0 1 1− 1 0 0 1 with spin 1/2. Such particle always moves with the speed 1 of light. The equations Eq. (7) show a correspondence In the matrix notation ψ∗ψ is scalar and ψψ∗ is 2 2matrix (with Dirac notation: ψ ψ and ψ ψ respectively). × between such spinor and 4D null vector (Fig. 2). This h | i | ih |

2 vector can be also rewritten by using : t 01 0 i 10 σ = ,σ= ,σ= x 10 y i −0 z 0 1      − (9) V=T1+Xσx + Yσy +Zσz, 0 1 0 j = ψ∗ψ Vi = 2Tr(σiV)=Tr(σiψψ∗)=ψ∗σiψ; ψ∗γ ψ j σ= σ ,σ ,σ :(T, X, Y, Z )=(ψ∗ψ,ψ∗σψ) { x y z} { } light cone 3.2 Massive particle x Massive charged particle with spin 1/2 like an electron is described by two Weyl spinors and has four complex y components:

ψ0 Figure 3: Massive particle ϕR 2 ψ1 ψ = ϕR,ϕL C ; ψ = (10) ϕL ∈ ψ2    ψ The representation is finite dimensional.  3  • It is possible to consider such massive particle as two The representation is unitary in a definite norm. qubits: • It can be considered as some mathematical reasons for: ψ = c 00 + c 01 + c 10 + c 11 (11) 00| i 01| i 10| i 11| i Using of quantum field theory (QFT) instead of sys- The first index is similar to and for each ϕR,ϕL. • The other one corresponds to|↑i discrete|↓i coordinate trans- tems with finite number of states. formation like spatial reflection: P :(t, ~x) (t, ~x). Necessity of a consideration of different kinds of in- It is also possible to build a vector by→ using− the 4D • teracting quantum fields. spinor and 4 4 Dirac matrices γµ. It is 4D vector of × current Fig. 3 : The relativistic physics have both these properties. We µ 0 µ j = ψ∗γ γ ψ (12) can consider (QED) as an ex- 01 0 σ ample. γ0 = ,γ= − . (13) 10 σ 0 It is not quite compatible with such properties of usual     with always positive: model of quantum computation as fixed size of registers and gates, one kind of qubits,etc.. 0 2 2 2 j = ψ∗ψ = ψ = ϕ + ϕ (14) i| i| || R|| || L|| but j0 is not LorentzP invariant. The Lorentz invariant 4 Quantum field theory and com- scalar is 0 ψ∗γ ψ = ϕR∗ ϕL + ϕL∗ ϕR (15) putations In articles about quantum computers Feynman [2, 3] has 3.3 Representations of Lorentz group used one of usual tools of a QFT — annihilation and We have used very simple construction of a qubit, but any creation operators a and a∗: other constructions also have limitations because a repre- 00 01 sentation of Lorentz group cannot satisfy contemporary a = ,a= , 10 ∗ 00 two following conditions: 2 (16)   10  N a∗a= 2A“no-go” result for bounded quantum networks. ≡ 00  

3 ψ0 = U ψ with Fermi relation for the anticommutator:     a∗,a a∗a+aa∗ = 1 (17) { }+ ≡ These operators are used for describing of usual quan- tum gate in [3], but this approach has more wide scope. Figure 4: Nonrelativistic gate This method has a resemblance with secondary quantiza- tion in a QFT. The gate can be built as some electro-magnetic device. From point of view of QED it is described as an inter- 4.1 Secondary action of two quantum fields and we should not split the processes on q-gates and qubits. The usual formula of sec- In a QFT wave functions are operators [9]. Let us consider ondary quantization is Ψ0 = ˆ ˆΨ (Fig. 5). Here Ψ, Ψ0 as an example: Uψ,A describe occupation numbers ,andψˆis wave operator for electron (positron), and Aˆ for photons. The wave oper- ˆ = ipx + ipx (18) ψp cpe− cp∗e ators for particle are included in and can form many nonlinear expressions. They correspondU to Feynman dia- There cp and cp∗ are operators of annihilation and cre- ˆ grams. Such description is linear in respect of Ψ, Ψ0, but ation of the particle with 4–momentum p and so ψ is an ˆ ˆ operator. There is Bose relation for the commutator: not on ψ, A.

ˆ ˆ Ψ0 =Ψψ,A [c∗,c] c∗c cc∗ = 1 (19) U − ≡ − γ

4.2 States and operators e

The operators cp and cp∗ act in some auxiliary Hilbert space and functions like Eq. (18) have more direct physical meaning than states in this space. The quantum field Figure 5: Relativistic gate of is described by some expression similar to Eq. (18)3. The matrices Eq. (16) are used for presentation of quan- 4.3 Algebraic and matrix notation tum gates in [3], but it should be mentioned that in rela- The relations Eq. (17) and Eq. (19) describe one particle. tivistic physics there is no sharp division between q-gates If we have a few particles then the full set of relations is: and q-states due to formulae like Eq. (18). This property of a QFT has some analogy with func- ak,ak + = a∗,a∗ + =0 tional style of programming in modern computer science { 0} { k k0} (20) ak,a∗ + =δkk [10]. In both cases there is no essential difference between { k0} 0 data (states) and functions (operators). A function can for particles like electrons (Fermi statistic, half-integer be used as data for some other function. spin) and [ck,ck ] =[c∗,c∗ ] =0 For example, let us consider an electron as the model 0 − k k0 − (21) [ck,c∗ ] =δkk of a qubit. In nonrelativistic quantum theory of compu- k0 − 0 tation a q-gate can change state of the qubits ψ0 = Uψ for particles like photons (Bose statistic, integer spin). (Fig. 4). Here are wave vectors of quantum system ψ,ψ0 The equations Eq. (16), Eq. (17) show representation (‘qubits’) and is an operator of the gate. U of operators with Fermi relations for one particle. The 3The main difference is commutational relations Eq. (17) for elec- matrix representations of Eq. (20) for many particles are trons and Eq. (19) for photons. more complicated.

4 The relations for Bose particles Eq. (19), Eq. (21) are [7] Charles Bennett, and Computa- impossible to express by using finite-dimensional matrices tion, Physics Today 4810, (October 1995), 24–30. because for any two matrices A, B: [8] and Wolfgang Rindler, Spinors and Space– Vol. 1, Cambridge University Press (1986). T race(AB BA)=0 = [A,B] =1 (22) − ⇒ − 6 [9] Nicolai Bogoliubov and Dmitri Shirkov, Introduction to Due to such properties of algebras of commutators the the Theory of Quantized Fields, Wiley (1980). presentation by using formal expressions with operators [10] John Backus, The algebra of functional programs: func- of annihilation and creation [3] instead of matrices can be tional level reasoning, linear equations, and extended def- more convenient in quantum theory of computation from initions, Lecture Notes in Computer Science 107, (1981), the point of view of relativistic physics. 1–43.

5 Conclusion

In nonrelativistic quantum theory of computation it was necessary only to point number of states 2n for description of qnbit. In relativistic theory there are many special cases. The charged and neutral, massive and massless particles etc. should be described differently.

Acknowlegments

The author is grateful to organizers of the PhysComp96, especially to Tommaso Toffoly, and to Physics Depart- ment of Boston University for support and hospitality.

References

[1] Paul Benioff, Quantum Mechanical Hamiltonian Models of Discrete Processes That Erase Their Own Histories: Application to Turing Machines, Internal Journal of The- oretical Physics 21 (1982), 177–201. [2] , Simulating Physics with Computers, Internal Journal of Theoretical Physics 21 (1982), 467– 488. [3] Richard Feynman, Quantum-Mechanical Computers, 16 (1986), 507–531. [4] David Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceed- ings of Royal Society of London A 400, (1985), 97–117. [5] David Deutsch, Quantum computational networks, Pro- ceedings of Royal Society of London A 425, (1989), 73– 90. [6] Benjamin Schumacher, Quantum Coding, Physical Re- view A51(1995), 2738–2747.

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