Simulation and Programming of Quantum Computers Roland Rudiger¨ [email protected] Lecture given at Department of Transportation, University of Applied Sciences and Department of High Performance Computing { Center of Logistics and Expert Systems GmbH, Salzgitter, Germany April 25, 2003 Outline . Quantum computation|basic concepts . Algorithms and simulation . Quantum programming languages . Realization of QCs|present state Quantum computation|basic concepts Introductory remarks and example General remarks on quantum mechanics . Feynman: \I think I can safely say that nobody understands quan- tum mechanics." . Bohr: \Anyone who is not shocked by quantum theory has not understood it." . \Quantum mechanics is a mathematical framework or set of rules for the construction of physical theories." (Nielsen & Chuang [NC00, p. 2]) . framework is highly counter-intuitive: physical states and processes are described in a state space terminology . main differences to classical physics: { superposition principle (At any given time a quantum system can be \in more than one state".) but: Statements in natural language like this might be mis- leading and inadequate (and sometimes wrong). { statistical interpretation (The outcomes of measurements obey probabilistic laws.) . in QM: typically 3 steps: preparation of a state (unitary) time evolution ! ! measurement . the problem: there is no direct intuitive interpretation of the notion of \state": states comprise the statistics of measurement outcomes . a pragmatic advice to students: \Shut up and calculate." An example 1 . spin-2 particle: a potential realization of a 2 level system (a \qubit") . experimental experience: for every given space direction there are exactly two possible measurement outcomes: . ~, where ~ = h , h = Planck's constant ±2 2π . state space: 2-dim Hilbert-space: = C2 H possible choice of basis: ~eup;~edown system states after a measurement of the electron spin in z-direction . conventional notation: z , z (Dirac's \ket"-vectors, the second j" i j# i half of a \bra"-(c)-\ket") or (in context of quantum computation): 0 = z , 1 = z j i j" i j i j# i . general state: = α 0 + β 1 , where α 2 + β 2 = 1, α, β C j i j i j i j j j j 2 ~ . spin operator: S~ = (S ;S ;S ) = ~σ x y z 2 . The components of ~σ = (σx; σy; σz) are the Pauli-matrices (also denoted by (X; Y; Z)): 0 1 0 i 1 0 σx = σy = − σz = 1 0 i 0 0 1 − . spin operators in terms of the basis ( 0 ; 1 ): j i j i ~ Sx = ( 0 1 + 1 0 ) 2 j ih j j ih j ~ Sy = ( i 0 1 + i 1 0 ) 2 − j ih j j ih j ~ Sz = ( 0 0 1 1 ) 2 j ih j − j ih j . eigenvalues: ~ and eigenstates: ±2 of S : = 1 ( 0 + 1 ); = 1 ( 0 1 ) x x p2 x p2 j" i 1 j i j i j# i 1 j i − j i of Sy : y = ( 0 + i 1 ); y = ( 0 i 1 ) j" i p2 j i j i j# i p2 j i − j i of Sz : z = 0 ; z = 1 j" i j i j# i j i . Suppose the system will be prepared in state z : j" i arbitrary space-direction ~n, ~n = 1: j j z n ~n = (sin θ cos ')~x + (sin θ sin ')~y + (cos θ)~z y x determine solutions of (~n S~) s = s s : structure of this eq.: ×× × = s × · j i j i ×× × × An elementary calculation yields: ~ s+ = 2 θ θ i' s+ = ~n = cos 0 + sin e 1 j i j" i 2j i 2 j i ~ s = 2 − − θ θ i' s = ~n = sin 0 + cos e 1 j −i j# i − 2j i 2 j i some typical situations|overview and summary: observable values prep. new probability (diagonal representation) state state ~ ~ ~ 2 Sx = x x x x + z x z x = 0:5 2 j" ih" j − 2 j# ih# j 2 j" i j" i jh" j" ij ~ 2 z x z x = 0:5 − 2 j" i j# i jh" j# ij ~ ~ ~ 2 Sy = y y y y + z y z y = 0:5 2 j" ih" j − 2 j# ih# j 2 j" i j" i jh" j" ij ~ 2 z y z y = 0:5 − 2 j" i j# i jh" j# ij ~ ~ ~ 2 Sz = z z z z + z z z z = 1 2 j" ih" j − 2 j# ih# j 2 j" i j" i jh" j" ij ~ 2 z z z z = 0 − 2 j" i j# i jh" j# ij ~ ~ ~ 2 2 θ S~ ~n = ~n ~n ~n ~n + z ~n z ~n = cos · 2 j" ih" j − 2 j# ih# j 2 j" i j" i jh" j" ij 2 ~ 2 2 θ z ~n z ~n = sin − 2 j" i j# i jh" j# ij 2 Postulates of quantum mechanics adopted from Nielsen & Chuang [NC00, p. 80] (and simplified) . Postulate 1: Associated to any isolated physical system is a Hilbert space (the state space); state vector: is a unit vector (a \ray") . Postulate 2: Evolution of a closed quantum system: described by a unitary transformation 0 = U ( 0 = statet=t ; = statet=t ) j i j i j i 2 j i 1 . Postulate 2': Time evolution of the state of a closed quantum system: described by the Schr¨odinger equation, d 0 1 i~ j i = H or (t) = exp (( i=~)Ht) (0) dt j i j i − j i @ U A | {z } H: Hermitian operator, the \Hamiltonian" of the closed system. Postulate 3': (Projective) measurements: also: von Neumann measurement, measurement of the first kind: { Projective measurement: described by an observable, M, a Her- mitean operator. (eigenvalues are real) This observable can be written as: M = mPm (= m m m ), where X Pm j ih j m Pm: projector onto the eigenspace of M with eigenvalue m { The possible outcomes of the measurement correspond to the eigenvalues, m. Probability: 2 Pr(getting result m) = Pm (= m m = m ) h j j i h j ih j i jh j ij { state of the quantum system immediately after the measure- ment: P m m iα after measurement = j i (= hmj i m = e m ). j i Pm jh j ijj i j i ph j j i Remark: Projective measurements are nowadays considered as a description of the measurement process which is not adequate in all situations. Postulate 4: State space of a composite physical system (the \and" in quantum physics (Penrose [Pen94])): tensor product of the state space of the component physical systems. formally: 1 2 n j i ⊗ j i ⊗ · · · ⊗ j i . Remarks: { The postulates / the interpretation may seem strange and odd. { Isn't there any simpler theory? { Local realistic theories seem closer to common sense. but: they predict a \Bell correlation" β with 2 β +2 − ≤ ≤ (one variant of Bell's inequalities) { QM predicts and experiments by Aspect et al. (1982) and Tittel et al. (1998) show that these inequalities are violated: 2p2 2:0 β 0:0 2:0 −2p2 − 0 π=4 π=2 θ { The experiments confirm quantum physics. { Therefore: Local realistic theories must be ruled out. Quantum networks . some definitions (D. Deutsch, as quoted in [EHI00]) { A qubit is a system in which the Boolean states 0 and 1 are represented by a prescribed pair of normalized and mutually orthogonal quantum states labeled as 0 ; 1 . fj i j ig { A quantum register is a collection of n qubits. { A quantum logic gate is a device which performs a fixed unitary operation on selected qubits in a fixed period of time. { A quantum network is a device consisting of quantum logic gates whose computational steps are synchronized in time. example: 1 qubit ( = C2): H preparation / :: measurement / state j i :: observable |U{z(2) } states are = α 0 + β 1 , where α 2 + β 2 = 1 j i j i j i j j j j . example: 3 qubits ( = C2 C2 C2): :::::::: H 0 ⊗::::::::⊗1 0 B :::::::: C preparation / j i B C measurement / B :::::::: C 1 B C B :::::::: C state j i B C observable B :::::::: C 2 B :::::::: C B C j i @ :::::::: A | U{z(8) } { state of the system is = 2 1 0 j i j i ⊗ j i ⊗ j i { e.g.: 0 1 1 = 011 = 3 ! j i ⊗ j i ⊗ j i j i j i 1 1 1 = 111 = 7 ! j i ⊗ j i ⊗ j i j i j i basis: ! 000 = 0 001 = 1 010 = 2 011 = 3 j i j i j i j i j i j i j i j i 100 = 4 101 = 5 110 = 6 111 = 7 j i j i j i j i j i j i j i j i any advantage so far, compared with classical systems? NO! ! but: look at superpositions: 1 0 + 1 1 1 = 1 011 + 111 = 1 3 + 7 ! p2j i j i⊗j i⊗j i p2j i j i p2j i j i a superposition of 8 numbers (which can possibly be pro- ! cessed simultaneously) 1 1 1 0 + 1 0 + 1 0 + 1 p2j i j i ⊗ p2j i j i ⊗ p2j i j i 1 = 3 000 + 001 + 010 + 011 p2 j i j i j i j i + 100 + 101 + 110 + 111 j i j i j i j i . state space tends to become large: H { state space of a n qubit system: = C2n H { this is a problem for (classical) simulators and { the key to potential successful applications of quantum com- puters . example: n = 5 qubits: :::::::::::::::::::::::::::::::: 0 :::::::::::::::::::::::::::::::: 1 :::::::::::::::::::::::::::::::: B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C 0 B C B :::::::::::::::::::::::::::::::: C j i B C B :::::::::::::::::::::::::::::::: C 1 B C B :::::::::::::::::::::::::::::::: C preparation / j i B C measurement / B :::::::::::::::::::::::::::::::: C 2 B C B :::::::::::::::::::::::::::::::: C state j i B C observable B :::::::::::::::::::::::::::::::: C 3 B :::::::::::::::::::::::::::::::: C B C j i B :::::::::::::::::::::::::::::::: C B C 4 B :::::::::::::::::::::::::::::::: C B C j i B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C B C B :::::::::::::::::::::::::::::::: C @ :::::::::::::::::::::::::::::::: A | unitary 32 32{z matrix U (32) } × .