Quantum Computation. Rudolf Gross.

Transcription

1 Quantum Computation Rudolf Gross Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften and Technische Universität München Edgar Lüscher-Seminar Gymnasium Zwiesel April 2015

2 Research Campus Garching ESO Plasma Physics MPQ Astrophysics Extraterrestr. Physics ZAE LRZ Informatics Mathematics Mechanical Engineering FRM II GRS Walther-Meißner-Institute Physics-Department /RG - 2

3 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 3

4 First general-purpose computing device Charles Babbage ( ) conceptualized and invented the first mechanical computer in the early 19th century in 1837 he conceives the calculating machine Analytical Engine only part of the machine was completed before his death Science Museum London Science and Society Picture Library engine incorporated (i) an arithmetic logic unit, (ii) a control flow, and (iii) integrated memory first design for a general-purpose computer that could be described in modern terms as Turing-complete /RG - 4

5 Turing machine (1936) is a hypothetical (mathematical) device that manipulates symbols on a strip of tape according to a table of rules can be adapted to simulate the logic of any computer algorithm RosarioVanTulpe Alan Mathison Turing ( ) /RG - 5

6 The first electromechanical computers Konrad Zuse was building the first binary digital computer Z1 in 1938 The first programmable electromechanical computer Z3 was completed in 1941 Zuse also developed the first algorithmic programming language called Plankalkül /RG - 6

7 First fully automatic, digital computer Replica of Zuse's Z3 (German Science Museum, Munich) /RG - 7

8 Programmable machines John von Neumann was proposing the EDVAC computer in 1945 he was introducing the concept of a computer that is controlled by a stored program /RG - 8

9 Digital electronic programmable computers with vacuum tubes 30 tons, 200 kw electric power, over 18,000 vacuum tubes, 1,500 relays, and hundreds of thousands of resistors, capacitors, and inductors, 6 operators, 160 m² space Colossus (1943) was the first electronic digital programmable computing device (Max Newman) US-built ENIAC (Electronic Numerical Integrator and Computer) was the first electronic programmable computer built in the US (John Mauchly, J. Presper Eckert) /RG - 9

10 Semiconductor Integrated Circuits Intel 2 nd generation Core i7 chip: 3.4 GHz, 32nm process technology (1.4 Mio. transistors) /RG - 10

11 Modern supercomputers LRZ Munich: peak performance: 3.6 PetaFLOPS (=10 15 Floating Point Operations Per Second) phase 2: cores, Haswell Xeon processor E v /RG - 11

12 From mechanical to quantum mechanical IP superconducting Qubit 20 µm Intel dual-core 45 nm (2007) WMI technology physics Enigma (1940) vacuum tubes ENIAC (1946) first transistor (1947) Bardeen, Brattain, & Shockley /RG - 12

13 Development of Hardware Platform for QIP Systems Collaborative Research Center 631 Cluster of Excellence NIM Semiconductor Quantum Dots CeNS Superconducting Qubits Trapped Atoms and Ions Al F F WSI 2 µm WMI MPQ /RG - 13

14 Superconducting Quantum Computer Vesuvius 3, 512 qubits, operated at T = 30 mk /RG - 14

15 ... Solid State Circuits Go Quantum today multi electron, spin, fluxon, photon devices near future single/few electron, spin, fluxon, photon devices quantum confinement tunneling classical description quantifiable, but not quantum Intel PTB 65 nm process 2005 single electron transistor /RG - 16

16 ... Solid State Circuits Go Quantum multi electron, spin, fluxon, photon devices superposition of states entanglement quantized em-fields today near future far future single/few electron, spin, fluxon, photon devices quantum electron, spin, fluxon, photon devices classical description quantifiable, but not quantum quantum description Intel PTB WMI 65 nm process 2005 single electron transistor superconducting qubit 2 µm /RG - 17

17 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 18

18 What is computation?... a procedure that transforms input information to an output result by a sequence of simple elementary operations algorithm if there exists an algorithm to solve a given problem, then it can be run on a universal Turing machine efficiency of algorithm measured by computational complexity /RG - 19

19 Computational Complexity... is the study of the resources (time, memory, energy,...) required to solve computational problems example: time for adding and multiplying two n-digit integer numbers using primary school algorithm main distinction: addition: t = α n multiplication: t = β n 2 multiplication is more complex than addition precise result for multiplication by classical computer: O n logn log (logn) (Schönhage, 1971) problems that can be solved using polynomial resources: P (e.g. t n k, k = const.) e.g. multiplication: O n logn loglogn problems that can be solved using resources that are superpolynominal: NP (e.g. t k n ) e.g. factorization of an n-digit integer: exp O n 1/3 log n 2/3 (general number field sieve GNFS algorithm) /RG - 20

20 Complexity Classes... are still under debate NP NPC P or P = NP P = polynomial time NP = superpolynomial time NPC = NP-complete (problem in NP is NPC if any problem in NP is polynomially reducible to it) believed to be right believed to be wrong /RG - 21

21 Time - n o of operations # Integer Factorization 989 complexity of a problem polynomial (P): t #op n k non-polynomial (NP): t #op k n (n: # of digits) time # of operations (#op) classical computer n o of digits exp exp(n) n Nn 22 integer factorization is NP problem on a classical computer algorithm is known, but too slow /RG - 23

22 Integer Factorization 989 complexity of a problem polynomial (P): t #op n k non-polynomial (NP): t #op k n (years) bits bits classical computer exp n miniaturization limit (n: # of digits) time # of operations (#op) bits year of fabrication integer factorization is NP problem on a classical computer algorithm is known, but too slow /RG - 24

23 Integer Factorization 989 complexity of a problem polynomial (P): t #op n k non-polynomial (NP): t #op k n (n: # of digits) time # of operations (#op) (minutes) quantum computer (100 MHz) 512 bits 1024 bits 2048 bits 4096 bits n o of bits integer factorization is NP problem on a classical computer exponential speed-up due to quantum algorthm (Shor) /RG - 25

24 Interacting Quantum System 989 example: N interacting spins, S = 1 2,.. for N = 1000: dimension of Hilbert space: > number of atoms in universe Richard Feynman (1981):...trying to find a computer simulation of physics, seems to me to be an excellent program to follow out...and I'm not happy with all the analyses that go with just the classical theory, because nature isn t classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem because it doesn't look so easy /RG - 26

25 Integer Factorization 989 =?? /RG - 28

26 Integer Factorization 989 = /RG - 29

27 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 30

28 Binary Arithmetics... are used because arithmetical rules are simple binary addition table a b s c s: sum c: carry over addition multiplication (29) (29) (21) (21) (50) (609) /RG - 31

29 The Classical Shannon Bit elementary unit of classical information 0 or 1 or or or classical bits can be copied copying machine (important difference to quantum bits) Claude Elwood Shannon ( ) /RG - 32

30 What is Information? H 2 p information is a general concept similar to the concept of entropy or energy, appearing in many forms: mechanical, thermal, electrical,... can be packed into many equivalent forms: 0,1,, good morning, guten Morgen information is physical (Landauer 1991) - ink on paper - charge on capacitor - currents in leads - spins - polarization of photons -... Claude E. Shannon (1948) information content of variable x appearing with probability p x : I p x = log a 1/p x = log a p x example: binary alphabet, a = 2, x 1 x 2 = 00, 01, 10,11; p x1 = p x2 = p = 1/2 I p = k i=1 log unit: 1 Shannon (sh) = k H p = k = 2 number of bits binary entropy function H 2 p p(x = 1) /RG - 33

31 What is Information? example: - we consider the character string Honolulu with k = 8 characters - the alphabet ist Z = H, o, n, l, u - with probabilities p H = 1 8, p o = 1 4, p n = 1 8, p l = 1 4, p u = 1 4 we calculate the total information by using the log basis a = 2 to get the result in the unit of bits: I = 8 i=1 log 2 p i = = 18 bit we need 18 bit to optimally code the word Honolulu in a binary basis /RG - 34

32 Elementary Logic Gates in any computation: n-bit input is tranferred into l-bit output f: 0, 1 n 0, 1 l can be decomposed into sequence of elementary logical operations logical (one-bit and two-bit) gates: AND, OR, XOR, NOT, NAND, NOR, XNOR, FANOUT, SWAP one-bit gates I: Y = A A Y two-bit gates AND: A B A B Y NOT: Y = A A Y ANSI/IEEE Std 91/91a-1991 OR: A B A B Y /RG - 35

33 Universal Logic Gates any logical function f: 0, 1 n 0, 1 l can be constructed by a universal set of elementary gates: (i) AND, OR, NOT, FANOUT (ii) NAND, FANOUT A A A the FANOUT gate is acting as a copying machine for classical bits we will see later that this gate cannot be realized for a quantum computer: no cloning theorem /RG - 36

34 Universal Logic Gates the AND gate is not logically reversible AND: Y = A B A B Y therefore, the (non-reversible) AND gate throws away or erases information Landauer showed that erasing a bit of information results in energy dissipation Landauer principle /RG - 37

35 Information and Energy Landauer principle (1961): each time a single bit of information is erased, the amount of energy dissipated into the environment is at least k B T ln 2 equivalently, we may say that the entropy of the environment is increased by at least k B ln 2 example: computer with 10 9 gates operated at 3 GHz clock speed at 300 K dissipates at least Rolf William Landauer (February 4, 1927 April 28, 1999) P = ln 2 10 mw R. Landauer, "Irreversibility and heat generation in the computing process," IBM Journal of Research and Development, vol. 5, pp , /RG - 38

36 Reversible Computation most of the classical logic gates are irreversible we cannot recover the input from a given output the Boolean operators erase a bit of information energy dissipation is unavoidable (Landauer principle) is it possible to do reversible computation without energy consumption? AND A B Y Yes, replace irreversible gates by reversible generalization Identity and NOT gate are reversible important reversible two-bit gate is CNOT (reversible XOR) A B additional three-bit gate required: Toffoli gate (C-CNOT) A = A B = A B A + B mod2 CNOT A B A B C. H. Bennett, "Logical reversibility of computation," IBM Journal of Research and Development, vol. 17, no. 6, pp , /RG - 39

37 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 40

38 What do we understand by»quantumquantum«is an amount of energy that can no longer be subdivided quantum hypothesis of Max Planck (1900): E = h ν light quanta are required to heat up 1 liter of H 2 O to 100 C quantum physics: theories, models and concepts based on the quantum hypothesis of Max Planck quantum mechanics and theory of relativity: foundations of modern physics microcosm macrocosm quantum jump: transition between two quantum states /RG - 41

39 »Quantum« /RG - 42

40 Excursion to the quantum world quantum objects are wave and particle at the same time diffraction of light at double-slit diffraction of electrons at double-slit /RG - 45

41 Excursion to the quantum world quantum tunneling tunneling of a wave packet quantum superposition uncertainty relation a physical system e.g. an electron exists partly in all its particular theoretically possible states simultaneously e.g. Schrödinger cat /RG - 46

42 Weird Quantum World /RG - 47

43 Weird Quantum World /RG - 48

44 Quantum Entanglement entangled quantum states: two or more particles can show nonlocal correlations such that their quantum states can no longer be described independently Ψ e = 1 2 ( 0 A0 B + 1 A 1 B ) (entangled state) Ψ Ψ A Ψ B Ψ H A H B Ψ s = 1 2 ( 0 A1 B + 1 A 1 B ) (separable state) = 1 2 ( 0 A + 1 A ) 1 B Ψ = Ψ A Ψ B Ruth Bloch, Entanglement II, bronze, 27" (2000) quantum correlation: measurements of observables of entangled particles are correlated decoherence: entanglement is broken through the interaction with the environment /RG - 49

45 EPR Paradox Can the quantum mechanical description of the physical reality be considered complete? Are there hidden variables? A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935) Ψ = 1 2 ( P AQ B Q A P B ) P = 1 ( + ) ; Q = 1 ( ) 2 2 measurement in basis { P, Q } Einstein (1935): spuky action at a distance Schrödinger (1935): entanglement A B A B Bell (1964): principle of locality is in conflict with quantum theory J.S. Bell, Physics 1, (1964) /RG - 50

46 Entangled Dices A B Ψ = A B A B /RG - 51

47 Bell s Inequality (a simple logic exercise) we consider a collection of objects with parameters A, B, and C: e.g. A = male, B = taller than 1.8 m, C = blue eyes (classroom example) we proof that the number of objects which have parameter A but not parameter B plus the number of objects which have parameter B but not parameter C is greater than or equal to the number of objects which have parameter A but not parameter C: # A, not B + # B, not C #(A, not C) this relationship is called Bell's inequality J.S. Bell, Physics 1, (1964) John Stewart Bell ( ) /RG - 52

48 Bell s Inequality (a simple logic exercise) Proof (has nothing to do with quantum mechanics): we assert that # A, not B, C + # not A, B, not C 0 pretty obvious, since either no group members have these combinations or some members do we add # A, not B, not C + # A, B, not C on both sides # A, not B, C + # A, not B, not C + # not A, B, not C + # A, B, not C 0 + # A, not B, not C + # A, B, not C # A, not B # B, not C # A, not C since either C or not C must be true since either A or not A must be true since either B or not B must be true # A, not B + # B, not C #(A, not C) q.e.d. no other assumptions made than (i) logic is a valid way to reason (ii) parameters A, B, C exist whether they are measured or not (there is a reality separate from its observation) /RG - 53

49 Bell s Inequality (applied to electron spin) Bell s inequality # A, not B + # B, not C #(A, not C) #, + #, #(, ) spin directions: A: θ = 0 = B: θ = 45 = C: θ = 90 = not A: θ = 180 = not B: θ = 135 = not C: θ = 90 = measurement of #,, #,, #(, ) e-gun measurement of # of electrons with,,, or, gives 50% for each projection but if we try to measure and at the same time, we have a problem: only 15% are (and 85% would be ), if we have measured before the preceeding measurement of # irrevocably changes # in the same way: the measurement of # irrevocably changes # in classroom example this would mean that measuring the gender would change their height: pretty weird but true for electron spins /RG - 54

50 Bell s Inequality (applied to electron spin) measurement of spin directions #, : if we have measured in the first measurement, the probability to find in the second is. 2 = cos = #, : if we have measured in the first measurement, the probability to find in the second is. 2 = cos 135 = #(, ): if we have measured in the first measurement, the probability to find in the second is 2 = cos = 0.5 arbitrary spin state: Ψ = cos Θ 2 + eiφ sin Θ 2 #, + #, = #, = 0. 5 violation of Bell s inequality first experiments demonstrating violation: Clauser, Horne, Shimony and Holt in 1969 using photon pairs /RG - 55

51 Violation of Bell s Inequality assumptions made in deriving Bell s inequality (i) logic is a valid way to reason (ii) parameters A, B, C exist whether they are measured or not electrons have spin in a given direction even if we do not measure it there is a reality separate from its observation hidden variables exist (iii) no information can travel faster than the speed of light locality hidden variables are local violation of Bell s inequality is in conflict with local realism!!...but what if logic is not valid? /RG - 56

52 Relevance of Quantum Phenomena Will quantum effects be part of our everyday life? Yes, they already are! interesting applications quantum technologies quantum computation quantum communication quantum simulation quantum metrology /RG - 57

53 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 58

54 The Quantum Bit classical bit (two distinct states) 0 or 1 or or or decisive quantum bit (arbitrary superposition of two quantum states computational basis) Ψ = α 0 + β 1 with α 2 + β 2 = 1 Ψ = α + β indecisive measurement ( collapse of wave function) Ψ = α + β with probability α 2 observer /RG - 59

55 The Quantum Bit Bloch sphere representation (geometrical picture of qubit) 0 or g ground state Ψ = cos Θ eiφ sin Θ 2 1 = cos Θ 2 e iφ sin Θ 2 1 or e excited state /RG - 60

56 The Quantum Bit quantum bit (mathematical picture of qubit): - two-level quantum system whose state is represented by a ket lying in a 2D Hilbert space H, which has the orthonormal basis 0, 1 - each ket can be thought of as a column vector 0 = 1 0 and 1 = 0 1 Ψ = α 0 + β 1 = α β 0 1 = α β tensor product of Hilbert spaces: H = H 1 H 2 H n state in 2 n -dimenisonal Hilbert space: Ψ = Ψ 1 Ψ 2 Ψ n = Ψ 1 Ψ 2 Ψ n example: H = H 1 H = 01 = 0 1 = = = /RG - 61

57 Measuring the State of a Qubit example: consider a 2-dim. quantum system in state Ψ = α 0 + β 1 what happens if we measure Ψ in the basis ± = ± 1? - we first express Ψ in the basis ± : Ψ = α+β α β thus measurement of Ψ in the basis ± yield two possible results: Ψ Ψ probability = α+β 2 2 probability = α β 2 2 note: we cannot completely control the outcome of the measurement /RG - 62

58 Representing Integers let H 2 be a 2-dim. Hilbert space with orthonormal basis 0, 1 then, H = 0 n 1 H 2 is a 2 n -dim. Hilbert space with induced orthonormal basis 0 00, 0 01, 0 10, 0 11,, 1 11 we represent the integer m with binary expansion m = n 1 j=0 m j 2 j, m j = 0 or 1, j as the ket m = m n 1 m n 2 m 1 m 0 example 23 = /RG - 63

59 The No Cloning Theorem quantum bit cannot be copied copying machine proof: - assume that there is a unitary operator U producing copies of A and B U A blank ] = AA and U B blank ] = BB - however, the quantum copying machine fails in copying state C = 1 ( A + B ) 2 U C blank ] = 1 ( AA + BB ) CC 2 remark: - cloning is inherently nonlinear - quantum mechanics is inherently linear quantum replicators do not exist /RG - 64

60 Massive Parallelism example: Ψ i = 1 2 ( ) for i = 1,2,3,, n then Ψ 0 Ψ 2 Ψ n 1 = n 1 i=0 = 1 2 = 1 2 n n 1 ( ) 2 ( ) ( ) ( ) ( ) = 1 2 n 2 n 1 a=0 a the n-qubit register contains all n-bit binary numbers simultaneously!!! n classical bits can store a single integer I, the n-qubit quantum register can be prepared in the corresponding state I of the computational basis, but also in a superposition /RG - 65

61 Massive Parallelism: Deutsch s Problem classical machine: 0 1 x f(x) f(0) = 0 or 1 f(1) = 0 or 1 we are satisfied to know whether f x = const (f(0) = f 1 ) or balanced (f 0 f 1 ) we have to run the machine twice to find this out quantum machine: 0 1 x y f( x y f x ) machine flips the second qubit if f acting on the first qubit is 1, and does not do anything if f acting on the first qubit is 0 we can determine if f x is constant or balanced by using the quantum black box twice. Can we get the answer by running the quantum black box just once? ( Deutsch s problem ) choose the input state to be a superposition of 0 and 1 x x 1 2 f(x) 1 f(x) = x 1 f(x) we have isolated the function f in an x-dependent phase /RG - 66

62 Massive Parallelism: Deutsch s Problem we also prepare the first qubit x in superposition state f(0) f(1) we perform a measurement that projects the first onto the basis ± = ± 1 we will always obtain + if the function is balanced and if it is constant the classical computer has to run the black box twice to distinguish a balanced function from a constant function, but a quantum computer does the job in one go! suppose we are interested in global properties of a function that acts on N bits, a function with 2 N possible arguments to compute a complete table of values of f(x) we have to calculate f exactly 2 N times (completely infeasible for N 1) with the quantum machine we can choose the input register to be in a state 1 N which requires to compute f x only once!! 2 speedup by massive quantum parallelism /RG - 67

63 Entangled Qubits U 0 A 1 B unitary transformation 0 A 1 B 1 A 0 B 2 not entangled separable entangled not separable definition: if a pure state Ψ H A H B can be written in the form Ψ = Ψ A Ψ B where Ψ i is a pure state of the i th subsystem, it is said to be separable /RG - 68

64 Observing Entangled Qubits 0 A 1 B 1 A 0 B 2 observes only the blue qubit whoosh!! 0 A 1 B with probability 1/2 1 A 0 B with probability 1/2 no longer entangled separable /RG - 69

65 Elementary Logic Gates classical computer: - n classical bits form a register of size n - sequence of elementary operations (e.g. AND, NOT) produce a given logic function quantum computer: - n quantum bits form a quantum register of size n - sequence of elementary operations (QUANTUM GATES) produce a given logic function typical sequence for quantum computation: initialization: manipulation: readout: prepare the quantum computer in a well-definded initial state apply elementary quantum gates to manipulate quantum state perform a quantum measurement at the end of the algorithm /RG - 70

66 Quantum Processor: Principle of Operation Qubit Bloch sphere 0 e g U 1 1-Qubit-Gate 1 Ψ = cos Θ eiφ sin Θ 2 1 M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000) /RG - 72

67 Quantum Processor: Principle of Operation Qubit 2-Qubit-Gate (C-NOT) e g e g e g g e readout U 1 U 1 U 1 1-Qubit-Gate M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000) /RG - 73

68 Single Qubit Gates single qubit gates: - rotate the state vector on the Bloch sphere - are represented by unitary matrices: UU = I Hadamard gate: - maps the basis state 0 to 1 ( ) and 1 to 1 ( 0 1 ) represents a rotation of π/2 about the axis 1 ( x + z) 2 H = A B B phase shift gate: - leaves the basis state 0 unchanged, maps 1 to e iφ 1 - equivalent to tracing a horizontal circle on the Bloch sphere by φ radians R φ = e iφ /RG - 74

69 Two Qubit Gates SWAP gate: - swaps two qubits SWAP = A B B A 00 = 10 = = 11 = CNOT gate: - performs the NOT operation on the second qubit only when the first qubit is 1 and otherwise leaves it unchanged CNOT = A B A B = A B A + B mod /RG - 75

70 Universal Quantum Logic Gates classical computation: any logical function can be constructed by a universal set of elementary gates: NAND FANOUT quantum computation:any logical function can be decomposed into one-qubit and two-qubit CNOT gates H, R φ, A CNOT A A /RG - 76

71 Gate Errors quantum computation is performed by a sequence of quantum gates applied to some initial state Ψ 0 : Ψ n = n i=1 U i Ψ 0 the unitary operations form a continuous set and any realistic implementation involves some error (operator V i slightly differing from perfect U i ): Ψ i = U i Ψ i 1 Ψ i = Ψ i + E i = V i Ψ i 1 error E i = (V i U i ) Ψ i 1 after n iterations: Ψ n Ψ n < n ε V i U i sup < ε (sup norm of operator V i U i ) unitary errors accumulate at worst linear with length of computation this takes place for systematic errors, for stochastic errors we expect a n growth /RG - 77

72 Quantum Decoherence qubits are coherent superpositions of two computational basis states: Ψ = cos Θ eiφ sin Θ 2 1 quantum decoherence is the loss of coherence or ordering of the phase angles between the components in the quantum superposition due to interaction with the environment (unobservable quantum degrees of freedom) Uni Erlangen example: two wave packets interfere to form interference fringes (left pattern) interaction with the fluctuating environment (wavy orange lines) blurs the interference pattern (right pattern) decoherence produces a gradual crossover between wave-like phenomena (interference) and particle-like behavior (classical, localized particles with well-defined trajectories) /RG - 78

73 Quantum Decoherence quantum decoherence occurs when a quantum system interacts with its environment in a thermodynamically irreversible way entanglement with the environment due to finite coupling quantum decoherence can be viewed as the loss of information from a quantum system into the environment the dynamics of the isolated quantum system is non-unitary, although the combined system plus environment evolves in a unitary fashion the dynamics of the quantum system alone is irreversible quantum decoherence represents a challenge for quantum computers, since they rely heavily on the undisturbed evolution of quantum states decoherence has to be managed, in order to perform quantum computation /RG - 79

74 A Few Words on Nomenclature quantum decoherence: originates from the quantum effect of the environment, entanglement with the environment dephasing: describes the effect that coherences, i.e. the off-diagonal elements of the density matrix, get reduced in a particular basis, namely the energy eigenbasis of the system dephasing may be reversible if it is not due to decoherence, as revealed, e.g. in spinecho experiments. phase averaging: a classical noise phenomenon entering through the dependence of the unitary system evolution on external control parameters which fluctuate examples: (i) vibrations of an interferometer grating, (ii) fluctuations of the classical magnetic field empirically, phase averaging is often hard to distinguish from decoherence dissipation: energy exchange with the environment leading to thermalization usually accompanied by decoherence /RG - 80

75 Quantum Processor at Work algorithm J.-S. Tsai, Proc. Jpn. Acad., Ser. B 86 (2010) /RG - 81

76 Quantum Algorithm quantum algorithm: sequence of simple elementary logic gate operations important difference to classical computation: the algorithm performed by the quantum computer may be a probabilistic algorithm if we run exactly the same program twice we obtain different results because of the randomness of the quantum measurement process the quantum algorithm actually generates a probability distribution of possible outputs example: in fact, Shor s factoring algorithm is not guaranteed to succeed in finding the prime factors it just succeeds with a reasonable probability that s okay though because it is easy to verify whether the factors are correct /RG - 82

77 A Quantum Computation E. Knill, Nature 463, 441 (2010) a. initialize qubits in state 00 b. generate superposition state Ψ = a 00 + a 01 + a 10 + a 11 with a = 0.25 c. apply quantum algorithm making use of quantum parallelism d. exploit interference to concentrate amplitudes on the marked configuration e. perform quantum measurement (in the shown case the outcome is deterministic and reveals the location of the mark at 10 ) /RG - 83

78 Contents a brief history of computation... from mechanical to quantum mechanical information processing computational complexity classical computation the weird world of quantum mechanics quantum computation quantum computers summary... where we are and where we hope to go /RG - 84

79 What do we need to build a quantum computer? Qubits: physical medium (hardware platform) that can support quantum systems with two distinguishable states One- and two-qubit gates: techniques to precisely manipulate and control the qubit state without introducing unwanted computational errors Coherence: adequate isolation of the qubits from the environment to avoid decoherence of quantum states Quantum error correction: method to correct for unavoidable computational errors Readout process: fast single-shot readout process with high fidelity Quantum algorithm: suitable sequence of simple elementary logic gate operations (software) /RG - 85

80 Hardware Platforms... there are many physical systems participating in the game trapped ions (may soon be used for quantum simulation) atoms in cavities: cavity & circuit QED systems superconducting quantum circuits (current runner up) nitrogen vacancies in diamond nuclear & electron spins optical systems: photons mechanical systems: phonons electrons on superfluid helium experimental techniques for manipulation and control are often demanding /RG - 86

81 The Nobel Prize in Physics 2012 Serge Haroche David J. Wineland The Nobel Prize in Physics 2012 was awarded jointly to Serge Haroche and David J. Wineland "for groundbreaking experimental methods that enable measuring and manipulation of individual quantum systems" /RG - 87

82 Light-Matter Interaction study of light matter interaction on a fundamental quantum level Cavity QED consequences of the quantum nature of light on light-matter interaction quantum mechanical control and manipulation of light and matter basis for quantum information technology & metrology /RG - 88

83 Cavity & Circuit QED cavity QED natural atom in optical cavity Rempe group circuit QED solid state circuit = artificial atom in µ-wave cavity WMI e.g. Kimble and Mabuchi groups at Caltech Rempe group at MPQ Garching,. e.g. Wallraff (ETH), Martinis (UCSB), Schoelkopf (Yale), Nakamura (Tokyo), Gross (Garching),. advantages of solid state systems: - design flexibility - tunability & manipulation - strong & ultrastrong light-matter interaction achievable - scalability /RG - 89

84 ... Solid State Circuits Go Quantum today near future far future multi electron, spin, fluxon, photon devices single/few electron, spin, fluxon, photon devices quantum electron, spin, fluxon, photon devices classical description quantifiable, but not quantum quantum description Intel PTB WMI 65 nm process 2005 single electron transistor superconducting qubit 2 µm /RG - 90

85 ... Towards Quantum Electronics conventional electronic circuits quantum electronic circuits Kondensator Spule Widerstand Diode - superposition states - entanglement superconducting flux quantum bit (superposition of clockwise and anticlockwise circulating persistent currents) /RG - 92

86 Superconducting Quantum Switch M. Mariantoni et al. Phys. Rev. B 78, (2008) A. Baust et al., Phys. Rev. B 91, (2015); arxiv: /RG - 97

87 Circuit-QED: Energy Scales resonator solid state atom w r w ge superconducting circuit QED nanomechanical systems ω r /2π ω ge /2π Hz ω r /2π Hz 1 GHz 50 mk ħω r J ultra-low temperature experiments ultra-sensitive µ-wave experiments /RG - 99

88 mk Technology for SC Quantum Circuits /RG - 100

89 Optical mk temperature 1 GHz ~ 50 mk ħω r ~ J /RG - 101

90 Optical mk temperature 1 GHz ~ 50 mk ħω r ~ J /RG - 102

91 One- and Two-Qubit Gates have sufficiently accurate quantum gates been demonstrated? no, and this is one of the key as-yet-unmet challenges present consensus/believe: for practical scalability, the probability of error introduced by the application of quantum gates must be less than 10-4 requirements for qubit-state initialization and measurement are more relaxed: /RG - 103

92 Moore s Law for Qubit Lifetime coherence time of superconducting qubits has been improved dramatically within only a decade superconducting qubits M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013) /RG - 104

93 Quantum Error Correction does the analog nature of configuration amplitudes (as opposed to classical digital computers) cause problems? no do the quantum gates have to be increasingly accurate as the number of gates is growing? why? no it is possible to digitize computations arbitrarily accurately by applying quantum error correction strategies error correction removes effects of computational errors and decoherence processes this requires relatively limited resources, provided that enough requirements for building a quantum computer are met /RG - 105

94 Superconducting Quantum Computer... towards superconducting quantum circuits, computers, simulators,... Photo credit: Erik Lucero Martinis UCSB and Google, superconducting quantum circuit with five Xmon qubits /RG - 106

95 Superconducting Quantum Computer /RG - 107

96 Superconducting Quantum Computer D-Wave One Systems /RG - 109

97 Superconducting Quantum Computer Photo: IBM IBM three-qubit chip: basis for a much larger quantum computer transmon qubit /RG - 110

98 Superconducting Quantum Circuit UCSB & chip with 9 X-mon qubits State preservation by repetitive error detection in a superconducting quantum circuit, J. Kelly et al., Nature 519, (2015) /RG - 111

99 Quantum Information Processing for what? Research Web Finances Optimization problems Graph theory problems Material science Pharmaceuticals Quantum chemistry Climate modeling Bioinformatics Weather predictions Image and pattern recognition Machine Learning Communication Advanced Search Risk modeling Trading strategies Financial forecasting Credit: istockphoto Google, IBM, D-Wave, Microsoft, Lockheed Martin, NASA, /RG - 112

100 QIP Perspectives When will we have a quantum computers, when will they outperform classical computers, which will be the best hardware platform,...?? long term predictions... I think there is a world market for maybe five computers. Thomas J. Watson, chairman of IBM, 1943 Whereas a calculator on the Eniac is equipped with vacuum tubes and weighs 30 tons, computers in the future may have only 1000 tubes and weigh only 1½ tons Popular Mechanics, March 1949 There is no reason anyone would want a computer in their home. Ken Olson, president, chairman and founder of DEC, are very likely to be wrong!! /RG - 113

101 Summary rapid progress in quantum information technology quantum computation is close to become reality quantum communication is at the horizon quantum simulation and quantum metrology are attracting growing interest quantum technology important for fundamental physics experiments...the future looks bright /RG - 114

102 The WMI team Thank you! /RG - 115