10.2 Introduction to quantum information processing

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1 AS-Chap Introduction to quantum information processing

10. Introduction to information processing AS-Chap.

physical (Landauer 1991) - Ink on paper - Charge on capacitor

2 10. Introduction to information processing AS-Chap Information General concept (similar to energy) Many forms: Mechanical, thermal, electric,... Can be packed into equivalent forms: 0 1 Information is physical (Landauer 1991) - Ink on paper - Charge on capacitor - Currents in leads - Spins - Information has uncertainty! Shannon entropy Measure of this uncertainty

10. Introduction to information processing AS-Chap. 10-3 Konrad Zuse (1945) Built the first binary digital computer ( Z1 ) in 1938.

3 10. Introduction to information processing AS-Chap Konrad Zuse (1945) Built the first binary digital computer ( Z1 ) in The first programmable electromechanical computer ( Z3 ) was completed in Zuse also developed the first algorithmic programming language called Plankalkül.

4 10. Introduction to information processing AS-Chap John von Neumann (1945) Proposed the EDVAC computer in Introduced the concept of a computer that is controlled by a stored program.

000 vacuum tubes Weight of 30 tons Space of 160 m² Required 6 operators Left to right: H.

5 10. Introduction to information processing The electronic numerical integrator and computer (ENIAC, 1946) Built at the University of Pennsylvania vacuum tubes Weight of 30 tons Space of 160 m² Required 6 operators Left to right: H.Wexler, J. von Neumann, M. H. Frankel, J. Namias, J. C. Freeman, R. Fjortoft, F. W. Reichelderfer, and J. G. Charney. AS-Chap. 10-5

growth in the number of transistors per integrated circuit

6 10. Introduction to information processing AS-Chap Moore s law In 1965, Moore observed an exponential growth in the number of transistors per integrated circuit Prediction: trend continues

7 10..1 Computational complexity AS-Chap Example: Is m a prime number? Complexity of a yes/no -problem Expressed in terms of required ressourses (memory space & computing time) as a function of the problem size N Prime number factorization N = log m (number of bits) L Logarithmic memory space O ln N PSPACE Polynomial memory space Chess computer P, QP Polynominal execution time Multiply two numbers O N NP EXP Not solvable in polynominal time Travelling salesman problem Exponential execution time Simple factorization ~ N/ Easy: 7919 x =? Hard: =? O ln N k O N k O k N Time - n o of operations n o of digits exp(n) N Known relations between the complexity classes L P NP PSPACE EXP

8 10..1 Computational complexity: Factorization AS-Chap Integer factorization NP hard on classical computer! bits Classical computer 10 3 Quantum computer (clock 100 MHz) (years) bits 51 bits miniaturization limit year of fabrication (minutes) bits 104 bits 048 bits 4096 bits n o of bits Algorithm is known, but it takes too long Exponential speedup by quantum algorithm (Shor)

10..1 Computational complexity: Factorization Integer factorization Best known classical algorithm (number field sieve) Required time exp ln m 1 3 ln ln m 3 Exponential in bit number N log m

9 10..1 Computational complexity: Factorization Integer factorization Best known classical algorithm (number field sieve) Required time exp ln m 1 3 ln ln m 3 Exponential in bit number N log m Factorization of 400-digit decimal takes years Peter W. Shor Shor quantum algorithm (Peter Shor, 1994) Required time ln m 3 Factorization of 400-digit decimal takes a few years Largest known prime number (013): digit decimal Quantum algorithms can reduce the execution time to human timescales, where classical algorithms would take forever in terms of these timescales! AS-Chap. 10-9

AS-Chap. 10-11 10..1 Computational complexity: Quantum simulation Example: N interacting spins (S = ½).

10 AS-Chap Computational complexity: Quantum simulation Example: N interacting spins (S = ½).. N = 1000 Dimension of Hilbert space = 1000 > number of atoms in universe Richard Feynman (1981): 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. Quantum simulation Encode the dynamics of a difficult-to-access quantum system in another quantum system, which can be more easily accessed Example Advantage Dirac dynamics of a relativistic electron in (nonrelativistic) trapped ion system (Nature 463, 68-71, 010) Less demanding than a universal quantum computer

11 10.. Improvement of IP systems Exploition of new functionalities field electronics opto electronics fluxonics mechatronics spintronics degree of freedom charge charge + optical degree of freedom charge + fluxonic degree of freedom charge + mechanical degree of freedom charge + spin degree of freedom RSFQ-Logic ultrafast AD-converters.. MRAM GMR read heads spin transistor.. AS-Chap. 10-1

electron, spin, fluxon, photon devices single

electron, spin, fluxon, photon devices classical

quantum description tunnel junctions gate WMI 1

12 10.. Improvement of IP systems AS-Chap today near future far future multi electron, spin, fluxon, photon devices single electron, spin, fluxon, photon devices quantum electron, spin, fluxon, photon devices classical description quantifiable, but not quantum quantum description tunnel junctions gate WMI 1 µm Intel WMI 65 nm process 005 single electron transistor superconducting qubit

13 10.. Improvement of IP systems Error-corrected quantum computing??? Scaling Enigma (1940) WMI Superconducting qubit (000) 9-qubit chip (UCSB/Google, 015) New physics Same physics Scaling Vacuum tubes ENIAC (1946) New technology First transistor (1947) Bardeen, Brattain, & Shockley Intel 45 nm chip (007) AS-Chap

14 AS-Chap Classical Quantum mechanical A p(t) x(t) B A p(t) x(t) B Point on a classical trajectory x(t) and p(t) can be measured simultaneously at arbitrary precisions No noise Δx = Δp = 0 Classical noise Probability distribution Nonnegative B 1 mv path V( r ) dt Quantum parallelism Superposition of states Entanglement of states Vacuum noise (quantum) Δx Δp ħ/ Quasiprobability distribution Can become negative (e.g., Fock states) A

15 AS-Chap Superposition of (basis) states A quantum degree of freedom is described through a wave function. Ruth Bloch, bronze, 7" (000) Entanglement of states Two quantum degrees of freedom can exhibit stronger correlations than any classical system.

16 AS-Chap (Quantum) Entanglement Uniquely quantum mechanical property of a composite system No/restricted knowledge on subsystems despite perfect knowledge on combined system Quantum correlations Entanglement between spatially separate subsystems Quantum teleportation Quantum communication Quantum illumination (radar)

17 AS-Chap Einstein-Podolsky-Rosen (EPR) paradoxon Example: Two spins Alice Bob Uncorrelated: 4 product states,,, Correlated: Linear combination of product states Entanglement Ψ = 1 ± Alice and Bob perform measurements on the entangled spins Suppose: Then: Alice first measures her spin and finds (50% chance) Bob always measures his spin (100%), although he may be far away from Alice Quantum mechanics is nonlocal! Repetition of the experiment Always the same result The two entangled spins are fully correlated

18 AS-Chap No-cloning theorem Classical bits can be copied easily: C C C Quantum bits (quantum states) cannot be copied No-cloning theorem Proof: Assume that there is a unitary transformation U producing copies of α and β U α0 = αα and U β0 = ββ However, the quantum copying machine fails in copying state γ0 = 1 ( α + β ) U γ0 = 1 ( αα + ββ ) γγ Combination of the EPR paradoxon and the no-cloning theorem Rescues the consistency between quantum mechanics and special relativity If Alice and Bob would share an entangled sequence of pairs containing segments of copies, they could communicate faster than light!

Teleportation Teleporting a quantum state (qubit) requires that Alice and Bob share an entangled state Ψ AB = 1 + ( EPR pair ) Teleportation protocol 1.

19 AS-Chap Quantum teleportation No-cloning theorem forbids copying state φ = a + b However, vanishing at one place and reappearing at another is allowed Teleportation Teleporting a quantum state (qubit) requires that Alice and Bob share an entangled state Ψ AB = 1 + ( EPR pair ) Teleportation protocol 1. Alice entangles her spin with the unknown state φ Alice. Alice measures what state her two spins are and tells Bob, which of the four possible results she has found Classical communication 3. Bob carries out the appropriate rotation of his spin by π Bob 4. As a result, Bob ends up with his spin in the state φ = a + b

20 AS-Chap Definition of a quantum bit Classical bit Deterministic, either in ground state g or in excited state e Quantum bit (qubit) Superposition of two computational basis states Ψ t = a t g + b t e a t, b t C with a(t) + b t = 1 All states can be visualized on the surface of a sphere Global phase unobservable Bloch sphere representation Ψ t = cos θ(t) e + e iφ(t) sin θ(t) g e z θ(t) φ(t) Ψ(t) y Bloch angles θ t Amplitude Energy, population φ(t) Phase Coherence x g

21 AS-Chap Linear algebra notation of operators and state vectors Qubit states can be written as vectors a e + b g a b 0 1 = a b Qubit operators (gates) can be written as matrices a e e + b g g + c e g + d g e a b c a 0 1 a c 1 0 = d b

22 AS-Chap Pseudo spin and Pauli matrices Ψ t = cos θ(t) e + e iφ(t) sin θ(t) g Pseudo spin Ψ equivalent to spin wavefunction in external magnetic field Unitary operations U Ψ expressed via the Hermitian Pauli spin matrices 1, σ x, σ y, σ z σ x σ y 0 i i 0 σ z g and e are the eigenvectors of σ z

23 AS-Chap Conventions: Pauli matrices and Bloch sphere σ x σ y 0 i i 0 σ z These definitons contain several conventions, such as The global scaling factor The positon of the minus sign in σ z Here, we show two examples with fixed σ z Physics convention g 0 1, e 1 0 Ground state energy negative (more physical ) Ψ t = cos θ(t) e + e iφ(t) sin θ(t) g Information theory (IT) convention M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press g 0 1 0, e Ground state energy positive ( unphysical ) Easily generalized, more logical Unless otherwise mentioned Physics convention! Ψ t = cos θ(t) g + e iφ(t) sin θ(t) e

24 AS-Chap Important states on the Bloch sphere Physics e z IT g z e g g e e i g g i e e + g e + i g y g + e g + i e y x g x e Ψ t = cos θ e + eiφ sin θ g Ψ t = cos θ g + eiφ sin θ e

25 AS-Chap Interpretation of the Pauli matrices σ x σ y 0 i i 0 σ z The Pauli matrices can expressed in terms of projection operators σ = g σ + = e e g Puts an excitation into the qubit Removes an excitation from the qubit e g e g σ x = σ + σ + σ y = i σ i σ + Induce transitions between g and e e g σ z = e e g g σ z gives the qubit population?? e g 1 = g g + e e Reflects normalization Projection-operator-based definitions change between conventions when matrix forms are fixed! σ x = g e + e g σ y = i e g i g e σ z = g g e e Fixed matrix form for σ ± are possible, but quite unintuitive Not in this lecture!

26 AS-Chap Quantum coherence Ψ t = cos θ(t) e + e iφ(t) sin θ(t) θ t Amplitude Energy, population φ(t) Phase Coherence g Ideal quantum system Completely isolated In reality, however, Environment must interact with Ψ t for control Uncontrolled interactions (noise) also exist Quantum effects (population oscillations, quantum interference, superpositions, entanglement) unobservable after characteristic time Decoherence time T dec After T dec, quantum effects have decayed to 1 e of their original level T dec is a time scale rather than a strict time Term decoherence originally only referred to phase Nowadays sloppily comprises both phase and amplitude effects

27 AS-Chap Energy and phase relaxation Ψ t = cos θ(t) Population e + e iφ(t) sin θ(t) θ t Amplitude Energy, population φ(t) Phase Coherence Energy relaxation time T 1 or T r k B T ħω ge decay from e to g Nonadiabatic (irreversible) processes Induced by high-frequency fluctuations (ω ω ge ) g Phase Pure dephasing time T φ Adiabatic (reversible) processes Induced by low-frequency fluctuations (ω 0) Often encountered: 1/f-noise Real measurements always contain T 1 -effects T 1 = T T 1 φ Nomenclature not very consistent in literature!

Definition of a single qubit gate Single qubit gate Unitary operation U on state Ψ Rotations on Bloch sphere + global phase Rotation matrices About x-axis R x α e iα σ x = cos α i sin α About y-axis

28 Definition of a single qubit gate Single qubit gate Unitary operation U on state Ψ Rotations on Bloch sphere + global phase Rotation matrices About x-axis R x α e iα σ x = cos α i sin α About y-axis R y α e iα σ y = cos α sin α i sin α cos α sin α cos α About z-axis R z α e iα σ z General unitary = e i α 0 α U = e iα R z β R y γ R z δ with α, β, γ, δ R Z-Y decomposition α is a global phase (unobservable) Other decompositions possible 0 e i AS-Chap. 10-9

29 AS-Chap Examples for 1-qubit gates: Examples for 1-qubit gates: NOT

30 Hadamard gate H is of particular importance Applied to one of the basis states g or e, it results in a superposition state of the basis states H 1 Physics convention = 1 σ x + σ z H = 1 e e g g + e g + g e e z e g H g = 1 ( e g ) H e = 1 ( e + g ) e + g x g y AS-Chap

31 AS-Chap Realization of a quantum two-level system Natural systems: Effective systems: E Natural spins, Orthogonally polarized light, Isolate two levels from a manifold structure Mechanisms: Energy separation, selection rules, Requires nonlinearity! 3 Other system states e g Two-level system

32 quantum optics trapped ions microscopic NMR s + meso- and macroscopic solid state devices CeNS NIST, Innsbruck, Munich contact F WSI Oxford, Stanford, IBM,MIT ENS Paris easily quantum, but difficult to scale WMI µm scalable, but not easily quantum AS-Chap

33 AS-Chap single particle states semiconductors global states superconductors nuclear spins of P in Si (Kane) superconducting flux, charge, phase, charge-phase qubits electron spins in quantum dots TU Delft...

34 AS-Chap Single-qubit system: Ψ = c c 1 Basis states (IT): Two-qubit system: Ψ = c c 10 + c c 4 11 Basis states (IT): Tensor product

35 AS-Chap Two-qubit operators: Tensor product (blockwise product) of single qubit operators

Two-bit-gates: NOT AND OR Spin Polarization Persistent

36 AS-Chap Classical bits 0,1 E.g., implementation by charge states Quantum bits Ψ t = cos θ e + eiφ sin θ g Implementation by any quantum two-level system 0: Q = 0 1: Q = Q 0 Basic gates: Single-bit-gate: Two-bit-gates: NOT AND OR Spin Polarization Persistent current Basic gates: Single-qubit-gate: Two-qubit-gates: rotations Hadamard C-NOT

37 AS-Chap Classical computer: Bits & gates Bits Capacitors Gates transistors V = 0 0 V > 0 1 V g = 0 V g > 0 closed open Example for a singlebit gate NOT Example for a two-bit gate AND

38 Universality theorem A small number of Single-bit gates (e.g., NOT) and two-bit gates allows for any manipulation on classical bits (universal set) Truth tables for various -bit gates (a,b) a b GATE f(a,b) Examples a OR b = a NAND a NAND (b NAND b) a AND b = a NAND b NAND (a NAND b) NOT a = a NAND a OR, AND, NOT also form universal set AS-Chap

39 AS-Chap Irreversible classical logic ( current): Logical irreversibility No ability to reconstuct the input from the output Physical irreversibility Heat dissipated during a gate operation Intimate connection Increase of entropy due to loss of 1 bit of information, ΔS = k B ln Dissipative heat Irreversibility Si-based computers Way above the ideal limit by a factor (transistor) 100 (DNA copying mechanism in human cell) Discussion somewhat academic

40 Reversible classical logic (Bennett, 1973) Conditions a b Absence of physical dissipation Number of output bits equals number of inputs Gate produces all input combinations at output a f(a,b) GATE GATE b a f(a,b) (a,b) (a, a NAND b) (a, a XOR b) (0,0) (0,1) (0,0) (0,1) (0,1) (0,1) (1,0) (1,1) (1,1) (1,1) (1,0) (1,0) CNOT gate (reversible XOR) Intrinsically irreversible Logically reversible AS-Chap

41 AS-Chap Three-qubit gates (Bennett, 1973) Universality Reversible logic requires three-bit gates Toffoli gate Controlled CNOT gate (CCNOT) Applies a NOT to bit c when both bit a and bit b are 1 a b c Some two-bit gates are intrinsically irreversible (e.g., NAND) Third bit required to store input safely

42 AS-Chap Quantum gates Intrinsically reversible Any irreversible manipulation would be associated with heat dissipation Destruction of quantum coherence Universal set of gates E.g., single qubit rotations and CNOT Three-qubit Toffoli gate only needed for quantum error correction Represented by unitary transformations - Normalization: length of state vector (on Bloch sphere) stays constant - Reversibility requires that matrix can be inverted - Complex matrix elements, since components of spinor are complex

43 AS-Chap SWAP gate: resp. interchange of a and b

44 AS-Chap Universal quantum processor: Required elements qubit (two-level quantum system) qubit gates (e.g., C-NOT) (controlled interactions) e g e g e g g e read out U 1 single qubit gate U 1 U 1

45 AS-Chap initialization 0> 0>... 0> preparation of superposition states example: 3 bit system computational steps unitary transformations 1 bit gates bit gates program parameter 0> + 1> 0> + 1>... 0> + 1> Y> = a 000> + b 001> + c 010> + d 100> + e 011> + f 101> + g 110> + h 111> quantum algorithm e.g. factorization (Shor) database search (Grover) quantum error correction read out final state e.g. Shor, Steane

46 DiVincenzo criteria for scalable QIP Qubits: The system has to provide a well defined two-level quantum system Preparation of the initial state: It must be possible to prepare the initial state with sufficient accuracy Decoherence: The phase coherence time must be long enough to allow for a sufficiently large number (typically >10 4 ) of coherent manipulations Quantum gates: There must be sufficient control over the qubit Hamiltonian to perform the necessary unitary transformations, i.e., single- and two-qubit operations Quantum measurement: For read-out of the quantum information a quantum measurement is needed Scalability: There should be the possibility to increase to number of qubits D. DiVincenzo, The physical implementation of quantum computation, Fortschr. Phys. 48, 771 (000). AS-Chap

47 AS-Chap generation of entangled state - concurrence of up to 94% L. DiCarlo et al., arxiv: v

AS-Chap. 10-49 You are here! M. H. Devoret and R. J.

48 AS-Chap You are here! M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (013); DOI:10.116/science

10.2 Introduction to quantum information processing

10.2 Introduction to quantum information processing AS-Chap. 10-1 10. Introduction to quantum information processing AS-Chap. 10-10. Introduction to Information Processing Information General concept (similar to energy) Many forms: Mechanical, thermal,