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Jonathan Coslovsky March 2008 Introduction The qubit Calculation (gates) Decoherence and error correction Applications Experimental implementations Experimental progress timescale Summery Silicon microprocessor chip – Since 1960s. The IBM System 360/20 1966 Moore’s law – The number of transistors placed on an integrated circuit is increasing exponentially. Polynomial computing tasks – P. Non-Polynomial computing tasks-NP. Example: Factorization: 15 3x5 91 7x13 703 ?x? 19x37 8876044532898802067 ???????x??????? Answer: 5915587277 x 1500450271 Why is this important? The principle of RSA. Solution #1: wait for a few years for a better (classical) computer. ◦ Problem: Today: L< 1μm. (L-Transistor size) Quantum effects become important: L~λ~nm. (λ-de Broglie wavelength of electrons). Individual atom size: L~a0~Ǻ. Moore’s law will eventually break down. Solution #2: Find a new computing technique. The idea: Take advantage of QM phenomena, such as superposition and entanglement. Is it possible? 1994-Shor, P algorithm for factoring numbers. Classical bit: 0 or 1. voltage on a transistor, magnetization of a ferromagnetic material, or intensity of a pulse of light. Qubit: Superposition in a two state system 01 0 Where E |1> c0 0 c1 1 . |0> 3 qubits register: c000 000 c001 001 c010 010 c011 011 c100 100 c101 101 c110 110 c111 111 Vector representation: c000 c 001 c 111 i , j , k 0,1 An N-qubit register is described by 2N (complex) amplitudes. 2 cijk 1 Quantum system Physical property 0 1 Photon Linear polarization Horizontal Vertical Photon Circular polarization Left Right Nucleus Spin Up Down Electron Spin Up Down Tow-level atom Excitation state Ground state Excited state Josephson junction Electric charge N Cooper pairs N+1 Cooper pairs Superconducting loop Magnetic flux Up Down A single qubit can be represented on a Bloch sphere: cos( ) 0 e sin( ) 1 2 i 2 1 0 Single bit-NOT gate Input bit Output bit 0 1 1 0 Two bits-C-NOT gate Control Input Target Input Target Output 0 0 0 0 1 1 1 0 1 1 1 0 But what is the meaning in a qubit? Quantum gate Matrix representation Bloch sphere representation NOT(X) 0 1 1 0 Rotation of π about the x-axis Z 1 0 0 1 Rotation of π about the z-axis 1 1 1 1 1 2 Rotation of π about the z-axis, rotation of π/2 about the y-axis Hadamard (H) The controlled-NOT gate: i.e. U CNOT 1 0 0 0 0 0 1 0 0 0 0 1 1 0 U CNOT 0 0 0 c00 c00 0 c01 c01 1 c10 c11 0 c11 c10 0 1 0 0 0 0 0 1 0 0 1 0 Example: ControlInput Target Input 0 Step 1: Step 2: 1 1 1 1 1 1 1 H 0 1 1 0 2 2 1 1 1 0 1 0 00 10 2 2 1 0 ' 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 00 11 1 2 1 2 0 2 0 0 1 Any qubit gate can be formed by combining C-NOT gates with single-qubit operations. Example: 1 0 0 0 U SWAP 0 0 1 0 0 1 0 0 0 0 0 1 Toffoli gate: Every unitary transformation U can be represented as a rotation over the Bloch Sphere. Every single-qubit operator U can be decomposed to: U e Rz (3 ) Ry ( 2 ) Rz (1 ) i For Example: Rx ( ) Rz ( / 2) Ry ( ) Rz ( / 2) X Rz ( / 2) Ry ( ) Rz ( / 2) Lets consider a 2-level atom: ◦ Rotation is achieved by a pulse. ◦ is set by the optical phase of the pulse. ◦ Pulse area: 01 01 Dipole moment 0 (t ) Electric field 0 (t )dt 11 Assume q A A , qB B , AB A B A ' A , B ' B , π-pulse at ωB’ Input Output 00 01 10 00 01 11 11 10 A ' B ' 01 A 10 00 B Control qubit – Excitation level of a cooled 9Be+ ion in a harmonic trap. Target qubit – 2 hyperfine levels of the ion. Interactions with environment decoherence loss of information. In order to increase decoherence time, typically low temperatures are used. Quantum systems are fragile. We need quantum error correction algorithms. Error rates are proportional to the ratio of operating time to decoherence time. Dephasing time: T2. Number of operations: T2 Nop Top System T2(s) Top(s) Nop Nuclear spin 104 10-3 107 Ion trap 100 10-6 (e) 106 Excitation (quantum dot) 10-9 (e) 10-12 (e) 103 Electron spin (quantum dot) 10-7 10-12 105 Superconducting flux qubit 10-8 (e) 10-10 (e) 102 (e) The idea: determine if a single bit function f(x) is constant or balanced. Classical computer-2 calls. Quantum computer-1 call. F(0) F(1) F1 0 0 constant F2 1 0 Balanced F3 0 1 Balanced f4 1 1 constant 0 0,1 1 0 1 2 1 y H q2 0 1 2 x H q1 1 2 U f 1 1 1 0 1 0 1 0, 0 0,1 1, 0 1,1 2 2 1 0, f (0) 0,1 f (0) 1, f (1) 1,1 f (1) 2 2 1 0, f (0) 0,1 f (0) 1, f (1) 1,1 f (1) 2 If f is constant: f(0)=f(1) 2 constant x 1 0, f (0) 0,1 f (0) 1, f (0) 1,1 f (0) 2 1 = 0 1 f (0) 1 f (0) 2 constant 1 H 0 1 0 2 2 1 0, f (0) 0,1 f (0) 1, f (1) 1,1 f (1) 2 If f is balanced: f(1)=1⊕f(0) 2 balanced x 1 0, f (0) 0,1 f (0) 1,1 f (0) 1, f (0) 2 1 = 0 1 f (0) 1 f (0) 2 balanced 1 H 0 1 1 2 Quantum search algorithm. For example: The London telephone directory: Holmes, Sherlock 221b Baker Street 123 456 It is easy to find Holmes’ telephone number, but, difficult to find the telephone 123 456. Classical computer ~ NData/2 op. Quantum computer ~ Sqrt(NData). 0 0 1 0 2 ... 0 1 0 1 1 1 0 1 0 1 ... 0 1 1 2 2 2 2 2 1 2 1 N N 2N x x 0 1 N Data N Data 1 x x 0 1 1,1,1...,1 2 0, 0,1, 0,..., 0 N Data N 1 N Grover operator: 1. 2. 3. 4. Oracle. Hadamard gate. CPS-conditional phase shift. Hadamard. ◦ Oracle: a gate that checks if we have the desired solution: f ( x) O x 1 x 1 x is the solution, f ( x) otherwise 0 The three other steps (H, CPS,H) perform an ‘inversion about the mean’. Example: Ndata=4, N=2 qubits. Applying oracle: 0 00 1 2 1 1 1 1 1 0 1 1 1 0 2 1 2 2 00 01 10 11 2 1 2 1 Suppose: target=10. 1 2 1 1 2 1 1 Inversion about the mean: 3 0,0,1,0 Mean This Algorithm can be used as a password cracker! Fourier transform operation: Classical computer (FFT) ~ n2n op. for N=2n numbers. Quantum Computer ~ n2 op. But the result of QFT is stored as amplitudes, it can not be read. But QC can find periodicity. 1994-Peter Shor – can be used to factorize large numbers. Is RSA encryption in danger? Largest prime number today (March 2008): 232582657-1. In order to factorize a 1000 bit number, between 1012 and 1018 qubits are required. Until today, systems of only a few qubits have been demonstrated. RSA is probably safe for the next few decades. C.C. -> Memory required is exponential in system size. N two-level system -> 2N amplitudes. A relatively small molecule of 53 atoms requires 253~9x1015 bits~1 Petabyte=106 Gigabytes. Q.C. -> N qubits required. Unfortunately, on making the measurements, we would only obtain N bits of information. P ( L) e n( L ) e L / L0 L / L0 Pi ( L, N ) e L / NL0 L ni ( L, N ) e L / NL0 n( L, N ) Ne L / NL0 Nmin ( L) L / L0 n( L, Nmin ) L / L0 e1 D. DiVincenzo: requirements for the system: 1. Scalable physically to increase the number of qubits . 2. Possible to prepare an initial state. 3. Decoherence time >> operation time. 4. Single- and two-qubit quantum gates must be demonstrated. 5. Possible to measure the state of each individual qubit. An optional solution for the scaling up problem is using a network of small quantum computers, thus creating a larger one. This uses “flying qubits”. Flying qubits 1995: Quantum C-NOT gate. 1998: 2 and 3 qubit NMR QC. Execution of Grover’s algorithm. 2000: 5 and 7 qubit NMR QC. Execution of order finding. 2001: Execution of Shor’s algorithm (15 was factored). 2005: Qubyte. 2006: 12 qubit QC. Is Moore’s law valid on a QC? # of qubits QC of 1000 qubits expected on 2032. QC of 1 GigaQubyte only on 2127 # of qubits (logarithmic sacle) y = 4E-145e0.167x 10 1 1996 1998 2000 2002 Year 2004 2006 2008 Quantum computation is in some cases more efficient than classical computation. QC is possible, at least in principle. Small QC systems have already been demonstrated. The main goal now is to scale up the systems.