12/3/09 How to make a Quantum Computer Shor’s Algorithm for factoring n   1) Pick a number q (with small prime factors) such that 2n 2 ≤ q ≤ 3n 2 € 2) Pick a random integer x that is coprime to n 3) Repeat steps labeled (a) through (g) order lo g(q) times, using the same random number x each time 1 12/3/09   (a) Create a quantum memory register and partition the qubits into two sets, called Register1 and Register2   If the qubits in Register1 are in the state reg1 and those in Register2 are in the state reg2, we represent the joint state of both registers as (decimally) reg1,reg2 €   (b) Load Register1 with all integers in the range 0 to q-1 and load Register2 with all zeros   The state of the complete register is q−1 ψ = 1 ∑ a,0 q a= 0 € 2 12/3/09   (c) Apply exploiting quantum parallelism the transformation xa mod n to each number in Register1 and Place the results in Register2 q−1 1 ψ = a, x a mod n ∑ q a= 0 €   (d) Measure the state of Register2 obtaining some result k   This has the effect of projecting out the state of Register1 to be a superposition of just those values of a such that   xa ψ = mod n=k 1 A ∑ a',k a' ∈A A={a’: xa mod n=k} and ||A|| is the number of elements in this set   Where € 3 12/3/09   How to find the period r of xa mod n=k? will compute the Fourier transform of |a’>   Fourier transform can be represented by an unitary operator (Quantum Fourier transform)   We •  Unitary because of the Parseval’s identity   States corresponding to integer multiplies of the inverse period, and these close to them have a higher value (greater amplitude)     Integer multiplies of the inverse period are λi/r After a measurement we get some number in Zq c i λi ≈ q r λ1 λ2 λ3 λ4 , , , , r r r r   We determine r by continuous fraction € 4 12/3/09   xr=1 mod n •  If r is an even number, then x r = 1 mod n  r 2  x 2  = 1 mod n    r 2  x 2  −1 = 0 mod n    r 2 2  x 2  −1 = 0 mod n    r  r   x 2 −1 x 2 + 1 = 0 mod n    € product (xr/2-1)(xr/2+1) is some integer multiple of n   Dividing (xr/2-1)(xr/2+1) by n results in a reminder of zero   One of the terms (xr/2-1)(xr/2+1) must have a nontrivial factor in common with n   The   gcd((xr/2-1),n) and gcd((xr/2+1),n) 5 12/3/09   From the found samples determine the period r by continuous fraction 121 , 61 , 184 , 182 , 61 , 122 , 0 , 121 , 0 , 121 , 181 , 61 →r=4   gcd(72-1,15)=3 €   gcd(72+1,15)=5   15=5*3   Any real quantum computer is going to incur kinds of errors caused by myriad physical processes such as decoherence, cosmic radiation, and spontaneous emission   Difficulties in maintaining a state   Preserving entangled particles until they are needed for quantum teleportation 6 12/3/09   Mach-Zehnder interferometer is a particularly simple device for demonstrating interference by division of amplitude   A light beam is first split into two parts by a beam splitter and then recombined by a second beam splitter Mach-Zehnder Interferometer Detector Mirror Detector Half-silvered mirror Light source Half-silvered mirror Mirror 7 12/3/09   Only one photon is emitted   Several experiments are repeated   The path the photon chooses ↑or → is represented by superposition half mirror H acts like a Hadamard operator   The 1 1 → + ↑ 2 2  1 1  H → + ↑= →  2 2  H→ = € Mach-Zehnder Interferometer Detector 0% Mirror Light source Activated, one photon emited Half-silvered mirror Half-silvered mirror Detector 100% Mirror 8 12/3/09 Mach-Zehnder Interferometer Detector 50% Mirror Light source Activated, one photon emited Half-silvered mirror Measurment Half-silvered mirror     Detector 50% Mirror Many candidates how to build a quantum computer The large number of candidates shows explicitly that the topic, in spite of rapid progress, is still in its infancy     D-Wave Systems Inc. claims to be the world’s first — and only — provider of quantum computing systems designed to run commercial applications http://www.dwavesys.com/ •  However, since D-Wave Systems has not released the full details many experts in the field have expressed skepticism 9 12/3/09   2001, IBM Test-Tube Quantum Computer   (Isaac Chuang and Costantino Yannoni)   Seven-qubit quantum computer that solved factorization of the number 15 using the Shor's Algorithm   Custom-designed molecules in a test tube representing 7 qubits 10 12/3/09 Heteropolymer-Based   Heteropolymer-Based Quantum Computers   Idea behind the heteropolymer computer is to use a linear array of atoms as memory cells   Each atom can be either in an excited or grounded state   This gives the basis for a binary arithmetic   Software consists of a sequence of laser pulses of particular frequencies that induce transitions of particular frequencies that induce transitions in certain atoms of the polymer 11 12/3/09   A molecular digital computer that relies on transitions among energy levels in atoms to switch states   Each atom has three energy levels   State 0 is the ground state   represents   State bit 0 1 is a meta stable state   represents bit 1   State 2 is a rapidly decaying exited state either to 0 or 1 12 12/3/09 Ion Trap-Basded   The Cirac-Zoller scheme uses a linear array of trapped ions as the basis for quantum memory register •  The trapping is arranged by electromagnetic fields, logical states of the qubits encoded in the energy states of the individual ions and the vibration states between the ions       Each ion is considered as a 2-state system containing a ground state and excited state The ions are arranged in a linear array such that each ion can be irradiated with light from a laser Laser pulses have the effect of exciting specific transitions in specific ions allowing the array to be placed in arbitrary superposed states 13 12/3/09 NMR-Based   Adapt Nuclear Magnetic Resonance techniques to accomplish the basic operations of a quantum computer   Consists of a test-tube sized sample of some liquid, with each molecule of this liquid acting as an independent quantum memory register   We would not measure the observables of a register   Measure the ensemble average of all the nuclear spins in the sample 14