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DNA Computing DNA Computing • Can we use DNA to do massive CS252 computations? Graduate Computer Architecture – Organisms do it Lecture 28 – DNA has very high information density: » 4 different base pairs: Esoteric Computer Architecture • Adenine/Thymine • Guanine/Cytosine DNA Computing & Quantum Computing » Always paired on opposite strands Energetically favorable – Active operations: » Copy: Split strands of DNA Prof John D. Kubiatowicz apart in solution, gain 2 copies » Concatenate: eg: http://www.cs.berkeley.edu/~kubitron/cs252 GTAATCCT will combine XXXXXCATT with AGGAYYYYY » Polymerase Chain Reaction (PCR): amplifies region of molecule ©1999 Access Excellence between two marker molecules @ the National Health Museum 5/11/2009 cs252-S09, Lecture 28 2 DNA Computing and Hamiltonian Path Even more promising uses of DNA • Given a set of cities and costs between them (possibly directed • Self-assembly of components paths): – DNA serves as substrate – Find shortest path to all cities – Attach active elements in middle of components. – Simpler: find single path that – Final step – metal deposited over DNA visits all cities • DNA Computing example is latter version: – Every city represented by unique 20 base-pair strand – Every path between cities represented by complementary pairs: Active Region 10 pairs from source city, 10 pairs from destination – Shorter example: AAGT for city 1, TTCG for city 2 Path 1->2: CAAA DNA Bonding Active Region Will build: AAGTTTCG • Other interesting structures could be built ..CAAA.. – Dump “city molecules” and “path molecules” into testtube. Select and amplify paths of right length. Analyze for result. – Been done for 6 cities! (Adleman, ~1998!) 5/11/2009 cs252-S09, Lecture 28 3 5/11/2009 cs252-S09, Lecture 28 4 Use Quantum Mechanics to Compute? Quantization: Use of “Spin” • Weird but useful properties of quantum mechanics: North – Quantization: Only certain values or orbits are good » Remember orbitals from chemistry??? – Superposition: Schizophrenic physical elements don’t quite know Spin ½ particle: Representation: whether they are one thing or another (Proton/Electron) |0> or |1> • All existing digital abstractions try to eliminate QM – Transistors/Gates designed with classical behavior – Binary abstraction: a “1” is a “1” and a “0” is a “0” • Quantum Computing: South Use of Quantization and Superposition to compute. • Interesting results: • Particles like Protons have an intrinsic “Spin” – Shor’s algorithm: factors in polynomial time! when defined with respect to an external – Grover’s algorithm: Finds items in unsorted database in time magnetic field proportional to square-root of n. – Materials simulation: exponential classically, linear-time QM • Quantum effect gives “1” and “0”: – Either spin is “UP” or “DOWN” nothing between 5/11/2009 cs252-S09, Lecture 28 5 5/11/2009 cs252-S09, Lecture 28 6 Kane Proposal II (First one didn’t quite work) Now add Superposition! • The bit can be in a combination of “1” and “0”: – Written as: = C0|0> + C1|1> Single Spin – The C’s are complex numbers! – Important Constraint: |C |2 + |C |2 =1 Control Gates 0 1 • If measure bit to see what looks like, 2 Inter-bit – With probability |C0| we will find |0> (say “UP”) 2 Control Gates – With probability |C1| we will find |1> (say “DOWN”) • Is this a real effect? Options: Phosphorus – This is just statistical – given a large number of protons, a fraction 2 Impurity Atoms of them (|C0| ) are “UP” and the rest are down. – This is a real effect, and the proton is really both things until you • Bits Represented by combination of proton/electron spin try to look at it • Reality: second choice! • Operations performed by manipulating control gates – There are experiments to prove it! – Complex sequences of pulses perform NMR-like operations • Temperature < 1° Kelvin! 5/11/2009 cs252-S09, Lecture 28 7 5/11/2009 cs252-S09, Lecture 28 8 A register can have many values! Spooky action at a distance • Implications of superposition: • Consider the following simple 2-bit state: = C |00>+ C |11> – An n-bit register can have 2n values simultaneously! 00 11 – Called an “EPR” pair for “Einstein, Podolsky, Rosen” – 3-bit example: • Now, separate the two bits: = C000|000>+ C001|001>+ C010|010>+ C011|011>+ C100|100>+ C101|101>+ C110|110>+ C111|111> • Probabilities of measuring all bits are set by Light-Years? coefficients: – So, prob of getting |000> is |C |2, etc. 000 • If we measure one of them, it instantaneously sets other one! – Suppose we measure only one bit (first): – Einstein called this a “spooky action at a distance” 2 2 2 2 » We get “0” with probability: P0=|C000| +|C001| +|C010| +|C011| – In particular, if we measure a |0> at one side, we get a |0> at the other Result: = (C000|000>+ C001|001>+ C010|010>+ C011|011>) (and vice versa) 2 2 2 2 » We get “1” with probability: P1=|C100| +|C101| +|C110| +|C111| • Teleportation Result: = (C100|100>+ C101|101>+ C110|110>+ C111|111>) – Can “pre-transport” an EPR pair (say bits X and Y) • Problem: Don’t want environment to measure – Later to transport bit A from one side to the other we: » Perform operation between A and X, yielding two classical bits before ready! » Send the two bits to the other side – Solution: Quantum Error Correction Codes! » Use the two bits to operate on Y » Poof! State of bit A appears in place of Y 5/11/2009 cs252-S09, Lecture 28 9 5/11/2009 cs252-S09, Lecture 28 10 Model: Operations on coefficients + measurements Security of Factoring • The Security of RSA Public-key cryptosystems Input Output Unitary depends on the difficult of factoring a number N=pq Complex Classical (product of two primes) Transformations Measure State Answer – Classical computer: sub-exponential time factoring – Quantum computer: polynomial time factoring • Basic Computing Paradigm: • Shor’s Factoring Algorithm (for a quantum computer) – Input is a register with superposition of many values Easy » Possibly all 2n inputs equally probable! 1) Choose random x : 2 x N-1. – Unitary transformations compute on coefficients Easy » Must maintain probability property (sum of squares = 1) Hard 2) If gcd(x,N) 1, Bingo! » Looks like doing computation on all 2n inputs simultaneously! 3) Find smallest integer r : xr 1 (mod N) – Output is one result attained by measurement Easy • If do this poorly, just like probabilistic computation: Easy 4) If r is odd, GOTO 1 – If 2n inputs equally probable, may be 2n outputs equally probable. 5) If r is even, a x r/2 (mod N) (a-1)(a+1) = kN – After measure, like picked random input to classical function! Easy – All interesting results have some form of “fourier transform” computation being Easy 6) If a = N-1 GOTO 1 done in unitary transformation 7) ELSE gcd(a ±1,N) is a non trivial factor of N. 5/11/2009 cs252-S09, Lecture 28 11 5/11/2009 cs252-S09, Lecture 28 12 r ION Trap Quantum Computer: Finding r with x 1 (mod N) Promising technology Top \ \ \ k \ Cross- k/ 1 / k/ x / Sectional k k View r 1 \ w\ w r y / x / w 0 y r 1 • IONS of Be+ trapped in w\ oscillating quadrature field Quantum – Internal electronic modes of IONS ()x / used for quantum bits Fourier w 0 – MEMs technology Transform 0 1 k – Target? 50,000 ions r r r – ROOM Temperature! • Finally: Perform measurement • Ions moved to interaction regions – Find out r with high probability – Ions interactions with one another Top View moderated by lasers – Get |y>|aw’> where y is of form k/r and w’ is related Proposal: NIST Group 5/11/2009 cs252-S09, Lecture 28 13 5/11/2009 cs252-S09, Lecture 28 14 Ion Trap Quantum Computer Ballistic Movement Network Two-Qubit Gate • Major Components - Data = an ion One-Qubit Two-Qubit Gate Gate - Gate = a location Q1 Q2 Q3 • Ballistic Movement R R - Apply pulse sequences to electrodes One-Qubit R Two-Qubit Gate R Gate - Electrostatic forces move ion Q4 Q5 - Intersections similar, but more complicated pulse sequences Memory Cell Interconnection Memory Cell Network Q1 Q2 • Problem: Noise accumulation! 5/11/2009 cs252-S09, Lecture 28 15 5/11/2009 cs252-S09, Lecture 28 16 Noise Accumulation from Movement Movement Option 2: Teleportation 1.0E-02 Source Location3. Transmit two Target Location classical bits 1.0E-03 D Entanglement 2. Local 4. Local Ops 1.0E-04 1.0E-4 inital error Ops E1 E2 D?D 1.0E-5 inital error 1.0E-05 1.0E-6 inital error 1. Generate EPR pair Qubit Error 1.0E-06 1.0E-7 inital error 1.0E-8 inital error • Goal: Transfer the state, not the data ion 1.0E-07 • Problem: EPR pairs become noisy 1.0E-08 • Teleportation Benefits 0 16324864 Distance Moved in Gates - Error Correction of data (arbitrary state): ~100 ms Purification of EPR pair (known state): ~120 µs • Noise may increase error by factor of 100 - Pre-distribution of EPR pairs 5/11/2009 cs252-S09, Lecture 28 17 5/11/2009 cs252-S09, Lecture 28 18 EPR Pair Distribution Network Setting Up a Teleportation Link • Purification = Amplification of EPR pair link One-Qubit Two-Qubit - Two EPR pairs One “purer” pair, one junk pair Gate Gate - Chance of failure • Need to send multiple pairs Q2 R Q3 R EPR Pair One-Qubit Two-Qubit GeneratorsR STRONGEREPR Qubits EntanglementEPR Qubits Gate R Gate Q4 Q5 P G P Memory Cell Interconnection Memory Cell Recycled Qubits Recycled Qubits Q1 Network For Data Teleportation 5/11/2009 cs252-S09, Lecture 28 19 5/11/2009 cs252-S09, Lecture 28 20 Chained Teleportation Quantum Network Architecture Teleportation Teleportation T G T G T G T P P P P G G G G Gate Gate Gate Gate T G T G T G T G T G T T G T G T G T P • Adjacent T nodes linked for teleportation P P P P P Gate Gate Gate Gate • Positive
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