Quantum Information & Quantum Computation

Transcription

1 CS9A, Sprig 5: Quatum Iformatio & Quatum Computatio Wim va Dam Egieerig, Room 59

2 Admiistrivia Do the exercises. Aswers will be posted at the ed of the week. Midterm examiatio will be Thursday, April 8 Ope book, ope everythig. Bookstore will start returig books o April 5. Other questios?

3 Thigs that have come up Kow how to take tesor products of vectors. Mid the orderig of qubits for quatum gates: Example: CNOT betwee two bits I both cases: mid the orderig of the dimesios i the vector/matrix otatio.

4 This Week Wrap-up of the quatum circuit model of efficiet quatum computatio. Effect of partial measuremets o superpositios. Small quatum algorithms.

5 Clea Reversible Computatio With CCNot gates, we ca implemet NOT ad AND. If we keep old memory aroud, ay classical circuit fuctio F ca be implemeted efficietly as U F : x,, # x,g x,f(x) (which is a classical trasform). By copyig the output F(x) ad ruig the circuit U F i reverse, we ca erase the garbage bits g x : x,g x,f(x), # x,g x,f(x),f(x) # x,,,f(x). I sum: x,, # x,f(x), ca be implemeted efficietly as log as we have clea -qubits aroud. Also i superpositio: Σ x x,, # Σ x x,f(x),.

6 Last Week s Questio Why ca we copy the F(x) bit ad ru the circuit U F i reverse to clea up the work space? Reaso: U F : x,, # x,g x,f(x) implemets a classical trasformatio that does ot create superpositios. If we have U F as a circuit, we ca also apply it to a superpositio of states. Geeral clea computatio: x {,} α a x, α x,f(x) x x x {,}

7 Power of Reversible Computatio We showed that the requiremet of reversibility does ot chage (sigificatly) the efficiecy of our computatios: Reversible Computatio = Geeral Computatio. But what about the efficiecy of implemetig geeral quatum trasformatios? We have to look at what it meas to efficietly implemet a computatio that uses quatum superpositios.

8 Closeess of States We kow that uitary trasformatios are ier product preservig. Hece the agle betwee two states ψ ad ψ is the same as the agle betwee C ψ ad C ψ after we applied the circuit C to them. If states are close, they remai close. Measure of closeess: Fidelity F ( ψ, φ ) = ψ φ If F( ψ, ψ ), the the states are close. If F( ψ, ψ ), the the states are far away. Close states lead to ear idetical probability distributios whe measured.

9 How Close? Take two states ψ ad φ with fidelity F. Measurig the states i the computatio basis {,} gives two probability distributios p ad q respectively. If the states are close (F ), the p ad q have to be close as well. How close? F p q F s {,} s s Quatum states that are close i terms of their fidelity behave i all respects almost the same.

10 Approximate Q-Computig If F(ψ,ψ ), the havig ψ istead of ψ is equally good whe performig computatios. If our ideal quatum circuit produces the outcome state ψ, the a approximate circuit that produces ψ also solves the computatioal problem. (If i doubt, ru the computatio several times ad take the majority of the outcomes.) Just as states ca be close, so ca gates ad circuits. For the task of quatum computatio it is sufficiet to implemet the wated uitary trasformatio approximately.

11 Uiversal Q-Computig If we use a small set of stadard gates {CCNOT, H, R z } the we ca implemet (approximately) ay possible uitary trasformatio U Â D D. Moreover, if a circuit is build usig a differet set of gates {G,,G r }, the we ca approximate this circuit efficietly usig the {CCNOT, H, R z } set. (You do this by fidig the proper replacemet circuits for G,,G r.) Other sets of gates are also possible. It does ot really matter which gates you use to study quatum circuit complexity.

12 Moder Church-Turig Thesis Whatever we ca build i the lab, we will be able to simulate it efficietly usig our quatum circuit model. By studyig quatum circuit complexity we are studyig the itrisic computatio complexity of problems i the quatum mechaical world as-we-kow it. Note that complexity theory does deped o the fact that Nature is ot classical (factorig, discrete log, ).

13 Geeral Set Up Iput size Cosider a fuctio F:{,} T {,} m We wat to kow some properties of F F is easy to compute, but {,} is too big. Quatum Approach: Create a superpositio of F(x) values by calculatig F oce o a superpositio: x {,} α a x, x,f(x) x x x {,} The, do somethig quatum smart with this state. α

14 Partial Measuremets What happes to Σ x α x x,f(x) if we measure the F(x) part of the register, but ot the x-part? Compare the two cases: ( + ) ( + ) a ( ( + + ) ) outcome "" outcome "" (, +, ) a,, outcome "" outcome "" Iformal: The state collapses accordig to the measuremet outcome, but ot more tha that.

15 Partial Measuremets II More formal descriptio of Measuremets Cosider a Boolea measuremet o a superpositio: x {,} α x,f(x) x Rewrite the state accordig to the Boolea values. α F(x) = x x, + αx x, F(x) = Depedig o the outcome, the state collapses to oe of the two outcomes, with probability Σ x α x (sum over approriate x values F(x)= or F(x)=).

16 Partial Measuremets III Eve more Formal Descriptio of Measuremet Let M be the set of measuremet outcomes, each quatum state φ ca be writte as φ = m M β m ψ m, m Whe measurig the M quatity: - We observe m M with probability β m - State collapses as φ # ψ m,m Note that this ψ m ca still be a superpositio. The state ψ m,m is agai properly ormalized.

17 Commo Computatioal Settig We create a superpositio of F:{,} values, where the amplitudes are uiform over all x {,}. After that we measure a F(x)=y value, such that x {,} x,f(x) a S For each y M this happes radomly with probability S y /, where S y = {x : F(x)=y}. y F(x) = y x,y You ca ot use this to fast search F(),F(),

18 CS9A, Sprig 5: Quatum Iformatio & Quatum Computatio Wim va Dam Egieerig, Room 59

19 Admiistrivia Remember: Midterm is ext Thursday, April 8 Ope book, ope otes. New hadout has bee posted (o measuremets). Do the exercises. New exercises ad aswers to old oes will be posted tomorrow (Friday). Tuesday: Q&A sessio of Midterm material.

20 Geeral Set Up Iput size Cosider a fuctio F:{,} T {,} m We wat to kow some properties of F F is easy to compute, but {,} is too big. Quatum Approach: Create a superpositio of F(x) values by calculatig F oce o a superpositio: x {,} α a x, α x,f(x) x x x {,} The, do somethig quatum smart with this state.

21 A First Example () Cosider a Boolea fuctio: F:{,} {,}, Implemeted by the uitary evolutio x,b # x,b F(x) for all x,b. Questio: F()=F()? Sigle Call Quatum Solutio: After calculatig F oly oce i superpositio we ca aswer the questio perfectly.

22 A First Example () Sigle Call Quatum Solutio: H Ha F. Apply Hadamard,Hadamard to, state. Apply F to superpositio Thus from, we get: = a = ( ( ) + ( F() ) ( ) ( + ) ( ) + ( ) ) F() [ ] F() F() ( ) + ( ) ( ) Phase-Flip Trick ( )

23 A First Example (3) Look at the left bit: If F() = F() we have : ( ) F() ( + ) ( ) If F() F() we have : ( ) F() ( ) ( ) ( + )/ ad ( )/ are orthogoal states Usig a Hadamard o the first bit we ca reliably distiguish betwee these two cases.

24 A Simple Example (4) Summary: Usig two qubits, a few Hadamards, a sigle applicatio of the fuctio F: x,b # x,b F(x) ad a fial measuremet we ca determie if F()=F() or ot. Classically you would eed two evaluatios of F to decide this problem. If evaluatig F is very expesive, the this might be a useful speed-up to solve the problem. Crucial igrediet: Phase-Flip Trick: F : x a ( ) F(x) x

25 Deutsch-Jozsa Algorithm Geeralizatio of the previous algorithm. Let F:{,} T {,} with either: F is costat : F( ) = = F( ), or F is balaced : 5% cases F(x)= ad 5% F(x)= Deutsch-Jozsa Algorithm decides this distictio with oly oe quatum-query F: x,b # x,b F(x). First create superpositio of x values ad apply Phase-Flip Trick with F(x) values to the appeded qubit state ( )/ =, yieldig: x {,} x a x {,} ( ) F(x) x

26 Deutsch-Jozsa Algorithm Depedig o whether F is costat or balaced, the (±)-phases i the superpositio are very differet. Geeralizatio of previous small example: apply Hadamard gates to the qubits. This gives ( ) x {,} F(x) x a F() ( ),..., aythig but,..., if if F costat F balaced If we measure these bits the for the outcomes: proves that F = costat; otherwise, F = balaced. Classically this requires / + queries.

27 Cetral Questio The crucial questio that we try to aswer i the theory of quatum algorithms is: For which fuctios F ca we determie which properties much faster tha classically? For which F/properties combiatios ca we use this ito a quatum algorithm that solves a relevat problem?

28 Quatum Query Results for Fuctio F:{,,N} {,} Searchig x:f(x)=? Grover s search: Θ( N) versus classical Ω(N) Parity F() + + F(N) mod? Classical: N; Quatum: ½ N Iterrogatio F(),,F(N)? (probabilistic) quatum: ½N+ N istead of N.