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Introduction to Quantum Computing Lecture 2 of 2

CS 497 Frontiers of Computer Science. Introduction to Quantum Computing Lecture 2 of 2. Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo.  10 .  00 . 0 1 . Recap of previous lecture. quantum states.

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Introduction to Quantum Computing Lecture 2 of 2

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  1. CS 497 Frontiers of Computer Science Introduction to Quantum ComputingLecture 2 of 2 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo January 16 & 18, 2007

  2. 10 00 01 Recap of previous lecture quantum states Quantum states on n qubits are 2n-dimensional unit vectors • The basic operations on them are: • unitary operations (rotations) • measurements, that project on • to the basis states quantum circuits • Quantum algorithms so far: • f(0)  f(1) [Deutsch] • one-out-of-four search

  3. one-out-of-Nsearch? Natural question: what about search problems in spaces larger than four (and without uniqueness conditions)? For spaces of size eight (say), the previous method breaks down—the state vectors will not be orthogonal Later on, we’ll see how to search a space of size N with O(N) queries ...

  4. Contents of lecture 2 • Preliminary remarks • Quantum states • Unitary operations & measurements • Subsystem structure & quantum circuit diagrams • Introductory remarks about quantum algorithms • Deutsch’s parity algorithm • One-out-of-four search algorithm • Shor’s period finding algorithm • Grover’s search algorithm • Concluding remarks

  5. Contents of lecture 2 • Preliminary remarks • Quantum states • Unitary operations & measurements • Subsystem structure & quantum circuit diagrams • Introductory remarks about quantum algorithms • Deutsch’s parity algorithm • One-out-of-four search algorithm • Shor’s period finding algorithm • Grover’s search algorithm • Concluding remarks

  6. y x r Period-finding Given:f :{0,1}n→T such that f is (strictly) r-periodic, with unknown period r Classically this is very hard in the general case—essentially it is as hard as finding a collision, which costs 2O(n) queries Yet Quantum algorithms can determine r very efficiently: with only O(1) queries to f This is the basis of Shor’s factoring algorithm ...

  7. Application of period-finding algorithm Order-finding problem ( factoring) Input:m (an n-bit integer) and a<msuch that gcd(x,m) = 1 Output: the minimum r> 0 such that ar=1(mod m) Example: let a = 4and m = 35 (note that gcd(4,35) = 1) • 41mod 35 = 4 • 42mod 35 = 16 • 43mod 35 = 29 • 44mod 35 = 11 • 45mod 35 = 9 • 46mod 35 = 1 • 47mod 35 = 4 • 48mod 35 = 16 • : Question: what is r in this case? Answer:r=6 The sequence is cyclic because the set Zm*={a {1,2,…, m1} : gcd(x,m) = 1} is a group under multiplication mod m

  8. Ua,m x1 x2 xn x1 x2 xn V H 0 0 0 f(x) Application of period-finding algorithm Order-finding problem ( factoring) Input:m (an n-bit integer) and a<msuch that gcd(x,m) = 1 Output: the minimum r> 0 such that ar=1(mod m) No classical polynomial-time algorithm is known for this problem—in fact, the factoring problem reduces to it Order-finding reduces to finding the period of the function f(x) =ax mod m, which can be computed in polynomial time A circuit computing the function f is substituted into the black-box:

  9. r r r r r r r k Sketch of period-finding algorithm Construct x|x,f(x) and then measure second register to get the following state: (k random) |k +|k+r +|k+2r +|k+3r + … +|k+(s1)r Measuring this state yields just a uniformly random value More is needed to extract r, a global property of f ...

  10. Quantum Fourier transform where  = e2i/N It’s a unitary operation on n qubits (an NN matrix, where N= 2n)

  11. r r r r r r r k s s s s s s Period inversion property Applying the quantum Fourier transform to the state |k +|k+r +|k+2r +|k+3r + … +|k+(s1)r yields the state |0 + k|s + 2k|2s + 3k|3s + … + (r1)k|(r1)s =e2i/r s=r1 Note: there is no longer an offset k (it’s now part of the “phase”) Measure to get multiple of s, from which r can be deduced

  12. H 4 H 8 4 H 16 8 4 H 32 16 8 4 H H m Computing the QFT Quantum circuit for F32: reverse order Gates: For F2n costs O(n2) gates

  13. Ua,m V H Quantumalgorithmfororder-finding Quantum Fourier transform after measuring these qubits, can calculate r using classical methods 0 H H 0 H H 4 0 H H 4 8 0 0 1 Ua,M x,y = x,axy modm (poly-size circuit) Number of gates for a constant success probability is: O(n2 log n loglog n)

  14. Two-dimensional periodicity Given: f :{0,1}n {0,1}n →T with a two-dimensional repeating pattern Goal: find a simple description of this periodic strucuture Quantum algorithms can also solve this very efficiently, and this is the basis of Shor’s discrete logarithm algorithm [1994]

  15. Hidden subgroup problem Let G be a known group and H be an unknown subgroup of G Let f : G→T have the property f(x)= f(y)iff xy1 H (i.e., x and yare in the same right coset of H) Problem: given a method for computing f,determine H Example:G = Sn , the symmetric group (permutations of {1,2,…, n}) Interesting fact: a fast algorithm for this leads to a fast algorithm for the graph isomorphism problem … alas no efficient quantum has been found for this version of HSP, despite significant effort by many people

  16. Contents of lecture 2 • Preliminary remarks • Quantum states • Unitary operations & measurements • Subsystem structure & quantum circuit diagrams • Introductory remarks about quantum algorithms • Deutsch’s parity algorithm • One-out-of-four search algorithm • Shor’s period finding algorithm • Grover’s search algorithm • Concluding remarks

  17. PSPACE NP co-NP 3-CNF-SAT factoring P Prelude: factoring vs. NP Is factoring an NP-hard problem? If so, then every problem in NP is solvable by a poly-time quantum algorithm! But factoring hasn’t been shown to be NP-hard Moreover, there is “evidence” that it is not NP-hard: factoring  NPco-NP If factoring is NP-hard then NP=co-NP

  18. Grover’s quantum algorithm makes only O(2n) queries x1 x1 Uf xn xn y y  f(x1,...,xn) Quantum search problem Given: a black box computing f : {0,1}n {0,1} Goal: find x {0,1}n such that f(x) = 1 Classically, using probabilistic procedures, order 2n queries are necessary to succeed—even with probability ¾ (say) Query: [Grover ’96]

  19. Applications of quantum search The function f could be realized as a 3-CNF formula: In fact, the search could be for a certificate for any problem in NP The resulting quantum algorithms appear to be quadratically more efficient than the best classical algorithms known

  20. reflection 2 reflection 1  Prelude to Grover’s algorithm: two reflections = a rotation Consider two lines with intersection angle : 2 2 1 1 Net effect: rotation by angle 2, regardless of starting vector

  21. D H H H x1 x1 H Uf xn xn y yf(x1,...,xn) Grover’s algorithm I Basic operations used: Query: Uf x(01)= (1) f(x)x(01) Diffusion: Costs only O(n) gates Hadamard:

  22. iteration 1 iteration 3 iteration 2 iteration 4 D D D D Uf Uf Uf Uf Grover’s algorithm II 0 H 0  • construct state H0...0 • repeatktimes: • apply DUfto state • 3. measure state, to get x{0,1}n, and check if f(x)=1

  23. Since DUf is a composition of two reflections, it is a rotation by 2, wheresin() 1/N target (assume unique) We want (2k+1)(1/N) /2, so set k  (/4)N target Grover’s algorithm III Algorithm: (DUf)kH0...0 2 2 2 H0...0 = xx 2 

  24. Contents of lecture 2 • Preliminary remarks • Quantum states • Unitary operations & measurements • Subsystem structure & quantum circuit diagrams • Introductory remarks about quantum algorithms • Deutsch’s parity algorithm • One-out-of-four search algorithm • Shor’s period finding algorithm and factoring • Grover’s search algorithm • Concluding remarks

  25. Conclusion Extended Chruch-Turing Thesis: any polynomial-time algorithm can be simulated by a probabilistic polynomial-time Turing machine Quantum computing challenges this thesis Either: • The Extended Chruch-Turing Thesis is false • Quantum mechanics as we understand it is false • There is a classical poly-time factoring algorithm There are many curious properties of quantum information in the context of computation, communication, and cryptography Waterloo has many people in the faculties of Mathematics, Science, and Engineering working in quantum computing (please see www.iqc.ca more information)

  26. Some possible project topics • Efficient “quantum proofs”: an example is the group non-membership problem • Nonlocal effects: apparently paradoxical tricks that can be performed with entangled states • Quantum error-correcting codes: important for physical implementations of quantum information processing • Alternate models of quantum computation: an example is the measurement-based model • Quantum walks: the quantum analogues of random walks, and their algorithmic applications • Quantum cryptography: information-theoretical security in a public-key setting

  27. THE END

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