Quantum Information

1. Quantum Information Stephen M. Barnett University of Strathclyde steve@phys.strath.ac.uk The Wolfson Foundation

2. Probability and Information • 2. Elements of Quantum Theory • 3. Quantum Cryptography • 4. Generalized Measurements • Entanglement • Quantum Information Processing • 7. Quantum Computation • 8. Quantum Information Theory 0 Motivation 1 Digital electronics 2 Quantum gates 3 Principles of quantum computation 4 Quantum algorthims 5 Errors and decoherence 6 Realizations?

3. CMOS Device Performance Device performance doubles roughly every 5 years!

4. P - solvable problems (computing time is polynomial in input size) Classical Deterministic Algorithm Factoring Discrete logarithm Quantum simulations ... Classical Probabilistic Algorithm Quantum Computing

5. Classical: O(N) = O(2n) Quantum: O(log2N) = O(n2) 2. Shor’s factoring algorithm N = pq find p and q given N Exponential speed up! What happens to RSA? What happens to money?? Naïve classical (trial): O(N1/2) = O(2n/2) Best known classical: O(2^[n1/3log2/3n]) Shor’s algorithm: O(polynomial[logN]) = O(polynomial n) Quantum algorithms: scaling of computing time with N~2n 1. F.T. to determine periodicities f(x+r) mod N = f(x) mod N findr

6. What is a computation? Generation of an output number (string of bits) based on an input number. “Black Box” or Computer …101101001… Output Input …000111010… How does the computer achieve this?

7. Physical bit - electrical voltage +5V = 1 0V = 0 0 1 1 0 NOT gate 6.1 Digital electronics Single bit operation

8. 0 0 0 1 0 0 1 1 0 1 0 1 AND gate 0 1 1 1 0 0 1 1 0 1 0 1 OR gate Two bit operations

9. 1 1 1 0 0 0 1 1 0 1 0 1 NAND gate A small set of gates (e.g. NAND, NOT) is universal in that any logical operation can be made from them. Two bit operations Not all the gates are needed

10. Hadamard H Phase S p / 8 T and many more 6.2 Quantum gates Single qubit operations

11. control bit target bit CNOT gate can make entangled states Two qubit operations - CNOT gate

12. We can break up any multi-qubit unitray transformation into a sequence of two-state transformations:

13. It follows that we can realise any multi-qubit transformation as a sequence of single-qubit and two-qubit unitary transformations. This is the analogue of the universality of NAND and NOT gates in digital electronics.

14. Exercise: Construct the Toffoli gate using just CNOT gates and single qubit gates. Try to use as few gates as possible. control bit 1 control bit 2 target bit The CNOT gate, together with one qubit gates are universal

16. Encode input onto qubit string Quantum evolution = unitary transformation Measurement gives output = computed function (hopefully!) 6.3 Principles of quantum computation A quantum computation is a (generalised) measurement

17. Constraints of unitarity? Consider the two bit map State overlap Problem. Our computation requires Quantum computation?

18. Exercise: Show that the states transformation is an allowed unitary transformation. Unitary evaluation of the function f a = input string …101101001... b = input string, usually set to “zero” …000000000...

19. We can show this by an explicit construction:

20. Parallel quantum computation Can input a superposition of many possible bit strings a. Output is an entangled stated with values of f (a) computed for each a.

21. Constant functions: Balanced functions: Deutsch’s algorithm f (A) A Black Box A black box that computes one of four possible one-bit functions: We wish to know if the function is constant or balanced. We can do this by performing two computations To give f (0) and f (1) . Can we do it in one step?

22. + for constant - for balanced are orthogonal states and so can be identified without error. A quantum computer allows solution in a single run:

23. Guaranteed classical solution in computations Quantum? Exponential speed up Suppose our box computes a one bit function of n bits and that this function is either constant or balanced. Balanced:0 or 1 for exactly half of the possible inputs Constant:0 or 1 independent of input Orthogonal states for constant or balanced functions so solution in ONE computation. Exponential speed up.

24. Classical: O(N) = O(2n) Quantum: O(log2N) = O(n2) 2. Shor’s factoring algorithm N = pq find p and q given N Naïve classical (trial): O(N1/2) = O(2n/2) Best known classical: O(2^[n1/3log2/3n]) Shor’s algorithm: O(polynomial[logN]) = O(polynomial n) 6.4 Quantum algorithms 1. F.T. to determine periodicities f(x+r) mod N = f(x) mod N findr 3. Grover’s search algorithm - searching a database Classical: O(N) Quantum: O(N1/2)

25. Factorisation algorithm Example: N = 15, m = 2 => FN(0) = 1 FN(1) = 2 FN(2) = 4 FN(3) = 8 FN(4) = 1 FN(5) = 2 … => r = 4 => mr/2 – 1 = 3 mr/2 + 1 = 5 Both OK Example: N = 15, m = 11 => FN(0) = 1 FN(1) = 11 FN(2) = 1 FN(3) = 11 … => r = 2 => mr/2 – 1 = 10 => GCD 5 mr/2 + 1 = 12 => GCD 3 Both OK N: Given big integer to be factorised m: Small integer chosen at random n = 0,1,2, … 1. Make the series FN(n) = mn mod N 2. Find the period r : FN(n+r) = FN(n) 3. The greatest common divisor of N and mr/2±1 divides N

26. Shor’s algorithm to factorise N 1.Find integers q and M such that: q = 2M > N2 and prepare two registers each containing M qubits. 2. Set the qubits in the first register in the state (|0> + |1>)/21/2 and those in the second in the state |0>.

27. 3. Choose an integer m at random and entangle the two registers so that This can be achieved by a unitary transformation (on a suitably programmed quantum computer) within polynomial time. 4. Fourier transform for register 1: 5. Measurement on register 1: => k = multiple of q/r is obtained with high probability => r = q/k

28. Phase error Bit flip error 6.5 Errors and decoherence Interaction with the environment introduces noise and causes errors

29. Bit flip error Phase error In this case In this case Deutsch’s algorithm

30. Probability that a given qubit has no error in time t Probability that none of n qubits has an error in time t Let t be the time taken to perform a gate operation. For an efficient algorithm we might need n2 operations. requires about 300 qubits. This gives Scaling The number of required gate operations tends to grow at least logarithmically in the n Decoherence is a real problem. We need efficient error correction!

31. This is a simultaneous eigenstate of with eigenvalue +1 in both cases. Quantum error-correction An error can make any change to a state so it is not obvious that error-correction is possible. The key idea, of course, is redundancy!

32. If a single spin-flip error occurs

33. We can, in fact correct any single-qubit error using the 7-qubit Steane code: “OK, I’m convinced. Where can I buy one?” All the states differ in least four qubits – they are also common eigenstates of 6 operators with eigenvalue +1. Any single-qubit error is detectable from a unique pattern of changes to these.

34. Ion-trap implementation - Cirac & Zoller, Wineland et al, Blatt et al. Single ion qubits coupled by their centre of mass motion

35. Centre of mass motion acts as a ‘bus’ We can entangle the ionic qubits using the centre of mass motion.

36. Blatt et al Innsbruck

37. Nuclear spins ( Vandersypen, Steffen, Breyta, Yannoni, Cleve, Chuang, July 2000 Physical Rev. Lett. ) • 5-spin molecule synthesized • First demonstration of a fast • 5-qubit algorithm • Pathway to 7-9 qubits

38. Quantum-dot array proposal

39. DiVincenzo’s criteria for implementing a quantum computer • Well defined extendible qubit array - stable memory • Preparable in the “000…” state • Long decoherence time (>104 operation time) • Universal set of gate operations • Single-quantum measurements D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” quant-ph/0002077.

40. Summary • Quantum information is radically different to its classical counterpart. This is because the superposition principle allows for many possible states. • Our inability to measure every property we might like leads to information security, but generalised measurements allow more possibilities than the more familiar von Neumann measurements. • Entanglement is the quintessential quantum property. It allows us to teleport quantum information AND it underlies the speed-up of quantum algorithms. • Quantum information technology will radically change all information processing and much else besides!