Quantum Computing

1. Quantum Computing Stephen Bartlett www.physics.usyd.edu.au/~bartlett

2. A Puzzle • Two rooms: • One room has three light switches • These are connected to three bulbs in the other room • You don’t know which bulbs are connected to which switches

3. A Puzzle • Condition: you’re only allowed to go into each room once • PROBLEM: how do we figure out which bulb is connected to which switch?

4. The mathematician’s problem • As a mathematical problem, there is no solution • I.e., there is no configuration for the switches (which you can only set once) that will give a unique matching of bulbs to switches when you observe the lights ? ? ? ? ? NO SOLUTION?

5. The physicist’s solution • As a physics problem, there is a solution • In the switch room: • Turn on two switches for a few minutes, then turn one off • In the bulb room: • See which bulb is on, feel which other bulb is hot Off Hot On Off On, then off On

6. Information is Physical - Rolf Landauer IBM Research

7. Information is... • ... abstract, but its use requires a physical representation • ... encoded in the symbols on a page, the registers of a computer, the neurons of a brain or the base-pairs in DNA • ... governed by the laws of physics!!! No information without representation!

8. A physicist’s view of computers ? ? = = Is there a fundamental difference between computers? What are their limitations, if any? Heat Energy PHYSICS Output Input 10101100101010001010101 0101001001001010101001

9. Information and physics Information is physical, and governed by the laws of physics Our best framework for physical theories is quantum mechanics Use quantum mechanics to describe information Quantum information! Quantum information investigates the processing, storage, and acquisition of information using quantum physics

10. Quantum computation • We can use quantum physics to solve mathematical problems Example: factor a 300-digit number M. Nielsen, Scientific American, Nov 2002 Shor’s quantum algorithm can factor numbers very quickly Difficulty of factorizing is the basis for modern cryptosystems used on the internet

11. Quantum Cryptography • Two remote parties can communicate securely by using the laws of quantum physics Quantum physics provides a powerful trade-off Information gain Disturbance

12. Quantum Algorithms

13. What is an algorithm? • Consider a problem where each instance has a solution • Example of a problem: Is an integer p a prime number? • The instance: a particular choice of integer • The solution: either yes or no (a decision problem) • Algorithm: a detailed step-by-step method for solving a problem • Example algorithm: a program PRIMALITY(p) that runs on a computer and gives yes or no for any input integer p Computer: a universal machine that can implement any algorithm Alan Turing

14. Example: discrete Fourier transform • Problem: for a given vector (xj), j=1,...,N, what is the discrete Fourier transform (DFT) vector • Algorithm: a detailed step-by-step method to calculate the DFT (yj) for any instance (xj) • With such an algorithm, one could: • write a DFT program to run on a computer • build a custom chip that calculates the DFT • train a team of children to execute the algorithm

15. +more? Computational complexity • Consider an algorithm that solves a given problem • Question: how much computing power do I need to execute this algorithm for a given input (instance) size? • Let N be an integer describing the size of our instance • Example:N could be the number of bits needed to write the input in memory • How does the number of steps in our algorithm depend on N? (Definition of “steps” is a bit arbitrary, but the choice doesn’t affect scaling)

16. Computational complexity of DFT • For the DFT, N could be the dimension of the vector • To calculate each yj, must sum N terms • This sum must be performed for N different yj • Computational complexity of DFT: requires N2 steps • DFTs are important ! a lot of work in optical computing (1950s,1960s) to do fast DFTs • 1965: Tukey and Cooley invent the Fast Fourier Transform (FFT), requires N logN steps • FFT much faster ! optical computing almost dies overnight

17. Complexity classes - P and NP All problems Naively categorise problems: • P: the set of problems with an algorithm that requires resources that are polynomial in the size of the problem • Problems in P are considered “solvable” • Not the whole story: an algorithm that scales as N100 is not easy in practice • Both DFT and FFT are in P but FFT requires fewer resources • NP: the set of problems for which a “guessed” solution can be checked using polynomial resources • Some problems in NP can be used for cryptography (data encryption, secure communication, etc.) NP P P = NP ?

18. RSA-129 1143816257578888676692357799761466120102182 96721242362562561842935706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 x 32769132993266709549961988190834461413177642967992942539798288533 Example: Factoring • Factoring: given a number, what are its prime factors? • Considered a “hard” problem in general, especially for numbers that are products of 2 large primes • Best factoring algorithm requires resources that grow exponentially in the size of the number (RSA-129 took 17 years) • Example: factor a 300-digit number • Best algorithm: takes 1024 steps • On computer at THz speed: 150,000 years • Difficulty of factoring is the basis of security for the RSA encryption scheme used, e.g., on the internet • Information security of interest to private and public sectors Example: 4633 = 41 x 113

19. Richard Feynman Problems a quantum system can solve ? P David Deutsch Quantum algorithms • Feynman (1982): there may be quantum systems that cannot be simulated efficiently on a “classical” computer • Deutsch (1985): proposed that machines using quantum processes might be able to perform computations that “classical” computers can only perform very poorly • Concept of quantum computer emerged as a universal device to execute such quantum algorithms

20. Factoring with quantum systems • Shor (1995): quantum factoring algorithm • To implement Shor’s algorithm, one could: • run it as a program on a “universal quantum computer” • design a custom quantum chip with hard-wired algorithm • find a quantum system that does it naturally! (?) Example: factor a 300-digit number Scientific American, Nov 2002

21. Implications • Information security and e-commerce are based on the use of NP problems that are not in P • must be “hard” (not in P) so that security is unbreakable • requires knowledge/assumptions about the algorithmic and computational power of your adversaries • Quantum algorithms (e.g., Shor’s factoring algorithm) require us to reassess the security of such systems • Lessons to be learned: • algorithms and complexity classes can change! • information security is based on assumptions of what is hard and what is possible ! better be convinced of their validity

22. How do quantum algorithms work? • What makes a quantum algorithm potentially faster than any classical one? • Quantum parallelism: by using superpositions of quantum states, the computer is executing the algorithm on all possible inputs at once • Dimension of quantum Hilbert space: the “size” of the state space for the quantum system is exponentially larger than the corresponding classical system • Entanglement capability: different subsystems (qubits) in a quantum computer become entangled, exhibiting nonclassical correlations • We don’t really know what makes quantum systems more powerful than a classical computer • Quantum algorithms are helping us understand the computational power of quantum vs classical systems

23. Implementations of Quantum Computing

24. Strong control i.e., strongly coupled to user Low noise i.e., an isolated, closed system Experimental QIP • Realising quantum information processing in a lab is extremely difficult • Requires two almost mutually-exclusive conditions: • Experimental effort: to gain strong, precise control over quantum systems that maintain their quantum nature

25. U Example 1: spin of electrons • The spin of an electron gives a quantum system • We have strong control over this spin using electric and magnetic fields • But through spin-spin interactions, a single electron spin interacts with every other electron nearby!

26. U? Example 2: polarised photons • The polarisation of a photon gives a quantum system • Photons in free space do not interact with each other (i.e., with electric or magnetic fields) • But how can we entangle two photons if we can’t interact them?

27. DiVincenzo criteria David DiVincenzo (IBM) – requirements for a quantum computer: • The machine must have a scalable collection of bits • It must be possible to initiate all of the bits to zero • The error rate should be sufficiently low • It must be possible to perform elementary logical operations between pairs of bits • Reliable readout of the final result must be possible Each bit must be individually addressable, and it must be possible to scale up to a large number of bits Decoherence times must be much longer than the gate operation times

28. Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atom Linear optics Nitrogen vacancies in diamond Electrons in liquid He Superconducting Josephson junctions charge qubits flux qubits phase qubits Quantum Hall qubits Coupled quantum dots spin, charge, excitons Spin spectroscopies, impurities in semiconductors Physical implementations Many sub-fields of physics have proposals for QC

29. Ion traps • Qubit: internal electronic state of atomic ion in a trap (ground and excited) • Coupling: use quantised vibrational mode along linear axis (phonons) • Single qubit gates: using laser Cirac and Zoller, Phys. Rev. Lett. (1995) The latest: Monroe group – UMich “T-Junction trap” Shuttling ions around corners

30. Linear optics • Qubit: polarisation of a single photon • Coupling: via measurement • Single-qubit gates: polarisation rotation = 1 Knill, Laflamme, Milburn, Nature (2001) = 0 The latest: Zeilinger group – UVienna “One-way” quantum computing with four qubits

31. Superconducting Josephson junctions Qubit: • Magnetic flux trapped in loop • Cooper pair charge on metal box • Charge-phase • Coupling: capacitive/inductive • Single-qubit gates: flux bias, charge on gate, current through junction Nakamura, Pashkin, Tsai, Nature (1999) The latest: Schoelkopf group – Yale Coherent coupling of a single photon to a superconducting qubit (Cooper pair box)

32. Nuclear magnetic resonance (NMR) • Qubit: nuclear spins of atoms in a designer molecule • Coupling and single-qubit gates: RF pulses tuned to NMR frequency Gershenfeld and Chuang, Science (1997) Silicon quantum computing • Qubit: • Nuclear spin of single P donor • Electron spin of single donor • Coupling: gate-controlled electron-electron interaction • Single-qubit gates: NMR pulse; gate bias in magnetic material Kane, Nature (1998)

33. Summary • Quantum computation requires precise control over isolated systems • Many possible physical realisations may lead to discoveries and advances in quantum computation • Are we at the turning point? • Recent theoretical results strongly suggest QC is feasible • Recent experimental developments suggest we might be there soon Australia is a major player UNSW, Melbourne and Queensland: experiment Queensland, Sydney, Macquarie, Griffith: theory

34. Quantum Cryptography

35. Cryptography • Alice wants to send a message to Bob, without an eavesdropper Eve intercepting the message • Public key cryptography (e.g., RSA): • security rests on assumptions about comp. complexity • vulnerable to attacks by a quantum computer! • Quantum mechanics provides a secure solution with quantum key distribution (QKD)

36. 11110 00100 11110 A B +11010 -11010 No transmitted information! 11110 00100 Private Key Cryptography • Private key cryptography can be provably secure • Alice has secret encoding key e, Bob has decoding key d • Protocol: message x, functions E(x,e) and D(y,d) s.t. • E.g.: one-time pad (e=d, random string as long as x) D(E(x,e),d) = x

37. 0110110011 Trusted courier? Problems with private keys • How are the private keys distributed? • Security rests on private keys being kept secret • Ideally, A and B wish to generate strings of random numbers secretly and nonlocally • Privacy amplification and information reconciliation can be applied to make near-perfect private keys

38. Eve receives a qubit that is either in or Measure in basis? Always gets right, leaves state in 50% chance will mistake for Collapses into basis Disturbance! Measure in basis? Similar result Using quantum mechanics • Information gain implies disturbance: • Any attempt to gain information about a quantum system must alter that system in an uncontrollable way • Example: non-orthogonal states of a qubit • Information gain by Eve causes an uncontrollable disturbance

39. a1 determines which basis a2 is an encoded bit in that basis BB84 QKD Protocol • 1984: Bennett and Brassard • Alice generates two random bits, a1,a2 • Alice prepares a qubit as follows: • Alice then sends the qubit to Bob

40. BB84 QKD Protocol • Bob receives the qubit • Bob chooses a random bit b1and measures the qubit as follows: • if b1=0, Bob measures in the basis • if b1=1, Bob measures in the basis obtaining a bit b2 • Alice and Bob publicly compare a1 and b1 • if they are the same (Bob measured in the same basis that Alice prepared) then a2=b2 • if they disagree, they discard that round This protocol is repeated(4+)ntimes

41. BB84 QKD Protocol • With high probability, Alice and Bob have 2n successes • To check for Eve’s interference: • Alice chooses n bits randomly and informs Bob • Alice and Bob compare their results for these n bits • If more than an acceptable number disagree, they abort ! evidence of Eve’s tampering (or a noisy channel) • Alice and Bob use the remaining n bits as a private key!

42. Summary of quantum crypto • Information is physical • Information gain implies disturbance: • Any attempt to gain information about a quantum system must alter that system in an uncontrollable way • Use this property to protect information • An eavesdropper’s attempt to gain information will alter the system and thus may be detected! • Future attempts to communicate securely or to protect private information in the midst of public decision may rely on quantum physics