Quantum Computing Stephen Bartlett

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

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?

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? ? ? ? ? ?

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 On On, then off Off OnHotOff

IBM Research

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!

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

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

Quantum computation We can use quantum physics to solve mathematical problems Shor’s quantum algorithm can factor numbers very quickly Difficulty of factorizing is the basis for modern cryptosystems used on the internet M. Nielsen, Scientific American, Nov 2002 Best classical algorithm: steps Shor’s quantum algorithm: steps On classical THz computer: 150,000 years On quantum THz computer: <1 second Example: factor a 300-digit number

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

Quantum Algorithms

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 Alan Turing Computer: a universal machine that can implement any algorithm

Example: discrete Fourier transform Problem: for a given vector (x j ), j=1,...,N, what is the discrete Fourier transform (DFT) vector Algorithm: a detailed step-by-step method to calculate the DFT (y j ) for any instance (x j ) 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

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) +more?

Computational complexity of DFT For the DFT, N could be the dimension of the vector To calculate each y j, must sum N terms This sum must be performed for N different y j Computational complexity of DFT: requires N 2 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

Complexity classes - P and NP 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 N 100 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.) P NP All problems P = NP ?

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 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 = x RSA-129

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 P Problems a quantum system can solve ? David Deutsch Richard Feynman

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! (?) Best classical algorithm: steps Shor’s quantum algorithm: steps On classical THz computer: 150,000 years On quantum THz computer: <1 second Example: factor a 300-digit number Scientific American, Nov 2002

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

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

Implementations of Quantum Computing

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 Low noise i.e., an isolated, closed system Strong control i.e., strongly coupled to user

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! 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? U?

DiVincenzo criteria David DiVincenzo (IBM) – requirements for a quantum computer: 1. The machine must have a scalable collection of bits 2. It must be possible to initiate all of the bits to zero 3. The error rate should be sufficiently low 4. It must be possible to perform elementary logical operations between pairs of bits 5. 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

Physical implementations 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 Many sub-fields of physics have proposals for QC

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

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

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

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) 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) Silicon quantum computing

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

Quantum Cryptography

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)

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) AB No transmitted information! D(E(x,e),d) = x

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 Trusted courier?

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 Eve receives a qubit that is either inor Measure in basis? 50% chance will mistake for Measure in basis? Similar result Always getsright, leaves state in Collapses into basisDisturbance!

BB84 QKD Protocol 1984: Bennett and Brassard Alice generates two random bits, a 1,a 2 Alice prepares a qubit as follows: Alice then sends the qubit to Bob bitsstate a 1 determines which basis a 2 is an encoded bit in that basis

BB84 QKD Protocol Bob receives the qubit Bob chooses a random bit b 1 and measures the qubit as follows:  if b 1 =0, Bob measures in the basis  if b 1 =1, Bob measures in the basis obtaining a bit b 2 Alice and Bob publicly compare a 1 and b 1  if they are the same (Bob measured in the same basis that Alice prepared) then a 2 =b 2  if they disagree, they discard that round This protocol is repeated (4+  )n times a1a1 b1b1 ?

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!

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