Quantum Computation 650 Spring 2009 Lectures 1-21 The World of Quantum Information Marianna Safronova Department of Physics and Astronomy February 10, 2009 Outline Quantum Information: fundamental principles (and how it is different from the classical one). Bits & Qubits Quantum weirdness: entanglement, superposition & measurement Logic gates & Quantum circuits Cryptography & quantum information A brief introduction to quantum computing Real world: what do we need to build a quantum computer/quantum network? Current status & future roadmap 1
Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. 2
Why Quantum Computers? Computer technology is making devices smaller and smaller reaching a point where classical physics is no longer a suitable model for the laws of physics. Bits & Qubits Fundamental building blocks of classical computers: BITS STATE: Definitely 0 or 1 Fundamental building blocks of quantum computers: Basis states: Quantum bits or QUBITS 0 and 1 Superposition: ψ = α 0 + β 1 3
Bits & Qubits Fundamental building blocks of classical computers: BITS Fundamental building blocks of quantum computers: Quantum bits or QUBITS STATE: Definitely 0 or 1 Basis states: 0 and 1 Qubit: any suitable two-level quantum system 4
Bits & Qubits: primary differences Superposition ψ = α 0 + β 1 Bits & Qubits: primary differences Measurement Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: Quantum calculation: number of parallel processes due to superposition QO FR 5
Bits & Qubits: primary differences Superposition Measurement ψ = α 0 + β 1 Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: we cannot just measure α and β and thus determine its state! We get either 0 or 1 with corresponding probabilities α 2 and β 2. α 2 2 + β = 1 The measurement changes the state of the qubit! Multiple qubits Hilbert space is a big place! - Carlton Caves 6
Multiple qubits Hilbert space is a big place! Two bits with states 0 and 1 form four definite states 00, 01, 10, and 11. Two qubits: can be in superposition of four computational basis set states. ψ = α 00 + β 01 + γ 10 + δ 11 - Carlton Caves 2 qubits 4 amplitudes 3 qubits 8 amplitudes 10 qubits 1024 amplitudes 20 qubits 1 048 576 amplitudes 30 qubits 1 073 741 824 amplitudes 500 qubits More amplitudes than our estimate of number of atoms in the Universe!!! Entanglement ψ = 00 + 11 2 Results of the measurement First qubit 0 1 Second qubit 0 1 ψ α β Entangled states 7
Quantum cryptography Classical cryptography Scytale - the first known mechanical device to implement permutation of characters for cryptographic purposes 8
Classical cryptography Private key cryptography How to securely transmit a private key? Key distribution A central problem in cryptography: the key distribution problem. 1) Mathematics solution: public key cryptography. 2) Physics solution: quantum cryptography. Public-key cryptography relies on the computational difficulty of certain hard mathematical problems (computational security) Quantum cryptography relies on the laws of quantum mechanics (information-theoretical security). 9
Quantum key distribution A quantum communication channel: physical system capable delivering quantum systems more or less intact from one place to another. Quantum mechanics: quantum bits cannot be copied or monitored. Any attempt to do so will result in altering it that can not be corrected. Problems Authentication Noisy channels Quantum logic gates 10
Logic gates Classical NOT gate Quantum NOT gate (X gate) A NOT A α 0 + β 1 X α 1 + β 0 A N O T A Matrix form representation 0 1 1 0 0 1 X = 1 0 The only non-trivial single bit gate α β X = β α More single qubit gates Any unitary matrix U will produce a quantum gate! Z 1 0 = 0 1 α 0 + β 1 α 0 β 1 Z Hadamard gate: H 1 1 1 = 2 1 1 α 0 + β 1 H 0 + 1 0 1 α + β 2 2 11
Single qubit gates,two-qubit gates, three-qubit gates How many gates do we need to make? Do we need three-qubit and four-qubit gates? Where do we find such physical interactions? Coming up with one suitable controlled interaction for physical system is already a problem! Universality: classical computation Only one classical gate (NAND) is needed to compute any function on bits! A B A A B A AND B NOT A A NAND B A B A AND B A NAND B 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 12
Universality: quantum computation How many quantum gates do we need to build any quantum gate? Any n-qubit gate can be made from 2-qubit gates. (Since any unitary n x n matrix can be decomposed to product of two-level matrices.) Only one two-qubit gate is needed! Example: CNOT gate Quantum parallelism I can look at both sides of my coin at a single glance 13
Quantum parallelism ( ) : { 0,1} { 0,1} f x x, y x, y f ( x) U f Superposition 0 + 1 2 0 x y U f x f(x) 0, f (0) + 1, f (1) 2 Single circuit just evaluated f(x) for both x=0 and 1 simultaneously! Quantum parallelism: a major problem So we can evaluate functions for all values of x at the same time using just one circuit! Need only n+1 qubits to evaluate 2 n values of x. But we still get only one answer when we measure the result: it collapses to x,f(x)!!! QO FR 14
Quantum parallelism: a major problem So we can evaluate functions for all values of x at the same time using just one circuit! Need only n+1 qubits to evaluate 2 n values of x. But we still get only one answer when we measure the result: it collapses to x,f(x)!!! We can t read out the result for specific value of x! With one qubit we get either f(0) or f(1) and we don t know which one in advance. This is why we need quantum algorithms! Quantum algorithms Unique features of quantum computation Superposition: n qubits can represent 2 n integers. Problem: if we read the outcome we lose the superposition and we can t know with certainty which one of the values we will obtain. Entanglement: measurements of states of different qubits may be highly correlated. 15
Quantum algorithms Strategy: Use superposition to calculate 2 n values of function simultaneously and do not read out the result until a useful result is expected with reasonably high probability. Use entanglement Current advantages of quantum computation Shor's quantum Fourier transform provides exponential speedup over known classical algorithms. Applications: solving discrete logarithm and factoring problems which enables a quantum computer to break public key cryptosystems such as RSA. Quantum searching (Grover's algorithm) allows quadratic speedup over classical computers. Simulations of quantum systems. 16
How to factor 15? Pick a number less then 15: 7 Calculate 7 n mod 15: Calculate n 7 n 15 A 7 n mod 15 1 7 1 7 2 49 45 4 3 343 330 13 4 2401 2400 1 R / 2 { ± } gcd 7 1,15 { } { } gcd 48,15 = 3, gcd 50,15 = 5 R=4 Shor s algorithm for N=15 Choose n such as 2 n <15: n=4 Choose y: y=7 Initialize two four-qubit register Create a superposition of states of the first register Compute the function f(k)=7 k mod 15 on the second register. Operate on the first register by a Fourier transform Measure the state of the first register: u=0, 4, 8, 12 are only non-zero results. ψ 0 = 0000 0000 Two cases give period R=4, therefore the procedure succeeds with probability 1/2 after one run. 17
Back to the real world: What do we need to build a quantum computer? Qubits which retain their properties. Scalable array of qubits. Initialization: ability to prepare one certain state repeatedly on demand. Need continuous supply of 0. Universal set of quantum gates. A system in which qubits can be made to evolve as desired. Long relevant decoherence times. Ability to efficiently read out the result. Real world strategy If X is very hard it can be substituted with more of Y. Of course, in many cases both X and Y are beyond the present experimental state of the art David P. DiVincenzo The physical implementation of quantum computation. 18
Quantum Computer (Innsbruck) P 1/2 D 5/2 quantum bit S 1/2 Courtesy of the R. Blatt s Innsbruck group Experimental proposals Liquid state NMR Trapped ions Cavity QED Trapped atoms Solid state schemes And other ones 19
The DiVincenzo criteria Quantum Information Processing with Neutral Atoms Quantum computing with neutral atoms 20
1. A scalable physical system with well characterized qubits: memory (a) Internal atomic state qubits: ground hyperfine states of neutral trapped atoms well characterized Very long lived! F=2 5s 1/2 6.8 GHz F=1 1 0 M F =-2,-1,0,1,2 87 Rb: Nuclear spin I=3/2 M F =-1,0,1 1. A scalable physical system with well characterized qubits: memory (b) Motional qubits : quantized levels in the trapping potential also well characterized 21
1. A scalable physical system with well characterized qubits: memory (a)internal atomic state qubits (b) Motional qubits Advantages: very long decoherence times! Internal states are well understood: atomic spectroscopy & atomic clocks. 1. A scalable physical system with well characterized qubits Optical lattices: loading of one atom per site may be achieved using Mott insulator transition. Scalability: the properties of optical lattice system do not change in the principal way when the size of the system is increased. Designer lattices may be created (for example with every third site loaded). Advantages: inherent scalability and parallelism. Potential problems: individual addressing. 22
2: Initialization Internal state preparation: putting atoms in the ground hyperfine state Very well understood (optical pumping technique is in use since 1950) Very reliable (>0.9999 population may be achieved) Motional states may be cooled to motional ground states (>95%) Loading with one atom per site: Mott insulator transition and other schemes. Zero s may be supplied during the computation (providing individual or array addressing). 3: A universal set of quantum gates 1. Single-qubit rotations: well understood and had been carried out in atomic spectroscopy since 1940 s. 2. Two-qubit gates: none currently implemented (conditional logic was demonstrated) Proposed interactions for two-qubit gates: (a) Electric-dipole interactions between atoms (b) Ground-state elastic collisions (c) Magnetic dipole interactions Only one gate proposal does not involve moving atoms (Rydberg gate). Advantages: possible parallel operations Disadvantages: decoherence issues during gate operations 23
Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Why Rydberg gate? Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Do not need to move atoms! FAST! 24
Local blockade of Rydberg excitations Excitations to Rydberg states are suppressed due to a dipole-dipole interaction or van der Waals interaction http://www.physics.uconn.edu/~rcote/ Rydberg gate scheme Rb 40p FAST! R Apply a series of laser pulses to realize the following logic gate: 1 5s 0 1 2 Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) 00 00 10 01 01 10 11 11 25
Rydberg gate scheme Rb 5s R 1 0 40p 1 2 [ 1] 2 [ 2] [ 1] π π π 00 00 00 00 01 01 01 01 10 R0 R0 10 11 R1 R1 11 4. Long relevant decoherence times F=2 Memory: long-lived states. 5s 1/2 6.8 GHz F=1 0 1 Fundamental decoherence mechanism for optically trapped qubits: photon scattering. Decoherence during gate operations: a serious issue. 5: Reading out a result Quantum jump method via cycling transitions. Advantages: standard atomic physics technique, well understood and reliable. 26
Experimental status All five requirements for quantum computations are implemented. Specifically two-qubit gates are demonstrated in several implementations. Several-qubit systems and search algorithm with toy data are demonstrated with NMR. Decoherence-free subspaces, noiseless subsystems, and active control are demonstrated. 27
Future Whatever is the future of the Quantum computation it is certainly a fascinating new field for human creativity, ingenuity and Our strive to learn more about the Universe. 28