COMPSCI 111/111G. Quantum Computing. Term 1, Prepared using LATEX Beamer by Cristian S.Calude 1 / 26

Transcription

1 COMPSCI 111/111G Quantum Computing Prepared using LATEX Beamer by Cristian S.Calude Term 1, / 26

2 Classical vs. Quantum Computers Quantum computing was first introduced by Yuri Manin in 1980 and Richard Feynman in A quantum computer is a computation device that makes direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Classical computer bit deterministic sequential Quantum computer qubit probabilistic parallel 2 / 26

3 The light switch game Figure 1: Example of light switch game 3 / 26

4 The light switch game Each light switch 1. has a number associated with it called bias, 2. and can be in two positions, ON = +1 and OFF = -1. Adding up all the switches bias values multiplied by their ON/OFF values gives a number. The objective of the game is to set the switches to get the lowest number. 4 / 26

5 The light switch game Switches biases are: +1, +0.2, +0.5, -0.7, -0.8, and Setting them, in order, to OFF, ON, OFF, OFF, ON, OFF, we get the number ( 1) (+1) + (+1) (+0.2) + ( 1) (+0.5) +( 1) ( 0.7) + (+1) ( 0.8) + ( 1) (+0.4) = / 26

6 The light switch game Did we get the lowest number? No, but almost. Switches OFF, OFF, OFF, ON, ON, OFF give the minimal number How did we guess it? Are we sure that -3.6 is the minimal number? Brute force approach: generate all 2 6 = 64 combinations of switches and choose the one which produces the lowest number. Find a better (smarter) algorithm to get the lowest number. 6 / 26

7 The light switch game Why do we need a better algorithm when the brute force gives us the solution? Number of light switches Number of combinations = = = >>> number of atoms in the universe 7 / 26

8 The light switch game A simple algorithm for the light switch game: Set all the switches with positive biases to OFF and all the switches with negative biases to ON and, finally, add up the result to get the lowest overall value. EASY, right? 8 / 26

9 The interacting light switch game A light switch game in which some pairs of switches have biases too. Figure 2: Example of interacting light switch 9 / 26

10 The interacting light switch game After setting switches to ON/OFF values, the new game s number is obtained by adding the sum of the switches bias values multiplied by their ON/OFF values, and the sum of pair of switches biases multiplied by the corresponding switches ON/OFF values. The goal is to set the switches to get the lowest number. 10 / 26

11 The interacting light switch game The interaction biases are: +0.2, -1, -0.3, -0.7, -0.7, +0.3, +0.8 and +0.1 (seven pairs have no interaction bias, i.e. their bias is 0). 11 / 26

12 The interacting light switch game Setting switches to OFF, OFF, OFF, ON, ON, OFF (the solution of the light switch game!) we get the number ( 1) (+1) + ( 1) (+0.2) + ( 1) (+0.5) + (+1) ( 0.7) +(+1) ( 0.8) + ( 1) (+0.4) + ( 1) ( 1) (+0.2) +( 1) ( 1) ( 0.3)+( 1) ( 1) ( 0.7)+( 1) (+1) ( 0.7) +( 1) (+1) (+0.3) + ( 1) (+1) ( 1) + (+1) ( 1) (+0.1). 12 / 26

13 The interacting light switch game Here is a simpler instance of the interacting light switch game: we have two switches with biases +1 and -0.8, respectively, and interaction bias -1. The brute force analysis shows the following: Switch 1 Switch 2 Switches sum Interaction sum Total sum ON ON ON OFF OFF ON OFF OFF The solution of the light switch game is OFF, ON, but this is not a solution for the interacting light switch game, which is OFF, OFF. 13 / 26

14 The interacting light switch game With the better algorithm not working in this case we are left with the brute force solution... Any better ideas? 14 / 26

15 A quantum approach to the interacting light switch game Quantum computers can help! The fundamental idea is to use qubits instead of bits and take advantage of the fact that a qubit can be into a superposition of states. What does superposition mean? A qubit is a two-state quantum-mechanical system. In contrast with the bit, which has to be in one state (+1) or the other (-1) of a two-state classical system, a qubit may be simultaneously in both states. Weird! Think that the qubit has not yet decided which state it wants to be in, hence, temporarily, it is in both states at the same time. 15 / 26

16 A quantum approach to the interacting light switch game Switches (with no biases) in Figure 2 can be represented in a quantum computer as in Figure 3: light switches are ON and OFF at the same time. Figure 3: Quantum representation of the interacting light switch 16 / 26

17 A quantum solution to the interacting light switch game Start with the representation in Figure 3 (only switches, no biases). All light switches are simultaneously ON and OFF, hence the solution (i.e. the correct ON/OFF settings for each switch) is represented in there, but temporarily hidden. Slowly turn on (read) all switch bias values. Simultaneously, slowly the quantum computer turns off the quantum superposition effect, so all switches slowly choose a classical state, either ON or OFF. At the end of the previous operation, each switch has chosen to be either ON or OFF (a solution is given). Quantum mechanics guarantees that the light switches settle into the right states (ON or OFF) to give the lowest overall number as required (the solution is correct). Readout the solution. 17 / 26

18 Adiabatic quantum computers The last step in the quantum solution seems like magic. How does this work? The reason is the adiabatic theorem (M. Born and V. Fock, 1928). A quantum mechanical system subjected to gradually changing external conditions an adiabatic process adapts itself such that it retains its initial character. 18 / 26

19 Adiabatic quantum computers Here is a simple (classical) example. An ideal pendulum in a vacuum with no friction is oscillating in a vertical plane. If the support is moved suddenly, the mode of oscillation of the pendulum will change. If the support is moved sufficiently slowly, the motion of the pendulum relative to the support will remain unchanged. A gradual change in external conditions allows the system to adapt, such that it retains its initial character. 19 / 26

20 Adiabatic quantum computers Q: Do adiabatic quantum computers exist? A: There are four quantum computers, all produced by the company called D-Wave Systems: one D-Wave One (2011) and five D-Wave Two (2013). Q: How does D-Wave assure that its computation is an adiabatic process? A: Among other things, by computing at a temperature below approximately C (almost 1 C colder than the coldest place in the universe, Boomerang Nebula). Q: D-Wave is a probabilistic machine, i.e. it may return wrong answers. Is this not a very bad thing? A: Not always. By returning multiple answers one can get important information about the confidence level of the quantum computer. 20 / 26

21 Adiabatic quantum computers Q: Is D-Wave a truly quantum computer, i.e. it really uses quantum effects? A: Probably yes, but the question is still open. Q: Is D-Wave capable of solving other problems except minimisations of problems where all variables take only 0-1 values? A: The problem is open. Q: Is D-Wave faster than the fastest classical computers? A: There are arguments for yes and arguments for no: it depends on how one measures the speed. No conclusive results yet. Q: Name an interesting problem solved with D-Wave. A: A Harvard University team of researchers presented results of the largest protein folding problem solved to date using D-Wave One in / 26

22 Adiabatic quantum computers Q: But then, D-Wave is a very, very limited machine, isn t it? A: No. Many problems including labelling images detecting and tracking objects in images extracting meaning from texts finding correlation in big databases improving natural language communication man-machine creating and testing scientific hypotheses... can be formulated as minimisations of pseudo-boolean functions with constraints. Q: How many qubits has D-Wave Two? A: / 26

23 Adiabatic quantum computers Q: What type of qubits does D-Wave use? A: D-Wave uses flux qubits pictured below. Flux qubit 23 / 26

24 Adiabatic quantum computers Flux qubit 24 / 26

25 Adiabatic quantum computers Q: Which companies bought D-Wave computers? A: Lockheed Martin (D-Wave One, D-Wave Two), Google and NASA (D-Wave Two), a US intelligence agency (D-Wave Two). Q: How much does a D-Wave cost? A: Not sure, probably around US$ 10, 000, 000. Q: What about D-Wave controversies? A: There are plenty as expected in a very audacious project. See for example the academic debate in the May 12, 2011 edition of Nature. Q: Are there non-adiabatic quantum computers? A: Yes, in theory they exist! For example, a quantum Turing machine is such a quantum computer. However, currently they cannot control more than a few qubits / 26

26 Adiabatic quantum computers Q: What is the size of the latest D-Wave quantum chip? A: Washington generation chips are 2,048 physical qubits. Q: Where can we find more information about D-Wave? A: Visit D-Wave. Q: Is there any research in this area done in the Computer Science Department? A: Yes, see Adiabatic Quantum Computing Challenges. 26 / 26