PPT ON QUANTUM COMPUTERS

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quantum Computers  Presentation Transcript

1.Quantum Computing

2.Introduction and History Data Representation Operations on Data Shor’s Algorithm Conclusion and Open Questions

3.What is a quantum computer? A quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level.

4.Introduction “I think I can safely say that nobody understands quantum mechanics” - Feynman 1982 - Feynman proposed the idea of creating machines based on the laws of quantum mechanics instead of the laws of classical physics. 1985 - David Deutsch developed the quantum turing machine, showing that quantum circuits are universal. 1994 - Peter Shor came up with a quantum algorithm to factor very large numbers in polynomial time. 1997 - Lov Grover develops a quantum search algorithm with O(vN) complexity

5.Introduction and History Data Representation Operations on Data Shor’s Algorithm Conclusion and Open Questions

6.Representation of Data - Qubits A bit of data is represented by a single atom that is in one of two states denoted by |0> and |1>. A single bit of this form is known as a qubit A physical implementation of a qubit could use the two energy levels of an atom. An excited state representing |1> and a ground state representing |0>.

7.Representation of Data - Superposition A single qubit can be forced into a superposition of the two states denoted by the addition of the state vectors: |?> = ? |0> + ? |1> Where ? and ? are complex numbers and |? | + | ? | = 1

8.Representation of Data - Superposition

9. Data Retrieval In general, an n qubit register can represent the numbers 0 through 2^n-1 simultaneously. Sound too good to be true?…It is! If we attempt to retrieve the values represented within a superposition, the superposition randomly collapses to represent just one of the original values.

10.Relationships among data - Entanglement Entanglement is the ability of quantum systems to exhibit correlations between states within a superposition. Imagine two qubits, each in the state |0> + |1> (a superposition of the 0 and 1.) We can entangle the two qubits such that the measurement of one qubit is always correlated to the measurement of the other qubit.

11.Introduction and History Data Representation Operations on Data Shor’s Algorithm Conclusion and Open Questions

12.Operations on Qubits - Reversible Logic Due to the nature of quantum physics, the destruction of information in a gate will cause heat to be evolved which can destroy the superposition of qubits.

13.Quantum Gates Quantum Gates are similar to classical gates, but do not have a degenerate output. i.e. their original input state can be derived from their output state, uniquely. They must be reversible. This means that a deterministic computation can be performed on a quantum computer only if it is reversible. Luckily, it has been shown that any deterministic computation can be made reversible.(Charles Bennet, 1973)

14.Quantum Gates - Hadamard Simplest gate involves one qubit and is called a Hadamard Gate (also known as a square-root of NOT gate.) Used to put qubits into superposition.

15.Quantum Gates - Controlled NOT A gate which operates on two qubits is called a Controlled-NOT (CN) Gate. If the bit on the control line is 1, invert the bit on the target line.

16.Example Operation - Multiplication By 2 We can build a reversible logic circuit to calculate multiplication by 2 using CN gates arranged in the following manner:

17.Quantum Gates - Controlled Controlled NOT (CCN) A gate which operates on three qubits is called a Controlled Controlled NOT (CCN) Gate. Iff the bits on both of the control lines is 1,then the target bit is inverted.

18.A Universal Quantum Computer The CCN gate has been shown to be a universal reversible logic gate as it can be used as a NAND gate.

19.Introduction and History Data Representation Operations on Data Shor’s Algorithm Conclusion and Open Questions

20.Shor’s Algorithm Shor’s algorithm shows (in principle,) that a quantum computer is capable of factoring very large numbers in polynomial time. The algorithm is dependant on Modular Arithmetic Quantum Parallelism Quantum Fourier Transform

21.Shor’s Algorithm - Periodicity An important result from Number Theory: F(a) = x mod N is a periodic function

22.Shor’s Algorithm - In Depth Analysis To Factor an odd integer N (Let’s choose 15) : Choose an integer q such that N < q < 2N let’s pick 256 Choose a random integer x such that GCD(x, N) = 1 let’s pick 7 Create two quantum registers (these registers must also be entangled so that the collapse of the input register corresponds to the collapse of the output register) Input register: must contain enough qubits to represent numbers as large as q-1. up to 255, so we need 8 qubits Output register: must contain enough qubits to represent numbers as large as N-1. up to 14, so we need 4 qubits

23.Shor’s Algorithm - Preparing Data Load the input register with an equally weighted superposition of all integers from 0 to q-1. 0 to 255 Load the output register with all zeros.

24.Shor’s Algorithm - Modular Arithmetic Apply the transformation x mod N to each number in the input register, storing the result of each computation in the output register.

25.Shor’s Algorithm - Superposition Collapse Now take a measurement on the output register. This will collapse the superposition to represent just one of the results of the transformation, let’s call this value c.

26.Shor’s Algorithm - Entanglement Now things really get interesting ! Since the two registers are entangled, measuring the output register will have the effect of partially collapsing the input register into an equal superposition of each state between 0 and q-1 that yielded c (the value of the collapsed output register.)

27.Shor’s Algorithm - QFT We now apply the Quantum Fourier transform on the partially collapsed input register. The fourier transform has the effect of taking a state |a> and transforming it into a state given by:

28.Shor’s Algorithm - QFT

29. Shor’s Algorithm - QFT The QFT will essentially peak the probability amplitudes at integer multiples of q/4 in our case 256/4, or 64. |0>, |64>, |128>, |192>, … So we no longer have an equal superposition of states, the probability amplitudes of the above states are now higher than the other states in our register. We measure the register, and it will collapse with high probability to one of these multiples of 64, let’s call this value p. With our knowledge of q, and p, there are methods of calculating the period (one method is the continuous fraction expansion of the ratio between q and p.)

30.Shor’s Algorithm - The Factors :) Now that we have the period, the factors of N can be determined by taking the greatest common divisor of N with respect to x ^ (P/2) + 1 and x ^ (P/2) - 1. The idea here is that this computation will be done on a classical computer.

31.Shor’s Algorithm - Problems The QFT comes up short and reveals the wrong period. This probability is actually dependant on your choice of q. The larger the q, the higher the probability of finding the correct probability. The period of the series ends up being odd

32.Introduction and History Data Representation Operations on Data Shor’s Algorithm Conclusion and Open Questions

33.Conclusion In 2001, a 7 qubit machine was built and programmed to run Shor’s algorithm to successfully factor 15. What algorithms will be discovered next? Can quantum computers solve NP Complete problems in polynomial time?