Quantum Computation and Quantum Information Part 1 of CS406 – Research Directions in Computing Dr. Rajagopal Nagarajan (biju@dcs) Assistant: Nick Papanikolaou.

Quantum Computation and Quantum Information

Part 1 of CS406 – Research Directions in Computing

Dr. Rajagopal Nagarajan (biju@dcs)

Assistant: Nick Papanikolaou (nikos@dcs)

Topics

What is quantum information processing? What does quantum mechanics make possible? What does quantum mechanics make impossible? When will quantum information processing be realized?

Moore’s Law

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.

Physics and Computation

Information is stored in a physical medium, and manipulated by physical processes.

The laws of physics dictate the capabilities of any information processing device.

Designs of “classical” computers are implicitly based in the classical framework for physics

Classical physics is known to be wrong or incomplete… and has been replaced by a more powerful framework: quantum mechanics.

A Quote

The nineteenth century was known as the machine age, the twentieth century will go down in history as the

information age. I believe the twenty-first century will be the quantum age.

--- Paul Davies, Professor Natural Philosophy, Australian Centre for Astrobiology

Small devices; quantum scale

The design of devices on such a small scale will require engineers to control quantum mechanical effects.

Allowing computers to take advantage of quantum mechanical behaviour allows us to do more than cram increasingly many microscopic components onto a silicon chip…

… it gives us a whole new framework in which information can be processed in fundamentally new ways.

Quantum mechanics and information

0 1

Any physical medium capable of representing 0 and 1 is in principle capable of storing any linear combination:

The states |0> and |1> are called computational basis states. Measurement of a qubit in a general state collapses it to one of the two basis states.

Quantum bits (Qubits)

A quantum bit is modelled by a vector in a two dimensional Hilbert space

The symbol for a quantum state is |ψ>

The two coefficients are complex numbers, and when squared they give the probability of obtaining state |0> or state |1>.

0 1

A Register of N qubits

The general state of n qubits is nx

x x}1,0{

α

The state is represented by a unit vector in an exponentially large Hilbert space!

where the x are complex numbers satisfying the normalization constraint:

.1}1,0{

2 nx

xα

Therefore, it seems exponentially hard to simulate n quantum particles on a classical computer (Feynman).

Quantum mechanics and information: Issues

How does this affect computational complexity? How does this affect information security? How does this affect communication complexity? How does quantum information help us better

understand physics?

How does this affect what is feasibly computable?

Which “infeasible” computational tasks become “feasible”?

How does this affect “computationally secure” cryptography?

What new computationally secure cryptosystems become possible?

Are quantum computers realistic?

If the quantum computers are a reasonable model of computation, and classical devices cannot efficiently simulate them, then the Strong Church-Turing thesis needs to be modified to state:

The answer seems to be YES!

A quantum computer can efficiently simulate any realistic model of computation.

photon source beamsplitter

detectors

A simple experiment in optics

• Consider a setup involving a photon source, a half-silvered mirror (beamsplitter), and a pair of photon detectors.

50%

50%

Simplest explanation: beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other.

Firing a photon into the device

100%

The simplest explanation is wrong!

The simplest explanation for this modified setup would still predict a 50-50 distribution:

full mirror

The weirdness of quantum mechanics

Classical probabilities

Consider a computation tree for a simple two-step (classical) probabilistic algorithm, which makes a coin-flip at each step, and whose output is 0 or 1:

2

1

2

1

2

1

2

1

2

1

0

1

0

1

The probability of the computation following a given path is obtained by multiplying the probabilities along all branches of that path… in the example the probability the computation follows the red path is

4

1

2

1

2

1

The probability of the computation giving the answer 0 is obtained by adding the probabilities of all paths resulting in 0:

21

41

41

2

1

|02

1

2

1

2

1

2

1

|1

|0

|1

2

1

Quantum probabilities

In quantum physics, we have probability amplitudes, which can have complex phase factors associated with them.

The probability amplitude associated with a path in the computation tree is obtained by multiplying the probability amplitudes on that path. In the example, the red path has amplitude 1/2, and the green path has amplitude –1/2.The probability amplitude for getting the answer |0 is obtained by adding the probability amplitudes… notice that the phase factors can lead to cancellations! The probability of obtaining |0 is obtained by squaring the total probability amplitude. In the example the probability of getting |0 is

02

1

2

12

Consider a modification of the experiment:

The simplest explanation for the modified setup would still predict a 50-50 distribution…

full mirror

Explanation of experiment

0 02

1

12

1

100%

0102

10

2

1

1012

11

2

1

Topics of Interest

Quantum circuits Quantum parallelism Quantum algorithms and applications Quantum cryptography Implementations of quantum computers

A quantum circuit provides an visual representation of a quantum algorithm.

00

00

time

quantum gatesinitial state measurement

Quantum Parallelism

Since quantum states can exist in exponential superposition, a computation of a function being performed on quantum states can process an exponential number of possible inputs in a single evaluation of f :

By exploiting a phenomenon known as quantum interference, some global properties of f can be deduced from the output.

12

0

n

ix xα f

12

0

)(n

ix xfα

Applications

• Efficient simulations of quantum systems

• Phase estimation; improved time-frequency and other measurement standards (e.g. GPS)

• Factoring and Discrete Logarithms

• Hidden subgroup problems

• Amplitude amplification

• and much more…

No Cloning Theorem

ψψψ 0There is no procedure that will copy or “clone” an

arbitrary quantum state, i.e.

Such an operation is not linear, and is not permitted by quantum mechanics.

We can copy all the elements of an orthogonal set of states, but when we extend this operation linearly, no other states will be correctly cloned.

0000

Using no-cloning to detect eavesdroppers

• Any attempts to produce pseudo-clones will be detected with significant probability. In general, any scheme to extract information about the state of a quantum system, will disturb the system in a way that can be detected with some probability.

This idea motived Wiesner to invent quantum money around 1970. His work was essentially ignored by the scientific community for a decade, until Bennett and Brassard built on these ideas to create quantum key distribution.

Quantum Information Security

We can exploit the eavesdropper detection that is intrinsic to quantum systems in order to derive new “unconditionally secure” information security protocols. The security depends only on the laws of physics, and not on computational assumptions.

Quantum key establishment (available now) Quantum random number generation (available now) Quantum money (requires stable quantum

memory)

Implementations

Why is it so hard to build quantum computers? How will they be built? When will we see quantum information processors?

Quantum Information is Fragile

• low energy • isolation from environment

• control of operations • superpositions are very fragile

0 1

106 eV

CLASSICAL|0

|1

10-6 eV

QUANTUM

Quantum Error Correction

… allows quantum computation in the presence of noise.

A quantum computation of any length can be made as accurate as desired, so long as the noise is below some threshold, e.g. P < 10-4.

• imperfections and imprecision are not fundamental obstacles to building quantum computers• this gives a criterion for scalability

guide for experimentalists benchmark for comparing technologies

Devices for Quantum Computing

• Atom traps• Cavity QED• Electron floating on helium• Electron trapped by surface acoustic waves• Ion traps• Nuclear magnetic resonance (NMR)

• Quantum optics

• Quantum dots

• Solid state

• Spintronics

• Josephson junctions

• and more…

What Implementations Look Like

The Bottom Line

What are the capabilities of quantum information processors?

What will be the impact of these capabilities? Which technologies will be realized and when?

When can we implement?

Quantum random number generators: now. Quantum key establishment: soon; some prototypes already

available Small scale quantum computers (e.g. needed for long distance

quantum communication): medium term Large scale quantum computers: medium--long term Precise times are hard to predict since we are in the early

stages and still trying a very broad range of approaches. Once we focus on technologies that show promise, expect progress to be very fast.

•Wireless Sensor Networks•Injectable Tissue Engineering•Nano Solar Cells•Mechatronics •Grid Computing •Molecular Imaging •Nanoimprint Lithography•Software Assurance •Glycomics •Quantum Cryptography