Quantum Computing

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Introduction to Computing

• Is currently done on your laptop today

• Numbers as we commonly use them are in decimal (base 10) format. Computers represent them in binary (base 2).

• Each computational operation is done one step at a time.

Decimal vs. Binary

Decimal Representation

103 102 101 100

1000 100 10 1

The number 14 1 in the 10’s and 4 in the 1’s

How do we represent this in another base?Add up all the digits to get the same value.

Binary Representation

23 22 21 20

8 4 2 1

14 = 8+4+2 → 1110 BITS!

Addition OperationDecimal: 5+4 = 9 Piece of cake in the way we think!

Binary: Represent 5 as a binary number 0101Represent 4 as a binary number 0100Add on a column by column (right to left) basis

1 = 1+0 0 = 0+00 = 1+1 with carry1 = 0+0 + carry

Final binary representation: 1001 (Decimal 9)

+ is one of a set of possible operations a computer can do. Others are -, *, /. Typically only one operation is done at a time. – Serial processing

How do we do this?Shown is an Intel processor capable of performing 1,000,000,000 (1 billion) mathematical operations per second!

It is composed of ~500,000,000 individual Transistors! (Transisitors have no moving parts and are turned on and

off by electrical signals)

The typical size of a transistor in the picture is about 0.04mm

0.4mm

How Far can this go?Scale of transistors rapidly reaching quantum dimensions.

1975 1980 1985 1990 1995 2000 2005104

105

106

107

108

109

number of transistors

year

Moore’s Law

•When the integrated microprocessor circuit was first made at Texas Instruments, Gordon Moore postulated that the number of transistors will doubles every 2 years.• The blue line shows the prediction. • The crosses show the actual numbers. • When the number of transistors goes down, so does the overall dimensions• Transistor size will approach quantum dimensions in ~6-10 years!• We had better be ready to embrace a new approach.

Quantum Principles• Quantum Uncertainty— states that

the position and velocity of a particle are unknown until observed.

• SUPERPOSITION: there is an equal probability that something is either in one state (1) or another (0). Thus, something is in both states, or between both states at the same time until observed.

• Quantum Nonlocality—Entanglement: When two particles share the same quantum state they are entangled. This means that two or more particles will share the same properties: for example, their spins are related. Even when removed from each other, these particles will continue to share the same properties.

Classical vs. Quantum PhysicsClassical Physics

• Predictable outcomes!

• Variables are continuous

• Take on finite known values

• Always has a known state

• Binary numbers in the form of switches that are on or off!

Quantum Physics

• Violate classical laws at a small scale (~h, Plank’s Constant)

• Variables are discrete |0> or |1>

• Can be in a superposition of values (representing all states simultaneously)

• State is intdeterminate until measured.

Bloch sphere shows all possible states Superposition

Quantum Binary SystemsCan we represent a binary system (bits) in

quantum rather than classical states?

Why not find a quantum mechanical system where we can use two ‘states’ to represent our binary numbers?

Qubits can be carried as atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor.

Entangled qubits allow multiple numbers to be represented simultaneously. |000>

QubitQuantum bit

Bit VS Qubit

• Classical Bit Qubit

• 0 or 1 0 or 1 or 1 and 0

• 101000 001 010 011 100 101 110 111

Today’s Computers• Turing Machine- theoretical device that

consists of tape of unlimited length that is divided into little squares. Each square can either hold a symbol (1 or 0) or be left blank.

• Today's computers work by manipulating bits that exist in one of two states: a 0 or a 1.

• 1 and 0’s are carried and turned on by states of electrical current

Quantum Computers• Quantum computers aren't limited to two states

like today’s computers. They encode information as quantum bits, or qubits, which can exist in superposition.

• Superposition- quantum computers can represent both 0 and 1 as well as everything in between at the same time.

• Qubits can be carried as atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor.

• Basically, a quantum computer can work on a million computations at once, while your desktop PC works on one.

How Quantum Computers Work

Superposition Numbers

Ψ= 1/√2(|> + |>) can be written asΨ= 1/√2(|0> + |1>) This can be expanded

|00> + |01> + |10> + |11>

Look! I just wrote down a binary representation where the numbers 0, 1, 2, and 3 are simultaneously represented. This is huge by comparison to a classical computer.

Physical Realization Quantum Dot

Quantum Dot (Single Electron Transistor SET) Use the Electron spin |>, |>

A zero is spin up 1 |>

A one is spin down 0 |>

Manipulate the spin using electric and magnetic fields

•Currently used effectively by Charlie Marcus at Harvard University.

•Current state of the art, 2 transistors talking to each other

Quantum Dot Now and Later

• Easy to make and operate• Small in size : we are dealing with ions,

atoms, electrons and photons• The system falls apart rapidly. That means

that the initial 0 or 1 in the system goes away. This phenomena is called decoherence and happens in about 1 microsecond.

• Compare to classical computation where the transistors can retain their state for years.(for example, how many of you have USB memory sticks?)

Physical Realization Trapped Ions

Trapped Ion use electron system states:

A binary zero is the lowest level or ground stateA one is a chosen higher level Manipulate the state using lasers tuned to the ‘transistion’ frequency in Strontium (408nm in the figure above)The NIST trapped Ion Storage Group currently holds the record of 7 qubits

Experiment

Trapped Ions Now and Later

• Difficult make and operate• Huge in scale, a full laboratory to make

7 systems• The system is very stable (many

seconds) but takes a very long time to impress a 0 or 1 on the system.

• The size and speed make it difficult to imagine a system like your laptop

Today’s Quantum Computer: The 16-qubit quantum computer

2007- Canadian startup company D-Wave demonstrated

a 16-qubit quantum computer (adiabatic computer), which solved a sudoku puzzle and other pattern matching problems.

The company claims it will produce practical systems by 2008 but skeptics believe practical quantum computers are still decades away.

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Photo courtesy © 2007D-Wave Systems, Inc.

D-Wave's 16-qubit quantum computer

Functional quantum computers will be valuable in factoring large numbers and be very useful for decoding and encoding secret information. No information on the internet would be secure.

Quantum computers could also be used to search large databases in a fraction of the time that it would take a conventional computer.

Other applications could include using quantum computers to study quantum mechanics, or even to design other quantum computers.

The Future of Quantum Computers

The University of Rochester &

Research in Quantum Computing

The U of R works with Cornell, Stanford, Harvard, and Rutgers in a collaborative effort for The Center for Quantum Information.

Of the nine professors involved, five come from UR:Carlos R. Stroud, Jr.Ian A. WalmsleyJoseph H. EberlyCarlos R. StroudNicholas P. Bigelow

At UR, there have been 36 peer-reviewed papers published, 13 manuscripts and 68 presentations given.

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Interesting AlgorithmFactoring Prime Numbers

• Prime numbers used in current day cryptography

• Peter Shor discovered quantum factoring algorithm

• Quantum time to factor • Classical time to factor

Peter Shor

€

O 2log N( )1

3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

O log N( )3

( )

Polynomial vs. Exponential time to calculate.

Exponential Speedup using quantum computer!

- Where a classical computer would take 5 trillion years to factor a 5,000 digit number, a quantum computer could finish in 2 minutes.

- Quantum computation is more powerful than classical computation because more can be computed in less time!!

Summary

• Classical computers reaching quantum scale - need alternative to transistor technology

• Quantum physics allows interesting but difficult alternative to computation