Quantum Computing Part 2: Computation Based on the paper: E. Rieffel & W. Polak, An introduction to Quantum Computing for Non-Physicists. ACM Computing.

Quantum Computing

Part 2: Computation

Based on the paper:

E. Rieffel & W. Polak, An introduction to Quantum Computing for Non-Physicists. ACM Computing Surveys, 32(3) 300-335, 2000.

And ppt by Melanie Mitchell, Portland State University

Bra/Ket notation

Ket:

• Column vector, used to describe quantum states

• E.g., can be written as

• In general,

Bra: • Conjugate transpose of

x

x

1,0 TT )1,0(,)0,1(

x

Tbabax ),(10

),( bax

Quantum bits (“qubits”)

A qubit is a unit vector in a two-dimensional complex vector space.

Assume a particular basis, denoted by

These basis states represent the classical bit values 0 and 1.

However, unlike classical bits, qubits can be in a

superposition of these two states, e.g.,

such that

1,0

10 ba

122 ba

• Suppose a qubit (a, b)T is measured with respect to the basis

Probability that will be measured is |a|2, and

Probability that will be measured is |b|2.

• When a qubit is measured, its state becomes the basis state that was measured.

• All information about the original superposition is lost.

0

1,0

1

Several qubits

• The system of n qubits "contain" 2n classical bits (basis states)

• Thus the potential of a quantum computer grows exponentially

• We can measure individual qubits in the multi-qubit system – For example, in a two-qubit system we can measure the state of

first or second qubit, or both

• The results of measurement are probabilistic

• After the measurement, the system collapses in the corresponding state

Example of using qubits: Quantum key distribution

(Bennett & Brassard, 1987)

• Solves problem of secure key distribution for private key cryptography

• Public key cryptography using RSA is great, but relies on factoring being a hard problem.

• Private key cryptography is in principle more secure, but relies on private key being securely transmitted.

• Quantum method gives perfect security in private key transmission

Alice wants to send a key to Bob by encoding each bit as the quantum state of a photon.

She secretly encodes each bit by randomly choosing one of two basis:

or

E.g.,

1

0

1 1 0 1 1 1 0 0 0 0

1

0

Bob measures the state of each photon he receives by randomly picking either basis, up-down (ud), or diagonal (diag).

Now Alice and Bob communicate what bases they used for each measurement. They save only the bits for which they used the same bases (approx. 50%). These bits form the private key.

1 1 0 1 1 1 0 0 0 0

What Alice sent

What Bob measured

Basis: ud diag ud diag ud ud diag ud diag diagMeasurement:

1 0 0 1 1 1 1 0 1 0

Bob measures the state of each photon he receives by randomly picking either basis, up-down (ud), or diagonal (diag).

Now Alice and Bob communicate what bases they used for each measurement. They save only the bits for which they used the same bases (approx. 50%). These bits form the private key.

1 1 0 1 1 1 0 0 0 0

What Alice sent

What Bob measured

Basis: ud diag ud diag ud ud diag ud diag diagMeasurement:

1 0 0 1 1 1 1 0 1 0

Now, suppose a third person, Eve, intercepts the stream of photons from Alice, measures their states, and sends new photons with the measured states to Bob.

She will choose the wrong basis 50% of the time.

So on the photons for which Bob chooses the same basis as Alice, 25% of them will be incorrect.

Bob and Alice can detect this error by sending sample “correct” (parity) bits to each other over open communication line. Thus they can detect an eavesdropper.

Multiple qubits• Suppose you have n particles whose states are each vectors in a 2-D vector

space (e.g., x, y position)

• How many dimensions do you need to describe complete state of the system?

– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)

– Complete state is simply a list of all the coordinates:

(x1, y1, x2, y2, ..., xn, yn)

– i.e., 2n dimensions

• Quantum case is different due to “entanglement”.

• Again, suppose you have n particles whose states are each vectors in a 2-D vector space.

– Particles have coordinates (x1, y1), (x2, y2), ..., (xn, yn)

– Complete state is more complicated, due to entanglement of states:

(x1x2x3...xn-1xn, x1x2x3...xn-1yn, x1x2x3...yn-1xn, ...)

– i.e., 2n dimensions

Example: n = 2

– Classical basis vectors: (1, 0), (0, 1)– Classical state of system: ( (1, 2), (2, 1) )

– Quantum basis vectors (Tensor Product):

(1, 0) (1,0) = (1, 0, 0, 0)

(1, 0) (0,1) = (0, 0, 1, 0)

(0, 1) (1,0) = (0, 1, 0, 0)

(0, 1) (0, 1)= (0, 0, 0, 1)

– Quantum state of system: Linear combination of those basis vectors

x

y (1, 2)

(2, 1)

Interpretation: In quantum system you now need four numbers to give the state of the two-particle system. Each dimension describes a particular entanglement of the original dimensions of the individual particles.

x

y (1, 2)

(2, 1)

In quantum computation, the “particles” are the qubits.

Each qubit has a state defined by two complex numbers:

A two-qubit state is defined by four complex numbers:

1

01 and

0

10

where,10 ba

1

0

0

0

11 ,

0

0

1

0

10 ,

0

1

0

0

01 ,

0

0

0

1

00

where,11100100

dcbaq

• Recall the notion of superposition:

– A single qubit is in a superposition of two basis states:

– A 3-qubit system is in a superposition of eight basis states:

where a1, a2, ..., an are complex numbers such that

• In general, an n-qubit system is in a superposition of 2n basis states.

10 ba

111110101100011010001000 87654321 aaaaaaaa

12

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1 aaaaaaaa

• This exponential increase in size of state space as n increases is why classical computers cannot efficiently simulate quantum systems.

• This was Feynman’s impetus for proposing quantum computers.

Measurement in entangled systems

• Example 1: The state is entangled:

– If no bits have yet been measured, the probability of measuring the first (or second) bit as is 0.5.

– However, if the second (or first) bit has been measured, the probability is 0 or 1, depending on what the second (or first) bit was measured to be.

)1100(2

1

0

• Example 2: The state is not entangled:

Can factor out first bit: it will always be measured as .

Measuring it does not affect the second bit.

)0100(2

1

0

EPR paradox

(Einstein, Podolsky, Rosen, “Can quantum-mechanical description of physical reality be considered complete?” (1935))

• Imagine a source that generates two maximally entangled particles:

• One particle is sent to Alice, and the other to Bob.

• Suppose Alice measures her particle and observes state .

• Then the combined state will now be . This means Bob will also measure .

• The change in state occurs instantaneously, no matter how far away Alice and Bob are from each other.

)1100(2

1

0

000

• Does this allow communication faster than the speed of light?

• EPR’s explanation: – Each particle has a “hidden” internal state that determines what the result of any

given measurement will be. – Called “local hidden variable theory”.– Einstein: “God does not play dice.”

• Problem with this explanation: – Doesn’t explain measurements with respect to a different basis. – Bell’s theorem: predicts an inequality that must be satisfied if hidden variable

theory is correct.– Experiments show it to be incorrect.

• Cause and effect explanation:

– Somehow Alice’s measurement causes the result found by Bob,

• Problems with this explanation:

– Violates relativity theory: Can set up experiment where one observer sees Alice do measurement first, while other observer sees Bob do measurement first. This does not change the results of the measurements.

– This also shows that Alice and Bob can’t use their particles to communicate faster than the speed of light.

– “All that can be said is that Alice and Bob will observe the same random behavior.”

• From Black et al., “Quantum computing and communications” (2002):

“The ugly truth is that general relativity and quantum mechanics are not consistent. That is, our current formulations of general relativity and quantum mechanics give different predictions for extreme cases. We assume there is a ‘Theory of Everything’ that reconciles the two, but it is still very much an area of thought and research.

Since relatively is not needed in quantum computing, we ignore this problem.”

Schor’s Algorithm

• Breaking RSA reduces to finding the prime factors of a large integer N

• Problem solved by period-finding

• E.g. sequence of integers : the powers of two.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …

• Look at the powers of 2 “mod 15″: in other words, the remainder when 15 divides each power of 2.

2, 4, 8, 1, 2, 4, 8, 1, 2, 4, …

• Taking the powers of 2 mod 15 gives a periodic sequence, whose period (i.e., how far you have to go before it starts repeating) is 4.

Euler in the 1760’s

• Let N be a product of two prime numbers, p and q, and consider the sequence

x mod N, x2 mod N, x3 mod N, x4 mod N, …

• Provided x is not divisible by p or q, the above sequence will repeat with some period that evenly divides (p-1)(q-1).

– e.g., if N=15, then the prime factors of N are p=3 and q=5, so (p-1)(q-1)=8. The period of the sequence was 4, which divides 8.

– If N=21, then p=3 and q=7, so (p-1)(q-1)=12. And indeed, the period was 6, which divides 12.

• If we could learn several random divisors of (p-1)(q-1) (e.g., by trying different random values of x),

– then with high probability we could put those divisors together to learn (p-1)(q-1) itself.

– once we knew (p-1)(q-1), we could then use some more little tricks to recover p and q, the prime factors we wanted.

Problem

Even though the sequence x mod N, x2 mod N, x3 mod N, x4 mod N, …

will eventually start repeating itself,

the number of steps before it repeats could be almost as large as N itself — and N might have hundreds or thousands of digits!

This is why finding the period doesn’t seem to lead to a fast classical factoring algorithm.

Quantum Computer Solution

Suppose we could create an enormous quantum superposition over all the numbers in our sequence:

x mod N, x2 mod N, x3 mod N, x4 mod N, …

There may be some quantum operation we could perform on that superposition that would reveal the period

To get this period-finding idea to work, we’ll have to answer two questions :

1. Using a quantum computer, can we quickly create a superposition over

x mod N, x2 mod N, x3 mod N, x4 mod N, …?

2. Supposing we did create such a superposition, how would we figure out the period?

1. Create a superposition over all integers r, from 1 up to N or so by using repeated squaring.

e.g. Suppose N=17, x=3, and r=14. Then the first step is to represent r as a sum of powers of 2:

r = 23 + 22 + 21.Then

xr = 314 = 3 23 + 22 + 21 = 3 23 3 22 3 21

Also, notice that we can do all the multiplications mod N, thereby preventing the numbers from growing out of hand at intermediate steps. This yields the result

314 mod 17 = 2.

We can create a quantum superposition over all pairs of integers of the form (r, xr mod N), where r ranges from 1 up to N

2. Given a superposition over all the elements of a periodic sequence, how do we extract the period of the sequence?

To get the period out, Shor uses something called the quantum Fourier transform, or QFT.

The QFT is a linear transformation (indeed a unitary transformation) that maps one vector of complex numbers to another vector of complex numbers.

It is a linear transformation that maps a quantum state encoding a periodic sequence, to a quantum state encoding the period of that sequence.

Another way to think about this is in terms of interference

Key point about quantum mechanics — the thing that makes it different from classical probability theory —

probabilities are always nonnegative

amplitudes in quantum mechanics can be positive, negative, or even complex.

amplitudes corresponding to different ways of getting a particular answer can “interfere destructively” and cancel each other out.

For all periods other than the “true” one, these contributions point in different directions and therefore cancel each other out.

Only for the “true” period do the contributions from different solutions all point in the same direction.

And that’s why, when we measure at the end, we’ll find the true period with high probability.

• The QFT closely related to the classical fast Fourier transform (FFT), but can be computed exponentially faster.

• The FFT takes as input a string of K complex numbers, xk, and produces as output another string of K numbers, yk, with

For an input string of K numbers, which repeat themselves with period r, the FFT produces an output string with period K/r, as illustrated in the following examples for the case of K = 8).

1 0 0 0 0 0 0 0 → 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 → 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 → 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 → 1 0 0 0 0 0 0 0

In case r divides K with a remainder, the inversion of the period is only approximate.

The FFT turns shifts in the input string into phase factors in the output string:

1 0 0 0 1 0 0 0 → 1 0 1 0 1 0 1 00 1 0 0 0 1 0 0 → 1 0 –i 0 –1 0 i 00 0 1 0 0 0 1 0 → 1 0 –1 0 1 0 –1 00 0 0 1 0 0 0 1 → 1 0 i 0 –1 0 –i 0

• The QFT performs exactly the same transformation as the FFT, but the complex numbers are stored in the amplitude and phase of the terms in a superposition.

Shor’s Quantum Factoring

• Quantum operation takes polynomial time O(N3)

• Best known traditional algorithm takes O(en1/3)– We have an exponential speed up!�

• The fastest supercomputer in the world can’t compete with a quantum computer for certain problems!– Many encryption algorithms rely on the difficulty of factoring �

large numbers.

– Motivation to build one!�

5-qubit example

• A 5-qubit example shows how quantum parallelism and use of the QFT make it possible to efficiently find the period of a function. For clarity, the states are written in decimal instead of binary notation, for example, |010 will be denoted |2 . ⟩ ⟩

• Consider two registers (groups of qubits) in which the first register contains three qubits, each initialized to an equal superposition of |0 ⟩and |1 , and with the second register consisting of two qubits set to |⟩0 . Suppressing normalization constants for clarity, the state of the ⟩system is thus written as

(|0 + |1 + |2 + |3 + |4 + |5 + |6 + |7 )|0⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟩ ⟩

Practical Requirements for ExperimentalImplementation

1. Need a system of qubits. 2. The qubits must be individually addressable and must

interact with each other to provide for a universal set of logic gates.

3. It must be possible to initialize them to a known state because the result of a computation generally depends on its input state.

4. Must be able to extract a computation result from the qubits by some measurement.

5. A long coherence time compared with an average logic gate's duration, such that many logic gates can be implemented within the coherence time.

Nuclear magnetic resonance quantumcomputing (NMR QC)

• Many atoms such as 1H, 13C, and 19F have a spin-1/2 nucleus.

• A nuclear spin-1/2 can be thought of as a tiny bar magnet spinning about its own axis, with two well-defined states; it can be aligned or anti-aligned with respect to an external magnetic field (spin up or spin down) and thus can represent logical zero and one.

• Since a nuclear spin is extremely small, it's a quantum mechanical object and can exist in a superposition of up and down.

• An atomic nucleus' spin can thus serve as a quantum bit.

• An NMR quantum computer consists of several individual atoms with a spin-1/2 nucleus.

Nuclear Magnetic Resonance

• RF pulse can effect the nuclear spin state

• Quantum NOT operation

– A properly timed and calibrated RF �pulse flips the spin state from spin up to down

• Halving the RF pulse can achieve a superposition dual 0/1 state

• Different qubits are ‘set’ to a value via a unique RF pulse at a specific frequency

J-coupling• J-coupling - indirect dipole dipole coupling

– ƒThe coupling between two nuclear spins due to the influence of bonding electrons on the magnetic field running between the two nuclei.

• Enables Quantum Conditional NOT

• Conditional on J-coupling between other nearby qubit state (0 or 1)

– ƒ A nearby qubit state cancels RF pulse

– ƒ J-coupling enables conditional not (CNOT)

– ƒ This operation enables a universal set

J-coupling

Alternate CNOT

1. Solid line |0> dashed line |1>

2. RF pulse rotates spin from +z to –y

3. Freely evolve for ½ J seconds (coupling)

• Rotated to either +x or –x based on other spin state

4. A 90o pulse on spin rotates back to +z or –z

A 5 Qubit Molecule

• A molecule with 5 Fluorine ½ spin molecules

• Each has a unique resonance frequency (f1, f2, f3, f4, f5)

• Allows each qubit to be individually addressed

• Frequency difference much larger than J coupling

Table of the relative chemical shifts of the 19F spins at 11.7 T [Hz], and the J couplings [Hz]

Initialize State of Qubits

• The thermal equilibrium of nuclear spin at room temperature is highly random

• Initialize all qubits by creating a effectively “pure” state

• Main conceptual breakthrough that made NMR quantum computing possible

Qubit Reading of State

• A specific read out pulse rotates spin to the +x –x axis

• The resulting counter spins create a time-varying magnetic field.

• This magnetic field is measured, and integrated to obtain a value.

– Positive was in |0> spin

– Negative was in |1> spin

• Learn the state of each spin or qubit in system

Schematic of NMR spectrometer

• Sample tube contains 1012 dissolved NMR molecules

• Placed in superconducting solenoid, high current, very strong Magnetic field

• RF coil for pulsing the molecules.

• Computer controls an amplifier for controlling RF pulse, and reading state

Quantum Experiment

Results

Experimentally measured spectra of spin 1, acquired after executing the order-finding algorithm, for four different cases. The respective permutations are shown in inset, with the starting element in bold. The markers above the spectra indicate the position of the lines in the full multiplet.