Quantum computing

Question 1

What is a T gate?

Answer 1

A pi/4 z rotation
1 0
0 e^-ipi/4

Question 2

What is a Toffoli gate?

Answer 2

A contolled- controlled not

Question 3

Why must a balance be acheived in reversible computing?

Answer 3

Reversible computing uses no energy but is infinitely slow.

Question 4

Quantum parallelism =

Answer 4

when an input state is in a superposition an operator will calculate the output for all parts of the superposition. n bits in the input register means 2^n calculations.

Question 5

The Divencenzo criteria

Answer 5

1. Qubits - must be well defined and scalable
2. Initialisation
3. Decorehence time - must be long
4. Universal logic - must be able to do single and two qubit operations
5. Measurement - must be qubit specific
6. Convert stationary and flying qubits
7. Transmit flying qubits faithmally

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Question 6

Deutch’s problem =

Answer 6

to determine the parity of an unknown function

Question 7

Sketch Deutch’s algorithm

Answer 7

Question 8

Sketch networks for the four possible binary functions (lower bit is output)

Answer 8

Question 9

Find the two qubit hadamard gate

Answer 9

1 1 1 1

1 -1 1 -1

1 1 -1 -1

1 -1 -1 1

Question 10

Simplify

H -.-H

|

H-X-H

Answer 10

Question 11

What is Grover’s problem?

Answer 11

Want to find the sole input that returns 1 in the fewest possible queries.

Question 12

Sketch Grover’s network in the case n=2

Answer 12

Note that the final 3 qubit gate has hollow circle - applies when both qubits are 0, not 1

Question 13

Why can’t a qubit be driven back to its correct value in the same way as a classical bit can be (3)?

Answer 13

1. Qubit can be in superposition - what would you drive it back to?
2. Driving process would be dissipative and thus non unitary.
3. Can’t measure state without destroying it - must only measure error.

Question 14

Differnce between error correction and decoherence free subspaces

Answer 14

Error correction works for non correlated errors, DFS for totally correlated errors.

Question 15

How does a decherence free subspace work?

Answer 15

Choose code-word eigenstates which are unaffected by errors likey to occur

ie to protect against Z errors use psi, to protect aginst X errors use phi bell states

Question 16

What is the function of the Deutsch-Jozsa algorithm?

Answer 16

Identifies whether a state is balanced or constant (but not mixed) in the fewest possible queries (can do it in one).

Question 17

Sketch the D-J network for n=2

Answer 17

Question 18

How are ions trapped in an ion trap quantum computer?

Answer 18

Trap ions using time-varying field (think saddle analogy).

One possible configuration is the linear Pauli trap - use strong fields to confine in X-Y plane and then allow ions to line up according to their mutual electrostatic repulsion.

Question 19

How are atoms trapped in an atomic quantum computer (3 techniques)

Answer 19

Optical mollasesses - use 6 lasers (2 along each axis) with a frequency below that of a transition. Only atoms travelling towards a laser will absorb the light and thus experience a change in momemtum. This also allows you to cool the atoms.

Dipole forces - treat atoms like prisms which have a refractive index. They will experience a force when a light is shone on them.

Optical eggbox - use standing waves. The wells represent areas of low potential. Wells are shallow -> atoms must be very cold.

Question 20

How are atoms initialised?

Answer 20

Use lasers to excite electrons away from all states except |0>. This relies on random relaxation. Don’t need to worry about stimulated emission as relaxation is rapid.

Question 21

How are ions intialised?

Answer 21

Same as atoms but also need to ensure you’re in vibronic ground state.

Use sideband cooling - use a laser tuned to decrease the vibrational energy and increase the electronic state.

Question 22

Why are hyperfine levels used as qubits in ions and trapped atoms?

Answer 22

They have a long spontaneous emission lifetime to a ground state.

Question 23

Why do ion traps have a short decherence time?

Answer 23

Very vulnerable to Coulomb force eg fluctuations in porential of confining electrons, charges trapped in nearby insulators.

Question 24

How are single qubit gates applied to atoms and ions?

Answer 24

Using lasers tuned to rotate a state using a Raman tarnsition.

Question 25

Why are single qubit gates hard to apply to atoms and how can this be improved (3)?

Answer 25

Because atoms are more tightly spaced. 1. Could apply the same gate to multiple qubits at onece 2. Could partially fill the lattice 3. Shift the energy levels of the different atoms using the AC stark effect.

Question 26

How are two qubit gates implemented in ions?

Answer 26

1. Apply a pi Rabi pulse to ion 1 tuned to convert its logical state to its vibrational state |01\> -\> |10\> and vice versa 2. This automatically means ion 2 is in the same vibrational stae since they are vibrationally coupled. 3. Can now apply a gate to ion 2 conditionally on its vibrational state ie to perform a Z gate use a pulse tuned to |11\> -\>|e0\> which will introduce a phase. 4. Finally apply a pi pulse to ion one to convert its vibrational and qubit infomation. This can be combined with other gates.

Question 27

How are two qubit gates implemented in atoms?

Answer 27

1. Use the m\_s = +- 1/2 states. They will interact with sigmal -+ light respectively. 2. Light can then be used to move the atoms. A collision between neighbouring atoms will only occur if they are in opposite states. 3. When they collide the Hamiltonian changes and the atoms pick up a phase. Specific phase can be controlled bu controlling the time. 4. This implements some sort of controlled phase gate.

Question 28

Why are massive entangled states interesting (2)?

Answer 28

Important for studying transition form quantum to classical regime. Cna be useful for error correction, quantum simulation and measuremnent based quantum computers

Question 29

How is readout achieved in ion and atom quantum computerts?

Answer 29

Using a cycling transition - use a laser tuned to a transition from the ground state to some excited state. Fluorescence is only observed if the qubit is in the ground state.

Question 30

Transition energy between NMR qubits

Answer 30

hbar gamma B gamma is the gyromagnetic ratio

Question 31

How can frequencies of NMR nuclei be diferent?

Answer 31

Use different nucear species Use different local environments - causes chemical shift

Question 32

How can we overcome the problem of not being able to observe a single photon from an NMR transition?

Answer 32

Use a dilute solution of small molecules

Question 33

Why are NMR solvents deuterated?

Answer 33

H-1 nuclei have spin 1/2, H-2 have spin 1

Question 34

Why can't NMR states be intialised by cooling?

Answer 34

Need very low temperatures due to tiny energy difference between levels - order 1mK

Question 35

How can initialisation be achieved in NMR quantum computers (3)?

Answer 35

1. Use a solid state system which can be cooled. 2. Use some method to obtain a non-equilibrium population of spin states. 3. Use pseudo-pure states in an ensemble system.

Question 36

Discuss decoherence in NMR quantum computers

Answer 36

Long decoherence lifetime of spin state but also have spontanteous emission and absorbtion from random fluctuations. Still gives around 1s. However, gates times in NMR computers are very long so this is relelvant. NMR computers use an ensemble state though so errrors cause a reduction in signal strength, not necessarily a wrong answer.

Question 37

How are single qubit gates implemented in NMR?

Answer 37

Using RF pulse for correct time.

Question 38

What is the Heisenberg form of the scalar coupling?

Answer 38

Question 39

What is the Hamiltonian for a two spin system?

Answer 39

Question 40

How does a spin echo work?

Answer 40

Allow the system to evolved under a Hamiltonian for t/2. Then reverse the spin using a NOT. The allow it to evolve for another t/2. Then NOT it again.

Question 41

Sketch how different terms in the Hamiltonian are isolated using spin echoes

Answer 41

Question 42

How can 2 qubit gates be implemented between non-adjacent qubits in an NMR quantum computer?

Answer 42

Using swap gates along a line to bring the two required qubits next to one another

Question 43

How are NMR qubits read?

Answer 43

0 and 1 are distinguished by their differnet relative phases in the detection signals. First a p/2 pulse must be applied to tip the spins as otherwise 0 and 1 will give no signal. The phases differ because 0 correspoponds to absorbtion and 1 to emission.

Question 44

Sketch the expected spectra for a homonuclear pair of NMR nuclei

Answer 44

Question 45

Sketch the expected spectra for a heteronuclear NMR pair

Answer 45

Question 46

Why do NMR measurements not destroy the superposition?

Answer 46

Becuase ensemble measurements are not projective to a single spin.

Question 47

3 advantages of projective measurements

Answer 47

1. Quantum error correction relies on projective measurement 2. Makes it easy to initialise a state - if you measure 1 simply no the state 3. Cause problems for algorithms which end in some superposition state - output is an average which can be hard to interpret.

Question 48

Advantages of ions for large scale computers (1)

Answer 48

Can trap thousands of ions whilst keeping space between them

Question 49

Disadvantages of ions (2)

Answer 49

1. Hard to implement logic gates in parallel in different quantum computers. Can solve using one laser per atom using a micromirror array 2. Can only implement one two qubit gate at a time because the vibrational data bus can only hold one qubit at a time. Can overcome using multiple ion traps each holding only a few ions and move ions or qubit information between them.

Question 50

Advantages of NMR

Answer 50

1. Easy to implement small computers

Question 51

Disadvantages of NMR (3)

Answer 51

1. Initialisation - can't be scaled up as excess population in pseudo pure states drops of as number of qubits increases 2. Lack of projective measurement makes error correction difficult. 3. Hard to scale as need lots of nuclei with different chemical environments/nuclei