Quantum computing Flashcards
What is a T gate?
A pi/4 z rotation
1 0
0 e^-ipi/4
What is a Toffoli gate?
A contolled- controlled not
Why must a balance be acheived in reversible computing?
Reversible computing uses no energy but is infinitely slow.
Quantum parallelism =
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.
The Divencenzo criteria
- Qubits - must be well defined and scalable
- Initialisation
- Decorehence time - must be long
- Universal logic - must be able to do single and two qubit operations
- Measurement - must be qubit specific
- Convert stationary and flying qubits
- Transmit flying qubits faithmally
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Deutch’s problem =
to determine the parity of an unknown function
Sketch Deutch’s algorithm
Sketch networks for the four possible binary functions (lower bit is output)
Find the two qubit hadamard gate
1 1 1 1
1 -1 1 -1
1 1 -1 -1
1 -1 -1 1
Simplify
H -.-H
|
H-X-H
What is Grover’s problem?
Want to find the sole input that returns 1 in the fewest possible queries.
Sketch Grover’s network in the case n=2
Note that the final 3 qubit gate has hollow circle - applies when both qubits are 0, not 1
Why can’t a qubit be driven back to its correct value in the same way as a classical bit can be (3)?
- Qubit can be in superposition - what would you drive it back to?
- Driving process would be dissipative and thus non unitary.
- Can’t measure state without destroying it - must only measure error.
Differnce between error correction and decoherence free subspaces
Error correction works for non correlated errors, DFS for totally correlated errors.
How does a decherence free subspace work?
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
What is the function of the Deutsch-Jozsa algorithm?
Identifies whether a state is balanced or constant (but not mixed) in the fewest possible queries (can do it in one).
Sketch the D-J network for n=2
How are ions trapped in an ion trap quantum computer?
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.
How are atoms trapped in an atomic quantum computer (3 techniques)
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.
How are atoms initialised?
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.
How are ions intialised?
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.
Why are hyperfine levels used as qubits in ions and trapped atoms?
They have a long spontaneous emission lifetime to a ground state.
Why do ion traps have a short decherence time?
Very vulnerable to Coulomb force eg fluctuations in porential of confining electrons, charges trapped in nearby insulators.
How are single qubit gates applied to atoms and ions?
Using lasers tuned to rotate a state using a Raman tarnsition.
Why are single qubit gates hard to apply to atoms and how can this be improved (3)?
Because atoms are more tightly spaced.
- Could apply the same gate to multiple qubits at onece
- Could partially fill the lattice
- Shift the energy levels of the different atoms using the AC stark effect.
How are two qubit gates implemented in ions?
- Apply a pi Rabi pulse to ion 1 tuned to convert its logical state to its vibrational state |01> -> |10> and vice versa
- This automatically means ion 2 is in the same vibrational stae since they are vibrationally coupled.
- 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.
- Finally apply a pi pulse to ion one to convert its vibrational and qubit infomation.
This can be combined with other gates.
How are two qubit gates implemented in atoms?
- Use the m_s = +- 1/2 states. They will interact with sigmal -+ light respectively.
- Light can then be used to move the atoms. A collision between neighbouring atoms will only occur if they are in opposite states.
- When they collide the Hamiltonian changes and the atoms pick up a phase. Specific phase can be controlled bu controlling the time.
- This implements some sort of controlled phase gate.
Why are massive entangled states interesting (2)?
Important for studying transition form quantum to classical regime.
Cna be useful for error correction, quantum simulation and measuremnent based quantum computers
How is readout achieved in ion and atom quantum computerts?
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.
Transition energy between NMR qubits
hbar gamma B
gamma is the gyromagnetic ratio
How can frequencies of NMR nuclei be diferent?
Use different nucear species
Use different local environments - causes chemical shift
How can we overcome the problem of not being able to observe a single photon from an NMR transition?
Use a dilute solution of small molecules
Why are NMR solvents deuterated?
H-1 nuclei have spin 1/2, H-2 have spin 1
Why can’t NMR states be intialised by cooling?
Need very low temperatures due to tiny energy difference between levels - order 1mK
How can initialisation be achieved in NMR quantum computers (3)?
- Use a solid state system which can be cooled.
- Use some method to obtain a non-equilibrium population of spin states.
- Use pseudo-pure states in an ensemble system.
Discuss decoherence in NMR quantum computers
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.
How are single qubit gates implemented in NMR?
Using RF pulse for correct time.
What is the Heisenberg form of the scalar coupling?
What is the Hamiltonian for a two spin system?
How does a spin echo work?
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.
Sketch how different terms in the Hamiltonian are isolated using spin echoes
How can 2 qubit gates be implemented between non-adjacent qubits in an NMR quantum computer?
Using swap gates along a line to bring the two required qubits next to one another
How are NMR qubits read?
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.
Sketch the expected spectra for a homonuclear pair of NMR nuclei
Sketch the expected spectra for a heteronuclear NMR pair
Why do NMR measurements not destroy the superposition?
Becuase ensemble measurements are not projective to a single spin.
3 advantages of projective measurements
- Quantum error correction relies on projective measurement
- Makes it easy to initialise a state - if you measure 1 simply no the state
- Cause problems for algorithms which end in some superposition state - output is an average which can be hard to interpret.
Advantages of ions for large scale computers (1)
Can trap thousands of ions whilst keeping space between them
Disadvantages of ions (2)
- Hard to implement logic gates in parallel in different quantum computers. Can solve using one laser per atom using a micromirror array
- 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.
Advantages of NMR
- Easy to implement small computers
Disadvantages of NMR (3)
- Initialisation - can’t be scaled up as excess population in pseudo pure states drops of as number of qubits increases
- Lack of projective measurement makes error correction difficult.
- Hard to scale as need lots of nuclei with different chemical environments/nuclei
