Lecture 6 Flashcards
What is the Church turing thesis? What is the strong version?
The church Turing thesis is that it doesn’t matter what computer or model of computation uses, all the same things can be computed, the strong version states they are all about as fast as each other (polynomial factors in length of the input, for comparable algorithms).
What do P and NP mean with regards to a problem L and turing machines?
L is in P if there is a deterministic turing machine that on input x accepts if x is in L and rejects if not, and whose running time is polynomial on the length of x.
L is in NP if there is a deterministic turing machine which given x is in L accepts for some string y (x, y), but if x is not in L never accepts (x,y) and whose running time is polynomial on the length of x
What are P/poly and BPP?
P/poly means there is a sequence of boolean circuits of polynomially bounded size in n such that when the length of x is n, the circuit is true if and only if x is in L.
BPP is when there is a polynomial time Turing machine and a polynomial p such that when the length of x is n, if r is a uniformly random bit string of lengthp(n), then the probability that M correctly identifies x on input (x, r) is at least 2/3.
What is BPP’s relation to other languages?
It is widely believed that P = BPP, but it has never been proven that BPP is a subset of EXP.
Why is P/poly a bit weird?
It contains undecidable languages, languages where there cannot possibly be an algorithm that always gives the correct answer.
What does quantum computing use instead of bits? Why are the possible states odd?
Qubits, these could be |0> or |1>, representing the possible state of a photon (vertical or horizontal polarization). It can also be in a superposition of both states, which is odd compared to standard bits.
What are quantum circuits? What are the main restrictions?
Quantum circuits have complexity class BQP, analagous to BPP with a P/poly twist. Gates are sued to apply operations to qubits, collections of qubits, and superpositions of collections of qubits. A computer is a turing machine which describes how to build these circuits to solve a problem.
Quantum gates can only apply unitary operations to their input, this means all computations could be run in reverse in absence of measurement and that it is impossible to duplicate data/inputs. This makes some standard gates, like and gates, quite difficult.
How do we make a quantum and gate? What does the hack for this mean in general?
take two registers |x> and |y> add a third input and make sure it’s initialized to |0>, then perform the operation:
(x, y, z) → (x, y, z ⊕ (x ∧ y)).
This need for control bits(x and y) and target bits (z) is very common, allowing all classical logic gates ot be simulated.
How does quantum cryptography work?
It is really about how keys are distributed, they are distributed via a quantum channel, any attempt to eavesdrop on this process is detectable, allowing for mitigation techniques to be used or the process restarted. Examples of mitigation techinques could be information reconcilliation or privacy amplification.
What is information reconcilliation with regard to quantum networks?
Used when Alice and Bob think they share a secret key but worry that some bits may not match, standard error correction techinques like block checksums can be used to find and correct some errors, leading to a resulting string which may be shorter than the original key, the communication for this is done in the clear, so some information about the key will be leaked.
What is privacy amplification with regard to quantum networks?
assume that Alice and Bob share a random key but worry Eve has some knowledge of it, if the knowledge is not that strong, i.e a random variable that is slightly correlated with the key, then Alice and Bob can use universal hash functions to map k to a shorter key so that the correlation becomes much weaker.
Why does Eve listening to a quantum channel disturb the signal?
if |x> and |y> and two non-orthogonal states, then no citcuit that accepts them on input lines(could be additional), and outputs them undisturbed can deriverany information about which was input(remaining output lines will be the same as each other), this combined with no ability to clone means that Eve must alter the data in order to inspect it.
What is the BB84 protocol?
A key distribution protocol for Quantum computing.
Signal from Alice to Bob is a sequence of photons and they want an m bit key.
Alice generates photons with either a vertical-horizontal polarization or a diagonal polarization, the correspondance from a qubit to a bit of the key is:
0 bit: vert-horiz |0> or diagonal |0> + |1>
1 bit:vert-horiz |1> or diagonal |0> - |1>
This correspondance will mean that if Bob measures a photon in that same basis it was generated then he gets its value. if he measures in the other basis it is a coin toss.
Alice then generates two bit strings a dna b of length (4 + x)n. She uses b and the correspondence to generate photons which encode a (if a bit of b is 0 she uses VH encoding, 1 if D encoding). Bob receives the photons, announces so publicly, and chooses his own random bit string b’ to try decode them. Alice then announces b to the world and Bob compares b and b’, announcing to the world a set of 2n bit-indices where he and Alice used the same basis at that position. (If there are not enough sames then restart).
Alice then chooses randomly and announces n of the 2n bit indices. Alice and Bob then publicly compare those n bits, if they disagree too often then transmission errors or Eve caused problems, so they abort and retry. If not and the error level is low enough then information reconcilliation and privacy amplification are done to get the key.
