CSI 789-004/PHYS 780-002: Quantum Complexity
Fall Semester 2004 Take-Home Mid-Term Exam
Issued 19th October and Due by Class Day 2nd November 2004
Answer any 5 out of the 6 questions. If you answer all 6 questions you get extra credit.
Question #1
a. Using the diagonalization technique show that the halting problem is Turing undecidable.
b. On the question of what can be measured in quantum mechanics? - we know the
“Traditional” approach to quantum measurements as:
“A quantum measurement is described by an observable, M, that is, a Hermitian operator
acting on the state space of the system. Measuring a system prepared in an eigenstate of
M gives the corresponding eigenvalue of M as the measurement outcome.”
- (For quoted text see reading reference for Question#1(b).)
“The question now presents itself – Can every observable be measured? The answer
theoretically is yes. In practice it may be very awkward, or perhaps even beyond the
ingenuity of the experimenter, to devise an apparatus which could measure some
particular observable, but the theory always allows one to imagine that the measurement
could be made.” - Paul A. M. Dirac
“The halting observable: Can we build a measuring device capable of measuring the
halting observable?
Yes: Would give us a procedure to solve the halting problem.
No: There is an interesting class of “superselection” rules controlling what observables
may, in fact, be measured.
Research Problem: Is the halting observable really measurable? If so, how? If not, why
not?”
- (For quoted text see reading reference for Question#1(b).)
Write your own solution of the Research Problem discussing both the ‘yes’ and ‘no’
answers.
Question#2
a. Define the following classical and quantum complexity classes and describe as many
relationships as you know among all the classes.
P, PSPACE, NP, NP-Complete, IP, ZPP, BPP, ZQP, BQP
