When Qubits Go Analog A Relatively Easy Problem in Quantum Information Theory Scott Aaronson (MIT)
Mar 26, 2015
When Qubits Go AnalogA Relatively Easy Problem in Quantum
Information Theory
Scott Aaronson (MIT)
Erik Demaine (motivated by a computational genetics problem): “Suppose a PSPACE machine can flip a coin with bias p an unlimited number of times. Can it extract an exponential amount of information (or even more) about p?”
Me: “I’m sure whatever the answer is, it’s obvious...”
Didn’t seem too likely there could be superpowerful “Advice Coins”
Indeed, Hellman-Cover (1970) proved the following...
Suppose a probabilistic finite automaton is trying to decide whether a coin has bias ½ or ½+. Then even if it can flip the coin an unlimited number of times, the automaton needs (1/) states to succeed with probability (say) 2/3.
Implies PSPACE/coin = PSPACE/poly
Bias=0.000000000000000110101111101
poly(n) advice bits another poly(n) advice bits
Yet quantum mechanics nullifies the Hellman-Cover Theorem!
Theorem: For any >0, can distinguish a coin with bias p=½ from a coin with bias p=½+ (with bounded error) using a single qutrit of memory.
0
10 1
2
Keep flipping the coin.
Whenever the coin lands heads, rotate /100 radians counterclockwise. Whenever it lands tails, rotate /100 radians clockwise.
Halt with probability ~2/100 at each time step, by measuring along the third dimension
Expected difference in final angle after halting, in p=½ vs. p=½+ cases: 1 radian
Standard deviation in angle:
So, could BQPSPACE/coin=ALL?
Theorem: No.
Proof: Let’s even let the machine run infinitely long; it only has to get the right answer in the limit
Let 0 = superoperator applied to our memory qubits whenever coin lands heads,
1 = superoperator when it lands tails
Then induced superoperator at each time step:
We’re interested in a fixed-point of p: a mixed state p such that
p = coin’s bias
Fixed-Points of Superoperators
Studied by [A.-Watrous 2008] in the context of quantum computing with closed timelike curves
Our result there: BQPCTC = PCTC = PSPACE
Quantum computers with CTCs have exactly the same power as classical computers with CTCs, namely PSPACE (or: “CTCs make time and space equivalent as computational resources”)
Key PointFixed-point p of a superoperator p can be expressed in terms of degree-2s rational functions of p, where s is the number of qubits
(Proof: Use Cramer’s Rule on 2s2s matrices)
Let ax(p) be the probability that the PSPACE machine accepts, on input x{1,...,N} and an advice coin with bias p
Then ax(p) is a degree-2s rational function of p
By calculus, a degree-2s rational function can cross the origin (or the line y=½) at most 22s times
To specify p, well enough to decide whether ax(p)½ for any x: suffices to say how many reals 0<q<p there are such that some ax(p) crosses the line y=½ at q
This takes log2(22sN)=s+1+log2(N) bits
So, coin can specify distinct functions
p
ax(p)
OK, how about a harder problem?Is there an oracle relative to which BQPPH?
Given oracle access to two Boolean functions,f,g:{-1,1}n{-1,1}n
Promised that either
(1)f and g are both uniformly random, or
(2)f,g were chosen by picking a random unit vector and letting f(x)=sgn(vx), g(x)=sgn(Hnvx)
Problem: Decide which
New candidate problem we should use for this:
“Fourier Checking”
I claim this problem is in BQP
then measure in the Hadamard basis
On the other hand, I conjecture the Fourier Checking problem is not in PH
I can show that any poly(n) bits of f(x) and g(x) are close to uniformly random. I conjecture that this suffices to put the problem outside PH (“Generalized Linial-Nisan Conjecture”)
NO WE CAN’T
PROVE QUANTUM LOWER BOUNDS