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