Quantum Computing and Hollywood

San Jose State UniversityApril 15, 2004

Scott AaronsonUC Berkeley

www.cs.berkeley.edu/~aaronson

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“Computers are useless. They only give you

answers.”

–Pablo Picasso

DUNCE

Not merely a false statement, but a pompous and asinine one

Computers CAN ask questions

More importantly, computers led to some of the deepest questions

ever asked

Could a machine be conscious?If you can recognize good ideas, can you also have them? (Does P=NP?)Can quantum parallelism be harnessed to solve astronomically hard problems?

Goal: Show that this question is ‘useful’ in Picasso’s sense

What Is Quantum Parallelism?In quantum mechanics, if an object can be in two states, call them |0 and |1, then it can also be in a superposition of those states:

|0+|1Here and are amplitudes: complex numbers satisfying ||2+||2=1

If we observe the object, we see |0 with probability ||2 and |1 with probability ||2

Also, the state “collapses” to whichever outcome we see

Example: Schrödinger’s cat

“But why can’t we say the cat is either alive or dead before we open the box, and we just don’t know which?”

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To modify a state 1|1++N|N, we can multiply the vector of amplitudes by a matrix that preserves

Such matrices are called unitary

Example:

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1N

ii

1 1 112 2 2

1 1 1 0

2 2 2

1 1 102 2 2

1 1 1 1

2 2 2

We’re seeing interference—cancellation of positive and negative amplitudes. This is the source of “quantum weirdness”

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Quantum ComputersA quantum state of n bits (“qubits”) takes 2n complex numbers to describe—

If n=1000, that’s a lot for Nature to keep track of!

The goal of quantum computing is to exploit this exponential parallelism in Nature

Problem: We only see one result when we measure

000 001 010 011

100 101 110 111

000 001 010 011

100 101 110 111

But maybe we could get the paths leading to wrong answers to interfere and cancel each other out…

Shor’s AlgorithmIn 1994, Peter Shor gave a quantum algorithm for factoring an N-digit number using only ~N2 steps(where a “step” is a unitary matrix acting only on 1 or 2 quantum bits)

Best known classical algorithm uses steps

So if you had a quantum computer, you could break RSA and other public-key cryptosystems

Heart of Shor’s algorithm is a “quantum Fourier transform” that finds the period of an exponentially long periodic sequence

1/ 3

~ 2N

Interesting

Grover’s Algorithm

In 1996, Lov Grover gave a quantum algorithm for searching an N-item list for a single “marked item,” using only

steps. Classically, of course, ~N steps are needed~ N

Marked item

So, Can We Build A Quantum Computer?

State of the art: In 2001, using a 7-qubit liquid NMR quantum computer, it was determined that with high probability

15 = 3 5Why is building a QC hard? DECOHERENCE—unwanted interaction with the environment

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Then, when we rotate the 1st qubit by 45°, no interference!

1st qubit interacts with “outside world” (2nd qubit)

Threshold Theorem (Aharonov & Ben-Or, Kitaev, Knill & Laflamme & Zurek): If the probability of decoherence per qubit per time step is below a threshold—about 10-4, depending on assumptions—then we can use error-correcting codes to do arbitrarily long quantum computations

So when will we see quantum computers? 20 years? 100 years? Never? Bet at ideosphere.com

Ridiculous!There must be some fundamental reason why quantum

computing is impossible!

QC of the sort that factors long numbers seems firmly rooted in science fiction … The present attitude would be analogous to, say, Maxwell selling the Daemon of his famous thought experiment as a path to cheaper electricity from heat. –Leonid Levin

[P]articularly after my discoveries … I strongly suspect that even if this is formally the case [that quantum computers offer exponential speedups], it will still not turn out to be a true representation of ultimate physical reality… –Stephen Wolfram

If quantum computing is impossible, then where exactly is our understanding of quantum mechanics wrong?

Crucial Question for Me

For more detailed discussions:A 2004, “Multilinear Formulas and Skepticism of Quantum Computing”A 2002, “Book Review on A New Kind of Science”

(Falsifying QM would be much more interesting than building a quantum computer!)

Stephen Wolfram and Gerard ‘t Hooft proposed alternatives to QM, but these fail to account for known experimental facts (Bell inequality violations)

So…We ought to worry now about the “quantum computing picture of reality”

Are there really billions of parallel universes with parallel you’s in them?

“To those who still cling to a single-universe worldview, I issue this challenge: explain how Shor's algorithm works”—David Deutsch

What exactly does “parallel universe” mean?

These movies don’t take their premise to its logical conclusion. Why can’t you learn from 2n-1 alternate realities instead of just one?

“Let a computer smear—with the right kind of quantum randomness —and you create, in effect, a ‘parallel’ machine with an astronomical number of processors … All you have to do is be sure that when you collapse the system, you choose the version that happened to find the needle in the mathematical haystack.”

From a scene in which the protagonist causes a computer to factor a huge number, by using his newfound ability to “postselect” quantum measurement results

Guess all numbers simultaneously using quantum parallelism. If you lose, kill yourself

Foolproof Way To Win The Lottery

(Also works for solving NP-complete problems, dating a supermodel, floating in midair…)

But Would Quantum Computers Really Work That Way?

See, quantum computers, by taking advantage of weird quantum phenomena which make no sense and no one understands but the numbers work out so shut the fuck up and take it—quantum computers are able to compute all possible computations at the same time, by existing simultaneously in an infinite number of parallel universes.—“Popular Eschatology”

Unlike a laboratory rat or an ordinary computer, which must probe the pathways one at a time, the quantum computer can simultaneously traverse every twist and turn and immediately converge upon the prize.—George Johnson, Slate

POPULAR SCIENCE WRITERS AGREE—YES!

Bennett-Bernstein-Brassard-Vazirani 1997

Can a quantum computer search a list of 2n items in only (say) n steps, as opposed to the 2n/2 steps needed by Grover’s algorithm?

Answer: No! Grover’s algorithm is optimal.

1|1 2|2 3|3 4|4 5|5 6|6 7|7 8|8

Someone must be screaming about a marked

item… too bad quantum mechanics is linear

9|9

DUDE!!! Everyone! The marked item! Over here!!!

My Actual Researchhas shown that the limitations of quantum

computers are rather far-reaching...

• Given a list of N numbers in which every number appears twice, it takes at least ~N1/5 steps to find a duplicate (improved by Yaoyun Shi to ~N1/3, which is tight). Means there’s no generic way to break crypto hash functions with a quantum computer.

• Given a function it takes at least ~2n/4 steps to find even a local minimum of fThis quantum result led to new classical lower bounds, and has been generalized to all graphs by Santha & Szegedy

: 0,1 ,n

f

My Actual Researchhas shown that the limitations of quantum

computers are rather far-reaching...

• The need to uncompute—that is, erase previous work in a reversibility-respecting way—can be a fundamental barrier to designing fast quantum algorithms

• In the “black box” setting, quantum computers can’t solve NP-complete problems efficiently even given a polynomial-size quantum advice state that depends on the input lengthIn complexity gobbledygook: there exists an oracle A such that NPABQPA/qpoly

Conclusions• The existence of countless parallel universes is a “terrible idea, except for the alternatives” (Steven Weinberg)

• But would quantum parallelism let us solve NP-complete problems by trying all possible solutions at once? No!

• The linearity of quantum mechanics prevents a “lone voice in the wilderness” from being heard. But if you and your parallel doppelgangers all contribute to solving a problem, you might learn the answer by interference…

“Computers are useless. They only give you answers.”—Picasso