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 Picassos 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 + |1 Here 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: Schrödingers cat But why cant we say the cat is either alive or dead before we open the box, and we just dont know which?
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: Were seeing interferencecancellation of positive and negative amplitudes. This is the source of quantum weirdness
Quantum Computers A quantum state of n bits (qubits) takes 2 n complex numbers to describe If n=1000, thats 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 But maybe we could get the paths leading to wrong answers to interfere and cancel each other out…
Shors Algorithm In 1994, Peter Shor gave a quantum algorithm for factoring an N-digit number using only ~N 2 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 Shors algorithm is a quantum Fourier transform that finds the period of an exponentially long periodic sequence Interesting
Grovers 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 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 5 Why is building a QC hard? DECOHERENCE unwanted interaction with the environment Then, when we rotate the 1 st qubit by 45°, no interference! 1 st qubit interacts with outside world (2 nd qubit)
Threshold Theorem (Aharonov & Ben-Or, Kitaev, Knill & Laflamme & Zurek): If the probability of decoherence per qubit per time step is below a thresholdabout 10 -4, depending on assumptionsthen 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 yous 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 dont take their premise to its logical conclusion. Why cant you learn from 2 n -1 alternate realities instead of just one?
Quarantine Let a computer smearwith 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 itquantum 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 AGREEYES!
Bennett-Bernstein-Brassard-Vazirani 1997 Can a quantum computer search a list of 2 n items in only (say) n steps, as opposed to the 2 n/2 steps needed by Grovers algorithm? Answer: No! Grovers 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 Research has 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 ~N 1/5 steps to find a duplicate (improved by Yaoyun Shi to ~N 1/3, which is tight). Means theres no generic way to break crypto hash functions with a quantum computer. Given a function it takes at least ~2 n/4 steps to find even a local minimum of f This quantum result led to new classical lower bounds, and has been generalized to all graphs by Santha & Szegedy
My Actual Research has shown that the limitations of quantum computers are rather far-reaching... The need to uncomputethat is, erase previous work in a reversibility-respecting waycan be a fundamental barrier to designing fast quantum algorithms In the black box setting, quantum computers cant solve NP-complete problems efficiently even given a polynomial-size quantum advice state that depends on the input length In complexity gobbledygook: there exists an oracle A such that NP A BQP A /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