Beyond Turing: Quantum and Adiabatic Computers

CS 301 Lecture, Dr. Lawlor

Ordinary Computer
Adiabatic Circuits
Quantum Computer
Get 'er done!
Substantially lower power use,
especially at low clockrate.
Speedups up to exponential:
e.g., search n values in sqrt(n) time
Data storage
1's and 0's (bits)
1's and 0's (bits)
Vector with axes 1 and 0 (qubits)
Not just 1 or 0: both at once!
No (uses energy = kT ln 2)
No (violates laws of physics)
Yes, except for "collapse" operation
CNOT, Hadamard rotate 45 degrees
Reversible Instructions
Reversible Quantum Operations,
and Irreversible Collapse
Square wave
Two trapezoidal waves
Limited by coherence time
Slowly, over next ten years
hard problems
Only helps at low clockrate
How many bits can you keep coherent?

Adiabatic circuits are based on a few interesting fundamental observations about modern circuit efficiency:
Saed Younis' 1994 MIT PhD Thesis outlines the basic adiabatic circuit model and its inherent power advantage, which is up to several hundredfold at sufficiently low clock rates.  These design principles have slowly been trickling into CPU designs piece by piece; still, most circuits are non-reversible and non-adiabatic today.  The path to a fully adiabatic computation, assuming we ever get there, will have to change the instruction set at some point, because tons of operations are destructive, like "mov" (which irrevocably destroys the destination), and will need to be replaced with "swap" (which doesn't destroy information).  Some future fully-adiabatic CPU will need reversible instructions, and in fact the compiler will probably need to generate entire "antifunctions" to undo the operation of each function.  To really be 100% reversible, the display would have to suck the final answer back out of your brain, so some degree of irreversibility is usually considered acceptable in designing real systems.

The energy kT ln 2 needed to erase one digit near room temperature is about 10-20 joules, which is... small.  If your machine is chewing through 10 billion bits per clock cycle, and does 10 billion clocks per second, this is one joule/second, or one watt of irreducible "bit erasure cost".  A typical desktop CPU is 30-100 watts, so this isn't a very big effect yet, but depending on how silicon fabrication scales up, it could become a show-stopper in the next ten years, and drive us toward reversible programs.

Once your program is fully reversible, you're actually halfway toward writing a quantum computer program!

Quantum Computers

Small things, like electrons, display several very odd mechanical properties with mystical sounding "quantum" names:

A quantum computer is based on "qubits", which you can think of as a dot in 2D space: the X axis means a "0", the Y axis means a "1".  Normal bits must lie on one axis or another, but qubits can live between the axes.  For example,
Since coordinates (0,0) means "not a 0, and not a 1", so we usually require the qubit to live on a unit sphere--it's either going to be a zero or going to be a one, so the probabilities must add up.  Tons of research groups have built one-bit quantum computers, but the real trick is entangling lots of bits without premature "collapse", and that seems to be a lot harder to actually pull off.

People started to get really interested in Quantum Computers when in 1994 Peter Shor showed a quantum computer could factor large numbers in polynomial time.  The stupid algorithm for factoring is exponential (just try all the factors!), and though there are smarter subexponential algorithms known, there aren't any non-quantum polynomial time algorithms known (yet).  RSA encryption, which your web browser uses to exchange keys in "https", relies on the difficulty of factoring large numbers, so cryptographers are very interested in quantum computers.

In 1996, Lov Grover showed an even weirder result, that a quantum search over n entries can be done in sqrt(n) time.  The matrix mathematics used to describe both these algorithms is pretty complex, and I have to admit I don't quite follow the description of either algorithm in detail, but the basic idea is:
  1. Initialize your quantum register with a superposition of 0 and 1: this hence contains every possible answer.
  2. Run a series of instructions to selectively amplify the answer you're looking for, or attenuate the answers you're not looking for.  For example, you can arrange "wrong answers" so they cancel each other out.  Each instruction must be a reversible operation, but in theory can be arbitrarily complex and there's no limit on the number of instructions.   However, in practice, the machine only works if you can keep the whole register entangled in a coherent superposition, accidental collapse or "decoherence" is currently the limiting factor in building big quantum computers.
  3. Finally, look at the register.  The act of looking will "collapse" to a particular set of 1's and 0's, hopefully representing the right answer.
At the moment, nobody has built a useful quantum computer, but there are lots of interesting experimental ones.  The biggest number a quantum computer has factored is 15 (=3*5, woo hoo).  The largest quantum computers actually built so far have only 8 bits, and only one register, which means the maximum theoretical speedup is 28=256 times faster than a normal machine.  But *if* the hardware can be scaled up, a quantum computer could solve problems that are intractable on classical computers.    Or perhaps there is some physical limitation on the scale of wavelike effects, and hence quantum computers will always be limited to a too-small number of bits or too-simple circuitry.  At the moment, nobody knows.

Roger Penrose has a theory that the human brain is actually a quantum computer.  This could explain how we're able to do some of the really amazing things we do, like recognize pictures of objects.  The deal is that it's easy for a computer to *simulate* a picture of an object (that is, object->picture is easy), but to *recognize* a picture of an object means searching over all possible objects, orientations, lighting, and so on (that is, picture->object is hard).  A quantum computer with sufficiently large registers (big enough to generate a complete image, so thousands of qubits) could in principle start with a superposition of all possible objects, and use a series of instructions to cancel out objects that are inconsistent with the current picture, finally collapsing out a plausible object that could have generated that picture.

There's a different and controversial "Adiabatic Quantum Computer" design used by British Columbia based quantum computer startup D-Wave that they hope will scale to solve large problems.  It is not well accepted whether this new design is workable, and there are serious doubts whether the 128-bit superconducting niobium hardware the startup has built is "really" a quantum computer.  They got a huge amount of press in 2007 and 2008, and are still in business, but were recently panned in IEEE Spectrum as "Does Not Quantum Compute". 

The future of quantum computers is currently in a superposition between two outcomes:
This superposition may collapse sometime in the next few years.  Or maybe not.