IMAGINE a computer that could compute without ever being switched on, a
magical machine calculating in stillness and silence, and churning out answers
without ever manipulating a single bit. Mathematician Richard Jozsa of the
University of Plymouth isn鈥檛 normally prone to crazy ideas, but this incredible
鈥渘on-computer鈥� is his brainchild. In his view, the quantum computer that you鈥檝e
heard so much about over the past few years might ultimately be famous not so
much for its startling speed as for its epic laziness.
鈥淭he mere fact that a quantum computer would give the answer if it were run,鈥�
he says, 鈥渃an be enough to get results even if the computer is, in fact, not
run.鈥� Jozsa claims that this apparently bizarre conclusion follows directly from
the peculiar logic of quantum theory. In everyday experience, effects follow
strictly from their causes: drop a wine glass onto the floor and it shatters. In
the quantum world, however, even the mere possibility that you could drop the
glass might be enough to make it shatter. 鈥淐auses鈥� that do not actually happen
can still produce 鈥渆ffects鈥�.
Brain teasing bombs
Bizarre or not, this kind of thinking has quite a few fans. Jozsa is the
latest into a field that spans the Israeli physicists Avshalom Elitzur and Lev
Vaidman and Oxford University鈥檚 Roger Penrose鈥攁ll of whom believe that
these counterfactual effects can be put to good intellectual use of one sort or
another. To show how, Elitzur and Vaidman first put forward a dramatic
theoretical brainteaser in 1993. Suppose you have a pile of bombs鈥攕ome
live, some duds. And suppose that the bombs are so sensitive that, if live, even
a single photon would make them explode. To make things explicit, Elitzur and
Vaidman envisioned bombs with mirrors attached to their triggers. If the mirror
of a live bomb were hit by a photon, it would recoil and set off the bomb. Duds,
with their mirrors stuck in place, would be unable to explode.
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Is there any way to identify a bomb with certainty as being live, without
also setting it off? Since you cannot interact with a live bomb in any way
without exploding it, there would seem to be no hope. But Elitzur and Vaidman
found a quantum solution. The trick involves a clever device鈥攁n
interferometer鈥攚hich splits beams of light in two, directs them along
separate paths, and then recombines them
(see Diagram). In quantum terms,
light is really made of tiny indivisible photons, and at the beam splitter, each
photon splits into a 鈥渟uperposition鈥� of two ghostly quantum states which travel
along the separate paths. After bouncing off mirrors, these phantoms meet again
at a second beam splitter, which undoes the work of the first by folding the
superposition back up into a single photon. The result of this strange procedure
is absolutely nothing鈥攖he re-formed photon is always a perfect copy of the
original, moving in the same direction, and entering the detector labelled A.
What has this device got to do with identifying a live bomb? Elitzur and
Vaidman wondered how it would operate if one of its mirrors were replaced by a
bomb鈥檚 trigger mirror. If the bomb is a dud鈥攕o that its mirror is fixed in
place鈥攖hen the ghostly photon that follows the lower path reflects
normally, the interferometer still does nothing, and the photon turns up in
detector A.
But with a live bomb, the mirror can move. This has no effect on the phantom
photon that runs along the upper path. But when the low-going phantom deflects
off the mirror, it causes the mirror to recoil. Heisenberg鈥檚 uncertainty
principle stipulates that the interaction between photon and mirror gives this
phantom a random kick. This is important: it alters the phantom travelling on
the lower path, and disrupts the recombination of the two phantom photons at the
beam splitter. As a result, with a live bomb, the photon will end up half the
time in detector A, and half the time in B.
One in four
So, according to Elitzur and Vaidman, you find a live bomb by watching
photons. You鈥檙e out of luck if the photon ends up in detector A. Both live bombs
and duds can send photons into A, so you cannot infer anything about the bomb.
But if the photon ends up in B, you know that the bomb was live. That鈥檚 the
conclusion. But what is the cost?
If the photon goes into detector B, there is a 50 per cent chance that it did
so by taking the lower path, and detonating the bomb. If so, you lose that bomb.
But there is also a 50 per cent chance that the photon got to B by the upper
path鈥攁nd this is the real counterfactual miracle. In this case, you know
the bomb is live, and yet it remains unexploded. You have learned about it,
without interacting with it.
If the pile of bombs is split evenly between live and dud, only 50 percent of
those tested will be live. In those cases, the instrument will tell you that the
bomb is live, without exploding it, only half the time. So, working out the
probabilities, you have a 1 in 4 chance of identifying a live bomb
non-explosively. In these cases, the mere fact that the mirror could have
recoiled if it had felt the photon鈥攅ven though it didn鈥檛鈥攍eads to a
tangible result.
This is the weird logic that Jozsa has applied to computing, using as an
example a quantum computer that tests a number to see if it is prime or not.
Amazingly, he has found that, in principle, similar counterfactual effects can
sometimes produce a result even if the machine never runs.
A computer consists of a switch, a set of program registers鈥攚hich hold
the input data and instructions for processing it鈥攁nd an output register
to hold the answer. To test whether a given number is prime, for example, the
program registers would hold both the number in question and the test program.
Turn the switch on, the program runs and the output register goes to 1 if the
number is prime, or remains at 0 if it isn鈥檛. In analogy with the bomb testing
rig, a prime number is like a live bomb, a non-prime is like a dud, and the
computer鈥檚 on-off switch plays the part of the photon.
In a typical run, Jozsa first sets the switch to off. Then he uses a switch
splitter (which acts like the beam splitter in the bomb test) to put the switch
into a superposition of on and off. In this superposition, the on and off
phantom computers don鈥檛 really travel on different paths in space, as do the
photon phantoms in the interferometer, but they follow different 鈥渃omputational
trajectories鈥�, so the diagram reflects an analogous process. The off phantom is
like the photon on the upper path, and the on phantom like the photon on the
lower. After waiting a time T鈥攋ust long enough for the computer to finish
if it were running鈥擩ozsa then uses a second switch splitter to fold the
on-off superposition back together. He then measures the state of the
switch.
Prime position
If the number really is a dud鈥攊t isn鈥檛 prime鈥攖hen the switch
always ends up off. Why? After the first splitter, the computer is in a
superposition of both on and off. But if the number is a dud, then both the off
and on phantoms leave the output register as it was鈥攁t 0. As a result, the
second switch splitter folds the unaltered phantoms together again to give the
original 鈥渞eal鈥� computer with its switch off. So if the number isn鈥檛 prime, the
two switch splitters in combination do nothing, and the computer will never be
found to be on (just as, in the bomb tester, a dud bomb always sent the photon
into detector A).
If the number is prime, however, then the on phantom of the computer will
suffer a real change鈥攊ts output register will go from 0 to 1. As in the
鈥渓ive bomb鈥� scenario, this spoils the subtle recombination of the phantom
computers at the second switch splitter. The switch splitters no longer combine
to do nothing. Instead, a measurement of the switch will find it to be off or on
with equal probability.
The counterfactual miracle occurs here if the switch is, after time T, found
to be on. Since a non-prime 鈥渄ud鈥� number always leaves the computer off, finding
the machine on implies that the number is prime. But just because the switch is
found to be on doesn鈥檛 mean that the computer was on all the while, and actually
made a computation. There is still only a 50-50 chance that the output register
will have changed. If it reads 1, then yes鈥攁 computation really was made.
But if the register reads 0, which it will half the time, then the computation
never ran. Instead, the machine remained off after the first splitter, and was
only flicked on at the second. In this case, you have your answer, but the
computer was off for nearly the entire time.
Feeding prime and non-prime numbers into such a machine in equal measure, it
will, like the bomb tester, learn the result for free one quarter of the time.
Of course, a computer that works only a quarter of the time may not be so
useful. But the odds can be improved considerably. Elitzur and Vaidman鈥檚 bomb
test had only a 1 in 4 chance of detecting a live bomb without exploding it. In
1995, however, a team of physicists led by Paul Kwiat atthe University of
Innsbruck showed that the detection probability can be increased as close as you
like to 1 out of 1 by using more elaborate bomb testers designed along similar
principles.
Jozsa has shown theoretically that an analogous trick can be done with
computing, and that, with near certainty, a quantum computer could verify that a
number is prime without ever doing any computation. But what if the number isn鈥檛
prime? Given a random number, could a quantum machine supply either answer,
prime or non-prime, in every case, still without doing any work? No one knows.
鈥淭hat may actually be impossible in principle,鈥� says Jozsa. 鈥淚鈥檝e put some
thought into the problem, and so have some other researchers, but we haven鈥檛 yet
worked out whether it can be done.鈥�
Pause for thought
If it can, then counterfactual effects would put a very peculiar spin on our
use of quantum computers, if and when they arrive. It would mean, says Jozsa,
鈥渢hat all computation could be done without computation鈥�, just by waiting a long
enough time so that the computer would have finished running if, in fact, it had
been running. Forget the need for electricity, or the hassle of burnt out
silicon chips. 鈥淲aiting time,鈥� he says, 鈥渨ould be the only essential resource
used up in a computation.鈥�
Right now, however, counterfactual computation remains a theoretical
curiosity that Jozsa hopes will stimulate much-needed lateral-thinking
approaches to quantum computing. Researchers working out ways to program quantum
computers tend to concentrate on methods adapted from classical computing. But
quantum mechanics opens up other possibilities. 鈥淭his result certainly shows
that there is more to quantum computing than just speeding things up,鈥� he
says.
The idea may stretch out in other directions as well. At Oxford University,
mathematical physicist Roger Penrose believes that Jozsa鈥檚 work may hint at a
link between quantum computing and human cognition. Putting the computational
power of the brain into a quantum mechanical context, Penrose speculates that
the neurons in the brain could cause some effects just through their inherent
potential to calculate, as well by actually doing those operations.
鈥淐ounterfactual computation could be playing a role in consciousness,鈥� he
suggests. 鈥淭hat kind of thing is often going on in one鈥檚 thinking, and it鈥檚
certainly a possibility that nature could make use of.鈥�
If you find that hard to believe, or have trouble accepting Jozsa鈥檚 idea that
computers could give results without ever being run, then ponder this: Paul
Kwiat has moved from Innsbruck to the Los Alamos National Laboratory, in New
Mexico. And just a few weeks ago, he and other physicists there demonstrated
experimentally that the very same counterfactual quantum effect can be used to
make pictures of objects without ever shining light on them. They were able to
detect the presence of an object by the fact that it would have absorbed or
scattered a photon, even though it never did. So prepare yourself for what may
be a bewildering future: quantum counterfactual logic really works.