Manual Reference Pages - QUANTUM::SUPERPOSITIONS (3)
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NAME
Quantum::Superpositions - QM-like superpositions in Perl
CONTENTS
VERSION
This document describes version 1.03 of Quantum::Superpositions,
released August 11, 2000.
SYNOPSIS
use Quantum::Superpositions;
if ($x == any($a, $b, $c)) { ... }
while ($nextval < all(@thresholds)) { ... }
$max = any(@value) < all(@values);
use Quantum::Superpositions BINARY => [ CORE::index ];
print index( any("opts","tops","spot"), "o" );
print index( "stop", any("p","s") );
BACKGROUND
Under the standard interpretation of quantum mechanics, until they are observed, particles exist only as a discontinuous probability
function. Under the Cophenhagen Interpretation, this situation is often visualized by imagining the state of an unobserved particle to be
a ghostly overlay of all its possible observable
states simultaneously. For example, a particle
that might be observed in state A, B, or C may
be considered to be in a pseudo-state where
it is simultaneously in states A, B, and C.
Such a particle is said to be in a superposition of states.
Research into applying particle superposition
in construction of computer hardware is already well advanced. The aim of such
research is to develop reliable quantum
memories, in which an individual bit is stored
as some measurable property of a quantised
particle (a qubit). Because the particle can be
physically coerced into a superposition of
states, it can store bits that are simultaneously
1 and 0.
Specific processes based on the interactions of
one or more qubits (such as interference, entanglement, or additional superposition) are
then be used to construct quantum logic
gates. Such gates can in turn be employed to
perform logical operations on qubits, allowing logical and mathematical operations to be
executed in parallel.
Unfortunately, the math required to design and use
quantum algorithms on quantum computers is painfully
hard. The Quantum::Superpositions module offers
another approach, based on the superposition of
entire scalar values (rather than individual qubits).
DESCRIPTION
The Quantum::Superpositions module adds two
new operators to Perl: any and all.
Each of these operators takes a list of values (states)
and superimposes them into a single scalar
value (a superposition), which can then be
stored in a standard scalar variable.
The any and all operators produce two distinct kinds of superposition. The any
operator produces a disjunctive superposition,
which may (notionally) be in any one of its
states at any time, according to the needs of
the algorithm that uses it.
In contrast, the all
operator creates a conjunctive superposition,
which is always in every one of its states
simultaneously.
Superpositions are scalar values and hence
can participate in arithmetic and logical operations just like any other type of scalar.
However, when an operation is applied to a
superposition, it is applied (notionally) in parallel to each
of the states in that superposition.
For example, if a superposition of states 1, 2, and 3 is
multiplied by 2:
$result = any(1,2,3) * 2;
the result is a superposition of states 2, 4, and
6. If that result is then compared with the
value 4:
if ($result == 4) { print "fore!" }
then the comparison also returns a superposition: one that is both true and false (since the
equality is true for one of the states of
$result and false for the other two).
Of course, a value that is both true and false is
of no use in an if statement, so some mechanism is needed to decide which superimposed boolean state should take precedence.
This mechanism is provided by the two types
of superposition available. A disjunctive superposition is true if any of its states is true,
whereas a conjunctive superposition is true
only if all of its states are true.
Thus the previous example does print
fore!, since the if condition is equivalent
to:
if (any(2,4,6) == 4)...
It suffices that any one of 2, 4, or 6 is equal to 4, so the condition
is true and the if block executes.
On the other hand, had the control statement
been:
if (all(2,4,6) == 4)...
the condition would fail, since it is not true
that all of 2, 4, and 6 are equal to 4.
Operations are also possible between two superpositions:
if (all(1,2,3)*any(5,6) < 21)
{ print "no alcohol"; }
if (all(1,2,3)*any(5,6) < 18)
{ print "no entry"; }
if (any(1,2,3)*all(5,6) < 18)
{ print "under-age" }
In this example, the string no alcohol is printed because the
superposition produced by the multiplication is the Cartesian product of
the respective states of the two operands: all(5,6,10,12,15,18).
Since all of these resultant states are less that 21, the condition is
true. In contrast, the string no entry is not printed, because not all
the product’s states are less than 18.
Note that the type of the first operand determines the type of the result of an operation.
Hence the third string — underage — is
printed, because multiplying a disjunctive
superposition by a conjunctive superposition
produces a result that is disjunctive:
any(5,6,10,12,15,18). The condition of
the if statement asks whether any of these
values is less than 18, which is true.
Composite Superpositions
The states of a superposition may be any kind
of scalar value — a number, a string, or a reference:
$wanted = any("Mr","Ms").any(@names);
if ($name eq $wanted) { print "Reward!"; }
$okay = all(\&check1,\&check2);
die unless $okay->();
my $large =
all( BigNum->new($centillion),
BigNum->new($googol),
BigNum->new($SkewesNum)
);
@huge = grep {$_ > $large} @nums;
More interestingly, since the individual states
of a superposition are scalar values and a superposition is itself a scalar value, a superposition may have states that are themselves
superpositions:
$ideal = any( all("tall", "rich", "handsome"),
all("rich", "old"),
all("smart","Australian","rich")
);
Operations involving such a composite superposition operate recursively and in parallel on each its states individually and then
recompose the result. For example:
while (@features = get_description)
{
if (any(@features) eq $ideal)
{
print "True love";
}
}
The any(@features) eq $ideal equality
is true if the input characteristics collectively
match any of the three superimposed conjunctive superpositions. That is, if the characteristics collectively equate to each of tall
and rich and handsome, or to both
rich and old, or to all three of
smart and Australian and rich.
Eigenstates
It is useful to be able to determine the list of
states that a given superposition represents.
In fact, it is not the states per se, but the
values to which the states may collapse — the
eigenstates that are useful.
In programming terms this is the
set of values @ev for a given superposition $s
such that any(@ev) == $s or
any(@ev) eq $s.
This list is provided by the eigenstates
operator, which may be called on any superposition:
print "The factor was: ",
eigenstates($factor);
print "Dont use any of:",
eigenstates($badpasswds);
Boolean evaluation of superpositions
The examples shown above assume the same meta-semantics for both
arithmetic and boolean operations, namely
that a binary operator is applied to the Cartesian product of the states of its two operands,
regardless of whether the operation is arithmetic or logical. Thus the comparison of two
superpositions produces a superposition of
1’s and 0’s, representing any (or all) possible
comparisons between the individual states of
the two operands.
The drawback of applying arithmetic metasemantics to logical operations is that it
causes useful information to be lost. Specifically, which states were responsible for the
success of the comparison. For example, it is
possible to determine if any number in the
array @newnums is less than all those in the
array @oldnums with:
if (any(@newnums) < @all(oldnums))
{
print "New minimum detected";
}
But this is almost certainly unsatisfactory, because it does not reveal which element(s) of
@newnum caused the condition to be true.
It is, however, possible to define a different
meta-semantics for logical operations between superpositions; one that preserves the
intuitive logic of comparisons but also gives
limited access to the states that cause those
comparsions to succeed.
The key is to deviate from the arithmetic view
of superpositional comparison (namely, that a
compared superposition yields a superposition of compared state combinations).
Instead, the various comparison operators are
redefined so that they form a superposition of
those eigenstates of the left operand that cause
the operation to be true. In other words, the
old meta-semantics superimposed the result
of each parallel comparison, whilst the new
meta-semantics superimposes the left operands of each parallel comparison that succeeds.
For example, under the original semantics,
the comparisons:
all(7,8,9) <= any(5,6,7) #A
all(5,6,7) <= any(7,8,9) #B
any(6,7,8) <= all(7,8,9) #C
would yield:
all(0,0,1,0,0,0,0,0,0) #A (false)
all(1,1,1,1,1,1,1,1,1) #B (true)
any(1,1,1,1,1,1,0,1,1) #C (true)
Under the new semantics they would yield:
all(7) #A (false)
all(5,6,7) #B (true)
any(6,7) #C (true)
The success of the comparison (the truth of
the result) is no longer determined by the values
of the resulting states, but by the number of
states in the resulting superposition.
The Quantum::Superpositions module treats logical
operations and boolean conversions in exactly this way.
Under these meta-semantics, it is possible to
check a comparison and also determine
which eigenstates of the left operand were
responsible for its success:
$newmins = any(@newnums) < all(@oldnums);
if ($newmins)
{
print "New minima found:", eigenstates($newmins);
}
Thus, these semantics provide a mechanism
to conduct parallel searches for minima and maxima :
sub min { eigenstates( any(@_) <= all(@_) ) }
sub max { eigenstates( any(@_) >= all(@_) ) }
These definitions are also quite intuitive, almost declarative: the minimum is any value
that is less-than-or-equal-to all of the other
values; the maximum is any value that is
greater-than-or-equal to all of them.
String evaluation of superpositions
Converting a superposition to a string produces
a string that encode the simplest set of eigenstates
equivalent to the original superposition.
If there is only one eigenstate, the stringification
of that state is the string representation.
This eliminates the need to explicitly apply the eigenstates
operator when only a single
resultant state is possible. For example:
print "lexicographically first: ",
any(@words) le all(@words);
In all other cases, superpositions are stringified
in the format: "all(eigenstates)" or
"any(eigenstates)".
Numerical evaluation of superpositions
Providing an implicit conversion to numeric (for situations where
superpositions are used as operands to an arithmetic operation, or as
array indices) is more challenging than stringification, since there is
no mechanism to capture the entire state of a superposition in a single
non-superimposed number.
Again, if the superposition has a single eigenstate, the conversion is just the standard conversion for that value. For instance, to output
the value in an array element with the smallest index in the set of indices @i:
print "The smallest element is: ",
$array[any(@i)<=all(@i)];
If the superposition has no eigenstates, there
is no numerical value to which it could collapse, so the result is undef.
If a disjunctive superposition has more than
one eigenstate, that superposition could collapse to any of those values. And it is convenient to allow it to do exactly that — collapse
(pseudo-)randomly to one of its eigenstates.
Indeed, doing so provides a useful notation
for random selection from a list:
print "And the winner is...",
$entrant[any(0..$#entrant)];
Superpositions as subroutine arguments
When a superposition is used as a subroutine
argument, that subroutine is applied in parallel to each state of the superposition and the
results re-superimposed to form the same
type of superposition. For example, given:
$n1 = any(1,4,9);
$r1 = sqrt($n1);
$n2 = all(1,4,9);
$r2 = pow($n2,3);
$r3 = pow($n1,$r1);
then $r1 contains the disjunctive superposition any(1,2,3), $r2 contains the conjunctive superposition all(1,64,729), and <$r3 >
contains the conjunctive superposition
any(1,4,9,16,64,81,729).
Because the built-in sqrt and pow functions
don’t know about superpositions, the module
provides a mechanism for informing them that their
arguments may be superimposed.
If the call to use Quantum::Superpositions
is given an argument list, that list specifies
which functions should be rewritten to handle
superpositions. Unary functions and subroutine
can be quantized like so:
sub incr { $_[0]+1 }
sub numeric { $_[0]+0 eq $_[0] }
use Quantum::Superpositions
UNARY => ["CORE::int", "main::incr"],
UNARY_LOGICAL => ["main::numeric"];
For binary functions and subroutines use:
sub max { $_[0] < $_[1] ? $_[1] : $_[0] }
sub same { my $failed; $IG{__WARN__}=sub{$failed=1};
return $_[0] eq $_[1] || $_[0]==$_[1] && !$failed;
}
use Quantum::Superpositions
BINARY => [main::max, CORE::index],
BINARY_LOGICAL => [main::same];
EXAMPLES
Primality testing
The power of programming with scalar superpositions is perhaps best seen
by returning the quantum computing’s favourite adversary: prime numbers.
Here, for example is an O(1) prime-number tester, based on naive
trial division:
sub is_prime
{
my ($n) = @_;
return $n % all(2..sqrt($n)+1) != 0
}
The subroutine takes a single argument ($n)
and computes (in parallel) its modulus with
respect to every integer between 2 and sqrt($n).
This produces a conjunctive superposition of
moduli, which is then compared with zero.
That comparison will only be true if all the
moduli are not zero, which is precisely the
requirement for an integer to be prime.
Because is_prime takes a single scalar argument, it can also be passed a superposition.
For example, here is a constant-time filter for
detecting whether a number is part of a pair
of twin primes:
sub has_twin
{
my ($n) = @_;
return is_prime($n) && is_prime($n+any(+2,-2);
}
Set membership and intersection
Set operations are particularly easy to perform using superimposable scalars.
For example, given an array of values
@elems, representing the elements of a set,
the value $v is an element of that set if:
$v == any(@elems)
Note that this is equivalent to the definition of
an eigenstate. That equivalence can be used to
compute set intersections. Given two disjunctive superpositions, $s1=any(@elems1)
and $s2=any(@elems2), representing two
sets, the values that constitute the intersection
of those sets must be eigenstates of both <$s1>
and $s2. Hence:
@intersection = eigenstates(all($s1, $s2));
This result can be extended to extract the
common elements from an arbitrary number
of arrays in parallel:
@common = eigenstates( all( any(@list1),
any(@list2),
any(@list3),
any(@list4),
)
);
Factoring
Factoring numbers is also trivial using superpositions.
The factors of an integer N are all
the quotients q of N/n (for all positive integers n < N) that are also integral. A positive
number q is integral if floor(q)==q. Hence the factors of a given number are computed by:
sub factors
{
my ($n) = @_;
my $q = $n / any(2..$n-1);
return eigenstates(floor($q)==$q);
}
Query processing
Superpositions can also be used to perform
text searches.
For example, to determine whether a given string
($target) appears in a collection of strings
(@db):
use Quantum::Superpositions BINARY => ["CORE::index"];
$found = index(any(@db), $target) >= 0;
To determine which of the database strings
contain the target:
sub contains_str
{
return $dbstr if (index($dbstr, $target) >= 0;
}
$found = contains_str(any(@db), $target);
@matches = eigenstates $found;
It is also possible to superimpose the target
string, rather than the database, so as to
search a single string for any of a set of targets:
sub contains_targ
{
if (index($dbstr, $target) >= 0)
{
return $target;
}
}
$found = contains_targ($string, any(@targets));
@matches = eigenstates $found;
or in every target simultaneously:
$found = contains_targ($string, all(@targets));
@matches = eigenstates $found;
AUTHOR
Damian Conway (damian@conway.org)
Now maintainted by Steven Lembark (lembark@wrkhors.com)
BUGS
There are undoubtedly serious bugs lurking somewhere in code this funky :-)
Bug reports and other feedback are most welcome.
COPYRIGHT
Copyright (c) 1998-2002, Damian Conway.
Copyright (c) 2002, Steven Lembark
All Rights Reserved.
This module is free software. It may be used, redistributed
and/or modified under the stame terms as Perl-5.6.1 (or later)
(see http://www.perl.com/perl/misc/Artistic.html).
perl v5.20.3 | QUANTUM::SUPERPOSITIONS (3) | 2003-04-22 |
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