[quantum] Geometric Algebra

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[quantum] Geometric Algebra

Tim Daly
Current quantum computing uses 2x2 (Pauli) operators
and (4x4) Dirac operators expressed as matrices.

But Hestenes
https://en.wikipedia.org/wiki/David_Hestenes
showed that these are simple subdomains of the Clifford
(aka Geometric) Algebra. Axiom implements the Clifford
algebra so it should be possible to re-formulate the
quantum algorithms using operations expressed in
this more general form.

It is possible that these more general Clifford forms would
improve clarity and expressiveness for quantum algorithms.
In particular, applying Clifford notation to the Bloch sphere
rotations seems like an interesting idea.

Also of interest is that the Hadamard gate, expressed as a
matrix [[1,1],[1,-1]], is fundamental quantum operator. But it
is also the method of constructing communication codes so
that multiple endpoints can fully share the same channel using
the full channel bandwidth. This leads to the thought that
quantum communication is intimately linked with these higher
A Geometric Algebra formulation of these matrix operations
provides a more general, coordinate-free language.

This leads to the need for more work on the Clifford domain.

Tim


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Re: [quantum] Geometric Algebra

Arthur Ralfs
Be aware that Hestenes has copyrighted some of his material.

Arthur


On 11/14/2016 01:11 AM, Tim Daly wrote:

> Current quantum computing uses 2x2 (Pauli) operators
> and (4x4) Dirac operators expressed as matrices.
>
> But Hestenes
> https://en.wikipedia.org/wiki/David_Hestenes
> showed that these are simple subdomains of the Clifford
> (aka Geometric) Algebra. Axiom implements the Clifford
> algebra so it should be possible to re-formulate the
> quantum algorithms using operations expressed in
> this more general form.
>
> It is possible that these more general Clifford forms would
> improve clarity and expressiveness for quantum algorithms.
> In particular, applying Clifford notation to the Bloch sphere
> rotations seems like an interesting idea.
>
> Also of interest is that the Hadamard gate, expressed as a
> matrix [[1,1],[1,-1]], is fundamental quantum operator. But it
> is also the method of constructing communication codes so
> that multiple endpoints can fully share the same channel using
> the full channel bandwidth. This leads to the thought that
> quantum communication is intimately linked with these higher
> codes. See
> http://www.uow.edu.au/~jennie/WEBPDF/2005_12.pdf
>
> A Geometric Algebra formulation of these matrix operations
> provides a more general, coordinate-free language.
>
> This leads to the need for more work on the Clifford domain.
>
> Tim
>
>
>
> _______________________________________________
> Axiom-developer mailing list
> [hidden email]
> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>


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Re: [quantum] Geometric Algebra

Bertfried Fauser-3
Hi Arthur,

  there were rumors about this when Rafal and I developed our Clifford
package (mid 90ies), so we registered it as original work with the US
Library of Congress (Rafal knows how he did that) and that makes it a
publicly available piece of software (and unpatentable). Though it
needs (alas) Maple to run, the software as it stands is open source
freely available and can be used (modified rewritten). Anyhow I will
patent natural numbers and all derived concepts, hey why not sets
(after Bourbaki all math is derivable from that :) )

@Tim:
  Martin Baker has written a new Grassmann and Clifford package for
FriCAS, which goes beyond what was done in AXIOM before. The
implementation of Clifford algebras in AXIOM is at best a proof of
concept. Rafal's and mine Clifford package have algorithms which in
certain situations can be proven to be optimal, and also special
(fast)
algorithms for diagonal (in suitable bases) bilinear (polar) forms (of
quadratic forms). Without such fast algorithms computations in
Clifford algebras of dimension 5 or higher are not feasible, even with
fast algorithms, the computation of a Clifford algebra multiplication
table
is not possible for dimension 8/9 upwards. Rafal and I investigated
how to parallelise the Clifford product, as Maple has some coarse
grained parallelism. We got 11 times faster code on a two core machine
(amazing isn't it?) and it showed to us that a much better design of
data structures brought more gain in speed that doing parallel
computations (2 versus 5.5 at best). Robotics people do computations
in 9 to 11 (base space) dimenional Clifford algebras and would also be
interested in fast software for doing that.
  While Hestenes was influential in the Geometric Algebra camp, he was
not the person who did this first. There is a continuous literature
going back to Hamilton, Grassmann and Clifford. Though Geometric
Algebra people do tend not to cite these papers.
  In the mid 90ies people tried to do quantum computing stuff using
Clifford algebras in the Hestenes (operator spinor) style. However
they ran into problems:
  When you tensor matrices not spinors, you pick up additional
dimensions, the proper tensor product is an amalgamated (central
product) one. People then used 'quantum correlators' (aka projection
operators) to fix this. However, in the end they just reconstructed
what people did with (spin-tensor) matrices anyhow. Unless there is
more insight or more abstraction (basis free Clifford algebras are
still a challenge to be done in a CAS) it might be not that fruitful?
Anyhow, a search for literature might be helpful. If I remember rightly
there was a special issue for Hestenes' 60ies birthday in Foundatons
of Physics, and in that you will find some of the work mentioned above
by Doran, Lasenby et al.

Unfortunately I do not have much time at the moment to help out, but if
question arise regarding algorithms related to Grassmann and Clifford
algebras etc I try to help.

Kind regards
BF.


--
% Dr Bertfried Fauser
%      Phone :  +49 7471 4031397   Mobile : +49 176 64094110

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Re: [quantum] Geometric Algebra

Arthur Ralfs
Bertfried,

Yours and Tim's private response has put my mind at ease, I guess.  My
concern was based on a book, Geometric Algebra by Dorst, Fontijne, and
Mann, recommended by Martin Baker, a few years ago.  When I got to
section 13.8 it was mentioned that some of the work was patented and not
freely available for commercial use.

Arthur

On 11/14/2016 09:43 AM, Bertfried Fauser wrote:

> Hi Arthur,
>
>   there were rumors about this when Rafal and I developed our Clifford
> package (mid 90ies), so we registered it as original work with the US
> Library of Congress (Rafal knows how he did that) and that makes it a
> publicly available piece of software (and unpatentable). Though it
> needs (alas) Maple to run, the software as it stands is open source
> freely available and can be used (modified rewritten). Anyhow I will
> patent natural numbers and all derived concepts, hey why not sets
> (after Bourbaki all math is derivable from that :) )
>
> @Tim:
>   Martin Baker has written a new Grassmann and Clifford package for
> FriCAS, which goes beyond what was done in AXIOM before. The
> implementation of Clifford algebras in AXIOM is at best a proof of
> concept. Rafal's and mine Clifford package have algorithms which in
> certain situations can be proven to be optimal, and also special
> (fast)
> algorithms for diagonal (in suitable bases) bilinear (polar) forms (of
> quadratic forms). Without such fast algorithms computations in
> Clifford algebras of dimension 5 or higher are not feasible, even with
> fast algorithms, the computation of a Clifford algebra multiplication
> table
> is not possible for dimension 8/9 upwards. Rafal and I investigated
> how to parallelise the Clifford product, as Maple has some coarse
> grained parallelism. We got 11 times faster code on a two core machine
> (amazing isn't it?) and it showed to us that a much better design of
> data structures brought more gain in speed that doing parallel
> computations (2 versus 5.5 at best). Robotics people do computations
> in 9 to 11 (base space) dimenional Clifford algebras and would also be
> interested in fast software for doing that.
>   While Hestenes was influential in the Geometric Algebra camp, he was
> not the person who did this first. There is a continuous literature
> going back to Hamilton, Grassmann and Clifford. Though Geometric
> Algebra people do tend not to cite these papers.
>   In the mid 90ies people tried to do quantum computing stuff using
> Clifford algebras in the Hestenes (operator spinor) style. However
> they ran into problems:
>   When you tensor matrices not spinors, you pick up additional
> dimensions, the proper tensor product is an amalgamated (central
> product) one. People then used 'quantum correlators' (aka projection
> operators) to fix this. However, in the end they just reconstructed
> what people did with (spin-tensor) matrices anyhow. Unless there is
> more insight or more abstraction (basis free Clifford algebras are
> still a challenge to be done in a CAS) it might be not that fruitful?
> Anyhow, a search for literature might be helpful. If I remember rightly
> there was a special issue for Hestenes' 60ies birthday in Foundatons
> of Physics, and in that you will find some of the work mentioned above
> by Doran, Lasenby et al.
>
> Unfortunately I do not have much time at the moment to help out, but if
> question arise regarding algorithms related to Grassmann and Clifford
> algebras etc I try to help.
>
> Kind regards
> BF.
>
>


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