Current quantum computing uses 2x2 (Pauli) operators and (4x4) Dirac operators expressed as matrices.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 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_______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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 > _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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 _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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. > > _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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