1. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Quantum Geometric Algebra ANPA Conference Cambridge, UK by Dr. Douglas J. Matzke matzke@ieee.org Aug 15-18, 2002

2. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Abstract Quantum computing concepts are described using geometric algebra, without using complex numbers or matrices. This novel approach enables the expression of the principle ideas of quantum computation without requiring an advanced degree in mathematics. Using a topologically derived algebraic notation that relies only on addition and the anticommutative geometric product, this talk describes the following quantum computing concepts: bits, vectors, states, orthogonality, qubits, classical states, superposition states, spinor, reversibility, unitary operator, singular, entanglement, ebits, separability, information erasure, destructive interference and measurement. These quantum concepts can be described simply in geometric algebra, thereby facilitating the understanding of quantum computing concepts by non-physicists and non-mathematicians.

3. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Overview of Presentation Co-Occurrence and Co-Exclusion Geometric Algebra G n Essentials Symmetric values, scalar addition and multiplication Graded N-vectors, scalar, bivectors, spinors Inner product, outer product, and anticommutative geometric product Qubit Definition is Co-Occurrence Standard and Superposition States, Hadamard Operator, Not Operator Reversibility, Unitary Operators, Pauli Operators, Circular basis Irreversibility, Singular Operators, Sparse Invariants and Measurement Eigenvectors, Projection Operators, trine states Quantum Registers Geometric product equivalent to tensor product, entanglement, separability Ebits and Bell/magic States/operators, non-separable and information erasure C-not, C-spin, Toffoli Operators Conclusions

4. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Co-Occurrence and Co-Exclusion Both of Mike Manthey’s concepts used heavily in this research Abstract Time Abstract Space Co-occurrence means states exist exactly simultaneously Co-exclusion means a change occurred due to an operator (0 means cannot occur) a = +a = ON and a = –a = not ON where a + a = 0

5. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Boolean Logic using +/* in G n Geometric Algebra is Boolean Complete Normal multiplication and mod 3 addition for ring {–1,0,1}, so can simplify to {–,0,+} and remove rows/columns for header value 0. Logic in G 2 = span {a, b} GA Mapping {+, –}GA Mapping {+, 0} Identity aa * 1 = a + 0 = a–1 – a = – (1 + a) NOT aa * –1 = – a–1 + a = – (1 – a) a XOR b– a b–1 + a b a OR ba + b – a b–1 – a – b + a b a AND b+1 – a – b – a b+1 + a + b + a b Also for any vector e: since e 2 =1 then e = 1/e –1–1 +1 0 Binary Values Z3Z3

6. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Geometric Algebra Essentials a b b a +a +b –a–a –b–b +a b –a b orientation G n=2 generates N=2 n : span{a, b} alsowith

7. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Number of Elements in G n Pascal’s Triangle ( Binomial) (1+a)(1+b)(1+c) = 1 + a + b + c + a b + a c + b c + a b c Graded: scalar, vector, bivector, trivector, …, n-vector for G n with N=2 n elements <multivector 0 + 1 + 2 + 3 + … + n Odd grade terms G n – = 1 + 3 + … Even Subalgebra G n + = 0 + 2 + … G 3 + are the quaternions: 1 + a b + a c + b c G n = G n + + G n – =

8. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Inner Product Calculation YY +1aba b+1aba b X +1 aba b X +10000 aa0a b0a0+10b bb–a b00b00+1–a–a a b 000 0b–a–a–1 Outer ProductInner Product and with vector variables w, x=a, y =b, z =c G 2 = span{a,b}: G 3 = span{a,b,c}: Only one non-zero term in sum for orthogonal basis set {a,b,c} only if X or Y are assigned vector x or y

9. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Qubit is Co-occurrence in G 2 Single Qubit: A = (±a0 ±a1) where Q 1 = G 2 = span{a0, a1} 4 elements & 3 4 = 81 multivectors +a1 –a1 +a0–a0 Classical: signs are opposite Superposition: signs are same A1A1 A0A0 A+A+ A–A– Row k a0a1 A 1 =A 0 =A + =A – = R0R0 ––00+– R1R1 –++–00 R2R2 +––+00 R3R3 ++00–+ Binary combinations of input states Anti-symmetric sums are classical states Symmetric sums are superposition states A 1 = R 1 – R 2 A 0 = R 2 – R 1 A + = R 0 – R 3 A – = R 3 – R 0

10. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Spinor is Hadamard Operator Start PhaseQubit State AEach Times SpinorResult = A S A End Phase Classical A 0 = +a0 – a1+a0 (a0 a1) = +a1A + = +a0 + a1 Superposed A 1 = –a0 + a1–a0 (a0 a1) = –a1A – = –a0 – a1 Superposed A + = +a0 + a1+a1 (a0 a1) = –a0A 1 = –a0 + a1 Classical A – = –a0 – a1–a1 (a0 a1) = +a0A 0 = +a0 – a1 Hadamard is the 90° phase or spinor operator S A = (a0 a1) NOT operator is 180° gate S A 2 = (a0 a1)(a0 a1) = –a0 a0 a1 a1 = –1 Therefore

11. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Unitary Pauli Noise States in G 2 CaseHilbert notationUse caseGA equivalent is (–1)= complement [a] (+ a0 – a1)(–1)  (– a0 + a1) [b] (– a0 + a1)(–1)  (+ a0 – a1) CaseHilbert notationUse casesGA equivalent is spinor S A = a0 a1 [a] [a]&[b] (– a0 + a1)(– a0 a1)  (+ a0 – a1) [b] [b]&[c] (+ a0 – a1 )(a0 a1)  (+ a0 + a1) [c] CaseHilbert notationUse casesGA equivalent is (–1 + S A ) = P A [a] [a]&[b] (+ a0 – a1)(–1 + a0 a1)  –a1 [b] + = Bit Flip: Phase Both Pauli operators -1, S A and P A are even grade!

12. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Reversible Basis Encodings: Standard, Dual, Pauli and Circular basis Label for RowStart StateDiag (–1 + a0 a1)Diag (a0)Diag (+ a0 – a1) classical 0 classical 1 + a0 – a1– a1(+1 + a0 a1)–1 – a0 + a1+ a1(–1 – a0 a1)+1 superposition + superposition – + a0 + a1+ a0(+1 – a0 a1)+ a0 a1 (random) – a0 – a1– a0(–1 + a0 a1)– a0 a1 (random) Label for BasisDiagonalsPauli = Ver/HorCircularDirect or Complex Reversible op. return to startV/Hor (1 + a0 a1)Cir (a0)Dir (– a0 + a1) +a0 a1 –a0 a1 +1 –1 +a1 –a1 +a0 –a0 Left Diagonal Classical= Right Diagonal =Superposition Pauli Vertical Pauli Horizontal Circular Direct ODD GRADE EVEN GRADE Ellipses are co-exclusions!

13. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Unitary Operators and Reversibility For multivector state X and multivector operator Y, If new state Z = X Y then Y is unitary if-and-only-if W = 1/Y = Y -1 exists such that Y W = Y Y -1 = 1 Therefore unitary operator Y is invertible/reversible: Z / Y = X Y / Y = X For unitary Y then requires det(Y)=±1 or |det(Y)| = 1 Trines are unitary: (Tr) 3 = 1 so 1/Tr = (Tr) 2 for Tr = (+1 ± a0 ± S A ) or (+1 ± a1 ± S A ) A 0 A 1 = 1 A – A + = 1

14. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Singular Operators in G n If 1/X is undefined then requires det(X) = 0, Since (±1±x) -1 is undefined then det(±1±x) = 0 and therefore X = (±1±x) is singular Singular examples: det( ± 1±a) = det( ± 1 ± b) = 0 Also fact that: det(X)det(Y) = det(XY), which means if factor X has det(X) = 0, then product (XY) also has det(XY) = 0. In G 2 : det(1±a)det(1±b) = det(1±a±b±a b) = 0

15. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Row Decode Operators R k are Singular Row k a0a1(–1)(1 – a0)(–1)(1 + a0)(–1)(1 – a1)(–1)(1 + a1) R0R0 ––+0+0 R1R1 –++00+ R2R2 +–0++0 R3R3 ++0+0+ Summation of R k  A0 – = R 0 + R 1 A0 + = R 2 + R 3 A1 – = R 0 + R 2 A1 + = R 1 + R 3 Denoted as Vector  [+ + 0 0][0 0 + +][+ 0 + 0][0 + 0 +] Row k a0a1(1–a0)(1–a1)(1–a0)(1+a1)(1+a0)(1–a1)(1+a0)(1+a1) R0R0 ––+000 R1R1 –+0+00 R2R2 +–00+0 R3R3 ++000+ State logic  R 0 = A0 – A1 – R 1 = A0 – A1 + R 2 = A0 + A1 – R 3 = A0 + A1 + Denoted as Vector  R 0 = [+ 0 0 0]R 1 = [0 + 0 0]R 2 = [0 0 + 0]R 3 = [0 0 0 +] R k are topologically smallest elements in G 2 and are linearly independent R 0 + R 1 + R 2 + R 3 = [+ + + +] = 1 Dual Vector Notation: Standard Algebraic Notation matrix diagonal

16. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Start States A Each start state A times each R k gives the answer A(1+a0)(1–a1)A(1–a0)(1+a1)A(1+a0)(1+a1)A(1–a0)(1–a1) A 0 = + a0 – a1 –1 + a1 = I + +1 + a1 = I – –a0 (+1 + a1)+a0 (–1 + a1) A 1 = – a0 + a1 +1 – a1 = I – –1 – a1 = I + –a0 (–1 – a1)+a0 (+1 – a1) A – = – a0 – a1–a0 (–1 + a1)+a0 (+1 + a1) +1 + a1 = I – –1 + a1 = I + A + = + a0 + a1–a0 (+1 – a1)+a0 (–1 – a1) –1 – a1 = I + +1 – a1 = I – End State  A’ => + a0 – a1A’ => – a0 + a1A’ => + a0 + a1A’ => – a0 – a1 Description  Classical States MeasurementSuperposition States Measurement Measurement and Sparse Invariants –1 + a1 = [+ 0 + 0] = I +0 +1 – a1 = [– 0 – 0] = I –0 –1 – a1 = [0 + 0 +] = I +90 +1 + a1 = [0 – 0 –] = I –90 ( I ± ) 2 = I + I – = – I + I + ~ +1 I – ~ –1

17. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Projection Operators P k and Eigenvectors E k Primary Tetrahedron (k=0–3)Dual Tetrahedron (=7–k) k =E k = R k –1P k = –R k R k = 1+E k k =E k = R k –1P k = –R k R k = 1+E k 0[0 – – –][– 0 0 0][+ 0 0 0]7[0 + + +][– + + +][+ – – –] 1[– 0 – –][0 – 0 0][0 + 0 0]6[+ 0 + +][+ – + +][– + – –] 2[– – 0 –][0 0 – 0][0 0 + 0]5[+ + 0 +][+ + – +][– – + –] 3[– – – 0][0 0 0 –][0 0 0 +]4[+ + + 0][+ + + –][– – – +] sum[0 0 0 0][– – – –][+ + + +]sum[0 0 0 0][– – – –][+ + + +] +a0 +a1 +a0 a1 - - - + + + 000 +a0 +a1 +a0 a1 - - - + + + 000 E0E0 E1E1 E2E2 E3E3 E5E5 E4E4 E7E7 E6E6 +a0 a1 +a0 +a1 - - - + + + +a0 +a1 +a0 a1 - - - + + + E0E0 E1E1 E2E2 E3E3 E5E5 E4E4 E7E7 E6E6 R k = –P k E k 2 = 1 E k R k = R k P k 2 = P k Idempotent!! E k = ± a0 ± a1 ± a0 a1

18. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM with A = (±a0 ±a1), B = (±b0 ±b1), C = (±c0 ±c1) then A B C = (±a0 ±a1)(±b0 ±b1)(±c0 ±c1) so A + B + = (+a0 +a1)(+b0 +b1) = a0 b0 + a0 b1 + a1 b0 + a1 b1 Geometric product replaces the tensor product Q q = G n=2q State Count: Total: 2 2q = 4 q Non-zero: 2 q Zeros: 4 q – 2 q A B C = 0 A 1 B 1 P A P B = a1 b1 = S 11 Row k State CombinationsIndividual bivector productsColumn Vector a0a1b0b1a0 b0a0 b1a1 b0a1 b1 A + B + A 0 B 0 R0R0 ––––+++++0 R3R3 ––++–––––0 R5R5 –+–++––+0– R6R6 –++––++–0+ R9R9 +––+–++–0+ R 10 +–+–+––+0– R 12 ++–––––––0 R 15 +++++++++0 Qubits form Quantum Register Q q

19. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Ebits: Bell/magic States and Operators Separable: A 0 B 0 (S A )(S B ) = A 0 (S A ) B 0 (S B ) = A + B + Non-Separable: A 0 B 0 (S A + S B ) = A + B 0 + A 0 B + = = –a0 b0 + 0 a0 b1 + 0 a1 b0 + a1 b1 = –a0 b0 + a1 b1 = S 00 + S 11 = B 0 B = (S A + S B ) B i±1 = ± B i B M = (S A – S B ) M i±1 = ± M i M B 0 = –S 00 + S 11 = B 1 = +S 01 + S 10 = B 2 = +S 00 – S 11 = B 3 = –S 01 – S 10 = M 0 = +S 01 – S 10 M 1 = –S 00 – S 11 M 2 = –S 01 + S 10 M 3 = +S 00 + S 11 Row k State CombinationsIndividual bivectors Output column a0a1b0b1–a0 b0a1 b1 R1R1 –––+––+ R2R2 ––+–++– R4R4 –+––––+ R7R7 –+++++– R8R8 +–––++– R 11 +–++––+ R 13 ++–+++– R 14 +++–––+ Valid states where exactly one qubit in superposition phase!! M 3 = B 2 (S 01 + S 10 ) Concurrent! B & M are Singular!

20. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM –=B3=–=B3= +S 10 –S 10 +S 01 –S 01 +S 11 –S 11 +S 00 –S 00 = B 0 ==M3=M3 M 1 = =B2= =B2= M 2 ==B1==B1= =M0 =M0 A 0/1 B 0/1 P AB A ± B ± P AB A 0/1 B ± P AB A ± B 0/1 P AB B 2 = I – and M 2 = I – and Interesting Facts about Ebits B and M are valid for Q q>2 as (S A ± S B ± S C ± …) – P A P B = B – (1+ S A S B ) = B + I + A B = B i + M j but B i M = M i B = 0, so A B B = B i+1 + 0 +0

21. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Cnot, Cspin and Toffoli Operators For Q 2 with qubits A and B, where A is the control: CNot AB = A 0 = (a0 – a1) where (A 0 ) 2 = –1 Cspin AB = = (–1 + A 0 ) = (–1 + a0 – a1) For Q 3 : qubits A, B & D where A & B are controls: Tof AB = CNot AD + CNot BD = A 1 + B 0 (concurrent!) = – a0 + a1 + b0 – b1 where (Tof AB ) 2 = 1 Row k State Combinations Active States A 0 B 0 D 0 (TOF AB ) a0a1b0b1d0d1 R 21 –+–+–+A 1 B 1 & D 1 – Inverted R 22 –+–++–A 1 B 1 & D 0 + R 41 +–+––+A 0 B 0 & D 1 + Identity R 42 +–+–+–A 0 B 0 & D 0 – Also for Q q P k 2q = P k E k x = 1 qx 12 26 380 4???

22. ANPA 2002: Quantum Geometric Algebra 8/15/2002 DJM Conclusions The Quantum Geometric Algebra approach appears to simply and elegantly define many of the properties of quantum computing. This work was facilitated tremendously by the use of custom tools that automatically maintained the GA anticommutative and topological rules in an algebraic fashion. Many thanks to Mike Manthey for all his inspiration and support on my PhD effort. Many questions and much work still remains.