Quantum Geometric Algebra

Transcription

1 ANPA 22: Quantum Geometric Algebra Quantum Geometric Algebra ANPA Conference Cambridge, UK by Dr. Douglas J. Matzke Aug 15-18, 22 8/15/22 DJM

2 ANPA 22: Quantum Geometric Algebra 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. 8/15/22 DJM

3 ANPA 22: Quantum Geometric Algebra 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 8/15/22 DJM

4 ANPA 22: Quantum Geometric Algebra Co-Occurrence and Co-Exclusion a a ON and a a not ON where a a a b b a c - d d - c c - d d - c Abstract Space c - d d - c (or can not occur) ( means cannot occur) Co-occurrence means states exist exactly simultaneously Co-exclusion means a change occurred due to an operator Abstract Time Both of Mike Manthey s concepts used heavily in this research 8/15/22 DJM

5 ANPA 22: Quantum Geometric Algebra Boolean Logic using /* in G n 1 1 * Normal multiplication and mod 3 addition for ring { 1,,1}, so can simplify to {,,} and remove rows/columns for header value. * If same then invert If diff then cancel NAND > NOR > If same then 1 If diff then -1 same XNOR same > differ XNOR differ > 1 Z 3 1 Binary Values Also for any vector e: since e 2 1 then e 1/e Logic ing 2 span{a, b} GA Mapping {, } GA Mapping {, } Identity a a * 1 a a 1 a (1 a) NOT a a * 1 a 1 a (1 a) a XOR b a b 1 a b a OR b a b a b 1 a b a b a AND b 1 a b a b 1 a b a b Geometric Algebra is Boolean Complete 8/15/22 DJM

6 ANPA 22: Quantum Geometric Algebra Geometric Algebra Essentials b b a ab ab i a b where geometric product is sum ab i cosθ of inner product (is a scalar) a b i sinθ and outer product (is a bivector) a a b b a G n2 generates N2 n : span{a, b} G 2 scalars {±1}, vectors {a, b}, and bivector {a b} then: With ab i (only orthonormal basis so are perpendicular) then a b b a (due to anti-commutative outer product) a 2 b 2 1 (due to inner product since collinear) bivector is spinor because: (right multiplication by spinor) a (a b) a a b b, and b (a b) a b b a spinor is also pseudoscalar I because: (a b) 2 a b a b a a b b (a) 2 (b) 2 1 NOT a b orientation so a b 1 NOT also x'rx Rwith R α βab, R α βab, α cos( θ /2), β sin( θ /2) a b orientation a b a b 8/15/22 DJM

7 ANPA 22: Quantum Geometric Algebra Number of Elements in G n Graded: scalar, vector, bivector, trivector,, n-vector for G n with N2 n elements (1a)(1b)(1c) 1 a b c a b a c b c a b c <multivector G n G n G n <A> <A> 1 <A> 2 <A> 3 <A> n Odd grade terms G n <A> 1 <A> 3 Even Subalgebra G n <A> <A> 2 G 3 are the quaternions: 1 a b a c b c Row n Col m n n 1 N 2 m 1 m Pascal s Triangle (Binomial) n 8/15/22 DJM

8 ANPA 22: Quantum Geometric Algebra Inner Product Calculation X X Y 1 a b a b and Y ( x y ) Z ( Y z) G 2 span{a,b}: G 3 span{a,b,c}: 1 1 a b a b Outer Product a a a b Y b b a b with vector variables w, xa, yb, zc wy i wi( a b ) ( wa i ) b - ( wb i ) a wz i ( wa i ) b c ( wb i ) a c ( wc i ) a b Only one non-zero term in sum for orthogonal basis set {a,b,c} XY XiY X Y a b a b X XY i 1 a b a b 1 Inner Product a 1 b Y b 1 a a b only if X or Y are assigned vector x or y b a 1 8/15/22 DJM

9 ANPA 22: Quantum Geometric Algebra Qubit is Co-occurrence in G 2 Single Qubit: A (±a ±a1) where Q 1 G 2 span{a, a1} 4 elements & multivectors A 1 a A a1 a1 A a A Superposition: signs are same Classical: signs are opposite a a1 A 1 a a1 A A a a1 A R Row k R 1 R 2 R 3 Binary combinations of input states Anti-symmetric sums are classical states A 1 R 1 R 2 a a1 a a1 A R 2 R 1 Symmetric sums are superposition states A R R 3 A R 3 R 8/15/22 DJM

10 ANPA 22: Quantum Geometric Algebra Spinor is Hadamard Operator Start Phase Qubit State A Each Times Spinor Result A S A End Phase Classical A a a1 A 1 a a1 a (a a1) a1 a (a a1) a1 A a a1 A a a1 Superposed Superposed A a a1 A a a1 a1 (a a1) a a1 (a a1) a A 1 a a1 A a a1 Classical Hadamard is the 9 phase or spinor operator S A (a a1) NOT operator is 18 gate S 2 A (a a1)(a a1) a a a1 a1 1 r p Therefore S A 1 NOT and generally θ θ / rand θ pθ 8/15/22 DJM

11 ANPA 22: Quantum Geometric Algebra σ σ 3 1 Unitary Pauli Noise States in G 2 Flip: Bit 1 1 Phase 1 1 Case [a] [b] Case [a] [b] [c] Hilbert notation σ 1 1 Use case [a] GA equivalent is ( 1) complement ( a a1)( 1) ( a a1) σ 1 1 [b] ( a a1)( 1) ( a a1) Hilbert notation σ σ 3 σ Use cases [a]&[b] [b]&[c] GA equivalent is spinor S A a a1 ( a a1)( a a1) ( a a1) ( a a1 )(a a1) ( a a1) σ 2 Both i i Case [a] [b] Hilbert notation σ 2 i σ 2 1 i 1 Use cases [a]&[b] GA equivalent is ( 1 S A ) P A ( a a1)( 1 a a1) a1 Pauli operators -1, S A and P A are even grade! 8/15/22 DJM

12 ANPA 22: Quantum Geometric Algebra Reversible Basis Encodings: Standard, Dual, Pauli and Circular basis Label for Row classical classical 1 superposition superposition Label for Basis Start State a a1 a a1 a a1 a a1 Diagonals Reversible op. return to start Diag ( 1 a a1) a1 a1 a a Pauli Ver/Hor V/Hor (1 a a1) Diag (a) (1 a a1) ( 1 a a1) (1 a a1) ( 1 a a1) Circular Cir (a) Diag ( a a1) 1 1 a a1 (random) a a1 (random) Direct or Complex Dir ( a a1) ODD GRADE Left Diagonal Classical a Pauli Vertical a1 a1 Right Diagonal Superposition a Pauli Horizontal 1 a a1 a a1 Circular 1 Direct Ellipses are co-exclusions! EVEN GRADE 8/15/22 DJM

13 ANPA 22: Quantum Geometric Algebra 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 A A 1 1 A A 1 Trines are unitary: (Tr) 3 1 so 1/Tr (Tr) 2 for Tr (1 ± a ± S A ) or (1 ± a1 ± S A ) 8/15/22 DJM

14 ANPA 22: Quantum Geometric Algebra Singular Operators in G n If 1/X is undefined then requires det(x), Since (±1±x) -1 is undefined then det(±1±x) and therefore X (±1±x) is singular Singular examples: det(±1±a) det(±1±b) Also fact that: det(x)det(y) det(xy), which means if factor X has det(x), then product (XY) also has det(xy). In G 2 : det(1±a)det(1±b) det(1±a±b±ab) X 1 * X ( ) 1 det( X ) 8/15/22 DJM T

15 ANPA 22: Quantum Geometric Algebra Row Decode Operators R k are Singular Row k a a1 ( 1)(1 a) ( 1)(1 a) ( 1)(1 a1) ( 1)(1 a1) R R 1 R 2 R 3 Summation of R k A R R 1 A R 2 R 3 A1 R R 2 A1 R 1 R 3 Denoted as Vector [ ] [ ] [ ] [ ] Row k a a1 (1 a)(1 a1) (1 a)(1a1) (1a)(1 a1) (1a)(1a1) R R 1 R 2 R 3 State logic R A A1 R 1 A A1 R 2 A A1 R 3 A A1 Denoted as Vector R [ ] R 1 [ ] R 2 [ ] R 3 [ ] Standard Algebraic Notation Dual Vector Notation: matrix diagonal R R 1 R 2 R 3 [ ] 1 R k are topologically smallest elements in G 2 and are linearly independent 8/15/22 DJM

16 ANPA 22: Quantum Geometric Algebra Measurement and Sparse Invariants Start States A A a a1 A 1 a a1 A a a1 A a a1 Each start state A times each R k gives the answer A(1a)(1 a1) A(1 a)(1a1) 1 a1 I 1 a1 I 1 a1 I 1 a1 I a ( 1 a1) a (1 a1) a (1 a1) a ( 1 a1) A(1a)(1a1) A(1 a)(1 a1) a (1 a1) a ( 1 a1) a ( 1 a1) a (1 a1) 1 a1 I 1 a1 I 1 a1 I 1 a1 I End State A > a a1 A > a a1 A > a a1 A > a a1 Description Classical States Measurement Superposition States Measurement I ~ 1 I ~ 1 I I (I ± ) 2 I 1 a1 [ ] I 1 a1 [ ] I 1 a1 [ ] I 9 1 a1 [ ] I 9 8/15/22 DJM

17 ANPA 22: Quantum Geometric Algebra Projection Operators P k and Eigenvectors E k Primary Tetrahedron (k 3) k E k R k 1 P k R k R k 1E k [ ] [ ] [ ] 1 [ ] [ ] [ ] 2 [ ] [ ] [ ] 3 [ ] [ ] [ ] sum [ ] [ ] [ ] a1 E E 6 a1 k sum Dual Tetrahedron (7 k) E k R k 1 [ ] [ ] [ ] [ ] [ ] a1 P k R k [ ] [ ] [ ] [ ] [ ] R k 1E k [ ] [ ] [ ] [ ] [ ] E E 6 a1 R k P k E k 2 1 E k R k R k P k 2 P k Idempotent!! E 2 E 4 E 2 E E3 a E 5 a a E 4 E 5 a a a1 E 1 E 7 a a1 a a1 E 1 E 7 a a1 E k ± a ± a1 ± a a1 PiP3 PP 1i 2 PiP4 P6iP5 7 8/15/22 DJM

18 ANPA 22: Quantum Geometric Algebra Qubits form Quantum Register Q q with A (±a ±a1), B (±b ±b1), C (±c ±c1) then A B C (±a ±a1)(±b ±b1)(±c ±c1) so A B (a a1)(b b1) a b a b1 a1 b a1 b1 Geometric product replaces the tensor product Row k State Combinations Individual bivector products Column Vector a a1 b b1 a b a b1 a1 b a1 b1 A B A B R R 3 R 5 R 6 R 9 R 1 R 12 R 15 Q q G n2q State Count: Total: 2 2q 4 q Non-zero: 2 q Zeros: 4 q 2 q A B C A 1 B 1 P A P B a1 b1 S 11 8/15/22 DJM

19 ANPA 22: Quantum Geometric Algebra Ebits: Bell/magic States and Operators Separable: A B (S A )(S B ) A (S A ) B (S B ) A B Non-Separable: A B (S A S B ) A B A B a b a b1 a1 b a1 b1 a b a1 b1 S S 11 B Row k State Combinations a a1 b b1 Individual bivectors a b a1 b1 Concurrent! Output column R 1 R 2 R 4 R 7 R 8 R 11 R 13 R 14 Valid states where exactly one qubit in superposition phase!! B (S A S B ) B i±1 ±B i B B S S 11 B 1 S 1 S 1 B 2 S S 11 B 3 S 1 S 1 Φ Ψ Φ Ψ M (S A S B ) M i±1 ±M i M M S 1 S 1 M 1 S S 11 M 2 S 1 S 1 M 3 S S 11 M 3 B 2 (S 1 S 1 ) B & M are 8/15/22 Singular! DJM

20 ANPA 22: Quantum Geometric Algebra Interesting Facts about Ebits B 2 I and M 2 I and B B I - 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, so A BB B i1 Φ S 11 A /1 B /1 P AB B M 3 S 1 A /1 B ± P AB M 2 B 1 Ψ S A ± B ± P AB S 11 S M 1 B 2 Φ S 1 A ± B /1 P AB Ψ B 3 B and M are valid for Q q>2 as (S A ± S B ± S C ± ) S 1 S 1 M 8/15/22 DJM

21 ANPA 22: Quantum Geometric Algebra Cnot, Cspin and Toffoli Operators For Q 2 with qubits A and B, where A is the control: CNot AB A (a a1) where (A ) 2 1 Cspin AB ( 1 A ) ( 1 a a1) CNot For Q 3 : qubits A, B & D where A & B are controls: Tof AB CNot AD CNot BD A 1 B (concurrent!) Row k R 42 a a1 b b1 where (Tof AB ) 2 1 a R 21 A 1 B 1 & D 1 R 22 A 1 B 1 & D R 41 A B & D 1 State Combinations a1 b b1 d d1 Active States A B & D A B D (TOF AB ) Inverted Identity Also for Q q P k 2q P k E kx 1 q x 2 6 8??? 8/15/22 DJM

22 ANPA 22: Quantum Geometric Algebra 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. 8/15/22 DJM