Transcript PhD Topic Proposal for Department of Electrical

ANPA 2002: Quantum Geometric Algebra
Quantum Geometric Algebra
ANPA Conference
Cambridge, UK
by Dr. Douglas J. Matzke
[email protected]
Aug 15-18, 2002
8/15/2002 DJM
ANPA 2002: 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.
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Overview of Presentation
• Co-Occurrence and Co-Exclusion
• Geometric Algebra Gn 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
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Co-Occurrence and Co-Exclusion
a+b=b+a
c-d
d-c
c-d|d-c
Abstract Space
a = +a = ON and
a = –a = not ON
where a + a = 0
c-d+d-c=0
(or can not occur)
(0 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
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Boolean Logic using +/* in Gn
+
0
1
–1
0
0
1
–1
1
1
–1
0
–1
–1
0
1
*
0
1
–1
0
0
0
0
1
0
1
–1
–1
0
–1
1
Normal multiplication and mod 3 addition
for ring {–1,0,1}, so can simplify to {–,0,+}
and remove rows/columns for header value 0.
+1
0
Z3
–1
Binary
Values
+
+
–
+
–
0
–
0
+
If same then invert
If d iff then cancel
*
+
–
+
+
–
–
–
+
If same then +1
If d iff then -1
+ NAND + => – same XNOR same => +
– NOR – => + differ XNOR differ => –
Logic inG2 = span{a, b}
GA Mapping {+, –}
GA Mapping {+, 0}
Identity a
a*1 =a+0=a
–1 – a = – (1 + a)
NOT a
a * –1 = – a
–1 + a = – (1 – a)
a XOR b
–ab
–1 + a b
a OR b
a+b–ab
–1 – a – b + a b
a AND b
+1 – a – b – a b
+1 + a + b + a b
Also for any vector e:
since e2=1 then e = 1/e Geometric Algebra is Boolean Complete
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Geometric Algebra Essentials
a
b
a
b
a  b  b  a
a b  a b  a  b where geometric product is sum
a b  cos
of inner product (is a scalar)
a  b  i sin  and outer product (is a bivector)
Gn=2 generates N=2n: span{a, b}
G2 = scalars {±1}, vectors {a, b}, and bivector {a b} then:
+b
With a b = 0 (only orthonormal basis so are perpendicular)
then a b = – b a
(due to anti-commutative outer product)
2
2
a =b =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
orientation
–a b
–a
+a
+a b
orientation
so a b  1  NOT
also x' = RxR with R     a b, R     a b,  cos( / 2),   sin( / 2)
–b
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Number of Elements in Gn
Graded: scalar, vector, bivector, trivector, …, n-vector for Gn with N=2n elements
(1+a)(1+b)(1+c) = 1 + a + b + c + a b + a c + b c + a b c <multivector
Gn = Gn+ + Gn– = <A>0 +
–
Odd grade terms Gn =
<A>1 + <A>3 + …
Even Subalgebra Gn+ =
<A>0 + <A>2 + …
G3+ are the quaternions:
1+ab+ac+bc
<A>1
Row  n
0
1
2
3
4
5
6
+
<A>2
+ <A>3 + … + <A>n
n
1      N  2n
m 1  m 
1
1
1 1
2
1 2 1
4
1 3 3 1
8
1 4 6 4 1
 16
1 5 10 10 5 1
 32
1 6 15 20 15 6 1
 64
Col  m
Pascal’s Triangle
n
(Binomial)
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Inner Product Calculation
Y = (x  y) and Z = (Y  z) with vector variables w, x=a, y=b, z=c
G2 = span{a,b}: w Y = w (a  b) = (w a)  b - (w b)  a
G3 = span{a,b,c}: w Z = (w a)  b  c  (w b)  a  c + (w c)  a  b
Only one non-zero term in sum for orthogonal basis set {a,b,c}
Outer Product
X Y
X
Y
+1
a
b
ab
+1
+1
a
b
ab
a
a
0
ab
0
b
b
–a b
0
ab
ab
0
0
Inner Product
X Y
Y
+1
a
b
ab
+1
0
0
0
0
a
0
+1
0
b
0
b
0
0
+1
–a
0
ab
0
b
–a
–1
X
XY  X Y  X  Y only if X or Y are assigned vector x or y
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Qubit is Co-occurrence in G2
Single Qubit:
A1
A = (±a0 ±a1)
A+ Superposition:
signs are same
–a0
where Q1 = G2 = span{a0, a1}
4 elements & 34 = 81 multivectors
A1 =
+a1
A0 =
A–
+a0
–a1
Classical: signs
A0 are opposite
A+ = a0 + a1 A – = a0 + a1
Row k
a0
a1
R0
–
–
0
0
+
–
R1
–
+
+
–
0
0
R2
+
–
–
+
0
0
R3
+
+
0
0
–
+
Binary combinations
of input states
a0 + a1
a0 + a1
Anti-symmetric sums
are classical states
A1 = R1 – R2
A0 = R2 – R1
Symmetric sums are
superposition states
A+ = R0 – R3
A– = R3 – R0
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Spinor is Hadamard Operator
Start Phase Qubit State A Each Times Spinor Result = A SA End Phase
Classical
Superposed
A0 = +a0 – a1
+a0 (a0 a1) = +a1
A+ = +a0 + a1
A1 = –a0 + a1
–a0 (a0 a1) = –a1
A– = –a0 – a1
A+ = +a0 + a1
+a1 (a0 a1) = –a0
A1 = –a0 + a1
A – = –a0 – a1
–a1 (a0 a1) = +a0
A0 = +a0 – a1
Superposed
Classical
Hadamard is the 90° phase or spinor operator SA= (a0 a1)
NOT operator is 180° gate SA2 = (a0 a1)(a0 a1) = –a0 a0 a1 a1 = –1
p
Therefore S A  1  NOT and generally r    / r and   p
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Unitary Pauli Noise States in G
Flip:
Bit
0
1

1 0
1  
Phase
1
3  
0
0

1
Both
i 

 i 0 
0
2  
Case Hilbert notation
Use case
2
GA equivalent is (–1)= complement
[a]
1 0  1
[a]
(+ a0 – a1)(–1)  (– a0 + a1)
[b]
1 1  0
[b]
(– a0 + a1)(–1)  (+ a0 – a1)
Case
Hilbert notation
Use cases
GA equivalent is spinor SA = a0 a1
[a]
3 1   1
[a]&[b]
(– a0 + a1)(– a0 a1)  (+ a0 – a1)
[b]
3 0  0
[c]
 3 1  1
[b]&[c]
(+ a0 – a1 )(a0 a1)  (+ a0 + a1)
Case
Hilbert notation
[a]
 2 0  i 0
[b]
 2 1  i 1
Use cases GA equivalent is (–1 + SA) = PA
[a]&[b]
(+ a0 – a1)(–1 + a0 a1)  –a1
Pauli operators -1, SA and PA are even grade!
+
=
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Reversible Basis Encodings:
Standard, Dual, Pauli and Circular basis
Label for Row
Start State
Diag (–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 Basis
Diagonals
Pauli = Ver/Hor
Circular
Direct or Complex
V/Hor (1 + a0 a1)
Cir (a0)
Dir (– a0 + a1)
Reversible op. return to start
Left Diagonal
Classical=
+a1
–a0
+a0 a1
Right Diagonal
=Superposition
+a0
Circular
Ellipses are
co-exclusions!
–1
+1
ODD
GRADE
EVEN
GRADE
Pauli
Vertical
–a1
Pauli
Horizontal
–a0 a1
Direct
8/15/2002 DJM
ANPA 2002: 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=XY/Y=X
For unitary Y then requires det(Y)=±1 or |det(Y)| = 1
A0 A1 = 1
A– A+ = 1
Trines are unitary: (Tr)3 = 1 so 1/Tr = (Tr)2
for Tr = (+1 ± a0 ± SA) or (+1 ± a1 ± SA)
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Singular Operators in Gn
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.
X 1  ( X * )T
1

det( X )
In G2: det(1±a)det(1±b) = det(1±a±b±a b) = 0
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Row Decode Operators Rk are Singular
Row k
a0
a1
(–1)(1 – a0)
(–1)(1 + a0)
(–1)(1 – a1)
(–1)(1 + a1)
R0
–
–
+
0
+
0
R1
–
+
+
0
0
+
R2
+
–
0
+
+
0
R3
+
+
0
+
0
+
Summation of Rk 
Denoted as Vector 
A0– = R0 + R1 A0+ = R2 + R3 A1– = R0 + R2 A1+ = R1 + R3
[+ + 0 0]
[0 0 + +]
[+ 0 + 0]
[0 + 0 +]
Row k
a0
a1
(1–a0)(1–a1)
(1–a0)(1+a1)
R0
–
–
+
0
0
0
R1
–
+
0
+
0
0
R2
+
–
0
0
+
0
R3
+
+
0
0
0
+
R0 = A0– A1–
R1 = A0– A1+
R2 = A0+ A1–
R3 = A0+ A1+
State logic 
Denoted as Vector 
(1+a0)(1–a1) (1+a0)(1+a1)
Standard
Algebraic
Notation
Dual
Vector
Notation:
matrix
diagonal
R0 + R1 +
R2 + R3 =
[+ + + +] = 1
R0 = [+ 0 0 0] R1 = [0 + 0 0] R2 = [0 0 + 0] R3 = [0 0 0 +]
Rk are topologically smallest elements in G2 and are linearly independent
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Measurement and Sparse Invariants
Each start state A times each Rk gives the answer
Start States A
A(1+a0)(1–a1)
A(1–a0)(1+a1)
A(1+a0)(1+a1)
A(1–a0)(1–a1)
A0 = + a0 – a1
–1 + a1 = I
+
+1 + a1 = I
–
–a0 (+1 + a1)
+a0 (–1 + a1)
A1 = – 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
–
A’ => + a0 – a1
A’ => – a0 + a1
End State 
Description 
Classical States Measurement
I + ~ +1
I – ~ –1
–1 + a1 = [+ 0 + 0] = I
–1 – a1 = [0 + 0 +] = I
+0
+90
A’ => + a0 + a1
A’ => – a0 – a1
Superposition States Measurement
I – = –I
+
(I ± )2 = I
+
+1 – a1 = [– 0 – 0] = I –0
+1 + a1 = [0 – 0 –] = I –90
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Projection Operators Pk and Eigenvectors Ek
Primary Tetrahedron (k=0–3)
Rk = –Pk
Dual Tetrahedron (=7–k)
k=
Ek = Rk–1
Pk = –Rk
Rk = 1+Ek
k=
Ek = Rk–1
Pk = –Rk
Rk = 1+Ek
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]
[– – – –]
Pk2 = Pk
[+ + + +]
Idempotent!!
+a1
E2
E0
000
000
+++
+++
+++
+a0
---
+a0
+a0
E5
E7
+a0 a1
+a1
+++
E3
+a0 a1
E1
Ek Rk = Rk
E6
E2
E4
---
E1
E0
E6
E
---3
+a0 a1
+a1
+a1
Ek2 = 1
---
E4
+a0
E5
E7
+a0 a1
Ek = ± a0 ± a1 ± a0 a1 P0 P3  P1 P2  P7 P4  P6 P5  0
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Qubits form Quantum Register Qq
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 
State Combinations
Row k
Individual bivector products
Column Vector
a0
a1
b0
b1
a0 b0
a0 b1
a1 b0
a1 b1
A+ B+
A0 B0
R0
–
–
–
–
+
+
+
+
+
0
R3
–
–
+
+
–
–
–
–
–
0
R5
–
+
–
+
+
–
–
+
0
–
R6
–
+
+
–
–
+
+
–
0
+
R9
+
–
–
+
–
+
+
–
0
+
R10
+
–
+
–
+
–
–
+
0
–
R12
+
+
–
–
–
–
–
–
–
0
R15
+
+
+
+
+
+
+
+
+
0
Qq =Gn=2q
State Count:
Total: 22q = 4q
Non-zero: 2q
Zeros: 4q – 2q
ABC=0
A1B1PAPB =
a1 b1 = S11
8/15/2002 DJM
ANPA 2002: Quantum Geometric Algebra
Ebits: Bell/magic States and Operators
Separable: A0 B0 (SA)(SB) = A0 (SA) B0 (SB) = A+ B+
Concurrent!
Non-Separable: A0 B0 (SA+ SB) = A+ B0 + A0 B+ =
= –a0 b0 + 0 a0 b1 + 0 a1 b0 + a1 b1
= –a0 b0 + a1 b1 = S00+ S11 = B 0
State Combinations
Individual bivectors
a0
a1
b0
b1
–a0 b0
a1 b1
Output
column
R1
–
–
–
+
–
–
+
R2
–
–
+
–
+
+
–
R4
–
+
–
–
–
–
+
R7
–
+
+
+
+
+
–
R8
+
–
–
–
+
+
–
R11
+
–
+
+
–
–
+
R13
+
+
–
+
+
+
–
R14
+
+
+
–
–
–
+
Row k
Valid states where exactly one qubit in superposition phase!!
B = (SA+ SB)
B i±1 = ±B i B
B 0 = –S00 + S11 =  

B 1 = +S01 + S10 = 

B 2 = +S00 – S11 = 

B 3 = –S01 – S10 = 
M = (SA – SB)
M i±1 = ±Mi M
M0 = +S01 – S10
M1 = –S00 – S11
M 2 = –S01 + S10
M 3 = +S00 + S11
M3 = B2 (S01 + S10)
8/15/2002
DJM
B & M are
Singular!
ANPA 2002: Quantum Geometric Algebra
Interesting Facts about Ebits
B 2 = I – and M 2 = I
and
–
B  B I-
–PA PB = B – (1+ SA SB) = B + I
 =B0 =

+S11
–S00
M1 =
A0/1 B0/1 PAB
=M3
+S00
A± B± PAB
+
A B = B i + M j but
B i M = M i B = 0,
so A B B = B i+1 ++00
M2 =
+S10
–S01

–S11
=B1= 
+S01
A± B0/1 PAB
–

=B2= 
=B3=
A0/1 B± PAB
–S10
B and M are valid for Qq>2 as (SA ± SB ± SC ± …)
=M0
8/15/2002 DJM

ANPA 2002: Quantum Geometric Algebra
Cnot, Cspin and Toffoli Operators
For Q2 with qubits A and B, where A is the control:
CNotAB = A0 = (a0 – a1) where (A0)2 = –1
CspinAB = CNot = (–1 + A0) = (–1 + a0 – a1)
For Q3: qubits A, B & D where A & B are controls:
TofAB = CNotAD + CNotBD = A1 + B0 (concurrent!)
= – a0 + a1 + b0 – b1 where (TofAB)2 = 1
State Combinations
a0
a1
b0
b1
d0
d1
Active
States
R21
–
+
–
+
–
+
A1 B1 & D1
–
R22
–
+
–
+
+
–
A1 B1 & D0
+
R41
+
–
+
–
–
+
A0 B0 & D1
+
R42
+
–
+
–
+
–
A0 B0 & D0
–
Rowk
A0 B0 D0 (TOFAB)
Inverted
Also for Qq
Pk2q = Pk
Ekx = 1
q
x
1
2
2
6
3
4
80
???
Identity
8/15/2002 DJM
ANPA 2002: 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.
8/15/2002 DJM