From 0c83cee10d079f5e1cd926bcf1e89b9afda7d7c3 Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Sun, 24 Apr 2022 14:52:36 -0400
Subject: [PATCH 1/7] Decomposition - symmetric and anti-symmetric

Decompose a quadratic equation into symmetric, antisymmetric parts using its transpose.
---
 everything_else/reproducing.m | 28 ++++++++++++++++++++++++++++
 1 file changed, 28 insertions(+)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index 8b03efc..7e6b415 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -349,6 +349,34 @@
 %% Pg. 55, SCM-BCR
 %% Pg. 56, Decomposition into symmetric and anti-symmetric parts
 
+% Anh Mai - Attempt for Topic: "Decomposition into symmetric and anti-symmetric parts"
+
+% Credit: Eric Huang: reproducing.m, 
+% Dr: Dattani:Book_About_Quadratization,
+% Kahl and Strandmark, "Generalized Roof Duality for Pseudo-Boolean Optimization,” International Conference on Computer Vision (2011).
+
+% Idea: Decompose a quadratic equation into symmetric and skew-symmetric parts. 
+% For example, A is matrix. 
+% Symmetric: A = Transpose(A).
+% Skew-symmetric: - A = Transpose(A)
+
+%Setting up the coefficients using Eric Huang's setup
+b = dec2bin(2^4-1:-1:0)-'0';
+b1 = b(:,1)
+b2 = b(:,2)
+b3 = b(:,3)
+b4 = b(:,4)
+
+% Take quadratic function as an example (Eq. 2 from the book)
+% f = f_sym + f_anti
+f=b1.*b2 + b2.*b3 + b3.*b4 - 4*b1.*b2.*b3;
+
+% f_sym = (1/2)*(f(b) + transpose f(b))
+f_sym = (1/2)*(f + f.')
+
+% f_anti = (1/2)*(f(b) - transpose f(b))
+f_anti = (1/2)*(f - f.')
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 QUANTUM GADGETS
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From ffe08617aab89af4c54f226fa37d599c11d33ea5 Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Sun, 24 Apr 2022 15:27:06 -0400
Subject: [PATCH 2/7] Update - Decomposition

Removed comments, all coefficients defined in one line.
---
 everything_else/reproducing.m | 22 +---------------------
 1 file changed, 1 insertion(+), 21 deletions(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index 7e6b415..6208a78 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -349,32 +349,12 @@
 %% Pg. 55, SCM-BCR
 %% Pg. 56, Decomposition into symmetric and anti-symmetric parts
 
-% Anh Mai - Attempt for Topic: "Decomposition into symmetric and anti-symmetric parts"
-
-% Credit: Eric Huang: reproducing.m, 
-% Dr: Dattani:Book_About_Quadratization,
-% Kahl and Strandmark, "Generalized Roof Duality for Pseudo-Boolean Optimization,” International Conference on Computer Vision (2011).
-
-% Idea: Decompose a quadratic equation into symmetric and skew-symmetric parts. 
-% For example, A is matrix. 
-% Symmetric: A = Transpose(A).
-% Skew-symmetric: - A = Transpose(A)
-
-%Setting up the coefficients using Eric Huang's setup
 b = dec2bin(2^4-1:-1:0)-'0';
-b1 = b(:,1)
-b2 = b(:,2)
-b3 = b(:,3)
-b4 = b(:,4)
+b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4)
 
-% Take quadratic function as an example (Eq. 2 from the book)
-% f = f_sym + f_anti
 f=b1.*b2 + b2.*b3 + b3.*b4 - 4*b1.*b2.*b3;
 
-% f_sym = (1/2)*(f(b) + transpose f(b))
 f_sym = (1/2)*(f + f.')
-
-% f_anti = (1/2)*(f(b) - transpose f(b))
 f_anti = (1/2)*(f - f.')
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From 571cb0faa44d0045d5919e4b98e01c6b3300cbe7 Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Sun, 24 Apr 2022 15:29:05 -0400
Subject: [PATCH 3/7] Update reproducing.m

---
 everything_else/reproducing.m | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index 6208a78..df143f6 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -350,12 +350,12 @@
 %% Pg. 56, Decomposition into symmetric and anti-symmetric parts
 
 b = dec2bin(2^4-1:-1:0)-'0';
-b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4)
+b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4);
 
 f=b1.*b2 + b2.*b3 + b3.*b4 - 4*b1.*b2.*b3;
 
-f_sym = (1/2)*(f + f.')
-f_anti = (1/2)*(f - f.')
+f_sym = (1/2)*(f + f.');
+f_anti = (1/2)*(f - f.');
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 QUANTUM GADGETS

From dd25ba832dc50a6c89eac0043fb24e77cb7f1874 Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Wed, 27 Apr 2022 13:43:48 -0400
Subject: [PATCH 4/7] Update - Apr 27 - Symm Decomp

Updated new function, coefficients and calculation for symmetry and antisymmetry
---
 everything_else/reproducing.m | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index df143f6..d9cee0b 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -352,10 +352,10 @@
 b = dec2bin(2^4-1:-1:0)-'0';
 b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4);
 
-f=b1.*b2 + b2.*b3 + b3.*b4 - 4*b1.*b2.*b3;
+f = b1 + 2*b1.*b2 - 4*b1.*b2.*b3 + 2*b1.*b2.*b3.*b4;
 
-f_sym = (1/2)*(f + f.');
-f_anti = (1/2)*(f - f.');
+f_sym = (1/2)*(f + (1 - f));
+f_anti = (1/2)*(f - (1 - f));
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 QUANTUM GADGETS

From 94437b0daa70fde11dfb601be5366e89cacb15df Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Fri, 29 Apr 2022 12:01:48 -0400
Subject: [PATCH 5/7] Update - Decomp - Include checks for accuracy

---
 everything_else/reproducing.m | 7 ++++++-
 1 file changed, 6 insertions(+), 1 deletion(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index d9cee0b..33221f2 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -353,10 +353,15 @@
 b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4);
 
 f = b1 + 2*b1.*b2 - 4*b1.*b2.*b3 + 2*b1.*b2.*b3.*b4;
-
 f_sym = (1/2)*(f + (1 - f));
 f_anti = (1/2)*(f - (1 - f));
 
+f_new = f_sym + f_anti;
+
+LHS=min(reshape(f_new,[]));
+RHS=min(reshape(f,[]));
+isequal(LHS,RHS);
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 QUANTUM GADGETS
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From 87d93fcecab1b58d2dce052b59104489eace870c Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Fri, 29 Apr 2022 17:21:45 -0400
Subject: [PATCH 6/7] Update - Decomp - Reshape

Corrected reshape functions.
---
 everything_else/reproducing.m | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index 33221f2..383eef1 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -353,13 +353,14 @@
 b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4);
 
 f = b1 + 2*b1.*b2 - 4*b1.*b2.*b3 + 2*b1.*b2.*b3.*b4;
+
 f_sym = (1/2)*(f + (1 - f));
 f_anti = (1/2)*(f - (1 - f));
 
 f_new = f_sym + f_anti;
 
-LHS=min(reshape(f_new,[]));
-RHS=min(reshape(f,[]));
+LHS=min(reshape(f_new,[],4));
+RHS=min(reshape(f,[],4));
 isequal(LHS,RHS);
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From adb37005d5ab3b8a7fcb2eeaaf7b2fcc3efed390 Mon Sep 17 00:00:00 2001
From: AnhMai-bit <71567193+AnhMai-bit@users.noreply.github.com>
Date: Wed, 4 May 2022 23:42:12 -0400
Subject: [PATCH 7/7] Decom - Update - May 4

Updated decomposition method, used NTR - KZFD and SFR-BCR-1.
---
 everything_else/reproducing.m | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/everything_else/reproducing.m b/everything_else/reproducing.m
index 383eef1..8088a99 100644
--- a/everything_else/reproducing.m
+++ b/everything_else/reproducing.m
@@ -349,19 +349,19 @@
 %% Pg. 55, SCM-BCR
 %% Pg. 56, Decomposition into symmetric and anti-symmetric parts
 
-b = dec2bin(2^4-1:-1:0)-'0';
-b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4);
+b = dec2bin(2^8-1:-1:0)-'0';
+b1 = b(:,1); b2 = b(:,2); b3 = b(:,3); b4 = b(:,4); ba = b(:,5); ba1 = b(:,6); ba2 = b(:,7); ba3 = b(:,8);
 
 f = b1 + 2*b1.*b2 - 4*b1.*b2.*b3 + 2*b1.*b2.*b3.*b4;
 
-f_sym = (1/2)*(f + (1 - f));
-f_anti = (1/2)*(f - (1 - f));
+f_new_symm = 2*(-5 + b1 + b2 + b3 + b4 - ba1 - 2*ba2 + 5*ba3).^2;
+f_new_anti = b1 + 2*b1.*b2 + 4*(2 - b1 - b2 - b3).*ba;
 
-f_new = f_sym + f_anti;
+f_new = f_new_anti + f_new_symm;
 
-LHS=min(reshape(f_new,[],4));
-RHS=min(reshape(f,[],4));
-isequal(LHS,RHS);
+LHS=min(reshape(f_new,16,[]))
+RHS=min(reshape(f,16,[]))
+isequal(LHS,RHS)
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 QUANTUM GADGETS