From 6c1f8e7ba66020052c6250fd4e58e0313989e2e0 Mon Sep 17 00:00:00 2001
From: Guillermo Alonso-Linaje <65235481+KetpuntoG@users.noreply.github.com>
Date: Wed, 6 Sep 2023 10:36:05 -0400
Subject: [PATCH] Fixing tutorial_apply_qsvt (#912)

This is a PR to solve [this
issue](https://github.com/PennyLaneAI/qml/issues/910).

---------

Co-authored-by: Jay Soni <jbsoni@uwaterloo.ca>
---
 demonstrations/tutorial_apply_qsvt.metadata.json |  2 +-
 demonstrations/tutorial_apply_qsvt.py            | 12 +++++++-----
 2 files changed, 8 insertions(+), 6 deletions(-)

diff --git a/demonstrations/tutorial_apply_qsvt.metadata.json b/demonstrations/tutorial_apply_qsvt.metadata.json
index ee9e3abcc8..76ed22a7f5 100644
--- a/demonstrations/tutorial_apply_qsvt.metadata.json
+++ b/demonstrations/tutorial_apply_qsvt.metadata.json
@@ -9,7 +9,7 @@
         }
     ],
     "dateOfPublication": "2023-08-22T00:00:00+00:00",
-    "dateOfLastModification": "2023-08-22T00:00:00+00:00",
+    "dateOfLastModification": "2023-09-05T00:00:00+00:00",
     "categories": [
         "Quantum Computing",
         "Algorithms",
diff --git a/demonstrations/tutorial_apply_qsvt.py b/demonstrations/tutorial_apply_qsvt.py
index bcba41884f..86b2381e9f 100644
--- a/demonstrations/tutorial_apply_qsvt.py
+++ b/demonstrations/tutorial_apply_qsvt.py
@@ -288,16 +288,17 @@ def loss_func(phi):
 #
 # .. math::
 #
-#    \hat{U}_{real}(\vec{\phi}) = \frac{1}{2} \ ( \hat{U}_{qsvt}(\vec{\phi}) + \hat{U}^{\dagger}_{qsvt}(\vec{\phi}) ).
+#    \hat{U}_{real}(\vec{\phi}) = \frac{1}{2} \ ( \hat{U}_{qsvt}(\vec{\phi}) + \hat{U}^{*}_{qsvt}(\vec{\phi}) ).
 #
-# Here we use a two-term LCU to define the quantum function for this operator:
+# Here we use a two-term LCU to define the quantum function for this operator. We obtain the complex
+# conjugate of :math:`\hat{U}_{qsvt}` by taking the adjoint of the operator block-encoding :math:`A^{T}`:
 
 
 def real_u(A, phi):
     qml.Hadamard(wires="ancilla1")
 
     qml.ctrl(sum_even_odd_circ, control=("ancilla1",), control_values=(0,))(A, phi, "ancilla2", [0, 1, 2])
-    qml.ctrl(qml.adjoint(sum_even_odd_circ), control=("ancilla1",), control_values=(1,))(A, phi, "ancilla2", [0, 1, 2])
+    qml.ctrl(qml.adjoint(sum_even_odd_circ), control=("ancilla1",), control_values=(1,))(A.T, phi, "ancilla2", [0, 1, 2])
 
     qml.Hadamard(wires="ancilla1")
 
@@ -306,8 +307,9 @@ def real_u(A, phi):
 #
 # Solving a Linear System with QSVT
 # ---------------------------------
-# Our goal is to solve the equation :math:`A \cdot \vec{x} = \vec{b}`. Let's begin by
-# defining the specific matrix :math:`A` and vector :math:`\vec{b}` :
+# Our goal is to solve the equation :math:`A \cdot \vec{x} = \vec{b}`. This method assumes
+# the matrix we will invert is hermitian. Let's begin by defining the specific matrix :math:`A`
+# and vector :math:`\vec{b}` :
 #
 
 A = np.array(