Fixed typos found by Megler, fancyfied index.

Author: Scinawa

index.Rmd

@@ -54,7 +54,7 @@ github-repo: "scinawa/quantumalgorithms.org"
 
 This is an [open source project accessible on GitHub](https://github.com/Scinawa/quantumalgorithms.org/) - only possible thanks to its many [contributors.](contributions-and-acknowledgements.html) The website is licensed under CC BY-NC-SA 4.0. We are searching for talented people and researchers to contribute: you can find a list of issues and enhancements that would improve this book further on the [issues list](https://github.com/Scinawa/quantumalgorithms.org/issues) for the GitHub repo, making recent material that keeps popping up in quantum information accessible to a larger audience.
 
-
+**MEMO: we have some funding for motivated contributors**
 
 The aim of this book is twofold:
 
@@ -132,18 +132,19 @@ https://stackoverflow.com/questions/41084020/add-a-html-block-above-each-chapter
 
 ## Changelog
 
-- August 2020: Migrated the old blog on bookdown
-- December 2020: Moved thesis in bookdown
-- January 2021: First system for typesetting algorithms, more appendix on linear algebra
-- February 2021: New subroutines for estimating $\ell_1$ norms.
-- March 2021: quantumalgorithms.org is proudly supported by the [Unitary Fund](https://unitary.fund/), and quantumalgorithms.org is a project of the [QOSF](https://qosf.org) mentorship program: 5 students started creating new content!
-- April 2021: Mobile version working, search functionality added, q-means, finding the minimum, new algo for dimensionality reduction, and factor score ratio estimation estimation.
-- June 2021: Quantum Monte Carlo algorithms, lower bounds techniques for query complexity of quantum algorithms, quantum algorithms for graph problems. The output of the mentorship program of the QOSF foundation!
-- January 2022: In the past months we improved the overall quality of the website (graphics, typos) and added some content (Deutsch-Josza, Bernstein-Vazirani, swap and Hadamard tests)
-- February 2022: We are happy to bring our swag (t-shirts, stickers, lanyards) to QIP. We added a whole chapter on perceptron algorithms and we doubled the chapter on quantum monte carlo algorithms with applications to finance and trace distance!
-- June 2022: We are partecipating to the UnitaryHack! We also presented this work at QNLP2022 where we distributed our swag! Now the website can be compiled also with the latest version of RStudio,pandoc,bookdown.
-- August 2022: Added in the appendix: polynomial approximation of 1/x, more on concentration inequalities (these were work started by contributors that discovered the project during the unitary.hack! ). Improved css as a bounty to the unitary.hack. Working (on a separate project) to improve chapter 3. 
-
+- **August 2020**: Migrated the old blog on bookdown
+- **December 2020**: Moved thesis in bookdown
+- **January 2021**: First system for typesetting algorithms, more appendix on linear algebra
+- **February 2021**: New subroutines for estimating $\ell_1$ norms.
+- **March 2021**: quantumalgorithms.org is proudly supported by the [Unitary Fund](https://unitary.fund/), and quantumalgorithms.org is a project of the [QOSF](https://qosf.org) mentorship program: 5 students started creating new content!
+- **April 2021**: Mobile version working, search functionality added, q-means, finding the minimum, new algo for dimensionality reduction, and factor score ratio estimation estimation.
+- **June 2021**: Quantum Monte Carlo algorithms, lower bounds techniques for query complexity of quantum algorithms, quantum algorithms for graph problems. The output of the mentorship program of the QOSF foundation!
+- **January 2022**: In the past months we improved the overall quality of the website (graphics, typos) and added some content (Deutsch-Josza, Bernstein-Vazirani, swap and Hadamard tests)
+- **February 2022**: We are happy to bring our swag (t-shirts, stickers, lanyards) to QIP. We added a whole chapter on perceptron algorithms and we doubled the chapter on quantum monte carlo algorithms with applications to finance and trace distance!
+- **June 2022**: We are partecipating to the UnitaryHack! We also presented this work at QNLP2022 where we distributed our swag! Now the website can be compiled also with the latest version of RStudio,pandoc,bookdown.
+- **August 2022**: Added in the appendix: polynomial approximation of 1/x, more on concentration inequalities (these were work started by contributors that discovered the project during the unitary.hack! ). Improved css as a bounty to the unitary.hack. Working (on a separate project) to improve chapter 3. 
+
+**NEW** Submit your solution of the exercises as PR on [GitHub](https://github.com/scinawa/quantumalgorithms.org) and get it published in the chapter on [Solutions](https://quantumalgorithms.org/solutions-to-exercises.html).
 
 
 

intro.Rmd

@@ -322,7 +322,7 @@ are [@dorn2008quantum], [@DeWolf], and [@kuperberg2011another].
 
 Le't s suppose that we have a quantum circuit of $n$ qubits, and for each qubit we have one Hadamard gate.
 What is the time complexity for running this circuit? Is it $O(n)$ or $O(1)$?. To answer this question we have to introduce discuss different ways for measuring the complexity of a quantum algorithm: the number of gates, the depth, and the number of queries to an oracle.
-The first two are a direct generalization of the measure of complexity of boolean circuits, where we measure the number of gates (serial time) or the depth of the circuit("parallel time"). The third measure of complexity is the query complexity, which is the number of times we use (or query) an oracle: a circuit or a data structure that we can use as a black box. In this section we will define more formally the query complexity of a quantum algorithm, and in the next chapter we will explain possible implementations of different kind of oracles. The *time* complexity, which is ultimately the most interesting measure of complexity of an algorithm for most practical purposes, is more tricky to define properly. If we assume that our hardware is capable of executing all the gates at the same depth in parallel, than the measure of time complexity becomes the depth of the quantum algorithm. We denote with $T(U)$ the time complexity needed to implement a quantum circuit implementing a unitary $U$, and this is measured in terms of **number of gates**, i.e. the size of the circuit (this is quite standard see for example the section the introduction in [@ambainis2022matching]). This is a concept that bears some similarity with the clock rate of classical CPUs. If our hardware is not capable of executing all the gates at the same depth in one clock, we might have to consider the time complexity as the number of gates (eventually divided by the number of gates that can be executed at the same time). As an important exception to this rule we have the query complexity, where we consider the cost of a single query of an oracle as $O(1)$, i.e. as any other gate. This is somehow justified because for certain oracles we have efficient (i.e. with depth polylogarithmic in the size of the oracle) implementations. For this, the (often non said) assumption is that the part of the quantum computer dedicated to run the main part of the circuit has to be evaluated with serial time (i.e. the number of gates), while the part of the circuit dedicated to executing the oracle has to be evaluated for its parallel time complexity. An example of this, which we will treat in more details in the next chapter is the QRAM and the QRAG gate. This assumption allows to justifiably conflate the query complexity of an algorithm as its time complexity.
+The first two are a direct generalization of the measure of complexity of boolean circuits, where we measure the number of gates (serial time) or the depth of the circuit (parallel time). The third measure of complexity is the query complexity, which is the number of times we use (or query) an oracle: a circuit or a data structure that we can use as a black box. In this section we will define more formally the query complexity of a quantum algorithm, and in the next chapter we will explain possible implementations of different kind of oracles. The *time* complexity, which is ultimately the most interesting measure of complexity of an algorithm for most practical purposes, is more tricky to define properly. If we assume that our hardware is capable of executing all the gates at the same depth in parallel, than the measure of time complexity becomes the depth of the quantum algorithm. We denote with $T(U)$ the time complexity needed to implement a quantum circuit implementing a unitary $U$, and this is measured in terms of **number of gates**, i.e. the size of the circuit (this is quite standard see for example the section the introduction in [@ambainis2022matching]). This is a concept that bears some similarity with the clock rate of classical CPUs. If our hardware is not capable of executing all the gates at the same depth in one clock, we might have to consider the time complexity as the number of gates (eventually divided by the number of gates that can be executed at the same time). As an important exception to this rule we have the query complexity, where we consider the cost of a single query of an oracle as $O(1)$, i.e. as any other gate. This is somehow justified because for certain oracles we have efficient (i.e. with depth polylogarithmic in the size of the oracle) implementations. For this, the (often non said) assumption is that the part of the quantum computer dedicated to run the main part of the circuit has to be evaluated with serial time (i.e. the number of gates), while the part of the circuit dedicated to executing the oracle has to be evaluated for its parallel time complexity. An example of this, which we will treat in more details in the next chapter is the QRAM and the QRAG gate. This assumption allows to justifiably conflate the query complexity of an algorithm as its time complexity.
 
 Last but not least, everything is made even more complicated by error correction. In fact, in some architecture and some error correction codes, the time complexity of the algorithm is well approximated by the number of $T$ gates of the circuit.
 
@@ -637,12 +637,11 @@ of interested with a certain level of accuracy and with a certain
 probability.
 
 ::: {.theorem #modified-hadamard-test-no-amplification name="Modified Hadamard test (no amplitude amplification)"}
-Assume to have access to a unitary $U_1$ that produces a state $U_1 \ket{0} = \ket{\psi_1}$ and a unitary $U_2$ that produces a state $\ket{\psi_2}$, where $\ket{\psi_1},\ket{\psi_2} \in \mathbb{R}^N$ for $N=2^n, n\in\mathbb{N}$. There is a quantum algorithm that allows to estimate the quantity $\braket{\psi_1|\psi_2}$ with additive precision $\epsilon$ using controlled applications of $U_1$ and $U_2$ $O(\frac{\log(1/\delta)}{\epsilon_2})$ times, with probability $1-\delta$
+Assume to have access to a unitary $U_1$ that produces a state $U_1 \ket{0} = \ket{\psi_1}$ and a unitary $U_2$ that produces a state $\ket{\psi_2}$, where $\ket{\psi_1},\ket{\psi_2} \in \mathbb{R}^N$ for $N=2^n, n\in\mathbb{N}$. There is a quantum algorithm that allows to estimate the quantity $\braket{\psi_1|\psi_2}$ with additive precision $\epsilon$ using controlled applications of $U_1$ and $U_2$ $O(\frac{\log(1/\delta)}{\epsilon^2})$ times, with probability $1-\delta$
 :::
 
 ::: {.proof}
-Create a state $\ket{0}\ket{0}\ket{0}$ where the first register is just an ancilla qubit, and the second and third register have $n$ qubits. Then, apply an Hadamard gate to the first qubit, so to obtain $\ket{+}\ket{0}\ket{0}$. Then, controlled on the first register being $0$, we apply the unitary $U_1$, and controlled on the register being $1$, we apply the unitary $U_2$. Then, we apply again the Hadamard gate on the ancilla qubit. The state that we obtain is the following:
-
+Create a state $\ket{0}\ket{0}$ where the first register is just an ancilla qubit, and the second register has $n$ qubits. Then, apply an Hadamard gate to the first qubit, so to obtain $\ket{+}\ket{0}\ket{0}$. Then, controlled on the first register being $0$, we apply the unitary $U_1$, and controlled on the register being $1$, we apply the unitary $U_2$. Then, we apply again the Hadamard gate on the ancilla qubit. The state that we obtain is the following:
 
 \begin{align}
 (H\otimes I ) \frac{1}{\sqrt{2}}\left( \ket{0}\ket{\psi_1} + \ket{1}\ket{\psi_2} \right) \\
@@ -684,7 +683,7 @@ receive the states from a third-party, i.e. via a communication
 protocol.
 
 ::: {.theorem #swap-test-no-amplification name="Swap test (no amplitude amplification)"}
-Assume to have access to a unitary $U_1$ that produces a state $U_1 \ket{0} = \ket{\psi_1}$ and a unitary $U_2$ that produces a state $\ket{\psi_2}$, where $\ket{\psi_1},\ket{\psi_2} \in \mathbb{R}^N$ for $N=2^n, n\in\mathbb{N}$. There is a quantum algorithm that allows to estimate the quantity $|\braket{\psi_1|\psi_2}|^2$ with additive precision $\epsilon$ using $U_1$ and $U_2$ $O(\frac{\log(1/\delta)}{\epsilon_2})$ times with probability $1-\delta$
+Assume to have access to a unitary $U_1$ that produces a state $U_1 \ket{0} = \ket{\psi_1}$ and a unitary $U_2$ that produces a state $\ket{\psi_2}$, where $\ket{\psi_1},\ket{\psi_2} \in \mathbb{R}^N$ for $N=2^n, n\in\mathbb{N}$. There is a quantum algorithm that allows to estimate the quantity $|\braket{\psi_1|\psi_2}|^2$ with additive precision $\epsilon$ using $U_1$ and $U_2$ $O(\frac{\log(1/\delta)}{\epsilon^2})$ times with probability $1-\delta$
 :::
 
 ::: {.proof}