update first week

Author: mhjensen

doc/pub/week1/html/week1-bs.html

@@ -25,7 +25,7 @@ o You may also find chapters 1-2 of Olivares useful, see URL:"https://link.sprin
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-These chpaters from different textbooks give you an overview of some of the central (often linear algebra) mathematical elements needed for quantum computing. The aim of these lecture notes  aim at giving you a basic overview of central mathematical elements. For more details we refer you to the above texts and the companion lecture notes.
+These chapters from different textbooks give you an overview of some of the central (often linear algebra) mathematical elements needed for quantum computing. These lecture notes  aim at giving you a basic overview of central mathematical elements. For more details we refer you to the above texts and the companion lecture notes.
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@@ -126,25 +126,23 @@ o PennyLane at URL:"https://pennylane.ai/ (tailored to machine learning)"
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 ===== What is quantum computing? =====
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-o It is computation (and more generally, information processing) based on the principles of quantum mechanics, rather than classical physics.
-o Quantum mechanics is a description of nature at its most fundamental level.
+It is computation (and more generally, information processing) based on the principles of quantum mechanics, rather than classical physics. Quantum mechanics is a description of nature at its most fundamental level.
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 !bblock
-o Formulated in the early 20th century to explain the behavior of subatomic particles.
-o QM, and its specializations (quantum field theory, quantum chromodynamics, many-body physics etc.), have been spectacularly successful in explaining microscopic physical phenomena.
+Formulated in the early 20th century to explain the behavior of subatomic particles. QM, and its specializations (quantum field theory, quantum chromodynamics, many-body physics etc.), have been spectacularly successful in explaining microscopic physical phenomena.
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 ===== The exponentiality of QM =====
 !bblock
-o After nearly a century of study, the best (classical) methods for predicting the behavior of general quantum systems require exponential resources.
+After nearly a century of study, the best (classical) methods for predicting the behavior of general quantum systems require exponential resources.
 o The state of $N$ particles requires at least $2^{N}$ numbers to describe.
 o For $N \sim 300$ (the number of particles in a single uranium atom),
   \[
     2^{300} \gg \approx 10^{82}\quad (\text{\# of atoms in the observable universe}).
   \]
-o Nature seems to be doing extravagant amounts of computation *behind the scenes*!
+Nature seems to be doing extravagant amounts of computation *behind the scenes*!
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@@ -224,8 +222,8 @@ o Option 2: Polynomial-time classical factoring algorithm.
 o Option 3: Quantum computers are more powerful than classical computers.
   * This would refute the Extended Church--Turing Thesis.
   * Church--Turing Thesis: A Turing machine can simulate any effective computation process.
-  * Extended Church--Turing Thesis: A probabilistic Turing machine can efficiently simulate any effective computation process.
-  * Option \#3 would violate the ECT (assuming factoring is hard for classical computers)!
+  * Extended Church-Turing (ECT) Thesis: A probabilistic Turing machine can efficiently simulate any effective computation process.
+  * Option 3 would violate the ECT (assuming factoring is hard for classical computers)!
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@@ -240,20 +238,20 @@ At least one of these must be true!
 
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 ===== Present day =====
-Quantum computing is extremely active and exciting:
+Quantum computing is an extremely active and exciting field:
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-o Engineering: primitive quantum computers are being built.
+o Qauntum engineering: quantum computers are being built.
 o Theoretical developments: new algorithms, new cryptographic protocols, new insights into how quantum computing compares to classical computing.
-o Deep connections to other sciences: condensed matter physics, quantum gravity, materials science, chemistry, computer science, pure mathematics.
+o Deep connections to other sciences: condensed matter physics, quantum gravity, materials science, chemistry, computer science, pure mathematics. Fundamental aspects of quantum mechanics.
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 !split 
 ===== Emerging quantum computers =====
 !bblock
-o Many players racing to build (scalable) quantum computers.
-o Different labs/companies are betting on different horses.
-o Superconducting qubits, ion traps, photonic systems, topological qubits, \dots
+o Many players (companies) racing to build (scalable) quantum computers.
+o Different labs/companies are betting on different quantum platforms.
+o Superconducting qubits, ion traps, photonic systems, topological qubits, and more
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@@ -275,7 +273,6 @@ o Not capable of running Shor’s algorithm, but should still be capable of solv
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 o In Fall 2019, Google announced that they had achieved “Quantum Supremacy” using their 53-qubit Sycamore quantum processor.
 o _Quantum supremacy:_ a moment when there is a convincing real-world demonstration of a quantum computing task that cannot feasibly be performed by a classical computer.
-o Sweet spot: 53 qubits just beyond the reach of Google’s compute clusters to simulate in a reasonable time. ($\sim 2^{53}$ ops)
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@@ -284,7 +281,7 @@ o Sweet spot: 53 qubits just beyond the reach of Google’s compute clusters to
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 o We are in the NISQ era, but we have compelling evidence that we can already perform super-classical tasks using these machines.
 o There is a _qubit race_ going on, but qubit count is not everything!
-o Scaling up (i.e. more, higher-quality qubits) is a tough engineering challenge, but no fundamental obstacles to building a QC with $10^{6}$ noiseless qubits.
+o Scaling up (i.e. more, higher-quality qubits) is a tough quantum engineering challenge, but no fundamental obstacles to building a QC with $10^{6}$ noiseless qubits.
 o "QUNorth":"https://qunorth.com/" in Denmark and "LUMIQ":"https://lumi-supercomputer.eu/europe-takes-a-quantum-leap-lumi-q-consortium-signs-contract/" as a European project
 o Still, large-scale fault-tolerant QCs look like they’re many years away.
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@@ -314,6 +311,175 @@ o more
 
 Connections with fundamental physics: Perhaps key to a theory of Quantum Gravity could be quantum entanglement, and quantum error correcting codes.
 
+!split
+===== The basic concepts =====
+
+Quantum technologies leverage principles of quantum mechanics to perform computations beyond classical capabilities.
+
+_Key Concepts:_
+!bblock
+o _Superposition:_ Qubits can exist in a combination of states.
+o _Entanglement:_ Correlation between qubits regardless of distance.
+o _Quantum Interference:_ Probability amplitudes interfere. Is a core tool in quantum algorithms. It is used to amplify the probability of finding the correct answer while suppressing incorrect ones.
+!eblock
+
+
+
+!split
+===== Quantum Speedups =====
+_Why Quantum?_
+!bblock
+o _Quantum Parallelism:_ Process multiple states simultaneously.
+o _Quantum Entanglement:_ Correlated states for richer information.
+o _Quantum Interference:_ Constructive and destructive interference to enhance solutions, is a core tool in quantum algorithms.
+!eblock
+
+!bblock
+_Grover's Algorithm:_
+!bt
+\[
+\text{Quantum Search Complexity: } O(\sqrt{N}) \text{ vs. } O(N)
+\]
+!et
+_Advantage:_ Speedups in high-dimensional optimization and linear algebra problems.
+!eblock
+
+!split
+===== Challenges and Limitations =====
+
+!bblock Quantum Hardware Limitations:
+o Noisy Intermediate-Scale Quantum (NISQ) devices.
+o Decoherence and limited qubit coherence times.
+!eblock
+
+!bblock Data Encoding:
+o Efficient embedding of classical data into quantum states.
+!eblock
+
+!bblock Scalability:
+o Difficult to scale circuits to large datasets.
+!eblock
+
+
+
+
+
+
+!split
+===== 1. Quantum Communication =====
+!bblock Quantum Teleportation:
+o Entanglement enables the transmission of quantum states using classical communication.
+o No need to send the physical quantum particle.
+!eblock
+
+!bblock Advantage:
+o Instantaneous state transfer within quantum mechanics constraints.
+o Quantum networks rely on entanglement for secure communication.
+!eblock
+
+
+!split
+===== 2. Quantum Cryptography =====
+!bblock Quantum Key Distribution:
+o Entanglement ensures secure communication.
+o Eavesdropping disturbs quantum states, revealing interception attempts.
+!eblock
+
+!bblock
+o Any measurement by a third party collapses the wavefunction.  
+o Ensures security based on quantum mechanics, not computational hardness.
+!eblock
+_Advantage:_ Unconditional security guaranteed by the laws of physics.
+
+
+!split
+===== 3. Quantum Computing =====
+
+!bblock Speedup in Quantum Algorithms:
+o Entanglement provides exponential state space.
+o Quantum parallelism arises from entangled qubits.
+!eblock
+
+Examples are  Grover's Algorithm
+!bt
+\[
+\mathcal{O}(\sqrt{N}) \text{ vs. } \mathcal{O}(N)
+\]
+!et
+and  Shor's Algorithm:
+!bt
+\[
+\text{Factoring in } \mathcal{O}((\log N)^3)
+\]
+!et
+
+!split
+===== 4. Quantum Metrology =====
+
+!bblock Quantum Metrology:
+o Uses entangled states for ultra-precise measurements.
+o Overcomes the classical shot-noise limit.
+!eblock
+
+!bblock
+o Quantum entanglement improves sensitivity beyond classical limits.
+!eblock
+
+
+!split
+===== Challenges of Quantum Entanglement =====
+!bblock Decoherence:
+o Entangled states are fragile.
+o Interaction with the environment collapses the wavefunction.
+!eblock
+
+!bblock Scalability:
+o Difficult to entangle large numbers of qubits.
+o Error correction requires complex protocols.
+!eblock
+
+!bblock Measurement Problem:
+o Measurement destroys entanglement.
+o Trade-off between information gain and entanglement preservation.
+!eblock
+
+
+!split
+===== What is Quantum Machine Learning? =====
+_Quantum Machine Learning (QML)_ integrates quantum computing with machine learning algorithms to exploit quantum advantages.
+
+!bblock Motivation:
+o High-dimensional Hilbert spaces for better feature representation.
+o Quantum parallelism for faster computation.
+o Quantum entanglement for richer data encoding.
+!eblock
+
+
+
+!split
+===== Di Vincenzo criteria =====
+
+!bblock Quantum computing requirements
+o  A scalable physical system with well-characterized qubit
+o  The ability to initialize the state of the qubits to a simple fiducial state
+o  Long relevant Quantum coherence times longer than the gate operation time
+o  A _universal_ set of quantum gates
+o  A qubit-specific measurement capability
+!eblock
+
+
+!split
+===== Quantum platforms =====
+
+!bblock
+o  Superconducting qubits use Josephson junction circuits operated at millikelvin temperatures
+o  Trapped-ion computers confine ions (for example $^{171}$Yb$^+$, $^{40}$Ca$^+$) in electromagnetic traps, encoding qubits in internal electronic states. Gates are implemented with laser or microwave fields coupling to vibrational modes.
+o  Photonic quantum computing employs photons (in dual-rail or  continuous-variable encodings) as qubits
+o  Spin qubits—realized in silicon quantum dots, donor atoms, or color centers—encode information in electron or nuclear spin states.
+o  Electrons on helium/neon and other
+!eblock
+
+
 
 
 
@@ -1552,16 +1718,15 @@ which written out in all its details reads
 
 
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-===== First exercise set for January 28 =====
+===== Exercise set for first week, to be discussed January 28 =====
 
 
-The last two exercises are meant to build the basis for
-the projects we will work on during the semester.  The first
+The exercises here are mostly meant as a reminder of basic linear algebra properties. We have also included some exercises which prepare for the first project. The first
 project deals with implementing the so-called
 _Variational Quantum Eigensolver_ algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
 
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-===== Ex1: Bell states  =====
+===== Exewrcise 1: Bell states  =====
 
 Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis
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@@ -1593,36 +1758,45 @@ and
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-===== Ex2: Entangled state  =====
+===== Exercise 2: Entangled state  =====
 
 Show that the state $\alpha \vert 00\rangle+\beta\vert 11\rangle$ cannot be written as the product of the tensor product of two states and is thus entangle. The constants  $\alpha$ and $\beta$ are both nonzero.
 
 
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-===== Ex3: Commutator identities  =====
+===== Exercise 3: Commutator identities  =====
 Prove the following commutator relations for different operators (marked with a hat)
 o $[\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}]$;
 o $[\hat{A},\hat{B}\hat{C}]= [\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}]$; and 
 o $[\hat{A},[\hat{B}\hat{C}]]= [\hat{B},[\hat{C},\hat{A}]]+[\hat{C},[\hat{A},\hat{B}]]=0$ (the so-called Jacobi identity).
 
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-===== Ex4: Pauli matrices =====
-o Set up the commutation rules for Pauli matrices, that is find $[\sigma_i,\sigma_j]$ where $i,j=x,y,z$.
+===== Exercise 4: Pauli matrices =====
+o Set up the commutation rules for the Pauli matrices, that is find $[\sigma_i,\sigma_j]$ where $i,j=x,y,z$.
 o We define $\bm{X}=\sigma_x$, $\bm{Y}=\sigma_y$ and $\bm{Z}=\sigma_z$. Show that $\bm{XX}=\bm{YY}=\bm{ZZ}=\bm{I}$.
 o Which one of the Pauli matrices has the qubit basis $\vert 0\rangle$ and $\vert 1\rangle$ as eigenbasis? What are the eigenvalues?
 
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-===== Ex5: Shared eigenvectors =====
+===== Exercise 5: Shared eigenvectors =====
 
 Prove that if two operators $\hat{A}$ and $\hat{B}$ commute they will share a basis of eigenstates
 
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-===== Ex6: One-qubit basis and  Pauli matrices  =====
+===== Exercise 6: One-qubit basis and  Pauli matrices  =====
 
-Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.
+Write a function (in Python for example) which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.
 
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-===== Ex7: Hadamard and Phase gates  =====
+===== Exercise 7: Hadamard and Phase gates  =====
+
+Apply the Hadamard and Phase matrices (or gates) to the same one-qubit basis states and study their actions on these states.
+Write a code which applies these matrices to the same one-qubit basis.
+
+
+
+
+
+
+
 
-Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.