Write TODOs for last three slides

.gitmodules

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+[submodule "lib/cel-slides-template-2025"]
+	path = lib/cel-slides-template-2025
+	url = git@gitlab.kit.edu:kit/cel/misc/cel-slides-template-2025.git

lib/cel-slides-template-2025

@@ -0,0 +1 @@
+Subproject commit 3e5094ffdc60e1a0550f7c42be8a720b5e9eb6c5

src/fundamentals/MA.bib

@@ -3,22 +3,57 @@
     title = {Designing fault-tolerant circuits using detector error models},
     url = {http://arxiv.org/abs/2407.13826},
     doi = {10.48550/arXiv.2407.13826},
-	abstract = {Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault-tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code's resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.},
+    abstract = {Quantum error-correcting codes, such as subspace,
+        subsystem, and Floquet codes, are typically constructed within
+        the stabilizer formalism, which does not fully capture the idea
+        of fault-tolerance needed for practical quantum computing
+        applications. In this work, we explore the remarkably powerful
+        formalism of detector error models, which fully captures
+        fault-tolerance at the circuit level. We introduce the detector
+        error model formalism in a pedagogical manner and provide several
+        examples. Additionally, we apply the formalism to three different
+        levels of abstraction in the engineering cycle of fault-tolerant
+        circuit designs: finding robust syndrome extraction circuits,
+        identifying efficient measurement schedules, and constructing
+        fault-tolerant procedures. We enhance the surface code's
+        resistance to measurement errors, devise short measurement
+        schedules for color codes, and implement a more efficient
+    fault-tolerant method for measuring logical operators.},
     number = {{arXiv}:2407.13826},
     publisher = {{arXiv}},
-	author = {Derks, Peter-Jan H. S. and Townsend-Teague, Alex and Burchards, Ansgar G. and Eisert, Jens},
+    author = {Derks, Peter-Jan H. S. and Townsend-Teague, Alex and
+    Burchards, Ansgar G. and Eisert, Jens},
     urldate = {2025-10-28},
     date = {2025-10-25},
     eprinttype = {arxiv},
     eprint = {2407.13826 [quant-ph]},
     keywords = {Quantum Physics, /s1, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/NLEMWTH8/Derks et al. - 2025 - Designing fault-tolerant circuits using detector error models.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/QEBN4FJT/2407.html:text/html},
+    file = {Preprint PDF:/home/andreas/Zotero/storage/NLEMWTH8/Derks
+        et al. - 2025 - Designing fault-tolerant circuits using detector
+        error
+    models.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/QEBN4FJT/2407.html:text/html},
 }
 
 @online{nielsen_quantum_2010,
-	title = {Quantum Computation and Quantum Information: 10th Anniversary Edition},
+    title = {Quantum Computation and Quantum Information: 10th
+    Anniversary Edition},
     shorttitle = {Quantum Computation and Quantum Information},
-	abstract = {One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.},
+    abstract = {One of the most cited books in physics of all time,
+        Quantum Computation and Quantum Information remains the best
+        textbook in this exciting field of science. This 10th anniversary
+        edition includes an introduction from the authors setting the
+        work in context. This comprehensive textbook describes such
+        remarkable effects as fast quantum algorithms, quantum
+        teleportation, quantum cryptography and quantum error-correction.
+        Quantum mechanics and computer science are introduced before
+        moving on to describe what a quantum computer is, how it can be
+        used to solve problems faster than 'classical' computers and its
+        real-world implementation. It concludes with an in-depth
+        treatment of quantum information. Containing a wealth of figures
+        and exercises, this well-known textbook is ideal for courses on
+        the subject, and will interest beginning graduate students and
+        researchers in physics, computer science, mathematics, and
+    electrical engineering.},
     titleaddon = {Cambridge Aspire website},
     author = {Nielsen, Michael A. and Chuang, Isaac L.},
     urldate = {2025-10-28},
@@ -26,40 +61,67 @@
     langid = {english},
     doi = {10.1017/CBO9780511976667},
     note = {{ISBN}: 9780511976667
-Publisher: Cambridge University Press},
+    Publisher: Cambridge University Press},
     keywords = {\#{FND}, \#{QM}, \#{QEC}},
-	file = {PDF:/home/andreas/Zotero/storage/2FGWZ5CC/Nielsen and Chuang - 2010 - Quantum Computation and Quantum Information 10th Anniversary Edition.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/RFPYY4AS/01E10196D0A682A6AEFFEA52D53BE9AE.html:text/html},
+    file = {PDF:/home/andreas/Zotero/storage/2FGWZ5CC/Nielsen and
+        Chuang - 2010 - Quantum Computation and Quantum Information 10th
+        Anniversary
+    Edition.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/RFPYY4AS/01E10196D0A682A6AEFFEA52D53BE9AE.html:text/html},
 }
 
 @thesis{klinke_neural_2025,
     location = {Karlsruhe},
-	title = {Neural Belief Propagation Ensemble Decoding of Quantum {LDPC} Codes},
+    title = {Neural Belief Propagation Ensemble Decoding of Quantum
+    {LDPC} Codes},
     institution = {{KIT}},
     type = {Bachelor's Thesis},
     author = {Klinke, Jeremi},
     date = {2025-09-26},
     keywords = {/s1, \#{QEC}},
-	file = {PDF:/home/andreas/Zotero/storage/ENJG2F8D/Klinke - Neural Belief Propagation Ensemble Decoding of Quantum LDPC Codes.pdf:application/pdf},
+    file = {PDF:/home/andreas/Zotero/storage/ENJG2F8D/Klinke - Neural
+        Belief Propagation Ensemble Decoding of Quantum LDPC
+    Codes.pdf:application/pdf},
 }
 
 @article{miao_quaternary_2025,
-	title = {Quaternary Neural Belief Propagation Decoding of Quantum {LDPC} Codes with Overcomplete Check Matrices},
+    title = {Quaternary Neural Belief Propagation Decoding of Quantum
+    {LDPC} Codes with Overcomplete Check Matrices},
     volume = {13},
     issn = {2169-3536},
     url = {http://arxiv.org/abs/2308.08208},
     doi = {10.1109/ACCESS.2025.3539475},
-	abstract = {Quantum low-density parity-check ({QLDPC}) codes are promising candidates for error correction in quantum computers. One of the major challenges in implementing {QLDPC} codes in quantum computers is the lack of a universal decoder. In this work, we first propose to decode {QLDPC} codes with a belief propagation ({BP}) decoder operating on overcomplete check matrices. Then, we extend the neural {BP} ({NBP}) decoder, which was originally studied for suboptimal binary {BP} decoding of {QLPDC} codes, to quaternary {BP} decoders. Numerical simulation results demonstrate that both approaches as well as their combination yield a low-latency, high-performance decoder for several short to moderate length {QLDPC} codes.},
+    abstract = {Quantum low-density parity-check ({QLDPC}) codes are
+        promising candidates for error correction in quantum computers.
+        One of the major challenges in implementing {QLDPC} codes in
+        quantum computers is the lack of a universal decoder. In this
+        work, we first propose to decode {QLDPC} codes with a belief
+        propagation ({BP}) decoder operating on overcomplete check
+        matrices. Then, we extend the neural {BP} ({NBP}) decoder, which
+        was originally studied for suboptimal binary {BP} decoding of
+        {QLPDC} codes, to quaternary {BP} decoders. Numerical simulation
+        results demonstrate that both approaches as well as their
+        combination yield a low-latency, high-performance decoder for
+    several short to moderate length {QLDPC} codes.},
     pages = {25637--25649},
     journaltitle = {{IEEE} Access},
     shortjournal = {{IEEE} Access},
-	author = {Miao, Sisi and Schnerring, Alexander and Li, Haizheng and Schmalen, Laurent},
+    author = {Miao, Sisi and Schnerring, Alexander and Li, Haizheng
+    and Schmalen, Laurent},
     urldate = {2025-10-28},
     date = {2025-02-05},
     eprinttype = {arxiv},
     eprint = {2308.08208 [quant-ph]},
-	note = {{TLDR}: This work proposes to decode {QLDPC} codes with a belief propagation ({BP}) decoder operating on overcomplete check matrices and extends the neural {BP} decoder, which was originally studied for suboptimal binary {BP} decoding of {QLPDC} codes, to quaternary {BP} decoders.},
-	keywords = {Quantum Physics, Computer Science - Information Theory, Mathematics - Information Theory, /unread, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/SJXAPQ9Z/Miao et al. - 2025 - Quaternary Neural Belief Propagation Decoding of Quantum LDPC Codes with Overcomplete Check Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/ZVHBIEHR/2308.html:text/html},
+    note = {{TLDR}: This work proposes to decode {QLDPC} codes with a
+        belief propagation ({BP}) decoder operating on overcomplete check
+        matrices and extends the neural {BP} decoder, which was
+        originally studied for suboptimal binary {BP} decoding of {QLPDC}
+    codes, to quaternary {BP} decoders.},
+    keywords = {Quantum Physics, Computer Science - Information
+    Theory, Mathematics - Information Theory, /unread, \#{QEC}},
+    file = {Preprint PDF:/home/andreas/Zotero/storage/SJXAPQ9Z/Miao
+        et al. - 2025 - Quaternary Neural Belief Propagation Decoding of
+        Quantum LDPC Codes with Overcomplete Check
+    Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/ZVHBIEHR/2308.html:text/html},
 }
 
 @article{roffe_quantum_2019,
@@ -69,7 +131,22 @@ Publisher: Cambridge University Press},
     url = {https://doi.org/10.1080/00107514.2019.1667078},
     doi = {10.1080/00107514.2019.1667078},
     shorttitle = {Quantum error correction},
-	abstract = {Quantum error correction protocols will play a central role in the realisation of quantum computing; the choice of error correction code will influence the full quantum computing stack, from the layout of qubits at the physical level to gate compilation strategies at the software level. As such, familiarity with quantum coding is an essential prerequisite for the understanding of current and future quantum computing architectures. In this review, we provide an introductory guide to the theory and implementation of quantum error correction codes. Where possible, fundamental concepts are described using the simplest examples of detection and correction codes, the working of which can be verified by hand. We outline the construction and operation of the surface code, the most widely pursued error correction protocol for experiment. Finally, we discuss issues that arise in the practical implementation of the surface code and other quantum error correction codes.},
+    abstract = {Quantum error correction protocols will play a
+        central role in the realisation of quantum computing; the choice
+        of error correction code will influence the full quantum
+        computing stack, from the layout of qubits at the physical level
+        to gate compilation strategies at the software level. As such,
+        familiarity with quantum coding is an essential prerequisite for
+        the understanding of current and future quantum computing
+        architectures. In this review, we provide an introductory guide
+        to the theory and implementation of quantum error correction
+        codes. Where possible, fundamental concepts are described using
+        the simplest examples of detection and correction codes, the
+        working of which can be verified by hand. We outline the
+        construction and operation of the surface code, the most widely
+        pursued error correction protocol for experiment. Finally, we
+        discuss issues that arise in the practical implementation of the
+    surface code and other quantum error correction codes.},
     pages = {226--245},
     number = {3},
     journaltitle = {Contemporary Physics},
@@ -77,23 +154,34 @@ Publisher: Cambridge University Press},
     urldate = {2025-11-04},
     date = {2019-07-03},
     keywords = {/s1, \#{FND}, \#{QEC}},
-	file = {Full Text PDF:/home/andreas/Zotero/storage/DW4EYDQ8/Roffe - 2019 - Quantum error correction an introductory guide.pdf:application/pdf},
+    file = {Full Text PDF:/home/andreas/Zotero/storage/DW4EYDQ8/Roffe
+        - 2019 - Quantum error correction an introductory
+    guide.pdf:application/pdf},
 }
 
 @misc{calderbank_quantum_1997,
     title = {Quantum Error Correction via Codes over {GF}(4)},
     url = {http://arxiv.org/abs/quant-ph/9608006},
     doi = {10.48550/arXiv.quant-ph/9608006},
-	abstract = {The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field {GF}(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.},
+    abstract = {The problem of finding quantum error-correcting codes
+        is transformed into the problem of finding additive codes over
+        the field {GF}(4) which are self-orthogonal with respect to a
+        certain trace inner product. Many new codes and new bounds are
+        presented, as well as a table of upper and lower bounds on such
+    codes of length up to 30 qubits.},
     number = {{arXiv}:quant-ph/9608006},
     publisher = {{arXiv}},
-	author = {Calderbank, A. R. and Rains, E. M. and Shor, P. W. and Sloane, N. J. A.},
+    author = {Calderbank, A. R. and Rains, E. M. and Shor, P. W. and
+    Sloane, N. J. A.},
     urldate = {2025-11-05},
     date = {1997-09-10},
     eprinttype = {arxiv},
     eprint = {quant-ph/9608006},
     keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/5IM4A6FA/Calderbank et al. - 1997 - Quantum Error Correction via Codes over GF(4).pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/LLQUFH76/9608006.html:text/html},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/5IM4A6FA/Calderbank et al. -
+        1997 - Quantum Error Correction via Codes over
+    GF(4).pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/LLQUFH76/9608006.html:text/html},
 }
 
 @article{shor_scheme_1995,
@@ -111,9 +199,14 @@ Publisher: Cambridge University Press},
     urldate = {2025-11-05},
     date = {1995-10-01},
     langid = {english},
-	note = {{TLDR}: In the mid-1990s, theorists devised methods to preserve the integrity of quantum bits{\textbackslash}char22\{\}techniques that may become the key to practical quantum computing on a large scale.},
+    note = {{TLDR}: In the mid-1990s, theorists devised methods to
+        preserve the integrity of quantum
+        bits{\textbackslash}char22\{\}techniques that may become the key
+    to practical quantum computing on a large scale.},
     keywords = {/s2, \#{FND}, \#{QEC}},
-	file = {PDF:/home/andreas/Zotero/storage/DG6QT7UX/Shor - 1995 - Scheme for reducing decoherence in quantum computer memory.pdf:application/pdf},
+    file = {PDF:/home/andreas/Zotero/storage/DG6QT7UX/Shor - 1995 -
+        Scheme for reducing decoherence in quantum computer
+    memory.pdf:application/pdf},
 }
 
 @article{divincenzo_fault-tolerant_1996,
@@ -122,7 +215,14 @@ Publisher: Cambridge University Press},
     issn = {0031-9007, 1079-7114},
     url = {http://arxiv.org/abs/quant-ph/9605031},
     doi = {10.1103/PhysRevLett.77.3260},
-	abstract = {We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced group-theoretic framework for unifying all known quantum codes.},
+    abstract = {We exhibit a simple, systematic procedure for
+        detecting and correcting errors using any of the recently
+        reported quantum error-correcting codes. The procedure is shown
+        explicitly for a code in which one qubit is mapped into five. The
+        quantum networks obtained are fault tolerant, that is, they can
+        function successfully even if errors occur during the error
+        correction. Our construction is derived using a recently
+    introduced group-theoretic framework for unifying all known quantum codes.},
     pages = {3260--3263},
     number = {15},
     journaltitle = {Physical Review Letters},
@@ -132,16 +232,39 @@ Publisher: Cambridge University Press},
     date = {1996-10-07},
     eprinttype = {arxiv},
     eprint = {quant-ph/9605031},
-	note = {{TLDR}: This work exhibits a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum error-correcting codes, derived using a recently introduced group-theoretic framework for unifying all known quantum codes.},
+    note = {{TLDR}: This work exhibits a simple, systematic procedure
+        for detecting and correcting errors using any of the recently
+        reported quantum error-correcting codes, derived using a recently
+    introduced group-theoretic framework for unifying all known quantum codes.},
     keywords = {Quantum Physics, /unread, \#{FND}, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/KNGHIXB3/DiVincenzo and Shor - 1996 - Fault-Tolerant Error Correction with Efficient Quantum Codes.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/6JURUG3K/9605031.html:text/html},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/KNGHIXB3/DiVincenzo and Shor -
+        1996 - Fault-Tolerant Error Correction with Efficient Quantum
+    Codes.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/6JURUG3K/9605031.html:text/html},
 }
 
 @misc{shor_fault-tolerant_1997,
     title = {Fault-tolerant quantum computation},
     url = {http://arxiv.org/abs/quant-ph/9605011},
     doi = {10.48550/arXiv.quant-ph/9605011},
-	abstract = {Recently, it was realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties of realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, thus making long computations impossible. A futher difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering the output of long computations unreliable. It was previously known that a quantum circuit with t gates could tolerate O(1/t) amounts of inaccuracy and decoherence per gate. We show, for any quantum computation with t gates, how to build a polynomial size quantum circuit that can tolerate O(1/(log t){\textasciicircum}c) amounts of inaccuracy and decoherence per gate, for some constant c. We do this by showing how to compute using quantum error correcting codes. These codes were previously known to provide resistance to errors while storing and transmitting quantum data.},
+    abstract = {Recently, it was realized that use of the properties
+        of quantum mechanics might speed up certain computations
+        dramatically. Interest in quantum computation has since been
+        growing. One of the main difficulties of realizing quantum
+        computation is that decoherence tends to destroy the information
+        in a superposition of states in a quantum computer, thus making
+        long computations impossible. A futher difficulty is that
+        inaccuracies in quantum state transformations throughout the
+        computation accumulate, rendering the output of long computations
+        unreliable. It was previously known that a quantum circuit with t
+        gates could tolerate O(1/t) amounts of inaccuracy and decoherence
+        per gate. We show, for any quantum computation with t gates, how
+        to build a polynomial size quantum circuit that can tolerate
+        O(1/(log t){\textasciicircum}c) amounts of inaccuracy and
+        decoherence per gate, for some constant c. We do this by showing
+        how to compute using quantum error correcting codes. These codes
+        were previously known to provide resistance to errors while
+    storing and transmitting quantum data.},
     number = {{arXiv}:quant-ph/9605011},
     publisher = {{arXiv}},
     author = {Shor, Peter W.},
@@ -150,14 +273,27 @@ Publisher: Cambridge University Press},
     eprinttype = {arxiv},
     eprint = {quant-ph/9605011},
     keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/CSLTPZU5/Shor - 1997 - Fault-tolerant quantum computation.pdf:application/pdf},
+    file = {Preprint PDF:/home/andreas/Zotero/storage/CSLTPZU5/Shor -
+    1997 - Fault-tolerant quantum computation.pdf:application/pdf},
 }
 
 @misc{gottesman_stabilizer_1997,
     title = {Stabilizer Codes and Quantum Error Correction},
     url = {http://arxiv.org/abs/quant-ph/9705052},
     doi = {10.48550/arXiv.quant-ph/9705052},
-	abstract = {Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.},
+    abstract = {Controlling operational errors and decoherence is one
+        of the major challenges facing the field of quantum computation
+        and other attempts to create specified many-particle entangled
+        states. The field of quantum error correction has developed to
+        meet this challenge. A group-theoretical structure and associated
+        subclass of quantum codes, the stabilizer codes, has proved
+        particularly fruitful in producing codes and in understanding the
+        structure of both specific codes and classes of codes. I will
+        give an overview of the field of quantum error correction and the
+        formalism of stabilizer codes. In the context of stabilizer
+        codes, I will discuss a number of known codes, the capacity of a
+        quantum channel, bounds on quantum codes, and fault-tolerant
+    quantum computation.},
     number = {{arXiv}:quant-ph/9705052},
     publisher = {{arXiv}},
     author = {Gottesman, Daniel},
@@ -166,7 +302,10 @@ Publisher: Cambridge University Press},
     eprinttype = {arxiv},
     eprint = {quant-ph/9705052},
     keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/JT582GBB/Gottesman - 1997 - Stabilizer Codes and Quantum Error Correction.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/5GCZHHTH/9705052.html:text/html},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/JT582GBB/Gottesman - 1997 -
+        Stabilizer Codes and Quantum Error
+    Correction.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/5GCZHHTH/9705052.html:text/html},
 }
 
 @article{gottesman_theory_1998,
@@ -174,7 +313,16 @@ Publisher: Cambridge University Press},
     volume = {57},
     url = {https://link.aps.org/doi/10.1103/PhysRevA.57.127},
     doi = {10.1103/PhysRevA.57.127},
-	abstract = {In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a theory of fault-tolerant operations on stabilizer codes based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-quantum-bit code.},
+    abstract = {In order to use quantum error-correcting codes to
+        improve the performance of a quantum computer, it is necessary to
+        be able to perform operations fault-tolerantly on encoded states.
+        I present a theory of fault-tolerant operations on stabilizer
+        codes based on symmetries of the code stabilizer. This allows a
+        straightforward determination of which operations can be
+        performed fault-tolerantly on a given code. I demonstrate that
+        fault-tolerant universal computation is possible for any
+        stabilizer code. I discuss a number of examples in more detail,
+    including the five-quantum-bit code.},
     pages = {127--137},
     number = {1},
     journaltitle = {Physical Review A},
@@ -183,16 +331,40 @@ Publisher: Cambridge University Press},
     urldate = {2025-11-06},
     date = {1998-01-01},
     note = {Publisher: American Physical Society
-{TLDR}: It is demonstrated that fault-tolerant universal computation is possible for any stabilizer code, including the five-quantum-bit code.},
+        {TLDR}: It is demonstrated that fault-tolerant universal
+        computation is possible for any stabilizer code, including the
+    five-quantum-bit code.},
     keywords = {/s1, \#{FND}, \#{QEC}},
-	file = {APS Snapshot:/home/andreas/Zotero/storage/BP7CHBIU/PhysRevA.57.html:text/html;Full Text PDF:/home/andreas/Zotero/storage/7E5TUIMN/Gottesman - 1998 - Theory of fault-tolerant quantum computation.pdf:application/pdf},
+    file = {APS
+        Snapshot:/home/andreas/Zotero/storage/BP7CHBIU/PhysRevA.57.html:text/html;Full
+        Text PDF:/home/andreas/Zotero/storage/7E5TUIMN/Gottesman - 1998 -
+    Theory of fault-tolerant quantum computation.pdf:application/pdf},
 }
 
 @misc{gottesman_introduction_2009,
-	title = {An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation},
+    title = {An Introduction to Quantum Error Correction and
+    Fault-Tolerant Quantum Computation},
     url = {http://arxiv.org/abs/0904.2557},
     doi = {10.48550/arXiv.0904.2557},
-	abstract = {Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over {GF}(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.},
+    abstract = {Quantum states are very delicate, so it is likely
+        some sort of quantum error correction will be necessary to build
+        reliable quantum computers. The theory of quantum
+        error-correcting codes has some close ties to and some striking
+        differences from the theory of classical error-correcting codes.
+        Many quantum codes can be described in terms of the stabilizer of
+        the codewords. The stabilizer is a finite Abelian group, and
+        allows a straightforward characterization of the error-correcting
+        properties of the code. The stabilizer formalism for quantum
+        codes also illustrates the relationships to classical coding
+        theory, particularly classical codes over {GF}(4), the finite
+        field with four elements. To build a quantum computer which
+        behaves correctly in the presence of errors, we also need a
+        theory of fault-tolerant quantum computation, instructing us how
+        to perform quantum gates on qubits which are encoded in a quantum
+        error-correcting code. The threshold theorem states that it is
+        possible to create a quantum computer to perform an arbitrary
+        quantum computation provided the error rate per physical gate or
+    time step is below some constant threshold value.},
     number = {{arXiv}:0904.2557},
     publisher = {{arXiv}},
     author = {Gottesman, Daniel},
@@ -201,7 +373,11 @@ Publisher: Cambridge University Press},
     eprinttype = {arxiv},
     eprint = {0904.2557 [quant-ph]},
     keywords = {Quantum Physics, /s1, \#{FND}, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/AGETMT4C/Gottesman - 2009 - An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/UHAPPP5S/0904.html:text/html},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/AGETMT4C/Gottesman - 2009 - An
+        Introduction to Quantum Error Correction and Fault-Tolerant
+        Quantum
+    Computation.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/UHAPPP5S/0904.html:text/html},
 }
 
 @article{calderbank_good_1996,
@@ -209,7 +385,16 @@ Publisher: Cambridge University Press},
     volume = {54},
     url = {https://link.aps.org/doi/10.1103/PhysRevA.54.1098},
     doi = {10.1103/PhysRevA.54.1098},
-	abstract = {A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2𝐻2(2t/n) where 𝐻2(p) is the binary entropy function -plog2p-(1-p)log2(1-p). Upper bounds on this asymptotic rate are given. © 1996 The American Physical Society.},
+    abstract = {A quantum error-correcting code is defined to be a
+        unitary mapping (encoding) of k qubits (two-state quantum
+        systems) into a subspace of the quantum state space of n qubits
+        such that if any t of the qubits undergo arbitrary decoherence,
+        not necessarily independently, the resulting n qubits can be used
+        to faithfully reconstruct the original quantum state of the k
+        encoded qubits. Quantum error-correcting codes are shown to exist
+        with asymptotic rate k/n=1-2𝐻2(2t/n) where 𝐻2(p) is the binary
+        entropy function -plog2p-(1-p)log2(1-p). Upper bounds on this
+    asymptotic rate are given. © 1996 The American Physical Society.},
     pages = {1098--1105},
     number = {2},
     journaltitle = {Physical Review A},
@@ -218,21 +403,34 @@ Publisher: Cambridge University Press},
     urldate = {2025-11-06},
     date = {1996-08-01},
     note = {Publisher: American Physical Society
-{TLDR}: The techniques investigated in this paper can be extended so as to reduce the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions.},
+        {TLDR}: The techniques investigated in this paper can be extended
+        so as to reduce the accuracy required for factorization of
+        numbers large enough to be difficult on conventional computers
+    appears to be closer to one part in billions.},
     keywords = {/s1, \#{FND}, \#{QEC}},
-	file = {APS Snapshot:/home/andreas/Zotero/storage/IK4DH994/PhysRevA.54.html:text/html;Full Text PDF:/home/andreas/Zotero/storage/RLKB7SKX/Calderbank and Shor - 1996 - Good quantum error-correcting codes exist.pdf:application/pdf},
+    file = {APS
+        Snapshot:/home/andreas/Zotero/storage/IK4DH994/PhysRevA.54.html:text/html;Full
+        Text PDF:/home/andreas/Zotero/storage/RLKB7SKX/Calderbank and
+        Shor - 1996 - Good quantum error-correcting codes
+    exist.pdf:application/pdf},
 }
 
 @book{griffiths_introduction_1995,
     title = {Introduction to Quantum Mechanics},
     isbn = {0-13-124405-1},
-	abstract = {Changes and additions to the new edition of this classic textbook include a new chapter on symmetries, new problems and examples, improved explanations, more numerical problems to be worked on a computer, new applications to solid state physics, and consolidated treatment of time-dependent potentials.},
+    abstract = {Changes and additions to the new edition of this
+        classic textbook include a new chapter on symmetries, new
+        problems and examples, improved explanations, more numerical
+        problems to be worked on a computer, new applications to solid
+    state physics, and consolidated treatment of time-dependent potentials.},
     publisher = {Prentice Hall},
     author = {Griffiths, David J.},
     date = {1995},
     langid = {english},
     keywords = {\#{FND}, \#{QM}, \#{MAT}},
-	file = {PDF:/home/andreas/Zotero/storage/ZLP4S5EB/Griffiths and Schroeter - 2018 - Introduction to Quantum Mechanics.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/V7B6J8YI/990799CA07A83FC5312402AF6860311E.html:text/html},
+    file = {PDF:/home/andreas/Zotero/storage/ZLP4S5EB/Griffiths and
+        Schroeter - 2018 - Introduction to Quantum
+    Mechanics.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/V7B6J8YI/990799CA07A83FC5312402AF6860311E.html:text/html},
 }
 
 @online{bradley_tensor_2018,
@@ -242,49 +440,107 @@ Publisher: Cambridge University Press},
     urldate = {2025-11-11},
     date = {2018-11-18},
     keywords = {\#{FND}, \#{MAT}},
-	file = {Snapshot:/home/andreas/Zotero/storage/JWTQ4W7G/the-tensor-product-demystified.html:text/html},
+    file =
+    {Snapshot:/home/andreas/Zotero/storage/JWTQ4W7G/the-tensor-product-demystified.html:text/html},
 }
 
 @misc{camps-moreno_toward_2024,
     title = {Toward Quantum {CSS}-T Codes from Sparse Matrices},
     url = {http://arxiv.org/abs/2406.00425},
     doi = {10.48550/arXiv.2406.00425},
-	abstract = {{CSS}-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A {CSS}-T code depends on a pair \$(C\_1, C\_2)\$ of binary linear codes \$C\_1\$ and \$C\_2\$ that satisfy certain conditions. We prove that \$C\_1\$ and \$C\_2\$ form a {CSS}-T pair if and only if \$C\_2 {\textbackslash}subset {\textbackslash}operatorname\{Hull\}(C\_1) {\textbackslash}cap {\textbackslash}operatorname\{Hull\}(C\_1{\textasciicircum}2)\$, where the hull of a code is the intersection of the code with its dual. We show that if \$(C\_1,C\_2)\$ is a {CSS}-T pair, and the code \$C\_2\$ is degenerated on \${\textbackslash}\{i{\textbackslash}\}\$, meaning that the \$i{\textasciicircum}\{th\}\$-entry is zero for all the elements in \$C\_2\$, then the pair of punctured codes \$(C\_1{\textbar}\_i,C\_2{\textbar}\_i)\$ is also a {CSS}-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum {LDPC} or {LDGM} {CSS}-T codes computationally.},
+    abstract = {{CSS}-T codes were recently introduced as quantum
+        error-correcting codes that respect a transversal gate. A {CSS}-T
+        code depends on a pair \$(C\_1, C\_2)\$ of binary linear codes
+        \$C\_1\$ and \$C\_2\$ that satisfy certain conditions. We prove
+        that \$C\_1\$ and \$C\_2\$ form a {CSS}-T pair if and only if
+        \$C\_2 {\textbackslash}subset
+        {\textbackslash}operatorname\{Hull\}(C\_1) {\textbackslash}cap
+        {\textbackslash}operatorname\{Hull\}(C\_1{\textasciicircum}2)\$,
+        where the hull of a code is the intersection of the code with its
+        dual. We show that if \$(C\_1,C\_2)\$ is a {CSS}-T pair, and the
+        code \$C\_2\$ is degenerated on
+        \${\textbackslash}\{i{\textbackslash}\}\$, meaning that the
+        \$i{\textasciicircum}\{th\}\$-entry is zero for all the elements
+        in \$C\_2\$, then the pair of punctured codes
+        \$(C\_1{\textbar}\_i,C\_2{\textbar}\_i)\$ is also a {CSS}-T pair.
+        Finally, we provide Magma code based on our results and
+        quasi-cyclic codes as a step toward finding quantum {LDPC} or
+    {LDGM} {CSS}-T codes computationally.},
     number = {{arXiv}:2406.00425},
     publisher = {{arXiv}},
-	author = {Camps-Moreno, Eduardo and López, Hiram H. and Matthews, Gretchen L. and {McMillon}, Emily},
+    author = {Camps-Moreno, Eduardo and López, Hiram H. and Matthews,
+    Gretchen L. and {McMillon}, Emily},
     urldate = {2025-11-13},
     date = {2024-06-04},
     eprinttype = {arxiv},
     eprint = {2406.00425 [cs]},
     keywords = {Computer Science - Information Theory, /unread, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/C634YE7N/Camps-Moreno et al. - 2024 - Toward Quantum CSS-T Codes from Sparse Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/TD3KFLFZ/2406.html:text/html},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/C634YE7N/Camps-Moreno et al. -
+        2024 - Toward Quantum CSS-T Codes from Sparse
+    Matrices.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/TD3KFLFZ/2406.html:text/html},
 }
 
 @misc{koutsioumpas_colour_2025,
     title = {Colour Codes Reach Surface Code Performance using Vibe Decoding},
     url = {http://arxiv.org/abs/2508.15743},
     doi = {10.48550/arXiv.2508.15743},
-	abstract = {Two-dimensional quantum colour codes hold significant promise for quantum error correction, offering advantages such as planar connectivity and low overhead logical gates. Despite their theoretical appeal, the practical deployment of these codes faces challenges due to complex decoding requirements compared to surface codes. This paper introduces vibe decoding which, for the first time, brings colour code performance on par with the surface code under practical decoding. Our approach leverages an ensemble of belief propagation decoders - each executing a distinct serial message passing schedule - combined with localised statistics post-processing. We refer to this combined protocol as {VibeLSD}. The {VibeLSD} decoder is highly versatile: our numerical results show it outperforms all practical existing colour code decoders across various syndrome extraction schemes, noise models, and error rates. By estimating qubit footprints through quantum memory simulations, we show that colour codes can operate with overhead that is comparable to, and in some cases lower than, that of the surface code. This, combined with the fact that localised statistics decoding is a parallel algorithm, makes {VibeLSD} suitable for implementation on specialised hardware for real-time decoding. Our results establish the colour code as a practical architecture for near-term quantum hardware, providing improved compilation efficiency for both Clifford and non-Clifford gates without incurring additional qubit overhead relative to the surface code.},
+    abstract = {Two-dimensional quantum colour codes hold significant
+        promise for quantum error correction, offering advantages such as
+        planar connectivity and low overhead logical gates. Despite their
+        theoretical appeal, the practical deployment of these codes faces
+        challenges due to complex decoding requirements compared to
+        surface codes. This paper introduces vibe decoding which, for the
+        first time, brings colour code performance on par with the
+        surface code under practical decoding. Our approach leverages an
+        ensemble of belief propagation decoders - each executing a
+        distinct serial message passing schedule - combined with
+        localised statistics post-processing. We refer to this combined
+        protocol as {VibeLSD}. The {VibeLSD} decoder is highly versatile:
+        our numerical results show it outperforms all practical existing
+        colour code decoders across various syndrome extraction schemes,
+        noise models, and error rates. By estimating qubit footprints
+        through quantum memory simulations, we show that colour codes can
+        operate with overhead that is comparable to, and in some cases
+        lower than, that of the surface code. This, combined with the
+        fact that localised statistics decoding is a parallel algorithm,
+        makes {VibeLSD} suitable for implementation on specialised
+        hardware for real-time decoding. Our results establish the colour
+        code as a practical architecture for near-term quantum hardware,
+        providing improved compilation efficiency for both Clifford and
+        non-Clifford gates without incurring additional qubit overhead
+    relative to the surface code.},
     number = {{arXiv}:2508.15743},
     publisher = {{arXiv}},
-	author = {Koutsioumpas, Stergios and Noszko, Tamas and Sayginel, Hasan and Webster, Mark and Roffe, Joschka},
+    author = {Koutsioumpas, Stergios and Noszko, Tamas and Sayginel,
+    Hasan and Webster, Mark and Roffe, Joschka},
     urldate = {2025-11-13},
     date = {2025-08-22},
     eprinttype = {arxiv},
     eprint = {2508.15743 [quant-ph]},
-	note = {{TLDR}: The results establish the colour code as a practical architecture for near-term quantum hardware, providing improved compilation efficiency for both Clifford and non-Clifford gates without incurring additional qubit overhead relative to the surface code.},
-	keywords = {Quantum Physics, Computer Science - Information Theory, /s1, \#{QEC}},
-	file = {Preprint PDF:/home/andreas/Zotero/storage/C4K3XG2S/Koutsioumpas et al. - 2025 - Colour Codes Reach Surface Code Performance using Vibe Decoding.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/76EXKH3S/2508.html:text/html},
+    note = {{TLDR}: The results establish the colour code as a
+        practical architecture for near-term quantum hardware, providing
+        improved compilation efficiency for both Clifford and
+        non-Clifford gates without incurring additional qubit overhead
+    relative to the surface code.},
+    keywords = {Quantum Physics, Computer Science - Information
+    Theory, /s1, \#{QEC}},
+    file = {Preprint
+        PDF:/home/andreas/Zotero/storage/C4K3XG2S/Koutsioumpas et al. -
+        2025 - Colour Codes Reach Surface Code Performance using Vibe
+    Decoding.pdf:application/pdf;Snapshot:/home/andreas/Zotero/storage/76EXKH3S/2508.html:text/html},
 }
 
 @article{koutsioumpas_automorphism_2025,
     title = {Automorphism Ensemble Decoding of Quantum {LDPC} Codes},
-	author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster, Mark and Browne, Dan E},
+    author = {Koutsioumpas, Stergios and Sayginel, Hasan and Webster,
+    Mark and Browne, Dan E},
     date = {2025-03-04},
     langid = {english},
     keywords = {/s1, \#{QEC}},
-	file = {PDF:/home/andreas/Zotero/storage/SHUGDAU8/Koutsioumpas et al. - Automorphism Ensemble Decoding of Quantum LDPC Codes.pdf:application/pdf},
+    file = {PDF:/home/andreas/Zotero/storage/SHUGDAU8/Koutsioumpas et
+        al. - Automorphism Ensemble Decoding of Quantum LDPC
+    Codes.pdf:application/pdf},
 }
 
 @article{geiselhart_automorphism_2021,
@@ -293,14 +549,68 @@ Publisher: Cambridge University Press},
     issn = {1558-0857},
     url = {https://ieeexplore.ieee.org/document/9492151},
     doi = {10.1109/TCOMM.2021.3098798},
-	abstract = {Reed–Muller ({RM}) codes are known for their good maximum likelihood ({ML}) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for {RM} codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation ({MBBP}) and may use any polar or {RM} decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation ({SC})-, {SC}-list ({SCL})- and belief propagation ({BP})-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-{ML} performance for the {RM}(3,7)-code (e.g., 0.04 {dB} away from the {ML} bound at {BLER} of 10−3) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of {RM} codes and {SC} decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.},
+    abstract = {Reed–Muller ({RM}) codes are known for their good
+        maximum likelihood ({ML}) performance in the short block-length
+        regime. Despite being one of the oldest classes of channel codes,
+        finding a low complexity soft-input decoding scheme is still an
+        open problem. In this work, we present a versatile decoding
+        architecture for {RM} codes based on their rich automorphism
+        group. The decoding algorithm can be seen as a generalization of
+        multiple-bases belief propagation ({MBBP}) and may use any polar
+        or {RM} decoder as constituent decoders. We provide extensive
+        error-rate performance simulations for successive cancellation
+        ({SC})-, {SC}-list ({SCL})- and belief propagation ({BP})-based
+        constituent decoders. We furthermore compare our results to
+        existing decoding schemes and report a near-{ML} performance for
+        the {RM}(3,7)-code (e.g., 0.04 {dB} away from the {ML} bound at
+        {BLER} of 10−3) at a competitive computational cost. Moreover, we
+        provide some insights into the automorphism subgroups of {RM}
+        codes and {SC} decoding and, thereby, prove the theoretical
+    limitations of this method with respect to polar codes.},
     pages = {6424--6438},
     number = {10},
     journaltitle = {{IEEE} Transactions on Communications},
-	author = {Geiselhart, Marvin and Elkelesh, Ahmed and Ebada, Moustafa and Cammerer, Sebastian and Brink, Stephan ten},
+    author = {Geiselhart, Marvin and Elkelesh, Ahmed and Ebada,
+    Moustafa and Cammerer, Sebastian and Brink, Stephan ten},
     urldate = {2025-11-13},
     date = {2021-07-21},
-	note = {{TLDR}: A versatile decoding architecture for {RM} codes based on their rich automorphism group is presented and the theoretical limitations of this method with respect to polar codes are proved.},
-	keywords = {/unread, \#{FND}, Belief propagation, belief propagation decoding, code automorphisms, Complexity theory, Encoding, ensemble decoding, Generators, Iterative decoding, list decoding, Maximum likelihood decoding, polar codes, Polar codes, Reed-Muller Codes, successive cancellation decoding, \#{CEC}},
-	file = {Full Text PDF:/home/andreas/Zotero/storage/KV3JR3MS/Geiselhart et al. - 2021 - Automorphism Ensemble Decoding of Reed–Muller Codes.pdf:application/pdf},
+    note = {{TLDR}: A versatile decoding architecture for {RM} codes
+        based on their rich automorphism group is presented and the
+        theoretical limitations of this method with respect to polar
+    codes are proved.},
+    keywords = {/unread, \#{FND}, Belief propagation, belief
+        propagation decoding, code automorphisms, Complexity theory,
+        Encoding, ensemble decoding, Generators, Iterative decoding, list
+        decoding, Maximum likelihood decoding, polar codes, Polar codes,
+    Reed-Muller Codes, successive cancellation decoding, \#{CEC}},
+    file = {Full Text
+        PDF:/home/andreas/Zotero/storage/KV3JR3MS/Geiselhart et al. -
+        2021 - Automorphism Ensemble Decoding of Reed–Muller
+    Codes.pdf:application/pdf},
 }
+
+@article{dirac_new_1939,
+    title = {A new notation for quantum mechanics},
+    volume = {35},
+    issn = {1469-8064, 0305-0041},
+    url =
+    {https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/new-notation-for-quantum-mechanics/4631DB9213D680D6332BA11799D76AFB},
+    doi = {10.1017/S0305004100021162},
+    abstract = {In mathematical theories the question of notation,
+        while not of primary importance, is yet worthy of careful
+        consideration, since a good notation can be of great value in
+        helping the development of a theory, by making it easy to write
+        down those quantities or combinations of quantities that are
+        important, and difficult or impossible to write down those that
+        are unimportant. The summation convention in tensor analysis is
+    an example, illustrating how specially appropriate a notation can be.},
+    language = {en},
+    number = {3},
+    journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+    author = {Dirac, P. a. M.},
+    month = jul,
+    year = {1939},
+    keywords = {/unread},
+    pages = {416--418},
+}
+

src/fundamentals/main.tex

@@ -61,17 +61,105 @@
 \chapter{Fundamentals}
 \label{ch:Fundamentals}
 
+\Ac{qec} is a field of research combining quantum mechanics and
+``classical'' communications engineering.
+This chapter provides the relevant theoretical background on both of
+these topics and subsequently, building on top of this, introduces the
+the fundamentals of \ac{qec}.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % TODO: Is Quantum Information Theory the correct title here? Would someth
 \section{Quantum Mechanics and Quantum Information Science}
 \label{sec:Quantum Mechanics and Quantum Information Science}
 
+% TODO: Should the brief intro to QC be made later on or here?
+%%%%%%%%%%%%%%%%
+\subsection{Core Concepts and Notation}
+\label{subsec:Notation}
+
+\ldots can be very elegantly expressed using the language of
+linear algebra.
+\todo{Mention that we model the state of a quantum mechanical system
+as a vector}
+The so called Bra-ket or Dirac notation is especially appropriate,
+having been proposed by Paul Dirac in 1939 for the express purpose
+of simplifying quantum mechanical notation \cite{dirac_new_1939}.
+Two new symbols are defined, \emph{bra}s $\bra{\cdot}$ and
+\emph{ket}s $\ket{\cdot}$.
+Kets denote ordinary vectors, while bras denote their Hermitian conjugates.
+For example, two vectors specified by the labels $a$ and $b$
+respectively are written as $\ket{a}$ and $\ket{b}$.
+Their inner product is $\braket{a\vert b}$.
+
+\red{\textbf{Tensor product}}
+\red{\ldots
+    \todo{Introduce determinate state or use a different word?}
+    Take for example two systems with the determinate states $\ket{0}$
+    and $\ket{1}$. In general, the state of each can be written as the
+    superposition%
+    %
+    \begin{align*}
+        \alpha \ket{0} + \beta \ket{1}
+        .%
+    \end{align*}
+    %
+    Combining these two sytems into one, the overall state becomes%
+    %
+    \begin{align*}
+        &\mleft( \alpha_1 \ket{0} + \beta_1 \ket{1} \mright) \otimes
+        \mleft( \alpha_2 \ket{0} + \beta_2 \ket{1} \mright) \\
+        = &\alpha_1 \alpha_2 \ket{0} \ket{0}
+        + \alpha_1 \alpha_2 \ket{0} \ket{1}
+        + \beta_1 \alpha_2 \ket{1} \ket{0}
+        + \beta_1 \beta_2 \ket{1} \ket{1}
+        % =: &\alpha_{00} \ket{00}
+        % + \alpha_{01} \ket{01}
+        % + \alpha_{10} \ket{10}
+        % + \alpha_{11} \ket{11}
+        .%
+    \end{align*}%
+    %
+    \ldots When not ambiguous in the context, the tensor product
+    symbol may be omitted, e.g.,
+    \begin{align*}
+        \ket{0} \otimes \ket{0} = \ket{0}\ket{0}
+        .%
+    \end{align*}
+}
+
+As we will see, the core concept that gives quantum computing its
+power is entanglement. When two quantum mechanical systems are
+entangled, measuring the state of one will collapse that of the other.
+Take for example two subsystems with the overall state
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \mleft( \ket{0}\ket{0} +
+    \ket{1}\ket{1} \mright)
+    .%
+\end{align*}
+%
+If we measure the first subsystem as being in $\ket{0}$, we can
+be certain that a measurement of the second subsystem will also yield $\ket{0}$.
+Introducing a new notation for entangled states, we can write%
+%
+\begin{align*}
+    \ket{\psi} = \frac{1}{\sqrt{2}} \left( \ket{00} + \ket{11} \right)
+    .%
+\end{align*}
+%
+
+\subsection{Projective Measurements}
+\label{subsec:Projective Measurements}
+
+%%%%%%%%%%%%%%%%
+\subsection{Quantum Gates}
+\label{subsec:Quantum Gates}
+
 \red{
     \textbf{Content:}
     \begin{itemize}
         \item Bra-ket notation
         \item The tensor product
-        \item Notation for entangled states
         \item Projective measurements (the related operators,
             eigenvalues/eigenspaces, etc.)
             \begin{itemize}