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‎doc/pub/week4/html/week4-bs.html‎

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@@ -40,6 +40,7 @@
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2,
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None,
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'plans-for-the-week-of-february-10-14'),
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('Readings', 2, None, 'readings'),
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('Gates, the whys and hows', 2, None, 'gates-the-whys-and-hows'),
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('Structure of the lecture', 2, None, 'structure-of-the-lecture'),
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('Part 1: Mathematical background',
@@ -99,11 +100,19 @@
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2,
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None,
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'part-2-specific-realizations-and-famous-gates'),
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('Spin Hamiltonian', 2, None, 'spin-hamiltonian'),
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('Field along the $z$-axis', 2, None, 'field-along-the-z-axis'),
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('Bringing back a state on the Bloch sphere',
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2,
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None,
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'bringing-back-a-state-on-the-bloch-sphere'),
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('Time evolution', 2, None, 'time-evolution'),
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('Final expression', 2, None, 'final-expression'),
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('Part 3: Fanous Quantum gates, circuits and simple algorithms '
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'(parts from last week)',
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'(repetition from last week)',
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2,
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None,
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'part-3-fanous-quantum-gates-circuits-and-simple-algorithms-parts-from-last-week'),
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'part-3-fanous-quantum-gates-circuits-and-simple-algorithms-repetition-from-last-week'),
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('Quantum circuits', 2, None, 'quantum-circuits'),
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('Single-Qubit Gates', 2, None, 'single-qubit-gates'),
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('Widely used gates', 2, None, 'widely-used-gates'),
@@ -156,7 +165,18 @@
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2,
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None,
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'more-on-rotation-operators'),
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('Code example', 2, None, 'code-example')]}
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('Code example', 2, None, 'code-example'),
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('Topics next week', 2, None, 'topics-next-week'),
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('Exercises this week: Hamiltonians and project 1',
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2,
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None,
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'exercises-this-week-hamiltonians-and-project-1'),
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('Rewriting in terms of Pauli matrices',
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2,
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None,
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'rewriting-in-terms-of-pauli-matrices'),
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('The interaction part', 2, None, 'the-interaction-part'),
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('Measurement basis', 2, None, 'measurement-basis')]}
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end of tocinfo -->
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<body>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-february-10-14" style="font-size: 80%;">Plans for the week of February 10-14</a></li>
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<!-- navigation toc: --> <li><a href="#readings" style="font-size: 80%;">Readings</a></li>
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<!-- navigation toc: --> <li><a href="#gates-the-whys-and-hows" style="font-size: 80%;">Gates, the whys and hows</a></li>
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<!-- navigation toc: --> <li><a href="#structure-of-the-lecture" style="font-size: 80%;">Structure of the lecture</a></li>
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<!-- navigation toc: --> <li><a href="#part-1-mathematical-background" style="font-size: 80%;">Part 1: Mathematical background</a></li>
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<!-- navigation toc: --> <li><a href="#initial-state-preparation" style="font-size: 80%;">Initial state preparation</a></li>
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<!-- navigation toc: --> <li><a href="#final-expression" style="font-size: 80%;">Final expression</a></li>
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<!-- navigation toc: --> <li><a href="#part-2-specific-realizations-and-famous-gates" style="font-size: 80%;">Part 2: Specific realizations and famous gates</a></li>
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<!-- navigation toc: --> <li><a href="#part-3-fanous-quantum-gates-circuits-and-simple-algorithms-parts-from-last-week" style="font-size: 80%;">Part 3: Fanous Quantum gates, circuits and simple algorithms (parts from last week)</a></li>
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<!-- navigation toc: --> <li><a href="#spin-hamiltonian" style="font-size: 80%;">Spin Hamiltonian</a></li>
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<!-- navigation toc: --> <li><a href="#field-along-the-z-axis" style="font-size: 80%;">Field along the \( z \)-axis</a></li>
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<!-- navigation toc: --> <li><a href="#bringing-back-a-state-on-the-bloch-sphere" style="font-size: 80%;">Bringing back a state on the Bloch sphere</a></li>
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<!-- navigation toc: --> <li><a href="#time-evolution" style="font-size: 80%;">Time evolution</a></li>
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<!-- navigation toc: --> <li><a href="#final-expression" style="font-size: 80%;">Final expression</a></li>
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<!-- navigation toc: --> <li><a href="#part-3-fanous-quantum-gates-circuits-and-simple-algorithms-repetition-from-last-week" style="font-size: 80%;">Part 3: Fanous Quantum gates, circuits and simple algorithms (repetition from last week)</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-circuits" style="font-size: 80%;">Quantum circuits</a></li>
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<!-- navigation toc: --> <li><a href="#single-qubit-gates" style="font-size: 80%;">Single-Qubit Gates</a></li>
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<!-- navigation toc: --> <li><a href="#widely-used-gates" style="font-size: 80%;">Widely used gates</a></li>
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<!-- navigation toc: --> <li><a href="#possible-ansatzes" style="font-size: 80%;">Possible ansatzes</a></li>
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<!-- navigation toc: --> <li><a href="#more-on-rotation-operators" style="font-size: 80%;">More on rotation operators</a></li>
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<!-- navigation toc: --> <li><a href="#code-example" style="font-size: 80%;">Code example</a></li>
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<!-- navigation toc: --> <li><a href="#topics-next-week" style="font-size: 80%;">Topics next week</a></li>
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<!-- navigation toc: --> <li><a href="#exercises-this-week-hamiltonians-and-project-1" style="font-size: 80%;">Exercises this week: Hamiltonians and project 1</a></li>
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<!-- navigation toc: --> <li><a href="#rewriting-in-terms-of-pauli-matrices" style="font-size: 80%;">Rewriting in terms of Pauli matrices</a></li>
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<!-- navigation toc: --> <li><a href="#the-interaction-part" style="font-size: 80%;">The interaction part</a></li>
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<!-- navigation toc: --> <li><a href="#measurement-basis" style="font-size: 80%;">Measurement basis</a></li>
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</ul>
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</li>
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<ol>
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<li> Reminder from last week on gates and circuits</li>
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<li> One-qubit and two-qubit gates, background and realizations
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<!-- o Entropy as a measurement of entanglement --></li>
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<li> One-qubit and two-qubit gates, background and realizations</li>
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<li> Simple Hamiltonian systems</li>
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<li> <a href="https://youtu.be/" target="_self">Video of lecture to be added</a>
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<!-- o <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2025/NotesFebruary12.pdf" target="_self">Whiteboard notes</a> --></li>
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</ol>
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<!-- !split -->
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<h2 id="readings" class="anchor">Readings </h2>
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<ol>
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<li> For the discussion of one-qubit, two-qubit and other gates, sections 2.6-2.11 and 3.1-3.4 of Hundt's book <b>Quantum Computing for Programmers</b>, contain most of the relevant information.</li>
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</ol>
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<!-- !split -->
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<h2 id="gates-the-whys-and-hows" class="anchor">Gates, the whys and hows </h2>
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<p>In quantum computing it is common to rewrite various unitary
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<h2 id="structure-of-the-lecture" class="anchor">Structure of the lecture </h2>
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<ol>
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<li> First we review some of the basic ways of representing the solution to the Schr\"odinger's equation, introducing the so-called intenraction, Heisenberg and Schr\"odinger prictures and unitary transformations.</li>
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<li> First we review some of the basic ways of representing the solution to the Schr\"odinger equation, introducing the so-called Interaction, Heisenberg and Schr\"odinger prictures and unitary transformations.</li>
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<li> Secondly, we present examples of physical processes and how they can be represented as unitary operations on a given state.</li>
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<li> These unitary transformations are then represented as gates. Setting gates together gives us a final circuit which can represent a specific physical system</li>
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</ol>
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<!-- !split -->
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<h2 id="part-2-specific-realizations-and-famous-gates" class="anchor">Part 2: Specific realizations and famous gates </h2>
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<p>This part is not yet ready and will be part of the whiteboard notes.
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I will try to get it ready by the end of Tuesday, Feb 11
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<p>Nuclear magnetic resonance (NMR) quantum computing is one of the several
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proposed approaches for constructing a quantum computer. It uses the
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spin states of nuclei within molecules as qubits. The quantum states
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are probed through the nuclear magnetic resonances, allowing the
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system to be implemented as a variation of nuclear magnetic resonance
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spectroscopy. NMR differs from other implementations of quantum
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computers in that it uses an ensemble of systems, in this case
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molecules, rather than a single pure state.
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</p>
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<p>You can read more about this at <a href="https://cba.mit.edu/docs/papers/98.06.sciqc.pdf" target="_self"><tt>https://cba.mit.edu/docs/papers/98.06.sciqc.pdf</tt></a></p>
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<!-- !split -->
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<h2 id="spin-hamiltonian" class="anchor">Spin Hamiltonian </h2>
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<p>In order to understand in terms of a given Hamiltonian how the
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different gates arise, we consider now the Hamiltonian of a nuclear
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spin in a magnetic field. Since the spin provides provides a magnetic
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dipole moment, a nucleus with a spin will interact with the magnetic
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field. The Haniltonian of a nucleus with spin interacting with a
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magnetic field \( \boldsymbol{B} \) is
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</p>
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$$
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H = -\boldsymbol{\mu}\boldsymbol{B},
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$$
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<p>with \( \boldsymbol{\mu}=\gamma\boldsymbol{S} \), \( \gamma \) being the so-called gyromagnetic ratio and \( \boldsymbol{S} \) the spin.</p>
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<!-- !split -->
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<h2 id="field-along-the-z-axis" class="anchor">Field along the \( z \)-axis </h2>
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<p>It is common to let the spin interact with a constant magnetic field
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along the \( z \)-axis. This gives an effecitve Hamiltonian
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</p>
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$$
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H_z = -\frac{\hbar\omega_L}{2}\sigma_z,
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$$
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<p>where \( \omega_L \) is the so-called Larmor precession frequency. This
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quantity includes also the constant magnetif field along the
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\( z \)-axis. For all practical purposes it suffices for us to have an
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expression of the Hamiltonian in terms of the Pauli-Z matrix.
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</p>
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<!-- !split -->
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<h2 id="bringing-back-a-state-on-the-bloch-sphere" class="anchor">Bringing back a state on the Bloch sphere </h2>
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<p>Suppose that our initial one qubit state (for example a spin-\( 1/2 \)
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nucleus for NMR studies) points along some arbitrary axis. As
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discussed during our second week, a point on the Bloch sphere can be
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represented as at time \( t=0 \)
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</p>
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$$
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\vert \psi(t=0) \rangle = \vert \psi(0) \rangle=\cos{(\frac{\theta}{2})}\vert 0\rangle +\exp{\imath\phi}\sin{(\frac{\theta}{2})}\vert 1\rangle.
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$$
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<!-- !split -->
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<h2 id="time-evolution" class="anchor">Time evolution </h2>
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<p>Since the hamiltonian is time-independent, the state \( \vert \psi(0)
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\rangle \), our system will evolve according to unitary transformation
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</p>
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$$
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\vert \psi(t) \rangle = U(t)\vert \psi(0) \rangle=\exp{\imath\omega_L t\sigma_z/2}\vert \psi(0) \rangle.
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$$
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<p>Inserting the Bloch sphere ansatz we have then</p>
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$$
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\vert \psi(t) \rangle=\exp{\imath\omega_L t\sigma_z/2}\cos{(\frac{\theta}{2})}\vert 0\rangle +\exp{\imath\omega_L t\sigma_z/2}\exp{\imath\phi}\sin{(\frac{\theta}{2})}\vert 1\rangle.
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$$
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<!-- !split -->
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<h2 id="final-expression" class="anchor">Final expression </h2>
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<p>In exercise 4 from the second week, we showed that, given \( \boldsymbol{A} \) an operator on a vector space satisfying \( \boldsymbol{A}^2=1 \) and \( \alpha \) any real constant, we had</p>
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$$
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\exp{\imath\alpha \boldsymbol{A}}=\sum_{n=0}^{\infty} \frac{(i\alpha)^n}{n!}\boldsymbol{A}^n=\boldsymbol{I}\cos{\alpha}+\imath\boldsymbol{A}\sin{\alpha}.
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$$
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<p>Using this result, we obtain</p>
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$$
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\vert \psi(t) \rangle=\exp{\imath\omega_L t/2}\cos{(\frac{\theta}{2})}\vert 0\rangle +\exp{-\imath\omega_L t/2}\exp{\imath\phi}\sin{(\frac{\theta}{2})}\vert 1\rangle.
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$$
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<!-- !split -->
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<h2 id="part-3-fanous-quantum-gates-circuits-and-simple-algorithms-parts-from-last-week" class="anchor">Part 3: Fanous Quantum gates, circuits and simple algorithms (parts from last week) </h2>
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<h2 id="part-3-fanous-quantum-gates-circuits-and-simple-algorithms-repetition-from-last-week" class="anchor">Part 3: Fanous Quantum gates, circuits and simple algorithms (repetition from last week) </h2>
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<p>Quantum gates are physical actions that are applied to the physical
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system representing the qubits. Mathematically, they are
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the angles \( \theta \) and \( \phi \). This will lead us to the so-called Variational Quantum Eigensolver to be discussed next week.
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</p>
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<!-- !split -->
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<h2 id="topics-next-week" class="anchor">Topics next week </h2>
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<ol>
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<li> We will extend the above one-qubit Hamiltonian to a two-qubit problem and analyze how we can set up its simulation</li>
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<li> Before we introduce the Variational Quantum Eigensolver (VQE), we need to discuss entropy as a measure of entanglement</li>
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<li> If we get time, we start our discussion of the VQE algorithm</li>
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</ol>
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<!-- !split -->
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<h2 id="exercises-this-week-hamiltonians-and-project-1" class="anchor">Exercises this week: Hamiltonians and project 1 </h2>
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<p>As an initial test, we consider a simple \( 2\times 2 \) real
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Hamiltonian consisting of a diagonal part \( H_0 \) and off-diagonal part
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\( H_I \), playing the roles of a non-interactive one-body and interactive
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two-body part respectively. Defined through their matrix elements, we
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express them in the Pauli basis \( \vert 0\rangle \) and \( \vert 1 \rangle \)
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</p>
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$$
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\begin{align*}
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\begin{split}
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H &= H_0 + H_I \\
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H_0 = \begin{bmatrix}
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E_1 & 0 \\
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0 & E_2
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\end{bmatrix}&, \hspace{20px}
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H_I = \lambda \begin{bmatrix}
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V_{11} & V_{12} \\
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V_{21} & V_{22}
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\end{bmatrix}
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\end{split}
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\end{align*}
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$$
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<p>Where \( \lambda \in [0,1] \) is a coupling constant parameterizing the strength of the interaction. </p>
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<!-- !split -->
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<h2 id="rewriting-in-terms-of-pauli-matrices" class="anchor">Rewriting in terms of Pauli matrices </h2>
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<p>Define</p>
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$$
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E_+ = \frac{E_1 + E_2}{2},\hspace{20px} E_- = \frac{E_1 - E_2}{2}
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$$
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<p>show that by combining the identity and \( Z \) Pauli matrix, this can be expressed as</p>
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$$
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H_0 = E_+ I + E_- Z
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$$
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<!-- !split -->
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<h2 id="the-interaction-part" class="anchor">The interaction part </h2>
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<p>For \( H_1 \) we use the same trick to fill the diagonal, defining \( V_+ = (V_{11} + V_{22})/2, V_- = (V_{11} - V_{22})/2 \). From the hermiticity requirements of \( H \), we note that \( V_{12} = V_{21} \equiv V_o \). Use this to simplify the problem to a simple \( X \) term. </p>
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$$
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H_I = V_+ I + V_- Z + V_o X
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$$
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<!-- !split -->
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<h2 id="measurement-basis" class="anchor">Measurement basis </h2>
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<p>For the above system show that the Pauli \( X \) matrix can be rewritten in terms of the Hadamard matrices and the Pauli \( Z \) matrix, that is</p>
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$$
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X=HZH.
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$$
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