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further update
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‎doc/pub/week6/html/week6-bs.html‎

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('Explicit expressions', 2, None, 'explicit-expressions'),
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('Plans for the week of February March 1',
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@@ -326,6 +342,10 @@
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<!-- navigation toc: --> <li><a href="#definitions" style="font-size: 80%;">Definitions</a></li>
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<!-- navigation toc: --> <li><a href="#the-hamiltonian-in-terms-of-pauli-boldsymbol-x-and-pauli-boldsymbol-z-matrices" style="font-size: 80%;">The Hamiltonian in terms of Pauli-\( \boldsymbol{X} \) and Pauli-\( \boldsymbol{Z} \) matrices</a></li>
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<!-- navigation toc: --> <li><a href="#how-do-we-perform-measurements" style="font-size: 80%;">How do we perform measurements?</a></li>
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<!-- navigation toc: --> <li><a href="#why-do-we-measure-on-one-qubit-first-consideration" style="font-size: 80%;">Why do we measure on one qubit? First consideration</a></li>
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<!-- navigation toc: --> <li><a href="#why-do-we-measure-on-one-qubit-second-consideration" style="font-size: 80%;">Why do we measure on one qubit? Second consideration</a></li>
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<!-- navigation toc: --> <li><a href="#why-do-we-measure-on-one-qubit-third-consideration" style="font-size: 80%;">Why do we measure on one qubit? Third consideration</a></li>
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<!-- navigation toc: --> <li><a href="#why-do-we-measure-on-one-qubit-fourth-consideration" style="font-size: 80%;">Why do we measure on one qubit? Fourth consideration</a></li>
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<!-- navigation toc: --> <li><a href="#explicit-expressions" style="font-size: 80%;">Explicit expressions</a></li>
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<!-- navigation toc: --> <li><a href="#plans-for-the-week-of-february-march-1" style="font-size: 80%;">Plans for the week of February March 1</a></li>
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@@ -2145,7 +2165,7 @@ <h2 id="the-hamiltonian-in-terms-of-pauli-boldsymbol-x-and-pauli-boldsymbol-z-ma
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<h2 id="how-do-we-perform-measurements" class="anchor">How do we perform measurements? </h2>
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<p>The above tensor products need to rewritten in terms of specific
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transformations so that we can perform the measumrents in the basis of
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transformations so that we can perform the measurements in the basis of
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the Pauli-\( \boldsymbol{Z} \) matrices. As we discussed earlier, we need to find
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a transformation of the form
21512171
</p>
@@ -2161,6 +2181,73 @@ <h2 id="how-do-we-perform-measurements" class="anchor">How do we perform measure
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<p>The implementation of these measurements will be discussed next week.</p>
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<!-- !split -->
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<h2 id="why-do-we-measure-on-one-qubit-first-consideration" class="anchor">Why do we measure on one qubit? First consideration </h2>
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<p>In quantum computing, measurements are typically performed on one
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qubit at a time due to a combination of theoretical, practical, and
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algorithmic considerations:
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</p>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> Adaptive Processing: Many quantum algorithms, such as quantum teleportation or error correction, require mid-circuit measurements. The outcomes determine subsequent operations, necessitating sequential measurements to adapt the circuit dynamically.</li>
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<li> Partial Information Extraction: Algorithms often need only specific qubits' results (e.g., in Shor's algorithm), making full-system measurements unnecessary.</li>
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</ol>
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</div>
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</div>
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<!-- !split -->
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<h2 id="why-do-we-measure-on-one-qubit-second-consideration" class="anchor">Why do we measure on one qubit? Second consideration </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> Collapse and Entanglement: Measuring a qubit collapses its state, potentially affecting entangled qubits. Sequential measurements allow controlled extraction of information while managing entanglement.</li>
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<li> Measurement Basis: Most algorithms use the computational basis (individual qubit measurements). Joint measurements in entangled bases are possible but require complex setups and are not always needed.</li>
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</ol>
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</div>
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</div>
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<!-- !split -->
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<h2 id="why-do-we-measure-on-one-qubit-third-consideration" class="anchor">Why do we measure on one qubit? Third consideration </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> Crosstalk and Noise: Simultaneous measurements risk disturbing neighboring qubits due to hardware imperfections, especially in noisy intermediate-scale quantum (NISQ) devices.</li>
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<li> Readout Constraints: Physical implementations (e.g., superconducting qubits) may have limited readout bandwidth, forcing sequential measurements.</li>
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</ol>
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</div>
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</div>
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<!-- !split -->
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<h2 id="why-do-we-measure-on-one-qubit-fourth-consideration" class="anchor">Why do we measure on one qubit? Fourth consideration </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<ol>
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<li> Qubit Reuse: Ancilla qubits (e.g., in error correction) are measured, reset, and reused, requiring sequential handling to avoid disrupting computational qubits.</li>
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</ol>
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</div>
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</div>
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<div class="panel panel-default">
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<div class="panel-body">
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<!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>While joint measurements are theoretically possible, the dominant practice of measuring one qubit at a time stems from algorithmic adaptability, hardware limitations, and the need to minimize quantum state disturbance. This approach balances efficiency, practicality, and the constraints of current quantum systems.</p>
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</div>
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</div>
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<!-- !split -->
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<h2 id="explicit-expressions" class="anchor">Explicit expressions </h2>
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<p>In order to perform our measurements, will then need the following operators \( \boldsymbol{U} \)</p>
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<p>where we have </p>
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$$
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\text{CX}_{10} = \begin{bmatrix}
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0 & 1 & 0 & 0 \\
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1 & 0 & 0 & 0 \\
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0 & 0 & 0 & 1 \\
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0 & 0 & 1 & 0 \\
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0 & 0 & 0 & 1
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0 & 1 & 0 & 0
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\end{bmatrix}.
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$$
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‎doc/pub/week6/html/week6-reveal.html‎

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@@ -2188,7 +2188,7 @@ <h2 id="the-hamiltonian-in-terms-of-pauli-boldsymbol-x-and-pauli-boldsymbol-z-ma
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<h2 id="how-do-we-perform-measurements">How do we perform measurements? </h2>
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<p>The above tensor products need to rewritten in terms of specific
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transformations so that we can perform the measumrents in the basis of
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transformations so that we can perform the measurements in the basis of
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the Pauli-\( \boldsymbol{Z} \) matrices. As we discussed earlier, we need to find
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a transformation of the form
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</p>
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<p>The implementation of these measurements will be discussed next week.</p>
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</section>
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<section>
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<h2 id="why-do-we-measure-on-one-qubit-first-consideration">Why do we measure on one qubit? First consideration </h2>
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<p>In quantum computing, measurements are typically performed on one
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qubit at a time due to a combination of theoretical, practical, and
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algorithmic considerations:
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</p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Algorithmic Requirements:</b>
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<p>
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<ol>
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<p><li> Adaptive Processing: Many quantum algorithms, such as quantum teleportation or error correction, require mid-circuit measurements. The outcomes determine subsequent operations, necessitating sequential measurements to adapt the circuit dynamically.</li>
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<p><li> Partial Information Extraction: Algorithms often need only specific qubits' results (e.g., in Shor's algorithm), making full-system measurements unnecessary.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="why-do-we-measure-on-one-qubit-second-consideration">Why do we measure on one qubit? Second consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Quantum Mechanical Principles:</b>
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<p>
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<ol>
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<p><li> Collapse and Entanglement: Measuring a qubit collapses its state, potentially affecting entangled qubits. Sequential measurements allow controlled extraction of information while managing entanglement.</li>
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<p><li> Measurement Basis: Most algorithms use the computational basis (individual qubit measurements). Joint measurements in entangled bases are possible but require complex setups and are not always needed.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="why-do-we-measure-on-one-qubit-third-consideration">Why do we measure on one qubit? Third consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Practical Hardware Limitations:</b>
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<p>
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<ol>
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<p><li> Crosstalk and Noise: Simultaneous measurements risk disturbing neighboring qubits due to hardware imperfections, especially in noisy intermediate-scale quantum (NISQ) devices.</li>
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<p><li> Readout Constraints: Physical implementations (e.g., superconducting qubits) may have limited readout bandwidth, forcing sequential measurements.</li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="why-do-we-measure-on-one-qubit-fourth-consideration">Why do we measure on one qubit? Fourth consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Resource Management:</b>
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<p>
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<ol>
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<p><li> Qubit Reuse: Ancilla qubits (e.g., in error correction) are measured, reset, and reused, requiring sequential handling to avoid disrupting computational qubits.</li>
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</ol>
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</div>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Conclusion:</b>
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<p>
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<p>While joint measurements are theoretically possible, the dominant practice of measuring one qubit at a time stems from algorithmic adaptability, hardware limitations, and the need to minimize quantum state disturbance. This approach balances efficiency, practicality, and the constraints of current quantum systems.</p>
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</div>
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</section>
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<section>
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<h2 id="explicit-expressions">Explicit expressions </h2>
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<p>In order to perform our measurements, will then need the following operators \( \boldsymbol{U} \)</p>
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<p>&nbsp;<br>
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$$
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\text{CX}_{10} = \begin{bmatrix}
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0 & 1 & 0 & 0 \\
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1 & 0 & 0 & 0 \\
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0 & 0 & 0 & 1 \\
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0 & 0 & 1 & 0 \\
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0 & 0 & 0 & 1
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0 & 1 & 0 & 0
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\end{bmatrix}.
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$$
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<p>&nbsp;<br>

‎doc/pub/week6/html/week6-solarized.html‎

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('Why do we measure on one qubit? First consideration',
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('Plans for the week of February March 1',
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2,
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<h2 id="how-do-we-perform-measurements">How do we perform measurements? </h2>
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<p>The above tensor products need to rewritten in terms of specific
2067-
transformations so that we can perform the measumrents in the basis of
2083+
transformations so that we can perform the measurements in the basis of
20682084
the Pauli-\( \boldsymbol{Z} \) matrices. As we discussed earlier, we need to find
20692085
a transformation of the form
20702086
</p>
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<p>The implementation of these measurements will be discussed next week.</p>
20822098

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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-do-we-measure-on-one-qubit-first-consideration">Why do we measure on one qubit? First consideration </h2>
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<p>In quantum computing, measurements are typically performed on one
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qubit at a time due to a combination of theoretical, practical, and
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algorithmic considerations:
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</p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Algorithmic Requirements:</b>
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<p>
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<ol>
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<li> Adaptive Processing: Many quantum algorithms, such as quantum teleportation or error correction, require mid-circuit measurements. The outcomes determine subsequent operations, necessitating sequential measurements to adapt the circuit dynamically.</li>
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<li> Partial Information Extraction: Algorithms often need only specific qubits' results (e.g., in Shor's algorithm), making full-system measurements unnecessary.</li>
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</ol>
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</div>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-do-we-measure-on-one-qubit-second-consideration">Why do we measure on one qubit? Second consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Quantum Mechanical Principles:</b>
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<p>
2123+
<ol>
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<li> Collapse and Entanglement: Measuring a qubit collapses its state, potentially affecting entangled qubits. Sequential measurements allow controlled extraction of information while managing entanglement.</li>
2125+
<li> Measurement Basis: Most algorithms use the computational basis (individual qubit measurements). Joint measurements in entangled bases are possible but require complex setups and are not always needed.</li>
2126+
</ol>
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</div>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-do-we-measure-on-one-qubit-third-consideration">Why do we measure on one qubit? Third consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Practical Hardware Limitations:</b>
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<p>
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<ol>
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<li> Crosstalk and Noise: Simultaneous measurements risk disturbing neighboring qubits due to hardware imperfections, especially in noisy intermediate-scale quantum (NISQ) devices.</li>
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<li> Readout Constraints: Physical implementations (e.g., superconducting qubits) may have limited readout bandwidth, forcing sequential measurements.</li>
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</ol>
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</div>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="why-do-we-measure-on-one-qubit-fourth-consideration">Why do we measure on one qubit? Fourth consideration </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Resource Management:</b>
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<p>
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<ol>
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<li> Qubit Reuse: Ancilla qubits (e.g., in error correction) are measured, reset, and reused, requiring sequential handling to avoid disrupting computational qubits.</li>
2151+
</ol>
2152+
</div>
2153+
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<div class="alert alert-block alert-block alert-text-normal">
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<b>Conclusion:</b>
2156+
<p>
2157+
<p>While joint measurements are theoretically possible, the dominant practice of measuring one qubit at a time stems from algorithmic adaptability, hardware limitations, and the need to minimize quantum state disturbance. This approach balances efficiency, practicality, and the constraints of current quantum systems.</p>
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</div>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="explicit-expressions">Explicit expressions </h2>
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<p>In order to perform our measurements, will then need the following operators \( \boldsymbol{U} \)</p>
@@ -2095,10 +2173,10 @@ <h2 id="explicit-expressions">Explicit expressions </h2>
20952173
<p>where we have </p>
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$$
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\text{CX}_{10} = \begin{bmatrix}
2098-
0 & 1 & 0 & 0 \\
20992176
1 & 0 & 0 & 0 \\
2177+
0 & 0 & 0 & 1 \\
21002178
0 & 0 & 1 & 0 \\
2101-
0 & 0 & 0 & 1
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0 & 1 & 0 & 0
21022180
\end{bmatrix}.
21032181
$$
21042182

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