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<h2 class="hd hd-2 unit-title">Quantum Fourier Transform (1 of 3)</h2>
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Quantum Fourier Transform I
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The Quantum Fourier transform is a quantum analogue of the discrete Fourier transform. Let [mathjaxinline]j[/mathjaxinline] be a [mathjaxinline]n[/mathjaxinline] bit string. We define [mathjaxinline]U_{QFT}[/mathjaxinline] by </p>
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<tr>
<td class="equation" style="width:80%; border:none">[mathjax]U_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</td>
</tr>
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<p>
where [mathjaxinline]jk[/mathjaxinline] in the exponent is the ordinary multiplication. QFT is a key submodule used in another important algorithm called the phase estimation algorithm, which we are going to see in the next problem. </p>
<p>
Compute the following. Choose the global phase so that the coefficient for [mathjaxinline]|{00}\rangle[/mathjaxinline] becomes real. Input <tt class="ttfamily">i</tt> for the imaginary number [mathjaxinline]i[/mathjaxinline]. </p>
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[mathjaxinline]U_{QFT}|{01}\rangle[/mathjaxinline] </p>
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[mathjaxinline]U_{QFT}\left[\frac{1}{\sqrt{2}}\left(|{00}\rangle +|{11}\rangle \right)\right][/mathjaxinline] </p>
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[mathjaxinline]U_{QFT}^{\dagger }|{+++}\rangle[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Quantum Fourier Transform (2 of 3)</h2>
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Quantum Fourier Transform II
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<p>
Continuing in the scenario of Part I of this problem, recall that the Quantum Fourier transform is a quantum analogue of the discrete Fourier transform, where [mathjaxinline]j[/mathjaxinline] is a [mathjaxinline]n[/mathjaxinline] bit string and [mathjaxinline]U_{QFT}[/mathjaxinline] is defined as </p>
<table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]U_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.6)</td>
</tr>
</table>
<p>
where [mathjaxinline]jk[/mathjaxinline] in the exponent is the ordinary multiplication. </p>
<p>
[mathjaxinline]k[/mathjaxinline] in the equation above can be written in the binary form as [mathjaxinline]k=k_1k_2\dots k_{n}[/mathjaxinline], which is not to be confused with multiplication. In other words, [mathjaxinline]k = \sum _{l=0}^{n-1} 2^{l} k_{n-l}[/mathjaxinline] where [mathjaxinline]k_ i[/mathjaxinline] corresponds to the [mathjaxinline]i[/mathjaxinline]th bit of [mathjaxinline]k[/mathjaxinline]. </p>
<p>
By separately considering the sum for each bit, [mathjaxinline]U_{QFT} |{j}\rangle[/mathjaxinline] can be written in the tensor product form [mathjaxinline]U_{QFT} |{j}\rangle = |{\phi _1}\rangle \otimes |{\phi _2}\rangle \otimes \dots |{\phi _ n}\rangle[/mathjaxinline]. Find the expression for [mathjaxinline]|{\phi _ m}\rangle[/mathjaxinline] in terms of [mathjaxinline]j[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and [mathjaxinline]n[/mathjaxinline]. Answer using the ket notation. You can input e^(i*pi) for [mathjaxinline]e^{i\pi }[/mathjaxinline] etc. Do not forget to explicitly include <tt class="ttfamily">*</tt> for multiplication e.g. <tt class="ttfamily">2*|0&gt;</tt> etc. </p>
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Quantum Fourier Transform III
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<p>
Continuing in the scenario of Part I of this problem, recall that the Quantum Fourier transform is a quantum analogue of the discrete Fourier transform, where [mathjaxinline]j[/mathjaxinline] is a [mathjaxinline]n[/mathjaxinline] bit string and [mathjaxinline]U_{QFT}[/mathjaxinline] is defined as </p>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]U_{QFT} |{j}\rangle = \frac{1}{\sqrt{2^{n}}}\sum _{k=0}^{2^ n-1} e^{\frac{2\pi i jk}{2^ n}}|{k}\rangle[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.10)</td>
</tr>
</table>
<p>
where [mathjaxinline]jk[/mathjaxinline] in the exponent is the ordinary multiplication. </p>
<p>
Complete the following circuit, which does the inverse QFT [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline] for [mathjaxinline]n=2[/mathjaxinline]. If you are not sure about the circuit for QFT, you may refer to the textbook or lecture. </p>
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Phase Estimation I
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<p>
Let [mathjaxinline]U[/mathjaxinline] be a unitary with [mathjaxinline]U|{u_ i}\rangle =e^{2\pi i \phi _ i}|{u_ i}\rangle[/mathjaxinline] where [mathjaxinline]0 \leq \phi _ i &lt;1[/mathjaxinline]. The phase estimation algorithm allows one to find out [mathjaxinline]\phi _ i[/mathjaxinline]. This submodule is widely used in numbers of important quantum algorithms, such as factoring algorithm, period finding algorithm etc. </p>
<p>
The following figure is a part of the circuit for the phase estimation algorithm. The controlled- [mathjaxinline]U[/mathjaxinline] are applied controlled on [mathjaxinline]t[/mathjaxinline] qubits, which we call first register, to the bottom qubits, which we call second register. After this part, [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline], the inverse QFT, is applied to the first register and piecewise measurements are made on the first register. </p>
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</div>
<p>
Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. </p>
<p>
Suppose we take [mathjaxinline]|{u_ j}\rangle[/mathjaxinline] for [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure. The state right before the inverse QFT takes the tensor product form. We write [mathjaxinline]\phi _ j = \psi _ j/2^ t[/mathjaxinline] so that [mathjaxinline]\psi _ j[/mathjaxinline] is an integer. What is the state of the [mathjaxinline]m[/mathjaxinline]th qubit from the top in the first register right before the inverse QFT? Answer using the ket notation in terms of [mathjaxinline]\psi _ j[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], [mathjaxinline]m[/mathjaxinline]. Input psi_j for [mathjaxinline]\psi _ j[/mathjaxinline]. </p>
<p>
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'jump_to_url': jump_to_url,
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'get_search_module': get_search_module,
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'get': get,
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<h2 class="hd hd-2 unit-title">Phase Estimation (2 of 3)</h2>
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Phase Estimation II
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<p>
Continuing in the same scenario as the previous part of this problem: </p>
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<img src="/assets/courseware/v1/69ac7a3b1b5277055e84101077f55eed/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3b_phase_estimation.png" width="400" style="scale:0.6"/>
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<p>
Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. </p>
<p>
Comparing the output of the QFT algorithm and the result of the previous problem, you notice that output of the inverse QFT gives [mathjaxinline]\psi[/mathjaxinline] as output. (Notice that the QFT circuit ends with reordering the qubits). Now, let's consider specific examples. Let [mathjaxinline]U=X[/mathjaxinline]. What is the smallest [mathjaxinline]t[/mathjaxinline] needed to exactly specify [mathjaxinline]\phi[/mathjaxinline]? If we take such [mathjaxinline]t[/mathjaxinline], and take [mathjaxinline]|{-}\rangle[/mathjaxinline] for the [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure, what is the state of the first register after the inverse QFT? Answer in the ket notation. This state should be [mathjaxinline]t_{min}[/mathjaxinline]-qubit state where [mathjaxinline]t_{min}[/mathjaxinline] is the answer you got. </p>
<p>
<p style="display:inline">minimum [mathjaxinline]t[/mathjaxinline]:</p>
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</p>
<p>
<p style="display:inline">state after [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">Phase Estimation (3 of 3)</h2>
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Phase Estimation III
</h3>
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<div class="problem">
<div>
<p>
Continuing in the same scenario as the previous part of this problem: </p>
<div class="figure">
<center>
<img src="/assets/courseware/v1/69ac7a3b1b5277055e84101077f55eed/asset-v1:MITx+8.370.2x+1T2018+type@asset+block/images_ps3b_phase_estimation.png" width="400" style="scale:0.6"/>
</center>
</div>
<p>
Let us suppose that [mathjaxinline]\phi _ i[/mathjaxinline] can be exactly specified with [mathjaxinline]t[/mathjaxinline] decimal digits in a way that [mathjaxinline]\phi _ i = 0.\phi _{i,1}\phi _{i,2}\dots \phi _{i,t}[/mathjaxinline]. Note that it is the decimal for the binary representation, e.g. [mathjaxinline]1/2 = 0.1, 1/4 = 0.01[/mathjaxinline] etc. </p>
<p>
Let [mathjaxinline]U=S[/mathjaxinline]. What is the smallest [mathjaxinline]t[/mathjaxinline] needed to exactly specify [mathjaxinline]\phi[/mathjaxinline]? If we take such [mathjaxinline]t[/mathjaxinline], and take [mathjaxinline]|{1}\rangle[/mathjaxinline] for the [mathjaxinline]|{u}\rangle[/mathjaxinline] in the figure, what is the state of the first register after the inverse QFT? Answer in the ket notation. This state should be [mathjaxinline]t_{min}[/mathjaxinline]-qubit state where [mathjaxinline]t_{min}[/mathjaxinline] is the answer you got. </p>
<p>
<p style="display:inline">minimum [mathjaxinline]t[/mathjaxinline]:</p>
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<div class="unanswered ">
<input type="text" name="input_Phase_Estimation_III_2_1" id="input_Phase_Estimation_III_2_1" aria-describedby="status_Phase_Estimation_III_2_1" value="" class="math"/>
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<span class="status unanswered" id="status_Phase_Estimation_III_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_Phase_Estimation_III_2_1" class="equation">`{::}`</div>
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<p>
<p style="display:inline">state after [mathjaxinline]U_{QFT}^{\dagger }[/mathjaxinline]</p>
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<div class="unanswered ">
<input type="text" name="input_Phase_Estimation_III_3_1" id="input_Phase_Estimation_III_3_1" aria-describedby="status_Phase_Estimation_III_3_1" value="" class="math"/>
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<span class="status unanswered" id="status_Phase_Estimation_III_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_Phase_Estimation_III_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_Phase_Estimation_III_3_1_dynamath" name="input_Phase_Estimation_III_3_1_dynamath"/>
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