Coherence Depletion in Quantum Algorithms
<p>In the case of <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> qubits, we plot the values of coherence with respect to the number of Grover iterations <span class="html-italic">k</span>: (<b>a</b>) the relative entropy of coherence <math display="inline"><semantics> <msubsup> <mi mathvariant="script">C</mi> <mi>r</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> in Equation (<a href="#FD13a-entropy-21-00260" class="html-disp-formula">13a</a>); (<b>b</b>) <math display="inline"><semantics> <mrow> <msubsup> <mi mathvariant="script">C</mi> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> <msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>′</mo> </msup> </msubsup> <mo>=</mo> <msub> <mo form="prefix">log</mo> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="script">C</mi> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <msubsup> <mi mathvariant="script">C</mi> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> being the <math display="inline"><semantics> <msub> <mi>l</mi> <mn>1</mn> </msub> </semantics></math>-norm of coherence in Equation (<a href="#FD13b-entropy-21-00260" class="html-disp-formula">13b</a>). The plots show the results with <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math> and 16 solutions, respectively. The minimal values indicate that an solution is found. As we can see, with the number of possible solutions increased, not only the number of Grover iterations needed decreases, but also the minimal value of coherence increases accordingly. See text and <a href="#entropy-21-00260-f002" class="html-fig">Figure 2</a> for more details.</p> "> Figure 2
<p>Minimal coherence of the system state with respect to the logarithm of the number of solutions <math display="inline"><semantics> <mrow> <msub> <mo form="prefix">log</mo> <mn>2</mn> </msub> <mi>M</mi> </mrow> </semantics></math>. As we can see, with the number of solutions <span class="html-italic">M</span> increased, the minimal value of coherence gets bigger which clearly indicates a superposition state consisting of more terms.</p> ">
Abstract
:1. Introduction
2. Resource Theory of Quantum Coherence
3. Grover’s Algorithm
- corresponds to the minimal values in Figure 1, namely the solution state .
- corresponds to the maximal values in Figure 1. Because of the square in this solution, there are actually two peaks close to each other (not quite visible if the number of solutions M is small). The right peak corresponds to the superposition state , while the left one corresponds to .
- corresponds to the local minimal values between the two peaks in Figure 1. Because the distance between these two peaks is exactly 1 and we are considering discrete operations, so this local valley has no physical meaning.
- means that there is no solution, i.e., .
4. Deutsch–Jozsa Algorithm
- If is a constant function, then
- If is a balanced function, then
5. Shor’s Algorithm/Quantum Order-Finding
6. Discussion
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Liu, Y.-C.; Shang, J.; Zhang, X. Coherence Depletion in Quantum Algorithms. Entropy 2019, 21, 260. https://doi.org/10.3390/e21030260
Liu Y-C, Shang J, Zhang X. Coherence Depletion in Quantum Algorithms. Entropy. 2019; 21(3):260. https://doi.org/10.3390/e21030260
Chicago/Turabian StyleLiu, Ye-Chao, Jiangwei Shang, and Xiangdong Zhang. 2019. "Coherence Depletion in Quantum Algorithms" Entropy 21, no. 3: 260. https://doi.org/10.3390/e21030260
APA StyleLiu, Y. -C., Shang, J., & Zhang, X. (2019). Coherence Depletion in Quantum Algorithms. Entropy, 21(3), 260. https://doi.org/10.3390/e21030260