1 Introduction
2 Methods: model and solution
2.1 Model of quantum system
2.2 Architecture of artificial neural network
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cell state: \( c_t = c_{t-1}\circ \text {f}_t + {\tilde{c}}_t\circ \text {i}_t, \)
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hidden state: \( s_t=\tanh (c_t)\circ \text {o}_t, \)
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input gate: \( \text {i}_t = \text {sigma}(x_tV^\text {i}+s_{t-1}W^\text {i}), \)
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forget gate: \( \text {f}_t =\text {sigma}(x_tV^\text {f}+s_{t-1}W^\text {f}), \)
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output gate \( \text {o}_t = \text {sigma}(x_tV^\text {o}+s_{t-1}W^\text {o}), \)
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candidate: \( {\tilde{c}}_t = \tanh (x_tV^\text {o}+s_{t-1}W^\text {o}), \)
3 Results: experiments
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time of evolution \(T=6\),
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number of intervals \(n=32\),
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control pulses in \([-1, 1]\).
3.1 Performance of the neural network
\(H_{\gamma } \)
|
\(\gamma \)
| |||
---|---|---|---|---|
0.2 | 0.4 | 0.6 | 0.8 | |
\(\gamma (\sigma _y\otimes \mathbb {1})\)
| .96 | .92 | .95 | .96 |
\(\gamma (0.2\sigma _x\otimes \mathbb {1}+0.8\sigma _y\otimes \mathbb {1})\)
| .97 | .94 | .97 | .97 |
\(\gamma (0.5\sigma _x\otimes \mathbb {1}+0.5\sigma _y\otimes \mathbb {1})\)
| .98 | .96 | .98 | .97 |
\(\gamma (0.8\sigma _x\otimes \mathbb {1}+0.2\sigma _y\otimes \mathbb {1})\)
| .99 | .97 | .98 | .99 |
\(H_{\gamma } \)
|
\(\gamma \)
| |
---|---|---|
0.6 | 0.8 | |
\(\gamma (\sigma _y\otimes \sigma _y)\)
| .98 | .90 |
\(\gamma (0.1\sigma _x\otimes \sigma _x+0.8\sigma _y\otimes \sigma _y+0.1\sigma _z\otimes \sigma _z)\)
| .98 | .98 |
\(\gamma (0.1\sigma _x\otimes \sigma _x+0.6\sigma _y\otimes \sigma _y+0.3\sigma _z\otimes \sigma _z)\)
| .98 | .98 |
\(\gamma (0.2\sigma _x\otimes \sigma _x+0.6\sigma _y\otimes \sigma _y+0.2\sigma _z\otimes \sigma _z)\)
| .96 | .97 |
\(\gamma (0.3\sigma _x\otimes \sigma _x+0.6\sigma _y\otimes \sigma _y+0.1\sigma _z\otimes \sigma _z)\)
| .98 | .98 |
3.2 Utilization of the approximation
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Step 1 Select a NCP vector from the testing set and generate the corresponding nnDCP.
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Step 2 If the fidelity between target operator and operator resulting from nnDCP is lower than \(90\%\), return to Step 1.
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Step 3 Select \(i\in \{0,\ldots ,n-1\}\), \(j\in \{x,z\}\), change (i, j) coordinate of NCP by fixed \(\varepsilon \)
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\(\bullet \) if \(\text {NCP}_{i,j} < 1-\varepsilon \), then \(\text {NCP}_{i,j} +\varepsilon \),
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\(\bullet \) if \(\text {NCP}_{i,j} > 1-\varepsilon \), then \(\text {NCP}_{i,j} -\varepsilon \), and calculate new collection of \(\text {nnDCP}\) vectors with elements \(\text {nnDCP}^{i,j}\), denoting that the element was obtained by perturbing jth component at the ith time slot.
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Step 4 If the fidelity between target operator and operator resulting from \(\text {nnDCP}^{i,j}\) is lower than \(90\%\), return to Step 3.
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Step 5 Calculate norm \(l_2\) (Euclidean norm) of difference between nnDCP and \(\text {nnDCP}^{i,j}\).