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manual/develop/en/html/_sources/algorithm/algorithms.rst.txt

@@ -183,7 +183,7 @@ Real-time evolution using iTPS (and other tensor network states) differs signifi
 
 One major difference is the size of the quantum entanglement of the target quantum state. In imaginary time evolution, as the evolution progresses towards the ground state, the quantum entanglement of the state does not become excessively large. Thus, the description by iTPS works well. However, in real-time evolution, typically (unless the initial state's iTPS is an eigenstate of the Hamiltonian), quantum entanglement can increase over time. To maintain the approximation accuracy of iTPS, it is necessary to increase the bond dimension of iTPS as the time gets longer. Naturally, increasing the bond dimension also increases computational costs, so with realistic computational resources, accurately approximating real-time evolution using iTPS is limited to short times. The applicable time range depends on the model, but for example, in spin models, the limit is often around a time :math:`t = O(1/J)` with respect to the typical interaction strength :math:`J`.
 
-Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment :math:`\delta t` of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the ``evolution`` section of the input file that is ultimately entered into tenes.
+Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment :math:`\delta t` of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the ``evolution`` section of the input file that is ultimately entered into TeNeS.
 
 Finite temperature simulation
 ===========================
@@ -262,7 +262,7 @@ Similarly to pure states, once the converged corner transfer matrices and edge t
 .. image:: ../../img/trace_Sz.*
    :align: center
 
-, the local magnetization :math:`\mathrm{Tr} (\rho S_i^z)` is calculated using the same diagram as :math:`\langle \Psi|S_i^z|\Psi\rangle`.
+the local magnetization :math:`\mathrm{Tr} (\rho S_i^z)` is calculated using the same diagram as :math:`\langle \Psi|S_i^z|\Psi\rangle`.
 
 Lastly, it is important to mention the drawbacks of approximation by iTPO. The density matrix of a mixed state is Hermitian and positive semidefinite, with non-negative eigenvalues. However, when approximating the density matrix with iTPO, this positive semidefiniteness is not guaranteed, and physical quantities calculated from the iTPO approximation might exhibit unphysical behavior, such as energies lower than the ground state energy. This is a problem of iTPO representation, and cannot be avoided just by improving the accuracy of CTMRG in expectation value calculation by increasing the bond dimension :math:`\chi`. To recover physical behavior, it is necessary to increase the bond dimension :math:`D` of iTPO to improve the approximation accuracy of the density matrix.
 

manual/develop/en/html/algorithm/algorithms.html

@@ -128,7 +128,7 @@ <h2><span class="section-number">6.4. </span>Real-time evolution by iTPS<a class
 <p>is approximated by iTPS, which allows for the calculation of approximate time evolution. The difference between imaginary and real-time evolution lies only in whether the coefficient of the Hamiltonian <span class="math notranslate nohighlight">\(\mathcal{H}\)</span> in the exponent is <span class="math notranslate nohighlight">\(-\tau\)</span> or <span class="math notranslate nohighlight">\(-it\)</span>, hence real-time evolution can also be computed using the same simple update and full update methods applied in imaginary time evolution, by employing the Suzuki-Trotter decomposition.</p>
 <p>Real-time evolution using iTPS (and other tensor network states) differs significantly from imaginary time evolution used for ground state calculation in two main aspects.</p>
 <p>One major difference is the size of the quantum entanglement of the target quantum state. In imaginary time evolution, as the evolution progresses towards the ground state, the quantum entanglement of the state does not become excessively large. Thus, the description by iTPS works well. However, in real-time evolution, typically (unless the initial state’s iTPS is an eigenstate of the Hamiltonian), quantum entanglement can increase over time. To maintain the approximation accuracy of iTPS, it is necessary to increase the bond dimension of iTPS as the time gets longer. Naturally, increasing the bond dimension also increases computational costs, so with realistic computational resources, accurately approximating real-time evolution using iTPS is limited to short times. The applicable time range depends on the model, but for example, in spin models, the limit is often around a time <span class="math notranslate nohighlight">\(t = O(1/J)\)</span> with respect to the typical interaction strength <span class="math notranslate nohighlight">\(J\)</span>.</p>
-<p>Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment <span class="math notranslate nohighlight">\(\delta t\)</span> of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the <code class="docutils literal notranslate"><span class="pre">evolution</span></code> section of the input file that is ultimately entered into tenes.</p>
+<p>Another difference is the characteristics of the physical phenomenon to be reproduced. When using imaginary time evolution to calculate the ground state, it is sufficient to reach the ground state after a sufficiently long evolution, so minor deviations from the correct path of imaginary time evolution are not a significant issue. On the other hand, in real-time evolution, there is often interest not only in the final state but also in the time evolution of the quantum state itself. To accurately approximate the path of time evolution, it is necessary to not only increase the bond dimension of iTPS but also to make the time increment <span class="math notranslate nohighlight">\(\delta t\)</span> of the Suzuki-Trotter decomposition sufficiently small. Depending on the situation, it may be more efficient to use higher-order Suzuki-Trotter decompositions. In TeNeS, it is possible to handle higher-order Suzuki-Trotter decompositions by editing the <code class="docutils literal notranslate"><span class="pre">evolution</span></code> section of the input file that is ultimately entered into TeNeS.</p>
 </section>
 <section id="finite-temperature-simulation">
 <h2><span class="section-number">6.5. </span>Finite temperature simulation<a class="headerlink" href="#finite-temperature-simulation" title="Permalink to this heading">¶</a></h2>
@@ -169,7 +169,7 @@ <h2><span class="section-number">6.5. </span>Finite temperature simulation<a cla
 <p>The computation cost of CTMRG for the corner transfer matrix representation with bond dimension <span class="math notranslate nohighlight">\(\chi\)</span> and iTPO with bond dimension <span class="math notranslate nohighlight">\(D\)</span> scales with <span class="math notranslate nohighlight">\(O(\chi^2 D^4)\)</span> and <span class="math notranslate nohighlight">\(O(\chi^3 D^3)\)</span>. Note that this computation cost is smaller compared to CTMRG for pure states with the same bond dimension <span class="math notranslate nohighlight">\(D\)</span>. The difference is due to the bond dimension of the tensor indicated by the black circle being <span class="math notranslate nohighlight">\(D^2\)</span> in pure state calculations, while <span class="math notranslate nohighlight">\(D\)</span> for mixed states. Correspondingly, the bond dimension <span class="math notranslate nohighlight">\(\chi\)</span> of the corner transfer matrices can be increased proportionally to <span class="math notranslate nohighlight">\(D\)</span>, i.e., <span class="math notranslate nohighlight">\(\chi \propto O(D)\)</span>. Under this condition, the computation cost of CTMRG becomes <span class="math notranslate nohighlight">\(O(D^6)\)</span>, and the required memory amount becomes <span class="math notranslate nohighlight">\(O(D^4)\)</span>. Thus, the computation cost of finite temperature calculations using iTPO is significantly lower than that of iTPS with the same <span class="math notranslate nohighlight">\(D\)</span>. It allows us to use larger bond dimensions <span class="math notranslate nohighlight">\(D\)</span> in finite temperature calculations.</p>
 <p>Similarly to pure states, once the converged corner transfer matrices and edge tensors are computed, <span class="math notranslate nohighlight">\(\mathrm{Tr} (\rho O)\)</span> can also be efficiently calculated. For example, when we define the tensor containing the operator as</p>
 <img alt="../_images/trace_Sz.png" class="align-center" src="../_images/trace_Sz.png" />
-<p>, the local magnetization <span class="math notranslate nohighlight">\(\mathrm{Tr} (\rho S_i^z)\)</span> is calculated using the same diagram as <span class="math notranslate nohighlight">\(\langle \Psi|S_i^z|\Psi\rangle\)</span>.</p>
+<p>the local magnetization <span class="math notranslate nohighlight">\(\mathrm{Tr} (\rho S_i^z)\)</span> is calculated using the same diagram as <span class="math notranslate nohighlight">\(\langle \Psi|S_i^z|\Psi\rangle\)</span>.</p>
 <p>Lastly, it is important to mention the drawbacks of approximation by iTPO. The density matrix of a mixed state is Hermitian and positive semidefinite, with non-negative eigenvalues. However, when approximating the density matrix with iTPO, this positive semidefiniteness is not guaranteed, and physical quantities calculated from the iTPO approximation might exhibit unphysical behavior, such as energies lower than the ground state energy. This is a problem of iTPO representation, and cannot be avoided just by improving the accuracy of CTMRG in expectation value calculation by increasing the bond dimension <span class="math notranslate nohighlight">\(\chi\)</span>. To recover physical behavior, it is necessary to increase the bond dimension <span class="math notranslate nohighlight">\(D\)</span> of iTPO to improve the approximation accuracy of the density matrix.</p>
 <p>As an alternative representation to avoid such unphysical behavior, a method has been proposed using purification of the density matrix, representing the purified density matrix with iTPO <a class="reference internal" href="#ref-purification"><span class="std std-ref">[Purification]</span></a>. However, in this case, the diagram appearing in the expectation value calculation becomes a double-layer structure similar to pure states. This structre requires a larger computational cost, and the manageable bond dimension <span class="math notranslate nohighlight">\(D\)</span> becomes smaller than in the direct iTPO representation.</p>
 <p class="rubric">References</p>