Dbi cost functions

doc/source/code-examples/applications-by-algorithm.rst

@@ -76,4 +76,13 @@ Diagonalization Algorithms
 .. toctree::
     :maxdepth: 1
 
-    tutorials/dbi/dbi.ipynb
+    tutorials/dbi/README.md
+
+    tutorials/dbi/dbi_cost_functions.ipynb
+    tutorials/dbi/dbi_gradient_descent_strategies.ipynb
+    tutorials/dbi/dbi_group_commutator_tests.ipynb
+    tutorials/dbi/dbi_scheduling.ipynb
+    tutorials/dbi/dbi_strategies_compare.ipynb
+    tutorials/dbi/dbi_strategy_Ising_model.ipynb
+    tutorials/dbi/dbi_strategy_Pauli-Z.ipynb
+    tutorials/dbi/dbi_tutorial_basic_intro.ipynb

doc/source/code-examples/applications-by-topic.rst

@@ -61,7 +61,6 @@ Quantum Physics
     tutorials/bell-variational/README.md
     tutorials/falqon/README.md
     tutorials/grover/README.md
-    tutorials/dbi/dbi.ipynb
 
 Quantum Machine Learning
 ^^^^^^^^^^^^^^^^^^^^^^^^

doc/source/code-examples/applications.rst

@@ -69,8 +69,6 @@ Quantum Physics
     tutorials/bell-variational/README.md
     tutorials/falqon/README.md
     tutorials/grover/README.md
-    tutorials/dbi/dbi.ipynb
-
 
 Quantum Machine Learning
 ^^^^^^^^^^^^^^^^^^^^^^^^

doc/source/code-examples/tutorials/dbi/README.md

@@ -0,0 +1 @@
+../../../../../examples/dbi/README.md

doc/source/code-examples/tutorials/dbi/dbi_cost_functions.ipynb

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+../../../../../examples/dbi/dbi_cost_functions.ipynb

doc/source/code-examples/tutorials/dbi/dbi_gradient_descent_strategies.ipynb

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+../../../../../examples/dbi/dbi_gradient_descent_strategies.ipynb

doc/source/code-examples/tutorials/dbi/dbi_group_commutator_tests.ipynb

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+../../../../../examples/dbi/dbi_group_commutator_tests.ipynb

doc/source/code-examples/tutorials/dbi/dbi_scheduling.ipynb

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+../../../../../examples/dbi/dbi_scheduling.ipynb

doc/source/code-examples/tutorials/dbi/dbi_strategies_compare.ipynb

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+../../../../../examples/dbi/dbi_strategies_compare.ipynb

doc/source/code-examples/tutorials/dbi/dbi_strategy_Ising_model.ipynb

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+../../../../../examples/dbi/dbi_strategy_Ising_model.ipynb

doc/source/code-examples/tutorials/dbi/dbi_strategy_Pauli-Z.ipynb

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+../../../../../examples/dbi/dbi_strategy_Pauli-Z.ipynb

doc/source/code-examples/tutorials/dbi/dbi_tutorial_basic_intro.ipynb

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+../../../../../examples/dbi/dbi_tutorial_basic_intro.ipynb

examples/dbi/README.md

@@ -0,0 +1,58 @@
+# Double-bracket quantum algorithms
+
+Qibo features a model implementing double-bracke quantum algorithms (DBQAs) which are helpful for approximating eigenstates based on the ability to run the evolution under the input Hamiltonian.
+
+More specifically, given a Hamiltonian $H_0$, how can we find a circuit which after applying to the reference state (usually $|0\rangle^{\otimes L}$ for $L$ qubits) will approximate an eigenstate?
+
+A standard way is to run variational quantum circuits. For example, Qibo already features  the `VQE` model [2] which provides the implementation of the variational quantum eigensolver framework.
+DBQAs allow to go beyond VQE in that they take a different approach to compiling the quantum circuit approximating the eigenstate.
+
+## What is the unitary of DBQA?
+
+Given $H_0$ we begin by assuming that we were given a diagonal and hermitian operator $D_0$ and a time $s_0$.
+The `dbi` module provides numerical strategies for selecting them.
+For any such choice we define the bracket
+$$ W_0 = [D_0, H_0]$$
+and the double-bracket rotation (DBR) of the input Hamiltonian to time $s$
+$$H_0(s) = e^{sW} H e^{- s W}$$
+
+### Why are double-bracket rotations useful?
+We can show that the magnitude of the off-diagonal norms will decrease.
+For this let us set the notation that $\sigma(A)$ is the restriction to the off-diagonal of the matrix A.
+In `numpy` this can be implemented by `\sigma(A) = A-np.diag(A)`. In Qibo we implement this as
+https://github.com/qiboteam/qibo/blob/8c9c610f5f2190b243dc9120a518a7612709bdbc/src/qibo/models/dbi/double_bracket.py#L145-L147
+which is part of the basic `DoubleBracketIteration` class in the `dbi` module.
+
+With this notation we next use the Hilbert-Schmidt scalar product and norm to measure the progress of diagonalization
+ $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s \langle W, [H,\sigma(H)\rangle+O(s^2)$$
+This equation tells us that as long as the scalar product $\langle W, [H,\sigma(H)\rangle$ is positive then after the DBR the magnitude of the off-diagonal couplings in $H_0(s)$ is less than in $H_0$.
+
+For the implementation of the DBR unitary $U_0(s) = e^{-s W_0}$ see
+https://github.com/qiboteam/qibo/blob/363a6e5e689e5b907a7602bd1cc8d9811c60ee69/src/qibo/models/dbi/double_bracket.py#L68
+
+### How to choose $D$?
+
+For theoretical considerations the canonical bracket is useful.
+For this we need the notation of the dephasing channel $\Delta(H)$ which is equivalent to `np.diag(h)`.
+ $M = [\Delta(H),\sigma(H)]= [H,\sigma(H)]= [\Delta(H),H]$
+ The canonical bracket appears on its own in the monotonicity relation above and gives an unconditional reduction of the magnitude of the off-diagonal terms
+ $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s ||M||^2+O(s^2)$$
+- the multi qubit Pauli Z generator with $Z(\mu) = (Z_1)^{\mu_1}\ldots (Z_L)^{\mu_L}$ where we optimize over all binary strings $\mu\in \{0,1\}^L$
+- the magnetic field $D = \sum_i B_i Z_i$
+- the two qubit Ising model $D  = \sum_i B_i Z_i + \sum_{i,j} J_{i,j} Z_i Z_j$, please follow the tutorial by Matteo and use the QIBO ising model for that with $h=0$
+
+
+### How to choose s?
+
+The theory above shows that in generic cases the DBR will have a linear diagonalization effect (as quantified by $||\sigma(H_0(s))||$).
+This can be further expanded with Taylor expansion and the Qibo implementation comes with methods for fitting the first local minimum.
+Additionally a grid search for the optimal step is provided for an exhaustive evaluation and hyperopt can be used for a more efficient 'unstructured' optimization; additionally simulated annealing is provided which sometimes outperforms hyperopt (and grid search), see example notebooks.
+The latter methods may output DBR durations $s_k$ which correspond to secondary local minima.
+
+
+
+
+
+[1] https://arxiv.org/abs/2206.11772
+
+[2] https://github.com/qiboteam/vqe-sun