Implementation of ADAPT-QAOA

[desireevl]

Summary

Here is a WIP of an implementation of ADAPT-QAOA related to issue #6441. Creating PR on behalf of @Elliot-Coupe @kaiterm

Details and comments

ADAPT-QAOA is a variation of the QAOA algorithm.
A previous version of this PR was closed as a new fork was made: #7131

• Implementation of ADAPT-QAOA

This commit implements 'ADAPT-QAOA', an iterative and problem-tailored variant of QAOA, introduced by Zhu et al., "An Adaptive Quantum Approximate Optimization Algorithm for Solving Combinatorial Problems on a Quantum Computer" (2022). For a pre-defined set or 'pool' of mixers and a given ansatz depth, ADAPT-QAOA updates the subsequent ansatz layer with the mixer operator that maximises the energy gradient with respect to the mixer pool. The functionality required to reproduce Zhu et al.'s results are contained in the class 'AdaptQAOA' which inherits QAOA, largely overriding many VQE methods and attributes.

The following parent methods and attributes have been overridden:

1. Attribute 'initial_point' modified to coincide with the values stipulated in the supplementary material of Zhu et al.:
   If 'initial_point' is unspecified by the user, parameters γ and β are randomly selected from uniform distributions over [1e-2, 1e-1] and [-pi/2, pi/2] respectively.

2. Attribute 'optimizer' now defaults to COBYLA:
   It was found that the optimisation step size of the default QAOA optimizer, 'Nelder-Mead', was dependent on the magnitude of the initial points. COBYLA was determined to have the best performance of all available derivative-free optimizers and thus chosen as the replacement default.

3. Method '_check_operator_ansatz':
   Updated to set the hidden attribute '_ansatz' as 'AdaptQAOAAnsatz'. Also sets the hidden, boolean attribute '_solution', to False; flagging that a solution to the eigen problem has not been found.

4. Method 'compute_minimum_eigenvalue' modified to update mixer operators on sequential layers:
   Given an initial state and while the counting variable '_reps' is smaller than the attribute 'max_reps', this function:
   i. Determines the mixer that maximises the energy gradient with respect to the mixer pool & denotes it as the local variable 'energy_norm'.
   ii. Checks: if 'energy_norm' is less than the function input 'threshold', then the algorithm stops, otherwise,
   iii. the ansatz is updated with the mixer with the highest energy gradient & the usual VQE method 'compute_minimum_eigenvalue' is called.
   iv. If the difference in the computed and the true ground state energy is smaller than or equal to the function input 'solution_tolerance', the algorithm stops.
   v. Otherwise the initial points are updated with the points returned by the optimizer & the above steps (1-4) repeat.

5. Method 'construct_circuit' modified to build the QAOA ansatz circuit from a list of custom mixer operators:
   The user may now input, 'mixer_operators', an ordered list of operators to be used at each ansatz layer, with a length equal to the desired circuit depth.

6. Ansatz setter modified for compatability requirements of 'compute_minimum_eigenvalue':
   The dunder-class variable '__ansatz' is used to update the ansatz mixer operator for the next layer from the setter call on line (345). This was done to ensure that the ansatz was iteratively updated within AdaptQAOA and otherwise selectively modified by the inherited class methods. The ansatz attributes/ parameters are all updated call to '_update_ansatz_params' (see note - below).

The following parent methods and attributes have been removed:

1. Attribute 'reps' removed and fixed to 1:
   This was a necessary change that coordinates with the functionality required 'AdaptQAOAAnsatz'. A description for this change can be found in the initial commit message of 'AdaptQAOAAnsatz'.

The following methods, attributes and functions have been added:

1. Function 'energy_grad_operator' added:
   Given a cost operator and the optimal parameter γ associated to the last ansatz layer, this function computes the operator-valued energy gradient with respect to the input mixer operator. Analytically this operator is denoted by the commutator of the cost and mixer operator, conjugated by the cost operator evolution evaluated at γ (see Zhu et al. for further details).

2. Method '_get_ansatz_expectation' added:
   Mirrors the essential functionality of VQE's 'construct_expectation', in "generating the ansatz circuit & expectation value measurement..", however with three modifications. (i) Generates the required expectation value from ansatz function input, rather than the class attribute 'ansatz'. (ii) method '_check_operator_ansatz' removed as re-verifying the cost operator is a redundant operation. (iii) Parameters are only assigned to the ansatz if a non-empty set is provided to the function input.

3. Method 'energy_gradient_expectation' added:
   For an input circuit ansatz and a list of optimised variational parameters, this function computes the expectation value of the energy gradient operator returned by 'energy_grad_operator'. The first conditional statement checks the number of parameters for the input mixer operator and, if required, updates the number of β-parameters/ initial points (denoted by the class parameter '_num_beta') to be optimised in the ansatz via the method '_update_points'. Following the computation of the mixer-cost energy gradient operator, the expectation value for the provided ansatz is computed with the method '_get_ansatz_expectation'.

4. Method 'compute_mixer_pool_energy_grads' added:
   For an input 'ansatz', this function returns the list energy gradients associated to the list of mixer operators defined by the 'mixer_pool' input. The specified ansatz is first assigned parameters, provided as a function input, then the energy gradients are computed in a loop over the mixer_pool. Expectation values are computed with the circuit sampler inherited by VQE, multiplied by the imaginary value 'i' and appended into a list. The function then returns the norm of all values in this list.

5. Property 'ground_state_energy' added:
   Attempts to compute the lowest energy eigenvalue of the cost operator using numpy's eigensolver. This is used in an early stoppage condition of 'compute_minimum_eigenvalue'.

6. Attribute 'mixer_pool' added:
   An optional custom mixer, or list of custom mixers that define the "pool" of mixing operators that the AdaptQAOA algorithm chooses optimal mixers from. If left unspecified, then it will default to the complete set of 'single', 'multi' and 'standard' QAOA mixers, as generated by the 'adapt_mixer_pool' function in AdaptQAOAAnsatz.

7. Attribute 'mixer_pool_type' added:
   An optional string that denotes the class of mixer pool generated by the 'adapt_mixer_pool' function in AdaptQAOAAnsatz. Valid choices include 'single', 'singular' and 'multi'.

8. Attribute 'max_reps' added:
   Denotes the maximum number of algorithm repetitions and overall maximum ansatz depth.

9. Method '_update_ansatz_params' added:
   If there are no '_mixer_operators' that have been defined in the private variable '_ansatz', (i.e. no optimal mixers have been found) this function sets the initial parameters/ conditions for the iterative eigensolver. Otherwise, the internal class variable 'ansatz' is updated with the mixer associated with the maximal energy gradient using 'construct_circuit'. Similarly, the number of cost and mixer variational parameters, respectively denoted by '_num_gamma' and '_num_beta' are updated the current circuit ansatz configuration.

10. Method '_update_points' added:
    This private method functions to update a set of internal class parameters that modify the current circuit ansatz's: initial points, number of initial points and number of variational parameters. The integer-valued input 'num_gamma', defines the number of γ-parameters. If 'num_gamma' is None/ not specified, it is updated to the internal class parameter '_num_gamma' that defines the number of γ-parameters for the subsequent ansatz layer. Similarly, the input 'num_beta' defines a list of integers that are the number of β-parameters for each ansatz layer. If 'num_beta' is None/ has not been set, it is updated to the internal class parameter '_num_beta', a list that defines the number of β-parameters for all rep + 1 ansatz layers.
    The private class variable '__user_ip', encapsulates the user-defined initial points. The initial values for γ and β are assigned to the respective class variables '_gamma_ip' and '_beta_ip' by indexing '__user_ip' with 'num_beta' and 'num_gamma'. If no initial points were defined, then '_gamma_ip' and '_beta_ip' are generated by the respective sampling of log-uniform and uniform distributions over [1e-2, 1e-1] and [-pi/2, pi/2] respectively. Finally, '_gamma_ip' and '_beta_ip' are concatenated into a list, denoted by the class variable '_init_point_rep' that defines the initial points for the given repetition/ circuit depth in the eigensolver.

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[desireevl]

Hi @Cryoris, you helped to review #7131 before it was closed. We were wondering if you would please be able to review this newer version? Thanks!

[woodsp-ibm]

@desireevl There is work underway to refactor all the existing algorithms over to use primitives - Sampler or Estimator accordingly. Algorithms taking a Quantum Instance and executing circuits via that will be deprecated shortly and eventually removed. QAOA will be redone shortly. Maybe take a look at Sampler and hopefully a QAOA supporting that will also be done shortly to look at too. The new algorithms support minimum eigensolvers will still be in qiskit.algorithms but instead of being in minimum_eigen_solvers folder they will be in a minimum_eigensolvers (no underscore in latter part now). I think we would not want to introduce any more algos based on the the old quantum instance. Indeed the work to do AdaptVQE has been temporarily put on hold while VQE is redone.

[woodsp-ibm]

Thank you for your interest in contributing to Qiskit. It has however been decided that the algorithms are to be moved out of Qiskit (see #10406) and the plan is a new community repository that will contain these going forwards. As such I am going to close this PR as it stands.

If you still have interest in persuing this then might I suggest a community project in the ecosystem for this. See https://qiskit.org/ecosystem/ and follow/click the "Join the ecosystem" for more information on what that would entail.

I will note it has been a while and things have changed quite a bit with the introduction of primitives as noted in a comment above when that work commenced.