Method of continuation for QAOA

The problem: It is difficult to optimize QAOA when there are many levels.

1. Optimize the whole circuit with m levels from randomly chosen 2m angles (bad approach)
2. Optimize the circuit for m-1levels, fix the angles and then optimize for m-th level. Repeat recursively (bad approach)
3. Optimize the circuit for m-1 levels, and then add two angles (based on some strategy) and optimize 2m angles again. can be improved by adding different two angles several times and choosing the best one
4. other methods (investigate)

Method of continuation for QAOA, cont.

New proposal

1. Choose somehow a large \(m\)
2. use the same circuit as for QAOA with mixer \(H_1\) and objective Hamiltonian \(H_2\)
3. \(\theta_0\) - random angles of length \(2m\)
4. for \(a = a_0, a_0 + \epsilon , a_0 + 2\epsilon , ..., 1\),
   1. get optimal angle \(\theta_{\rm opt}\}\) of the circuit according to objective Hamiltonian \((1-a) H_1 + a H_2\) starting with \(\theta_0\)
   2. \(\theta_0 \leftarrow \theta_{\rm opt}\)
5. \(\theta_0\) is the optimal angle.

Method of continuation for QAOA, cont.

Motivation

1. method of continuation (?) is a classical algorithm which  starts in the solution of a simple problem and gradually change it to a solution of the harder problem
2. At extreme case \(a = 0\) the optimal solution is (0,0,...,0) for objective Hamiltonian and arbitrary for mixer Hamiltonian. Hence for a close to 0 it should be simple to found a solution
3. Some motivation from adiabatic theorem??

Method of continuation for QAOA, cont.

Research problems

1. Can we, with the same number of layers, achieve lower energy states?
2. If not, can we reach it faster?
3. What are good choices of \(a_0\) (initial value) and \(\epsilon\) (step values)?

Thoughts

Almost optimal TSP Hamiltonian implementation

Quality measures of the encoding/implementation

1. number of qubits (noise)
2. the number of gates (noise)
3. the depth of the circuit (noise, time complexity)
4. number of measurements (time complexity)
5. number of parameterized gates (time complexity for gradient-based methods)
6. size of the effective space (time complexity/solution quality)

Secondary quality measures

1. volume
2. the difference of maximal and minimal energy

Almost optimal TSP Hamiltonian implementation

Minimal requirements for TSP for n cities

1. number of qubits: \(O(n \log n)\)
2. the number of gates: \(O(n^2)\)
3. the depth of the circuit: \(O(n^2/(n \log n))=n/\log n\)
4. number of measurements: \(n (\max_{i\neq j} W_{ij} - \min_{i\neq j} W_{ij})\) seems to be realistic, probably just \(O(n)\) for \(\max_{i,j}W_{ij} = O(1)\)
5. number of parameterized gates: \(O(n^2)\) seems realistic
6. size of the effective space: O(n!), also \(O(n^n)\) makes sense

Difference between \(O(n^n)\) and n! is only in \(O(\exp(n))\) which can be saved in \(O(n)\) qubits. This is why I consider \(n^n\) to be almost optimal

Almost optimal TSP Hamiltonian implementation

QAOA is worse in any aspect

NEW ENCODING

Green if optimal (up to polylog) - HAS TO BE CHECKED IF TRUE!

Similar work for Max-K-Cut: https://doi.org/10.1007/s42979-020-00437-z . Seems to be worse in many contexts

New encoding

The Hamitlonian

1. \(A_1\), \(A_2\) are penalty parameters
2. \([\varphi]\) is Iverson notation which gives one i \(\varphi\) is true, and zero otherwise
3. \(H_{\rm route\ cost}\) is a Hamiltonian which for valid \(b\) gives cost of the route

\([b\ \rm is\ permutation]\)

1. Let \(|b_1\rangle, \dots, |b_n \rangle\) be quantum registers for encoding cities in a binary way
2. Let \(|\#1\rangle,\ldots,|\#n\rangle \) be quantum registers which will store numbers of \(1,\dots, n\) in previous register
   1. for all \(b_i\) we add +1 to \(|\#k\rangle\) if city \(k\) is encoded in \(b_i\). This can be done through (easy to parallelize)
   2. storing the output of "is \(k\) in b_i" through "Toffoli" on a separate ancilla qubit \(|flag_i\rangle\)
   3. if the \(|flag_i\rangle\) is 1, then add 1 to \(|\#k\rangle\), then clean \(|flag_i\rangle\)
3. Apply XOR 1 on all \(|\# k\rangle\), and then NOT everywhere on it
4. Apply Toffoli conditioned on all \(|\#1\rangle, \dots,|\#n\rangle\) with output on \(|FLAG\rangle\)
5. Rotate \(|FLAG\rangle\) by \(R_z(\theta_i A_1)\)
6. uncompute 1.-4.

It's like in Grover, but with rotation except of \(Z\) gate

\([b\ \rm is\ permutation]\) (analysis)

1. \(O(n\log n)\) ancilla qubits required
2. \(O(n^2 {\rm poly}(\log n))\) gates (to be confirmed)
3. \(O(n\log n)\) depth (to be confirmed)

Parallelization: we have to make (or not) addition for each pair \((b_t, \#k)\). However this can be done as follows: first all pairs \((b_i,\#i)\), then \((b_i,\#(i+1))\), etc.

\([b_t\ \rm is\ valid]\)

Cone be done as in paper of me, Zoltan and Ola, which gives already \(O(n)\) complexity, but we can already start in \(\sum_{x=0}^{n-1}|x\rangle\) (obviously tensored to \(n\)-th power)

With proper choice of \(\alpha_{?}\) we should be able to generate any real-valued superposition

\(\log n-2\) ancilla qubits, \(O(n)\) depth, \(O(n {\rm poly}(\log n))\) no. gates

Then we can use GM-like oracle applied \(|b_i\rangle\)-wise and drop the space to \(n^n\)

Cost Hamiltonian (check!)

1. Let \(|b_1\rangle, \dots, |b_n \rangle\) be quantum registers for encoding cities in a binary way
2. Let \(|\#1\rangle,\ldots,|\#n\rangle \) be quantum registers which will store numbers of \(1,\dots, n\) in previous register
   1. for all \(b_i\) we add +1 to \(|\#k\rangle\) if city \(k\) is encoded in \(b_i\). This can be done through (easy to parallelize)
   2. storing the output of "is \(k\) in b_i" through "Toffoli" on a separate ancilla qubit \(|flag_i\rangle\)
   3. if the \(|flag_i\rangle\) is 1, then add 1 to \(|\#k\rangle\), then clean \(|flag_i\rangle\)
3. For each pair \((\#k, \#m)\) (can be parallelized)
   1. save the flag of "both are ones" to \(|FLAG\rangle\)
   2. Rotate \(|FLAG\rangle\) by \(R_z(\theta_i W_{k,m})\)
4. all but \(R_z\) rotation

This is less demanding version of permutation checking

Why this is interesting

1. We created an objective Hamiltonian simulation which is "almost the best of them all in every context I know"
2. It should give the same quality per layer as HOBO (which was the best of all I know), perhaps even better
3. The Hamiltonian-way encoding cannot reach such good quality measures values ("proof": if we want to encode the Hamiltonian in a way that we have different register for different timepoints, then between each pair of such registers we have to include whole information about \(W\), which results in \(O(n^3)\) gates in total (as we have \(n^2\) values in \(W\) and \(n\) pairs))

Questions two answer

• What is the exact form of Hamiltonian (cost route part not obvious)
• What about not-full connectivity of the hardware (linear or square) can we say something?
• Can we extend this to other algorithms? Max-k-Cut, VRP, job scheduling etc.