QAOA - the project

HOBO vs QUBO

deep but not so much qubits

short, but many qubits

\theta

QUBO

HOBO

Why is HOBO interesting?

  1. Uses less qubits,
  2. By this feasible space takes most of the space
\frac{n!}{2^{n^2}} \approx \sqrt{2\pi n}\frac{2^{n\log_2 n - n\log_2 \mathrm e}}{2^{n^2}} \approx \color{LimeGreen} O(2^{-n^2})
\frac{n!}{2^{n \log_2(n)}} \approx \sqrt{2\pi n}\frac{n^n/\mathrm e^{-n}}{n^n} \approx \color{yellow} O(e^{-n})

Q: How many of bit assignments are feasible one (proper routes) for TSP?

QUBO:

HOBO:

We spend less time on finding feasible space, which we can use on optimizing the solution

This is not that simple

Solution: change the encoding (all the time)

HOBO requires much deeper circuits.

But can we benefit from reducing the feasible space, even w/o the improvement on the number of qubits?

1011_2 \leftrightarrow 0000100000000000

The idea

\Huge \dagger

HOBO-size mixer

Change of the encoding (from binary to one-hot vector at inverse)

QUBO objective encoding

\theta
\theta'
|00\ldots0\rangle
|00\ldots0\rangle
|+\rangle
|\varphi(\theta,\theta')\rangle

Quality measures

  • Number of qubits is the number of the worst encoding
  • Depth is equal to sum of depths of each part
  • Number of measurement is tricky is as for objective Hamiltonian
\Huge \dagger
\theta
\theta'

depth

How to transform encoding?

\textrm{Looks like depth is $O(n)$ where $n$ is max number}

First bunch of tasks

  • Implement quantum circuit for  QUBO (use parallelism)
  • implement encoding transformation (binary to unary)
  • make notes
    • prove it works (should be simple)
    • compute the depth of the circuit (up to constant)
    • determine no. measurements

Test if QAOA works!

Simplifications

\Huge \dagger
\theta
\theta'

Change of the encoding is a permutation, but because of we do uncomputation, we can do it up to local phases

Depth 11

Depth 7

RCCX|111\rangle = \mathrm {e}^{\mathrm i \varphi} | 110\rangle

Mixing objective Hamiltonian

\Huge \dagger
\theta
\theta'
\theta'

Part of objective Hamiltonian in HOBO

Part of objective Hamiltonian in QUBO

Q: What is the objective Hamiltonian? probably sum of them

Affects depth and no. of measurements

Starting in W-state

\theta
\theta'

Q: How to generate W-state?

Problem with mixer Hamiltonian - how to define such?

|W \rangle = \frac{1}{\sqrt n}(|100\ldots0\rangle+|010\ldots0\rangle+|001\ldots0\rangle + |000\ldots1\rangle)
|W \rangle

Automatic testing

\Huge \dagger
\theta
\theta'
\theta'
| 0\rangle

We can make measurement to test, whether we still are at right place, and continue if only 0s are measured

Q1: Does it give any benefits? - test by simulation

Q2: Does it improve anything? - assumption on the strength of the noise

Q3: error correction?

Tools

  • git (probably GitHub for now)
  • Anaconda + Python + qiskit, or
  • Julia + Yao
  • TeX
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