Quantum Approximate Optimization Algorithm
How it works
|++\ldots + \rangle
\exp(-\mathrm i \theta_{i+1}\sum_i X_i)
\exp(-\mathrm i \theta_i H_{\rm obj})
Quantum evolutiom
i \leftarrow i+2
| \varphi(\theta) \rangle
Thetas are optimized (somehow)!
Quantum annealing
Step 1.: write as a HOBO
A_1\sum\limits_{t=1}^N H_{\rm valid} (b_t) + A_2\sum\limits_{t=1}^N \sum\limits_{t'=t+1}^N H_{\neq}(b_t,b_{t'}) + B\sum\limits_{\substack{i,j=1\\i\neq j}}^N W_{ij} \sum\limits_{t=1}^N
H_\delta (b_t,i)H_\delta (b_{t+1},j).
A sum of constraints and objective hamiltonians
Need to choose proper values A_1, A_2, B.
Quantum annealing
Step 2.: turn into a 2-local Ising model
H(b)=\sum_{I\subseteq \{1,\dots,n\}} \alpha_I \prod_{i\in I} b_i
H(s \cup s') = \sum_{i\neq j} J_{ij} s_is_j + \sum h_i s_i
H(s) =\sum_{I\subseteq \{1,\dots,n\}} \tilde \alpha_I \prod_{i\in I} s_i
b_i \leftarrow (1-s_i)/2
Alternatively turn to QUBO and then to 2-local Ising model
x_1x_2x_3 \leftarrow \min _z (zx_3 + B (x_1x_2 - 2 (x_1+x_2)z+3z)
non unique!
New qubits!
Quantum annealing
Step 3.: fit to interaction graph
H(s \cup s') = \sum_{i\neq j} J_{ij} s_is_j + \sum h_i s_i
J_{ij} != 0 iff s_i and s_j are connected - more qubits needed!
Quantum annealing
Conclusions
- D-Wave technology requires 2-local interactions following special topology (Chimera, Pegasus)
- Thus many qubits are required
- As long as we can write as HOBO (and we can fit it) we can run it (the whole procedure is known)
QAOA
Step 1.: write as a HOBO
A_1\sum\limits_{t=1}^N H_{\rm valid} (b_t) + A_2\sum\limits_{t=1}^N \sum\limits_{t'=t+1}^N H_{\neq}(b_t,b_{t'}) + B\sum\limits_{\substack{i,j=1\\i\neq j}}^N W_{ij} \sum\limits_{t=1}^N
H_\delta (b_t,i)H_\delta (b_{t+1},j).
A sum of constraints and objective hamiltonians
Need to choose proper values A_1, A_2, B.
QAOA
Step 2.: turn into a Ising model
H(b)=\sum_{I\subseteq \{1,\dots,n\}} \alpha_I \prod_{i\in I} b_i
H(s) =\sum_{I\subseteq \{1,\dots,n\}} \tilde \alpha_I \prod_{i\in I} s_i
b_i \leftarrow (1-s_i)/2
No extra qubits needed in principle!, however there may be more nonzero terms
QAOA
Step 2a.: how to implement mixer
\exp(-\mathrm i \theta_{i+1}\sum_i X_i)
\exp(-\mathrm i \theta_i H_{\rm obj})
\exp(-\mathrm i \theta_{i+1}\sum_i X_i) = \prod_i \exp(-\mathrm i \theta_{i+1}X_i)
Always commuting!
QAOA
Step 2b.: how to implement obj. Ham.
\exp(-\mathrm i \theta_{i+1}\sum_i X_i)
\exp(-\mathrm i \theta_i H_{\rm obj})
\exp(-\mathrm i \theta_{i}(\alpha_1 Z_1 Z_2 Z_3+ \alpha_2 Z_1Z_2 )) \\=
\exp(-\mathrm i \theta_{i}\alpha_1 Z_1 Z_2 Z_3 ) \exp(-\mathrm i \theta_{i} \alpha_2 Z_1 Z_2 )
Always commuting!
\theta_i\alpha_1
\theta_i\alpha_2
QAOA
Step 2b.: how to implement obj. Ham.
\exp(-\mathrm i \theta_{i+1}\sum_i X_i)
\exp(-\mathrm i \theta_i H_{\rm obj})
\exp(-\mathrm i \alpha Z Z \cdots Z)
Extra qubit needed but CNOT-s commutes!
Note that not all qubits may be connected for QC as well - but we can SWAP
QAOA
Step 3.: run the optimizer
Depending on classical algorithm we use we may need:
- Estimate energy (simple)
- Estimate gradient (harder but manageable)
- Estimate Hessian (super-hard)
The more derivatives we know - usually the better the algorithm works
Math
We need to estimate the energy of the state given Hamiltonian
|\varphi\rangle
Measurement
0100100...10
Hamiltonian
E_i
The mean of energies E_i is expectedly the correct energy
\langle \varphi | H |\varphi \rangle
Math
We may need to estimate the gradient
First way: calculate from the definition of gradient based on
(E_\theta)' \approx \frac{E_{\theta+\varepsilon} - E_{\theta-\varepsilon}}{2\varepsilon}
\mathbb E(E_\theta)' \neq \mathbb E \left ( \frac{E_{\theta+\varepsilon} - E_{\theta-\varepsilon}}{2\varepsilon} \right )
Problem:
One can choose a different linear combination of energies for different thetas, however then the time complexity grows
Math
Energy estimation techniques
1.\quad H = \sum_i \alpha_i P_i \implies \langle \varphi | H | \varphi \rangle = \sum_i\alpha_i \langle \varphi | P_i | \varphi \rangle
Hence we can estimate energy for each P_i independently.
2. Since all P_i commute, we can estimate energy based on bit-strings coming out from measurement.
\mathbb{P} (| X - \mathbb E X | \geq t ) \leq 2\exp\left (- \frac{2M t^2}{\max X-\min X} \right ).
NISQ assumptions
- QC are noisy (we are not able to implement exactly the unitary gate)
- Measurements are noisy (outputs, and by this energies are not always reliable,
- We don't have much qubits
QAOA simulations
Levels of simulation truth
1. Calculate |phi> and then classicaly calculate <phi|H|phi>
"fast", but completely unphysical
2. Calculate |phi>, then make measurements and estimate energy
perfect but super slow
3. Calculate (noisy) |phi>, then make (noisy) measurement and estimate energy
slow, physical, but not NISQ-y
Order of ZZ...Z
What kind of quality measures can we have?
- Number of qubits (super important!)
- Depth of the circuit (super important!)
- Number of Pauli terms (important!)
- noise robustness (important)
- Number of QC run/measurements
depth
Thank you!
Something extra
QAOA
By Adam Glos
