Space-efficient binary optimization for variational computing

Adam Glos

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences

Overview

Introduction to NISQ era quantum

combinatorial optimization

Making problems quantum accesible - QUBO and Ising model

Something more - HOBO

Future ideas?

The problem: TSP

given the cost between each cities, find the cheapest route such that the route goes through all cities and comes back

We need short, mathematical, and general description

Integer programming?

Binary optimization

where all variables are bits

Since every number can be represented by bits...

0-1 integer programming is quite general!

Quadratic Unconstrained Binary Optimization

f(b)=\sum_{i,j} Q_{i,j} b_ib_j+ C

Any NP-complete problem can be turned into QUBO

Every binary function can be turned into quantum polynomial, but not necessarily quadratic!

f:\{0,1\}^n \to \mathbb R

A\sum\limits_{v=0}^{N-1} \left( 1 - \sum\limits_{t=0}^{N-1} b_{tv} \right)^2 + A \sum\limits_{t=0}^{N-1} \left( 1 - \sum\limits_{v=0}^{N-1} b_{tv} \right)^2 + B \sum\limits_{\substack{v,w=0\\v\neq w}}^{N-1} W_{vw} \sum\limits_{t=0}^{N-1} b_{tv}b_{t+1,w}

\sum_{I\subseteq \{1,\dots,n\}} \alpha_I \prod_{i\in I} b_i

Lucas, Andrew. "Ising formulations of many NP problems." Frontiers in Physics 2 (2014): 5.

Ising model

Instead of 0-1, we have now -1/1

If HOBO model has finite order, then it is small AND corresponding Ising model is small

This is not always true for higher order models

b_1b_2\cdots b_n

b_i \leftarrow \frac{1-s_i}{2}

s_i \leftarrow Z_i

\sum_{I\subseteq \{1,\dots,n\}} \alpha_I \prod_{i\in I} b_i

\sum_{I\subseteq \{1,\dots,n\}} \tilde \alpha_I \prod_{i\in I} Z_i

\sum_{I\subseteq \{1,\dots,n\}} \tilde \alpha_I \prod_{i\in I} s_i

QUBO is transformed to 2-local Ising model

Quantum optimization algorithms

Quantum annealing

Different input for each algorithm!

Variational optimization (VQE, QAOA)

Quantum annealing

\sum_{i\neq j} J_{ij} s_is_j + \sum h_i s_i +C

Must be QUBO (quadratization sometimes needed)
many (qu)bits needed
graph embedding required

Variational Quantum Eigensolver

Can be a general binary optimization - we only need an efficient procedure for calculating energy!
just walking on a Hilbert space...
ansatz-dependent
designed for harder problems

Peruzzo, Alberto, et al. "A variational eigenvalue solver on a photonic quantum processor." Nature communications 5.1 (2014): 1-7.

QAOA - a special VQE

Unlike VQE, we have to implement the objective Hamiltonian

Still we can use higher order terms!

|++\ldots + \rangle

\exp(-\mathrm i r_{i}\sum_i X_i)

\exp(-\mathrm i p_i H_{\rm obj})

i \leftarrow i+1

| \varphi(p,r) \rangle

Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).

TSP - waste of qubits

We have n! possible routes

\log_2(n!) = \Theta(n \log n)

only N log(N) qubits!

QAOA - can we run it?

We have to implement the objective Hamiltonian

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).

H_{\neq}^{\rm HOBO}(b,b') = H_\delta^{\rm HOBO}(b,b') \coloneqq \prod_{k=1}^K (1- (b_k - b'_k)^2)

H_{\rm valid}^{\rm HOBO}(b_t) \coloneqq \sum_{k^0\in K_0} b_{t,k_0}\prod_{k=k^0+1}^{K-1} (1-(b_{t,k} - \tilde b_{k})^2)