Space-efficient binary optimization for variational computing

Adam Glos, Aleksandra Krawiec, Zoltán Zimborás

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences

The problem: TSP

given the cost between 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?

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 a polynomial, but not necessarily quadratic!

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

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

0,1 => 1,-1

finite order => small model

QUBO => 2-local Ising model

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

This is not true in general

b_1b_2\cdots b_n \to \frac{1}{2^n} (1-s_1) \cdots (1-s_n)

QAOA - a special gate-model QA

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).

b_1\cdots b_k

What if we have quite good quantum computers, but with very small number of qubits?

TSP - waste of qubits

We have n! possible routes

\log_2(n!) \sim n \log n

only n log(n) qubits! (but needs higher-order terms)

n^2

QAOA - can we run it?

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

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

HOBO of order ~log(n) but still a small number of terms!

A_1\sum\limits_{t} H_{\rm valid} (b_t) + A_2\sum\limits_{t} \sum\limits_{t'\neq t} H_{\neq}(b_t,b_{t'}) +

+ B\sum\limits_{\substack{v,w\\v\neq w}} W_{vw} \sum\limits_{t} H_\delta (b_t,v)H_\delta (b_{t+1},w)

(constraints)

(objective)

QAOA - can we run it? cont.

\exp(-\mathrm i \alpha Z_1 Z_2 Z_3) ?

\exp(-\mathrm i (\alpha_1 Z_1 Z_2 Z_3+ \alpha_2 Z_1Z_2 )) \\= \exp(-\mathrm i \alpha_1 Z_1 Z_2 Z_3 ) \exp(-\mathrm i \alpha_2 Z_1 Z_2 )

QAOA - can we run it? cont.

round robin-tournament

Is HOBO better?

N=3

N=4

N=3

N=5

N=4

Text

depth

no. qubits

n^2

n^3

n\log n

HOBO

QUBO

Text

Intermediate cases?

Good mixed approach

0 \rightarrow 01\,00

1 \rightarrow 10\,00

2 \rightarrow 11\,00

3 \rightarrow 00\,01

4 \rightarrow 00\,10

5 \rightarrow 00\,11

n^{1+2\alpha}

HOBO

QUBO

n^{1-\alpha}\log n

Conclusions

smaller number of qubits,
(probably) more efficient than QUBO + QAOA,
has smaller relative feasible space,
produces MUCH deeper circuits than QUBO!

Glos, A., Krawiec, A., & Zimborás, Z. (2020). Space-efficient binary optimization for variational computing. arXiv preprint arXiv:2009.07309.

Also true for Quantum Alternating Operator Ansatz

New idea !

Today, 15:40 GMT!

HOBO-size mixer

QUBO objective encoding

|+\rangle

\theta

\Huge \dagger

|\varphi(\theta,\theta')\rangle

\theta'

|00\ldots0\rangle

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