Variational Quantum Non-Orthogonal Optimization Pablo Bermejo1 2and Rom an Or us1 2 3 4 1Multiverse Computing Paseo de Miram on 170 E-20014 San Sebasti an Spain

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Variational Quantum Non-Orthogonal Optimization

Pablo Bermejo1, 2 and Rom´an Or´us1, 2, 3, 4

1Multiverse Computing, Paseo de Miram´on 170, E-20014 San Sebasti´an, Spain

2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebasti´an, Spain

3Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain

4Corresponding author: roman.orus@dipc.org

Abstract

Current universal quantum computers have a limited number of noisy qubits. Because of this, it is

diﬃcult to use them to solve large-scale complex optimization problems. In this paper we tackle this

issue by proposing a quantum optimization scheme where discrete classical variables are encoded in

non-orthogonal states of the quantum system. We develop the case of non-orthogonal qubit states,

with individual qubits on the quantum computer handling more than one bit classical variable.

Combining this idea with Variational Quantum Eigensolvers (VQE) and quantum state tomography,

we show that it is possible to signiﬁcantly reduce the number of qubits required by quantum hardware

to solve complex optimization problems. We benchmark our algorithm by successfully optimizing a

polynomial of degree 8 and 15 variables using only 15 qubits. Our proposal opens the path towards

solving real-life useful optimization problems in today’s limited quantum hardware.

Introduction.- Quantum computing is evolving rapidly

thanks to signiﬁcant advances in hardware. Current

Noisy Intermediate-Scale Quantum (NISQ) processors [1]

are starting to show promise in speciﬁc tasks, including

claims on quantum advantage [2,3], fault-tolerant gates

[4], and industrial applications for speciﬁc problems [5].

Nonetheless, NISQ devices are still quite limited at the

time of solving certain problems of relevance, such as

complex optimization problems. This is true with the

current implementation of quantum optimization algo-

rithms such as Variational Quantum Eigensolvers (VQE)

[6] and Quantum Approximate Optimization Algorithms

(QAOA) [7].

In this paper we attack this limitation by developing

new quantum optimization schemes where classical vari-

ables are not encoded in orthogonal qubit states, but

rather in other degrees of freedom of the quantum com-

puter. An example is the case in which individual qubits

are allowed to represent more than one classical bit vari-

able each, by considering the degrees of freedom of the

Bloch sphere [8,9]. As we shall see, this allows to signiﬁ-

cantly reduce the number of qubits needed to solve com-

plex discrete optimization problems in universal gate-

based quantum computers, thus boosting the practical

utility of NISQ devices for real-life problems.

The problem.- A key relevant problem for which NISQ

devices are very limited is optimization, which amounts

to the minimization of a cost function, typically under

some constraints which can also be included via, e.g.,

Lagrange multipliers. At the end of the day, an arbitrary

cost function Hof a discrete optimization problem takes

the form

H≡f(q0, q2,···qN−1),(1)

with qα= 0,1,··· , p −1 a discrete variable that can take

up to pdiﬀerent values, with α= 0,1,··· , N −1. For

example, if p= 2 for all α, then we have the case of an

optimization problem with usual bit variables, qα= 0,1

for all α. Finding the minimum of such a cost func-

tion, for Nbit variables, and for a discrete polynomial

of degree 2, is known as a Quadratic Unconstrained Bi-

nary Optimization (QUBO) problem and is well-known

to be NP-Hard. Higher-order polynomials, and higher-

dimensional variables, make the problem even harder.

In many optimization schemes for NISQ devices, cost

functions such as the one described above are typically

considered by ﬁrst boiling it down to bit variables. In

a binary encoding scheme, this implies that the original

variables qαare expressed in terms of m=⌈log2(p)⌉bits

each, i.e.,

qα=

m−1

i=0

2ixi,α ∀α. (2)

As an example, if p= 4, then we have that m= 2 and

the correspondence x0x1→qbetween the individual bits

and the original variable (for a given α) is [00 →0; 01 →

1; 10 →2; 11 →3]. With this decomposition in mind,

the cost function Hcan be written as

H=f(x0,0, x1,0,··· , xm−1,N−1),(3)

in terms of m×Nclassical bit variables.

As we can see, the number of bit variables quickly

explodes in classical optimization problems. It is not

surprising, therefore, that when solving such problems

on quantum computers, their capabilities are quite lim-

ited if we consider one qubit in the quantum register for

each bit variable of the cost function. Say, for instance,

that you run a VQE algorithm on a universal gate-based

quantum computer. The largest such machine built as of

today is the IBM System One with 127 supercoducting

qubits [10]. This implies that, with such an algorithmic

arXiv:2210.04639v2 [quant-ph] 29 Jun 2023

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VariationalQuantumNon-OrthogonalOptimizationPabloBermejo1,2andRom´anOr´us1,2,3,41MultiverseComputing,PaseodeMiram´on170,E-20014SanSebasti´an,Spain2DonostiaInternationalPhysicsCenter,PaseoManueldeLardizabal4,E-20018SanSebasti´an,Spain3IkerbasqueFoundationforScience,MariaDiazdeHaro3,E-48013Bilbao,Spai...

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