Variational Quantum Continuous Optimization a Cornerstone of Quantum Mathematical Analysis Pablo Bermejo1 2and Rom an Or us1 2 3

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Variational Quantum Continuous Optimization:

a Cornerstone of Quantum Mathematical Analysis

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

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

Here we show how universal quantum computers based on the quantum circuit model can handle

mathematical analysis calculations for functions with continuous domains, without any digitaliza-

tion, and with remarkably few qubits. The basic building block of our approach is a variational

quantum circuit where each qubit encodes up to three continuous variables (two angles and one

radious in the Bloch sphere). By combining this encoding with quantum state tomography, a vari-

ational quantum circuit of nqubits can optimize functions of up to 3ncontinuous variables in an

analog way. We then explain how this quantum algorithm for continuous optimization is at the basis

of a whole toolbox for mathematical analysis on quantum computers. For instance, we show how to

use it to compute arbitrary series expansions such as, e.g., Fourier (harmonic) decompositions. In

turn, Fourier analysis allows us to implement essentially any task related to function calculus, includ-

ing the evaluation of multidimensional deﬁnite integrals, solving (systems of) diﬀerential equations,

and more. To prove the validity of our approach, we provide benchmarking calculations for many

of these use-cases implemented on a quantum computer simulator. The advantages with respect to

classical algorithms for mathematical analysis, as well as perspectives and possible extensions, are

also discussed.

I. INTRODUCTION

The ﬁeld of quantum computing is evolving rapidly.

What looked like science-ﬁction a few years ago, is now

starting to become a reality. Thanks to recent experi-

mental breakthroughs, we now can enjoy the ﬁrst com-

mercial quantum processors. Such machines are noisy

and have a limited number of qubits (i.e., they are Noisy

Intermediate-Scale Quantum - NISQ - devices [1]), and

are thus far from ideal. Following the trends in his-

tory, we should expect to have more powerful and noise-

resistant processors in the future. But, as of today, our

quantum computers are made of a small number of noisy

qubits.

Now, here comes a hard question: can we do some-

thing useful with these imperfect quantum machines?

The qualiﬁed answer to this question is yes. To put it in

perspective, think of the evolution of classical comput-

ers. One would never use a computer from the 1930’s

to develop, say, a speech recognition system. But those

machines were deﬁnitely useful at cracking some crypto-

graphic codes. The situation with current quantum com-

puters is, in a way, similar to that of classical computers

in the 30’s. Of course we cannot hope to use today’s

quantum processors for very complex tasks such as, say,

factoring a 617-digit number (as used in the RSA cyber-

security protocol). But they are certainly useful for other

tasks. And of course, what consumes mental energy is to

ﬁnd those tasks, where today’s quantum computers can

begin to oﬀer real, practical value. So, what can we al-

ready do with NISQ processors that is actually useful?

A good example where NISQ devices can start oﬀering

value is optimization. By developing hybrid quantum-

classical solutions, present-day quantum annealers can

handle hard optimization problems in real scenarios [2–

10]. Another good example is that of machine learning,

where few-qubit universal quantum systems are, at least,

as powerful as deep neural networks for supervised and

unsupervised learning [11,12], thanks to the use of vari-

ational quantum optimization methods [13,14].

In this paper we follow the above philosophy, and show

how few-qubit quantum computers can already imple-

ment arbitrary multi-dimensional function calculus in a

remarkably eﬃcient way. The basic building block of

our approach is a variational quantum algorithm to op-

timize functions with continuous domains. In our ap-

proach, each qubit in the quantum circuit encodes up

to three continuous variables in the parameters of the

Bloch sphere: two angles and one radius. By combining

this with single-qubit quantum tomography, a variational

optimization of the circuit parameters allows to ﬁnd the

extreme values of multidimensional functions in a purely-

analog and fast way.

Based on the above, we then show how our quantum

optimization algorithm can be used to implement, essen-

tially, any type of calculus on functions with continuous

domains. As a ﬁrst step, we show how to compute arbi-

trary series expansions of any function, and in particular

Fourier (harmonic) analysis, with the coeﬃcients of the

expansions calculated via continuous quantum optimiza-

tion. And, then, with this tool at our disposal, we explain

how this allows us to do pretty much any calculation, in-

cluding deﬁnite integrals of multidimensional functions,

solving (systems of) diﬀerential equations, and more.

The quantum optimization algorithm presented here

is the cornerstone of an eﬃcient quantum toolbox for

mathematical analysis on NISQ processors. To put it in

perspective, a quantum computer of 127 qubits, such as

the IBM-Q System One [15], could use our algorithms to

analyze 381-dimensional functions directly in the contin-

arXiv:2210.03136v1 [quant-ph] 6 Oct 2022

uum. Additionally, we also ﬁnd that the classical simula-

tion of these quantum algorithms is very fast, comparable

in performance to standard classical computational soft-

ware.

This paper is organized as follows: in Sec.II we present

the building block of our construction, namely, our quan-

tum algorithm for continuous optimization. In Sec.III we

explain diﬀerent applications of this algorithm, including

series expansions, Fourier analysis, deﬁnite integral cal-

culations, and diﬀerential equations, where we provide

also benchmarks and examples of such calculations via

simulation of a quantum computer. Finally, Sec.IV dis-

cusses the eﬃciency of our methods, and wraps up with

our conclusions and perspectives for future work.

II. CONTINUOUS QUANTUM OPTIMIZATION

Let us start by presenting our quantum algorithm for

optimizing multidimensional functions with continuous

domains. In the following, we discuss the basics of the

problem and the proposed quantum algorithm, which is

based on a variational quantum circuit with a peculiar

encoding of the function’s variables.

A. The problem

Without loss of generality, let us consider a real scalar

function f(~x) of mreal variables ~x = (x1, x2,··· , xm).

The domain of each variable xαis Dα⊆Rfor all

α= 1,2,··· , m. All the derivations that we explain in

this paper can be directly translated to more complex

scenarios such as, e.g., real and complex vector and ten-

sor ﬁelds of real and complex variables.

The statement of the problem is simple: ﬁnd the min-

imum of f(~x), i.e.,

min

~x f(~x).(1)

As opposed to a discrete optimization problem, which is

the usual case addressed by quantum computing, here

the variables used in the objective function are continu-

ous. At this stage, we do not demand more constraints

on the function or the domains. In typical applications,

though, one may focus on, e.g., continuous and diﬀer-

entiable functions. From the use-case perspective, con-

tinuous optimization is at the very core of many real-

world problems in mathematical science and engineering,

including the design of biomolecules, ﬁnancial portfolio

optimization, and ﬂuid dynamics, among many others.

B. Variable encoding

We now explain one of the main characteristics of our

quantum algorithm: the way we encode the continuous

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⇢

FIG. 1: Bloch sphere of one qubit. The mixed-state state ρ

is a point in the interior of the sphere and is characterized by

three continuous parameters (θ, φ, r) as speciﬁed in the text.

Pure states correspond to points in the surface of the sphere,

so that r= 1, and depend on two angular variables (θ, φ).

variables xαin the degrees of freedom of a quantum com-

puter.

To see the encoding in perspective, let us remind the

basic degrees of freedom of a single qubit. In general, a

qubit is described by a point of a Bloch sphere, see Fig.1.

Pure states of a qubit correspond to points on the surface

of the sphere, and can be parametrized by two angles,

which we call θ∈[0, π] and φ∈[0,2π). Accordingly,

the pure state |ψiof a qubit is given by the well-known

formula

|ψi= cos θ

2|0i+eiφ sin θ

2|1i,(2)

with {|0i,|1i} the orthogonal computational basis of the

Hilbert space. Similarly, mixed states of a qubit cor-

respond to points in the interior of the Bloch sphere.

These states correspond to the single-qubit conﬁgura-

tions where the qubit is entangled with some other quan-

tum system, and therefore the qubit’s reduced density

matrix is no longer a pure state. Mathematically, the

mixed state ρof a qubit is written as

ρ=1

2(I+~n ·~σ),(3)

where Iis the 2 ×2 identity matrix, ~n = (nx, ny, nz)

is a three-dimensional vector of real components, and

~σ = (σx, σy, σz) is a three-dimensional vector of the

three Pauli matrices σx, σyand σz. In this notation,

the qubit is in a pure state (i.e., in the surface of the

Bloch’s sphere) iﬀ |~n|= 1, in which case the vector

is given by ~n = (sin θcos φ, sin θsin φ, cos θ) in spheri-

cal coordinates, so that ρ=|ψihψ|. In addition, the

qubit is in a mixed state whenever |~n|<1, so that

~n =r(sin θcos φ, sin θsin φ, cos θ), with r < 1 being the

radial coordinate of a point inside the sphere.

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VariationalQuantumContinuousOptimization:aCornerstoneofQuantumMathematicalAnalysisPabloBermejo1,2andRomanOrus1,2,31MultiverseComputing,PaseodeMiramon170,E-20014SanSebastian,Spain2DonostiaInternationalPhysicsCenter,PaseoManueldeLardizabal4,E-20018SanSebastian,Spain3IkerbasqueFoundationforScience...

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Variational Quantum Continuous Optimization a Cornerstone of Quantum Mathematical Analysis Pablo Bermejo1 2and Rom an Or us1 2 3

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