Evolving Quantum Circuits Daniel Tandeitnik1and Thiago Guerreiro1y 1Department of Physics Pontical Catholic University of Rio de Janeiro Rio de Janeiro 22451-900 Brazil

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Evolving Quantum Circuits

Daniel Tandeitnik1, ∗and Thiago Guerreiro1, †

1Department of Physics, Pontiﬁcal Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900, Brazil

We develop genetic algorithms for searching quantum circuits, in particular stabilizer quantum

error correction codes. Quantum codes equivalent to notable examples such as the 5-qubit perfect

code, Shor’s code, and the 7-qubit color code are evolved out of initially random quantum circuits.

We anticipate evolution as a promising tool in the NISQ era, with applications such as the search

for novel topological ordered states, quantum compiling, and hardware optimization.

I. INTRODUCTION

Natural selection is a unifying idea in biology [1].

Through evolution and competition, systems can nav-

igate the complexity landscape [2] from small-sized

molecules and molecule sets [3] to complex molecular

machines [4] to living organisms [5]. Through natural

selection, complex designs such as eyes and brains can

come into existence in the universe [6], and artiﬁcial evo-

lution has been employed in the laboratory to produce

molecules with desired optical response properties [7].

In view of that, an interesting engineering question is

whether one can exploit evolution as a design tool for

complex technology where human intuition and the stan-

dard deductive method might encounter diﬃculties or

perhaps even fail.

A natural ﬁeld in which human intuition often encoun-

ters diﬃculties is quantum information science. Devising

innovative means of producing complex quantum states

and algorithms [8] outside the scope of known quantum

information primitives [9] is a challenging task [10], espe-

cially under the limitations imposed by present-day quan-

tum hardware [11]. This motivates the main question of

this work: Can artiﬁcial selection be employed as an ef-

fective tool to design useful quantum circuits?

Computer-assisted searches for new physical phenom-

ena [12] and laws of nature [13–15] comprise a very

timely research topic, with notable examples including

the search for new quantum optics experiments [16–18],

resourceful states in quantum metrology [19–21], ground

states in condensed matter systems [22], the study of non-

locality [23], entanglement and Bell inequalities [24,25]

and the automated discovery of autonomous quantum

error correction systems [26] . Closely related to these

developments, tools from artiﬁcial intelligence, genetic

algorithms and competition have also been employed in

chemistry, on the search for new pathways to organic

molecules and closed autocatalytic reaction sets [27], the

eﬃcient training of neural networks for image classiﬁca-

tion [28] and the evolution of deep learning algorithms

through competition [29]. Here, we propose harvesting

some of these tools and ideas within the context of quan-

tum computing and quantum algorithms.

∗tandeitnik@aluno.puc-rio.br

†barbosa@puc-rio.br

Hilbert space is large [30–32] and – similarly to the

vast and complex landscape of the biosphere – evolution

may oﬀer an eﬃcient path to navigate its complexity.

To test this idea, we apply genetic algorithms (GA) to

the search for quantum circuits. As much as these ideas

could beneﬁt from a full scale quantum computer, there

are not many such devices readily available for use at the

present time [33–36]. We therefore focus on stabilizer

circuits [37], which are eﬃciently simulable on classical

computers [38,39]. The stabilizer formalism is the natu-

ral language of quantum error correction [37], leading us

to evolve quantum error correction codes (QECCs) [40].

Within the framework of stabilizer QECCs, we will

demonstrate that evolution, given the appropriate ﬁt-

ness landscape, can successfully produce known exam-

ples of error correcting codes, notably the Perfect 5-qubit

code [41], Shor’s 9-qubit code [42], the 7-qubit color code

[43,44] and novel examples of QECCs. These are ar-

guably simple textbook examples but, as we will point

out, the sample space (modulo equivalences) of circuits

in which such codes live is so large that random search

becomes prohibitive, thus proving the principle that evo-

lution eﬃciently drives the search.

Artiﬁcial selection might oﬀer valuable opportunities

in the current era of NISQ devices [45] in which elemen-

tary quantum gates are costly and noisy. Generically,

random stabilizer circuits are capable of generating good

codewords for QECCs [46], but evolution can go beyond

typicality in guiding the search for simple, low-depth cir-

cuits more amenable to noisy devices. Device speciﬁcity

can also be taken into account by devising ﬁtness land-

scapes in terms of the complexity geometry metric [47],

which penalizes gates according to hardware characteris-

tics [48].

At present, our artiﬁcial selection algorithm can be

compared to the evolution of simple bacteria in controlled

laboratory conditions where the ﬁtness landscape is sim-

ple and well understood [49]. With more complex ﬁt-

ness functions and the addition of quantum hardware

we anticipate improvements and systematic means of de-

vising complex quantum circuits in various applications

beyond stabilizer circuits and QECCs including but not

limited to learning unitaries [50], quantum compiling [51–

53], and hardware speciﬁc tailor-made circuits [33,48].

We highlight that while evolution may be complemen-

tary to known circuit optimisation schemes [52], its main

strength relies on the possibility of creating novel and

arXiv:2210.05058v1 [quant-ph] 11 Oct 2022

creative quantum circuits. All scripts used in this work

are available in the GitHub repository [54].

This paper is organized as follows. In Section II we in-

troduce the GA applied to quantum circuits. Section III

is dedicated to a simple application of the GA, demon-

strating its capabilities within a well-understood context.

Next, we apply the tools of evolution to the search for

QECCs in Section IV. We conclude with a brief discus-

sion and outlook.

II. GENETIC ALGORITHM

For about 4.28 billion years [55], millions of living

species go through a continuous optimization process,

what Darwin termed the evolution of species [56]. The

evolutionary process occurs through the interaction of

populations of species with their habitat, whose function

is to select individuals with a greater aptitude to grow,

thrive, and reproduce. Mathematically, one can allude to

the environment as a ﬁtness function whose domain and

codomain are individuals of a population and a measure-

ment of how well an individual is adapted to its habitat,

respectively. The ﬁtness function then becomes a cost

function of the natural selection optimization problem.

A key point of evolution is its robustness to noise, an

often missing feature in standard optimization methods

[57]. The environment is not static: through natural un-

predictable events, it transforms in a chaotic fashion lead-

ing to an ever-changing ﬁtness landscape. Inspired by the

evident success of life in thriving through evolution in dy-

namic, noisy environments, researchers have been using

algorithms simulating natural selection to solve mathe-

matical optimization problems as early as 1957 [58–64],

broadly termed evolutionary algorithms [65].

In this section we make a brief introduction to the

biological evolutionary process emphasizing its genetic

point of view. Next, GAs to evolve Cliﬀord quantum

circuits are introduced.

A. The evolutionary process

Given a population of individuals belonging to the

same species, one may divide its evolution process into

four stages [57]: reproduction, mutation, competition,

and selection. In broad terms, reproduction is the trans-

mission of genetic material from parents to their oﬀ-

spring. Mutations are minute stochastic errors that oc-

cur during the transmission of genes. Competition and

selection drift the population towards individuals better

adapted to the environment, where the ﬁttest individuals

reproduce passing along their genes to future generations,

while the unﬁtted perish removing their genes from the

gene pool. An evolutionary algorithm aims at emulating,

to some extent, this cycle for a population of potential

solutions to an optimization problem until a threshold is

reached.

The evolution process hinges on the representation of

individuals as their genetic material, commonly referred

to as the genotype, and a screening method that favors

certain genotypes over others. Further, the genotype is

a collection of DNA strands made of elementary build-

ing blocks from a ﬁnite set of possibilities, namely the

four nucleotides adenine, thymine, guanine, and cyto-

sine [66]. Considering sexual reproduction, the geno-

type design naturally leads to the genetic operations of

crossover and mutation. Crossover is a recombination

of the parent’s DNA, where whole segments of DNA are

interpolated to form the oﬀspring’s DNA at conception.

Crossover promotes the dispersion of good DNA blocks

among the population, as natural selection gradually re-

moves bad blocks, and mutation enables the search for

new solutions not reachable through recombination.

Given the genetic operators, we have only an aimless,

random walk through the genotype sample space due to

their stochastic nature. Natural selection is responsible

for drifting the walk in the direction of genotypes that

beget ﬁtter individuals. Natural selection results from

the interplay between the environment’s restrictions on

the individuals and the phenotype each one displays. The

phenotype is the genetic expression of the genotype, i.e.,

the set of physical attributes displayed by the individual

[67] — this set determines the individual’s probability of

reproduction.

A helpful way of visualizing the evolutionary optimiza-

tion process is with a ﬁtness landscape [68,69]. The ﬁt-

ness landscape is a mathematical construct that maps

genotypes to reproduction rates. The environment de-

ﬁnes a function in which the domain is the space of all

possible genotypes and whose codomain is the reproduc-

tion rate. Distances on the landscape are deﬁned as the

closeness of two genotypes, i.e., how similar are the phe-

notypes they spawn. Figure 1exempliﬁes a representa-

tion of an arbitrary ﬁtness landscape. The rationale is

that, at an initial time, the population genotype set is

centered at some point in the landscape. As the genera-

tions go by, the population stochastically drifts towards

the surface peaks due to natural selection and the genetic

operators. Naturally, the ﬁtness landscape in Figure 1is

a schematic representation, as actually producing such a

graph may be impossible due to the complex nature of

the problem at hand.

B. Genetic algorithms applied to Cliﬀord circuits

We now describe how we mimic nature’s evolutionary

process in a GA applied to Cliﬀord quantum circuits.

We note three main aspects we need to cover to build a

GA that simulates nature: (1) a genetic representation

of the tentative solutions, (2) a method to implement

the genetic operators of crossover and mutation, (3) a

selection mechanism that distinguishes between solutions

regarding the optimization problem.

To illustrate how we genetically represent quantum cir-

FIG. 1: Illustration of three distinct populations climb-

ing a hypothetical ﬁtness landscape. At an initial time,

each population starts at some low point of the land-

scape (colored circles) and stochastically rises towards

the peaks. The mutation rate regulates the size of each

step. The height of the surface represents the reproduc-

tion rate of a given genotype.

FIG. 2: A random quantum circuit and its genotype.

Each row of the genotype is a gate in ascending order

(from top to bottom) of application, with the columns

storing the operator and the indices of the target qubits.

The CNOT is abbreviated as C.

cuits in this work, consider the random circuit depicted

in Figure 2(a) and its genetic expression in Figure 2(b).

We genetically represent a circuit composed of tgates as

at×3 array whose rows are gates in ascending order of

application. The ﬁrst column stores the operator, and

the second and third the indices of the aﬀected qubits.

If the gate is a CNOT, the second (third) column is the

control (target) qubit index. For single-qubit gates the

third column is ignored.

Crossovers and mutations naturally become row oper-

ations by expressing quantum circuits as a matrix. There

is a freedom of choice in deﬁning how each genetic opera-

tor works. The most appropriate option often arises from

experimentation since the eﬃciency of a GA is strongly

dependent on the optimization problem itself [65]. For

instance, the crossover can be performed at one-point or

at multiple points [57,65,70]. Additionally, we may use

more than two parents at the reproduction step, thus

having more than two genotypes to crossover [71,72].

For simplicity, we worked solely with one-point crossovers

2 1

P 3

C 2 1

Parent A Parent B Offspring A Offspring B

FIG. 3: Example of a crossover between two genotypes.

The parents’ genotypes are divided at randomly chosen

points and their oﬀspring are built by stacking the pieces.

Offspring Mutaded

offspring

P 3

H 2

FIG. 4: Illustration of the three types of mutation that

may occur to a circuit. From top to bottom: the ﬁrst

gate is replaced by a Hadamard gate on qubit 2, a new

row in inserted between the second and third rows, and

an identity replaces the ﬁfth gate, hence deleting it.

between the genotypes of two parents. Hence, given two

arbitrary parents A and B, their genotypes are divided

at split points randomly selected according to a uniform

probability distribution. Oﬀspring A(B) is formed by

stacking the top portion of parent A(B) on top of the

bottom portion of parent B(A). Figure 3shows an exam-

ple of the crossover operation.

Mutations happen to the oﬀspring’s genotype after a

crossover procedure. With equal probabilities, it is cho-

sen whether (a) the mutation modiﬁes an existing gate

or (b) it inserts a new gate into the circuit. Then,

•If (a): a random row of the genotype is uniformly

selected to be altered. The row contents are over-

written by a new gate uniformly selected between

{I, H, P, CNOT}. If the identity is picked, the en-

tire row is deleted;

•If (b): a random insertion point on the genotype

is uniformly selected. A new row with a new uni-

formly selected gate chosen between {H, P, CNOT}

is inserted at the chosen point.

Figure 4shows an example displaying the three kinds of

mutations that can happen to an oﬀspring genotype.

Just as in nature, we also consider that each genotype

gives rise to a phenotype in GAs. We regard the pheno-

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EvolvingQuantumCircuitsDanielTandeitnik1,andThiagoGuerreiro1,y1DepartmentofPhysics,PonticalCatholicUniversityofRiodeJaneiro,RiodeJaneiro22451-900,BrazilWedevelopgeneticalgorithmsforsearchingquantumcircuits,inparticularstabilizerquantumerrorcorrectioncodes.Quantumcodesequivalenttonotableexamplessuc...

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Evolving Quantum Circuits Daniel Tandeitnik1and Thiago Guerreiro1y 1Department of Physics Pontical Catholic University of Rio de Janeiro Rio de Janeiro 22451-900 Brazil

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