The document describes a method for synthesizing quantum circuits using an island genetic algorithm. The algorithm uses an island model with multiple independent populations that occasionally exchange genetic information. Chromosomes represent candidate circuits as sequences of gates and qubits. The algorithm aims to optimize circuits to match a target unitary matrix by maximizing trace fidelity. Results found circuits for Hadamard coin, Toffoli gate, and Fredkin gate with high fidelity in under 5000 iterations.

1. Synthesis of Quantum Circuits
with an Island Genetic Algorithm
Fernando T. Miranda, Pedro Paulo Balbi, and
Pedro C.S. Costa

2. Synthesis of Quantum Circuits with an Island Genetic Algorithm
● The focus of the paper is finding a unitary decomposition based on the
circuit model of computation.
● From a specific gate set, the goal of this work is to find an equivalent
circuit, through the evolutionary search, for a given unitary U.
Objective Function
Given a unitary matrix U we search for the quantum circuit that produces this
matrix with the greatest possible fidelity

3. Trace fidelity:
This function quantifies the fidelity of the quantum circuit with respect to the
target unitary.
Ua : unitary given
Ut : the target unitary
n: number of qubits needed to describe Ua
Tr(.): trace operation of a given matrix
However, by the Solovay-Kitaev theorem, we know that the ideal solution cannot be achieved
Notice: when Ua = Ut, f = 1.
f ∊ [0,1]

4. Chromosome Representation
Each item from left-to-right represents a column of the circuit.
({1: (‘H’, -1, None)}, {1: (‘X’, 0, None)}, {2: (‘X’, 1, None)}, {1: (‘X’, 0, None)})
Target qubit - Operation - Controlled-qubit - Parameter
Here, the absence of control is indicated by -1 and the absence of
parameter is indicated by None. The indices are mapped from top to
bottom, with 0 indicating the first qubit.

5. Chromosome Representation
Each chromosome is made up of B unitary blocks, each one being a
circuit itself.
Types of blocks
Non-controlled block: single column block (depth = 1) without any control
points.
Controlled-block: single column block (depth = 1) with 1 or more control
points.
Helper Block: predefined block with depth greater than 1.

6. Chromosome Representation (example)
Non-controlled Blocks Controlled Blocks Helper Block

7. Evolutionary Algorithm
Classical Genetic Algorithms present the problem of premature
convergence:
● Due to the lack of continuous parameters to guide search.
● The exchange of genetic material tends to make individuals too similar.
● Reaching almost always a local maximum.
To overcome this problem an Island Model concept was adopted.

8. Island Model
In this type of model, different populations (sets of individuals) evolve
independently at first. Later, they exchange genetic material in a process
called migration.
This model is a parallel version of an ordinary GA, where there is a exchange of
information among different populations.
● Each population may follow different evolutionary path.
● This increases diversity and the effectiveness of the search.

9. Island Model (first step)
● g : number of populations.
● p : number of chromosomes of each population.
● Individuals are created with a number of blocks between 4 and 15 by
randomly generating each one of its blocks.
○ First, a block type is randomly chosen. For non-controller and
controlled blocks, gates and their positions are chosen uniformly at
random.
○ For helper blocks, the type of helper and the control wire is chosen
uniformly at random.

10. After that, evolution will proceed in the following way:
The first step occurs in isolation within each population
● We pick the leader (individual with the highest fitness) of each population.
● For every individual in the population a new offspring is created:
offspring = leader[:nl] + individual[:ni]
● Then, a single mutation is applied to the offspring
○ if fitness(individual) < fitness(offspring):
replace individual by offspring
○ else:
discard(offspring)
nl = max(1, floor(Bl x rl))
ni = max(1, floor(Bi x ri))
number of blocks in
leader / individual

11. 1st Crossover (example)
rl = 0.3
ri = 0.7
nl = max(1, floor(4 x rl)) = 1
ni = max(1, floor(3 x ri)) = 2
leader
individual

12. Crossover between Populations
● This operator has a vital role in providing genetic variability to the
evolutionary process. It happens as follow:
● For an individual l in population_i:
○ Pick individual m from population_j; { i ≠ j }
■ Create an offspring with min(B(l),B(m)) number of blocks
○ if fitness(l) < fitness(offspring):
replace individual by offspring
○ else:
discard(offspring)
number of blocks of l / m
Each block of the offspring will be
selected from the homologous
position of individuals l or m, with
equal probability.

13. Mutation
First, each individual will mutate but the block where the mutation happens is
randomly chosen.
The mutation mechanism for each type of block is described as follow:
● Non controlled block
● Controlled block
Random shuffling: randomly reassigns the control, target, and gates present to new positions
on the wire.
Removal of Control points: only available when there is more that one control point.
Gate change: pick a gate and change its operator.
If both random shuffling and gate change are available only one is applied.
replace the unitary by a random gate
● Random shuffling
● removal of control point
● gate change

14. Mutation
● Helper blocks
Examples of Mutation:
randomly pick a sub-block, and
mutate it according to its type

15. Stop Criterion
The algorithm is interrupted when the equation below is smaller or equal to 1x10^(-6). This
criterion is checked after the 2nd crossover process
Where f was defined previously as the trace fidelity function.

16. The whole process is
summarized in this
graph

17. Parameters for experiment
rl = 0.7
ri = 0.3
g = 20
p = 30
Maximum number of iterations: 10^4
Block sizes: 4 to 15
Quantum gate set available: {H,S,CNOT,X,X^(1/2),T} including the complex
conjugate of the gates T, S, and X^(1/2).
They also added extra gates like the one provided by IBM Quantum
Experience.

18. Results
Hadamard Coin for the Quantum Walker
ε < 1x10^(-6)
Found in less than 600 iterations
Eliminating redundancies like HH = I or ZZ = I

19. Results
Toffoli Gate
ε < 1x10^(-6)
Found in less than 1250 iterations
Eliminating redundancies like

20. Results
Fredkin Gate
ε < 1x10^(-16)
Found in less than 5000 iterations
Eliminating redundancies like

21. Conclusions
● The implementation was able to find the solutions in a number of iterations that
were either smaller or comparable to other analogous works.
● However, it doesn’t consider the cost of implementation for each gate.
● One important limitation of the present work relates to the maximum number of
qubits allowed: unitaries with 10 or more qubits will make the search considerably
slower, due to the exponential usage of memory. But this can be relatively easy
to remedy by making use of sparse data structures.
● The size of the unitary seems to affect the number of iterations until a solution is
found. The bigger the longer it takes.
● As we pointed out before, the solutions presented redundant circuit segments.
● A faster convergence can be achieved by introducing more high-level
components into the search space.
● This work provided the proof-of-principle that the island GA can be used for
circuit synthesis, regardless of the practical quantum device used.