Concrete Quantum Circuits to Prepare Generalized Dicke States on a

Quantum Machine

Shintaro Narisada

, Shohei Beppu

, Kazuhide Fukushima

and Shinsaku Kiyomoto

KDDI Research, Inc., Fujimino, Japan

Keywords:

Dicke State, Hamming Weight, Noisy Computation, Quantum Circuit, Quantum Computing, Generalization.

Abstract:

A Dicke state is a superposition of n-qubit with Hamming weight k, denoted by |D

i. Dicke states are fre-

quently employed to prepare input superpositions for quantum search algorithms (e.g., Grover search and

quantum walks) that solve combinatorial problems with a certain number





of candidate solutions. B

artschi

and Eidenbenz propose a concrete quantum circuit to construct the Dicke state |D

i with polynomial quantum

gates, and they generalize the circuit in terms of Hamming weight k to prepare a superposition of Dicke states.

Subsequently, Esser et al. present another quantum circuit to generate the Dicke state |D

i with polynomial

gates and a few auxiliary quantum bits. In this paper, we generalize Esser’s state preparation circuit to con-

struct a superposition of Dicke states. We conduct a concrete comparison with two generalized Dicke state

preparation circuits. We perform noisy simulations and experiments using real quantum machines from the

IBM quantum experience service (IBMQ). Both circuits successfully construct the generalized Dicke state

superposition using a noisy intermediate-scale quantum (NISQ) device, albeit somewhat affected by noise.

1 INTRODUCTION

The exponential hardness of combinatorial problems

is one of the foundations of the security of modern so-

ciety. With the advent of quantum computers, quan-

tum search algorithms to solve combinatorial prob-

lems such as the Grover algorithm (Grover, 1996),

quantum amplitude ampliﬁcation (QAA) (Brassard

et al., 2002) and quantum walks (Ambainis, 2004)

are being realized (Mandviwalla et al., 2018; Acasi-

ete et al., 2020). The input to these algorithms is nor-

mally a quantum state superposition associated with

the candidate solutions to a speciﬁc problem to be

solved. While uniform superposition of all 2

n-qubit

states can be prepared with n Hadamard gates, it is

known that constructing an arbitrary superposition re-

quires Θ(2

) quantum gates (Shende et al., 2006).

There are several studies of quantum circuits that ef-

ﬁciently prepare superpositions of a certain class of

states, including circuits to generate linear-size su-

perpositions called the |GHZ

i and |W

i states (Cruz

et al., 2019), and circuits to generate exponential-size

https://orcid.org/0000-0002-9399-9778

https://orcid.org/0000-0002-8220-9515

https://orcid.org/0000-0003-2571-0116

https://orcid.org/0000-0003-0268-0532

superpositions called the Dicke state |D

i probabilis-

tically (Childs et al., 2000; Chakraborty et al., 2014)

or deterministically (Kaye and Mosca, 2001; B

artschi

and Eidenbenz, 2019; Mukherjee et al., 2020; Esser

et al., 2021; B

artschi and Eidenbenz, 2022) in poly-

nomial time.

i is the superposition of all n-qubit states with

Hamming weight k of size





, which has a close re-

lationship to combinatorial problems such as the k-

vertex cover problem (Cook et al., 2020), extractive

summarization (Niroula et al., 2022) and syndrome

decoding (SD) problem (Esser et al., 2022; Perriello

et al., 2021; Chailloux et al., 2021), which is the

security basis of code-based cryptosystems. A gen-

eralized Dicke state preparation circuit to construct

a superposition of Dicke states |D

i for any k ≤ n

with polynomial gates is also proposed in (Kaye and

Mosca, 2001; B

artschi and Eidenbenz, 2019). For

certain combinatorial problems, the number of com-

binations for candidate solutions may be a sum of





for any 0 ≤ i ≤ n rather than





. For example, the

SD problem can be easier to solve if it is extended to

choose binary vectors with weight ≤ k than vectors

with weight k from n vectors.

In this paper, we generalize the Dicke state prepa-

ration circuit shown in (Esser et al., 2021) and pro-

pose another circuit to construct an equal superposi-

Narisada, S., Beppu, S., Fukushima, K. and Kiyomoto, S.

Concrete Quantum Circuits to Prepare Generalized Dicke States on a Quantum Machine.

DOI: 10.5220/0011618000003405

In Proceedings of the 9th International Conference on Information Systems Security and Privacy (ICISSP 2023), pages 329-338

ISBN: 978-989-758-624-8; ISSN: 2184-4356

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

329

tion of |D

i for any 0 ≤ i ≤ k with polynomial size

quantum gates. We implement the generalized cir-

cuit and B

artschi’s circuit using the Qiskit library and

give a comparison between two generalized Dicke

state preparation circuits. Quantum simulations of our

implementations in both noise-free models and noisy

circuit models are also conducted. We run our imple-

mentations on the quantum device from IBMQ and

achieved superposition for some generalized Dicke

states.

2 PRELIMINARIES

We write a qubit as |xi,x ∈{0,1}. A tensor product of

two qubits |xi and |yi is |xi⊗|yi or simply |xi|yi. A

tensor product of two identical qubits |xi is shortened

by |xi

⊗2

= |xi⊗|xi. Let |x

...x

i,x

∈ {0, 1},1 ≤

i ≤ n be an n-qubit quantum state. |x

i+1

...i

de-

notes the sequence of qubits starting at i. We may

treat a quantum state |x

...x

i as a binary string

= x

...x

for simplicity of explanation. The

Hamming weight of a binary string b

is wt(b

) =

|{i | x

= 1}|. A binary string of length n and Ham-

ming weight k is written by b

n,k

. The empty string is

denoted by ε. |ψ

i = c

00...0

|00...0i+ c

00...1

|00...1i+

... + c

11...1

|11...1i denotes the n-qubit superposition,

where

∑

,...,i

...i

= 1. Each |c

...i

can be con-

sidered as the existence probability p

...i

of the cor-

responding quantum state |i

...i

i. Measuring the n-

qubit superposition yields one n-qubit state |i

...i

with probability p

...i

. A Dicke state is deﬁned as

follows:

Deﬁnition 1. A Dicke state |D

i is the equal super-

position of all n-qubit states |xi with Hamming weight

wt(x) = k,

i =





∑

x∈{0,1}

,wt(x)=k

|xi. (1)

For example, |D

i =

√

(|100i+ |010i+ |001i).

A superposition of Dicke states is written by

∑

0≤i≤n

i, (2)

where α

∈ C, α

+ . .. + α

= 1 (Kaye and Mosca,

2001; Bastin et al., 2009). Hereafter, we consider

an equal superposition of Dicke states for arbitrary

weights. For an integer set K ⊆ [0, n], we will refer to

equal superpositions of Dicke states with Hamming

weights i ∈ K as a generalized Dicke state:

Deﬁnition 2. Generalized Dicke state |D

i for an in-

teger set K ⊆ [0,n] is the equal superposition of all

n-qubit states |xi with Hamming weight wt(x) ∈ K,

i =

∑

x∈{0,1}

,wt(x)∈K

|xi, (3)

where |C

| =

∑

i∈K





. For example, |D

i =

√

(|000i+ |100i+ |010i+ |001i + |111i) for K =

{0,1,3}. Using Equation 2, the generalized Dicke

state is a case when α





/|C

| for all i ∈K and

= 0 otherwise.

Any unitary operation on the quantum state |xi is

denoted by U|xi, where U is a unitary matrix. We will

introduce certain basic unitary operations on quantum

states.

• I: identity operator I =



1 0

0 1



, I|0i = |0i and

I|1i = |1i.

• X: single qubit gate deﬁned as X =



0 1

1 0



X|0i = |1i and X|1i = |0i as with the classical

NOT gate.

• CNOT (controlled-X): 2-qubits gate that applies

X to the target qubit when the control qubit is |1i

and does nothing when the control qubit is |0i.

• C

X (multi-controlled-X): (n+1)-qubits gate that

applies X to the target qubit when the control n-

qubit satisfy some quantum state |b

i and does

nothing when the control is not |b

i. When n = 2

and b

= 11, C

X is called a Toffoli gate.

• R

(θ): single qubit gate deﬁned by R

(θ) =



cos(θ/2) −sin(θ/2)

sin(θ/2) cos(θ/2)



. Note that since

(2cos

−1

(

√

p))|0i =

√

p|0i+

√

1 − p|1i, it is

helpful when we want to partition the prob-

ability of |0i into p and 1 − p. We also

use CR

(θ) (controlled-R

(θ)) gates and C

(θ)

(multi-controlled-R

(θ)) gates.

Thereafter, these basic gates are combined to achieve

our desired unitary operation. A tensor product of two

unitary operators U

and U

is U

⊗U

. A tensor prod-

uct of two identical unitary operators U is shortened

by U

⊗2

= U ⊗U. A quantum circuit is a sequence of

unitary operations starting from |00...0i. The purpose

of a quantum circuit is to obtain the desired quantum

state |ψi(e.g., corresponding to the solution of a com-

binatorial problem) with high probability when mea-

suring the qubits at the endpoint of the circuit.

The computational complexity of a quantum cir-

cuit (circuit complexity) is evaluated by its width and

depth. The width of a quantum circuit is the total

number of qubits required for the circuit. Depth is

ICISSP 2023 - 9th International Conference on Information Systems Security and Privacy

330

the longest path in the circuit. Auxiliary qubits (an-

cilla) are extra qubits that are often employed to sim-

plify gating operations. For example, ancilla qubits

are employed to store the carry of the quantum full

adder circuit.

3 DETERMINISTIC DICKE

STATE PREPARATION

We will explain previous deterministic Dicke state

preparation circuits proposed by B

artschi (B

artschi

and Eidenbenz, 2019) and Esser (Esser et al., 2021)

et al. The former uses no ancilla qubits, but its circuit

is somewhat complicated. The latter is intuitive con-

struction by using some ancilla qubits as a counter.

3.1 Dicke State Preparation Without

Ancilla Qubits

The central part of their circuit is the construction of

a unitary operator U

n,k

such that |00..00

|{z}

n−`

1..1

|{z}

i is an

input and |D

i is an output for any 0 ≤ ` ≤ k.

Deﬁnition 3 (U

n,k

). n-qubit unitary gate satisfying

n,k

|0i

⊗n−`

|1i

⊗`

= |D

i for all 0 ≤ ` ≤ k.

By setting ` ≤ k instead of ` = k, we can induc-

tively construct U

n,k

by using the following property

for the Dicke state.

Lemma 1. (Lamata et al., 2013; Moreno and Parisio,

2018) |D

i has the following inductive sum form:

n−1

`−1

i⊗|1i+

n −`

n−1

i⊗|0i. (4)

Intuitively, |D

n−1

`−1

i contains



n−1

`−1



strings, and

n−1

i contains



n−1



strings with equal existence

probabilities



n−1

`−1



−1

and



n−1



−1

, respectively. The

sum of |D

n−1

`−1

i⊗|1i and |D

n−1

i⊗|0i then contains



n−1

`−1





n−1







distinct strings. The coefﬁcients

of each term are intended to equalize the probability.

Both Dicke states |D

n−1

`−1

i and |D

n−1

i can be gen-

erated by the same unitary U

n−1,k

given the inputs

|0i

⊗n−`

|1i

⊗`−1

and |0i

⊗n−1−`

|1i

⊗`

, respectively. To

inductively construct U

n,k

, we need a unitary gate that

corresponds to the following operation that changes

(actually left shift) the last qubit |1i to |0i with prob-

ability

n−`

for inputs |0i

⊗n−`

|1i

⊗`

for all ` ≤ k

|0i

⊗n−`

|1i

⊗`

7→

|0i

⊗n−`

|1i

⊗`−1

|1i+

n −`

|0i

⊗n−`−1

|1i

⊗`−1

|0i. (5)

Note that we can disregard the ﬁrst n −k −1 qubits

since they are always 0. Next, we can consider a uni-

tary S

n,k

(referred to as split and cyclic shift in the

original paper) such that for the last k + 1 qubits it

performs a left shift operation with probability

n−`

for all ` ≤ k.

Deﬁnition 4 (S

n,k

). (k + 1)-qubits unitary gate for all

1 ≤ ` ≤ k satisfying

n,k

|0i

⊗k+1

=|0i

⊗k+1

n,k

|0i

⊗k+1−`

|1i

⊗`

|0i

⊗k+1−`

|1i

⊗`−1

|1i+

n −`

|0i

⊗k−`

|1i

⊗`−1

|0i

n,k

|1i

⊗k+1

=|1i

⊗k+1

One can construct the S

n,k

gate concretely by com-

bining the CNOT, CR

(θ) gate and C

(θ) gate:

Deﬁnition 5 (Building blocks of S

n,k

). S

n,k

is the con-

nection of k basic gates B

n,1

to B

n,k

n,1

:|00i

n−1

→|00i

n−1

|11i

n−1

→|11i

n−1

|01i

n−1

→

|01i

n−1

n −1

|10i

n−1

n,`

:|00i

n−`

|0i

→|00i

n−`

|0i

|01i

n−`

|0i

→|01i

n−`

|0i

|00i

n−`

|1i

→|00i

n−`

|1i

|11i

n−`

|0i

→|11i

n−`

|0i

|01i

n−`

|1i

→

|01i

n−`

|1i

n −`

|11i

n−`

|0i

where 2 ≤` ≤k. Namely, S

n,k

= B

n,k

n,k−1

...B

n,2

n,1

| {z }

k basic gates

Figure 1 shows the basic gate components of B

n,1

and

n,`

Using the inductive property U

i,i

= (U

i−1,i−1

⊗

⊗k−i

)·S

i,i−1

for 2 ≤i ≤k and U

i,k

= (U

i−1,k

⊗I

⊗n−i

)·

⊗i−k−1

⊗S

i,k

) for k + 1 ≤i ≤n, U

n,k

is decomposed

by S

n,k

gates as follows:

Lemma 2. (B

artschi and Eidenbenz, 2019) The fol-

lowing inductive construction of U

n,k

is consistent

with Deﬁnition 3.

Concrete Quantum Circuits to Prepare Generalized Dicke States on a Quantum Machine

331

𝑛 − 1

𝑛

𝑅

(2 cos

1/𝑛)

𝑛 − ℓ 𝑅

(2 cos

ℓ/𝑛)

𝑛

⋯

𝑛 − ℓ + 1

Figure 1: B

n,1

gate (left) and B

n,`

gate for 2 ≤ ` ≤ k (right).

n,k

∏

i=2

i,i−1

⊗I

⊗n−i

) ·

∏

i=k+1

⊗i−k−1

⊗S

i,k

⊗I

⊗n−i

For instance, U

5,3

|00111i = (S

2,1

⊗I

⊗3

) ·(S

3,2

⊗

⊗2

) · (S

4,3

⊗ I) · (I ⊗ S

5,3

)|00111i = |D

i. Since

|0i

⊗n−k

|1i

can be constructed from |00...0i by ap-

plying X gates, Dicke state |D

i can be prepared from

the initial state |00...0i. The circuit complexity is

shown as follows:

Theorem 1. (B

artschi and Eidenbenz, 2019) Dicke

states |D

i can be prepared with a circuit of width n

and depth O(n) using k X gates and one U

n,k

gate.

We can further reduce the number of

gates (Mukherjee et al., 2020) or the circuit depth to

O(k log

) for constant k (B

artschi and Eidenbenz,

2022). However, for the sake of circuit simplicity,

only the circuit from (B

artschi and Eidenbenz, 2019)

is considered in this paper.

3.2 Dicke State Preparation with

Ancilla Qubits

Esser’s circuit is also based on the inductive prop-

erty of the Dicke state (Lemma 1). However, un-

like B

artschi’s circuit, which determines the qubit se-

quence from right to left, Esser’s circuit determines

the quantum bit string in left-to-right order by consid-

ering Lemma 1 as follows:

i =

n −k

|0i|D

n−1

|1i|D

n−1

k−1

i = |0i

⊗n

,|D

i = |1i

⊗n

The lemma can be written like a Markov chain.

For example, |D

i =

|0i|D

i +

|1i|D

i =

|00i|D

i +

|01i|D

i +

|10i|D

i +

|11i|D

i =

(|0011i + |0101i + |0110i +

|1001i + |1010i + |1100i). Recall that the

(θ) gate can divide the existence prob-

ability of qubits to α

and 1 − α

(2cos

−1

(α))|0i = α|0i+

√

1 −α

|1i. In particu-

lar, when the i-th qubit is rotated, |b

i−1, j

i|D

n−(i−1)

k−j

i, j

i−1, j

i|0i|D

n−i

k−j

i +

1 −α

i, j

i−1, j

i|1i|D

n−i

k−j−1

is satisﬁed for probability α

i, j



n−i

k−j





n−i+1

k−j



(n −i + 1 −k + j)/(n −i + 1) and any binary pre-

ﬁx b

i−1, j

of length i −1 with weight j for 1 ≤ i ≤ n

and 0 ≤ j ≤ k −1. The probability distribution is

independent of the pattern of bit subsequence b

i−1, j

but depends on the length and weight of b

i−1, j

, and

thus, the number of required R

(θ) gates to divide the

probability is estimated to be O(nk).

To achieve such a Markov chain for |D

i on

a quantum circuit, we just need to keep track of

the Hamming weight j of the b

i−1, j

. We prepare

ancilla dlog(k + 1)e-qubits c to store the weight

as binary digits. When i = 1, c is initialized to

0 and b

0,0

= ε. Applying the R

(2cos

−1

(α

i, j

))

gate yields |b

i−1, j

(2cos

−1

(α

i, j

))|0i

→

√

i, j

i−1, j

i|0i

1 −α

i, j

i−1, j

i|1i

. Next,

c is incremented by 1 for the quantum state

i−1, j

i|1i

→ |b

i, j+1

i. We show a pseudocode

for constructing |D

i with auxiliary variable c in

Algorithm 1.

Algorithm 1: Preparation of |D

i with auxiliary variable c.

Input: integer k ≤ n, n-qubit

...x

i = |0i

⊗n

and dlog(k + 1)e

ancilla qubits to store c

Output: |D

1 c ← 0

2 for i ← 1, ... , n do

3 for j ← 0, ..., k −1 do

4 if c = j then

5 |0i

→ α

i, j

|0i

1 −α

i, j

|1i

6 if x

= 1 then

7 c ← c + 1

8 return |x

...x

Quantum Gates for Algorithm 1

Similar to U

n,k

, we construct a unitary opera-

tor A

n,k

such that |0i

⊗n+dlog(k+1)e

is an input and

i|0i

⊗dlog(k+1)e

is an output.

Deﬁnition 6 (A

n,k

). n-qubit unitary gate satisfying

n,k

|0i

⊗n+dlog(k+1)e

= |D

i|0i

⊗dlog(k+1)e

The building blocks of A

n,k

are C

X and C

(θ)

ICISSP 2023 - 9th International Conference on Information Systems Security and Privacy

332

𝑖

𝛼

! ,#

𝑛 + 1

⋯

|0 ⟩

⋯

|𝑘 − 1⟩

𝑛 + 2

⋯

𝑛 + log(𝑘 + 1)

⋯

𝛼

!,$ %&

Figure 2: R

gate (left) and N

gate (right) for 1 ≤ i ≤n. The R

(2cos

−1

(α

i, j

)) gate is abbreviated by α

i, j

gates. We prepare two quantum registers |x

...x

i and

...c

dlog(k+1)e

ito store the Dicke state and weight for

bit strings. For these two registers, the rotation unitary

and increment unitary N

are applied in alternating

order from 1 to n.

Deﬁnition 7 (Building blocks of R

). R

rotates the i-

th qubit |0i

according to the value of the ancilla qubit

for 1 ≤ i ≤ n and 0 ≤ j ≤k −1

:|0i

|ji

n+1

→



i, j

|0i

1 −α

i, j

|1i



|ji

n+1

where α

i, j

n−i+1−k+ j

n−i+1

. |ji

n+1

is the ancilla qubits

...c

dlog(k+1)e

i corresponding to the binary repre-

sentation of an integer j for 0 ≤ j ≤ k −1. R

is con-

structed by k C

(θ) gates shown in Figure 2 (left).

Deﬁnition 8 (Building blocks N

). N

increments the

ancilla qubit if |xi

= |1i

for 1 ≤ i ≤ n

:|0i

|ji

n+1

→|0i

|ji

n+1

|1i

|ji

n+1

→|1i

|j + 1i

n+1

can be constructed by k C

X gates in Figure 2

(right).

i can be generated using a circuit with ancilla

qubits, whose circuit complexity is computed as in the

next theorem.

Theorem 2. (Esser et al., 2021; da Silva and Park,

2022) Dicke states |D

ican be prepared with a circuit

of width n + dlog(k + 1)e and depth O(nk log k) using

one A

n,k

gate.

Note that the original circuit depth in (Esser

et al., 2021) is O(nk log k log log k) using 2dlog(k +

1)e ancilla qubits since the C

(θ) gate with

log(k + 1) control can be decomposed into a depth

O(logk log log k) elementary circuit with additional

dlog(k + 1)e ancilla. However, due to (da Silva and

Park, 2022), one can implement any n-controlled

single-qubit unitary gate C

U with O(n) depth and

O(n

) elementary gates without ancilla qubits. There-

fore, we removed the log log k term and dlog(k + 1)e

additional ancilla from the original theorem.

The circuit can also be a concrete instantiation of

the circuit in (Kaye and Mosca, 2001) with a smaller

number of ancilla qubits of size dlog(k + 1)e than

log(n/ε) of the original circuit, where ε  1 is a pre-

cision parameter.

4 GENERALIZED DICKE STATE

PREPARATION

In this section, we review the generalized Dicke

state preparation circuit proposed in (B

artschi and Ei-

denbenz, 2019). Based on the circuit, we general-

ize another Dicke state preparation circuit proposed

in (Esser et al., 2021) with the same circuit complex-

ity.

4.1 Dicke State Generalization Without

Ancilla Qubits

Algorithm 2: Prepare-|D

i without ancilla.

Input: n,k, empty quantum circuit qc

Output: qc constructs |D

1 qc.ry(2cos

−1

(β

,n)

2 for i ← 1, ... , k −1 do

3 qc.cry(2cos

−1

(β

),n −i + 1,n −i)

4 qc.U

n,k

// concat U

n,k

gate

5 return qc

First, we review the concrete construction of |D

for any integer set i ∈ K (B

artschi and Eidenbenz,

2019). To construct |D

i in polynomial gates, the

authors use the property of the U

n,k

gate such that

n,k

|0i

⊗n−`

|1i

⊗`

= |D

i for all 0 ≤ ` ≤ k. Recall

that the input of U

n,k

for the original algorithm is

|0i

⊗n−k

|1i

⊗k

. However, because U

n,k

accepts weights

` ≤k, U

n,k

also returns |D

ifor the input |0i

⊗n−`

|1i

⊗`

for all ` < k. Thus, to achieve a superposition of

Dicke states, one can set k ← maxK and input the

superposition of |00..00i, |00..01i, ..., |00..11..11

|{z}

to the U

n,k

gate. Since we want the equal superposi-

tion of Dicke states, the input superposition for U

n,k

is given by

∑

0≤i≤n

i, where α





/|C

Concrete Quantum Circuits to Prepare Generalized Dicke States on a Quantum Machine

333

|0i

⊕ ⊕

≤

4,2

Figure 3: Quantum circuit to prepare |D

≤

i =

∑

x∈{0,1}

,wt(x)≤2

|xi generated from Algorithm 2.

for all i ∈ K and α

= 0 otherwise. Such a super-

position can be constructed with one R

(θ) gate and

k −1 CR

(θ) gates. Algorithm 2 is a Qiskit-like pseu-

docode to generate a quantum circuit to prepare |D

where β

= (

/(1 −

∑

i−1

j=0

). qc.a represents the

append of gate a to circuit qc. The R

(θ) gate is de-

noted by ry(θ,i), where i is the target qubit to which

the gate is applied. The CR

(θ) gate is denoted by

cry(θ,i, j), where i is the control and j is the target

qubit. We show an example of the circuit for |D

with K = {0, 1,2} in Figure 3.

Theorem 3. (B

artschi and Eidenbenz, 2019) Gener-

alized Dicke states |D

i can be prepared with a cir-

cuit of width n and depth O(n) using one R

(θ) gate,

k −1 CR

(θ) gates and one U

n,k

gate with k ←max K.

4.2 Dicke State Generalization with

Ancilla Qubits

We show that Esser’s quantum circuit can also be gen-

eralized by considering the Markov chain for the sum

of combinations

∑

i∈K





by using the same building

blocks R

and N

. To do so, we set k ←maxK to store

the weight w in the ancilla qubits for all 0 ≤ w ≤ i,

i ∈ K. The angle value α

i, j

in block R

is replaced by

i, j

= β

i, j

/β

i−1, j

, (6)

where β

i, j

∑

`∈K,`≥j



n−i

`−j



. The entire algorithm

to construct |D

i is described in Algorithm 3. The

(θ) gate is denoted by mcry(θ, j,i), where j is

the control state and i is the target. Increment(i, qc)

in Line 7 appends N

unitary to qc. The correctness of

the Algorithm 3 is shown as follows.

Theorem 4. Algorithm 3 outputs a circuit to pre-

pare |D

i from |0i

⊗n+dlog(k+1)e

with a circuit of width

n + dlog(k + 1)e and depth O(nk log k) , where k ←

maxK.

Proof. In Line 3, Algorithm 3 rotates the 1st qubits

by R

(2cos

−1

(α

1,0

)):

|0i

⊗n

1,0

−−→α

1,0

|0i

⊗n

1 −α

1,0

|1i|0i

⊗n−1

1,0

0,0

|0i

⊗n

1,1

0,0

|1i|0i

⊗n−1

Algorithm 3: Prepare-|D

i with ancilla.

Input: n,k ← max K, qc with registers

|bi ← |0i

⊗n

,|ci ← |0i

⊗dlog(k+1)e

Output: qc constructs |D

1 for i ← 1, ... , n do

2 if i = 1 then

3 qc.ry(2cos

−1

(α

1,0

),1)

4 else

5 for j ← 0, ..., k −1 do

6 qc.mcry(2cos

−1

(α

i, j

),|ci = j, i)

7 qc ← Increment(i,qc)

8 return qc

⊕

|0i

⊕

≤

Figure 4: Quantum circuit to prepare |D

≤

i generated from

Algorithm 3.

since

1 −α

i, j

= β

i, j+1

/β

i−1, j

from







i−1





i−1

j−1



. After n rotations, we obtain

∑

n,w

∏

i=1

i,w

i−1,w

i−1

n,w

i =

∑

n,w

0,w

n,w

(7)

where w

= 0 and w

i−1

≤ w

i−1

+ 1. If w

∈ K,

n,w





= 1 and 0 otherwise. Since β

0,0

|, we obtain the equal superposition for all states

n,w

i, w

∈K with probability

. The circuit com-

plexity is the same as in (Esser et al., 2021) since

we only change the angle of the R

(θ) gate and C

gates.

The whole quantum circuit for |D

i with K =

{0,1,2} is presented in Figure 4.

5 EXPERIMENTS

To verify the performance of the generalized Dicke

state preparation circuits, we conduct several exper-

iments on both the quantum simulator and the ac-

tual quantum machine from IBM Quantum. All al-

gorithms are implemented in Qiskit.

ICISSP 2023 - 9th International Conference on Information Systems Security and Privacy

334

Figure 5: Probability distribution obtained from the noise-free simulated quantum circuit for 10

shots corresponding to

≥

i, |D

≤

i, and |D

{0,1,4}

Figure 6: Comparison of the noisy probability distribution obtained from Algorithm 2 and Algorithm 3 corresponding to

≤

5.1 Noise Free Simulation

First, we verify that Algorithm 2 and Algorithm 3

indeed construct generalized Dicke states using

the QASM quantum simulator under the condi-

tion that the circuit is noise-free. Circuit runs

and measurements were repeated numerous times

speciﬁed by shots. We observe whether the de-

sired probability distribution (superposition) is con-

structed by aggregating the quantum state measured

at each run. In our experiments, we built cir-

cuits that prepare |D

≥

i =

∑

x∈{0,1}

,wt(x)≥3

|xi,

≤

i =

∑

x∈{0,1}

,wt(x)≤2

|xi and |D

{0,1,4}

i =

∑

x∈{0,1}

,wt(x)∈{0,1,4}

|xi. The results of Algo-

rithm 3 for 10

shots are shown in Figure 5. The

ﬁgure shows that our algorithms indeed generate uni-

form superpositions |D

≥

i, |D

≤

i, and |D

{0,1,4}

i by

observing the distribution aggregated from the mea-

sured quantum state for each shot. For Algorithm 2,

we conﬁrm that the uniform distributions are correctly

constructed. Note that since Algorithm 2 and Algo-

rithm 3 are deterministic, no quantum state with un-

desired Hamming weights can be obtained as long as

the quantum circuit is noise-free.

5.2 Noisy Simulation

Next, we compared the probability distribution gath-

ered from Algorithm 2 and Algorithm 3 in the noisy

simulation model. In the experiment, the error rate

of a 1-qubit gate such as X is set to 10

−3

and that

of CNOT is set to 10

−2

. These values are plausible

and are derived from the error rates of IBM’s quan-

tum computers, such as Falcon (IBM, 2022). Fig-

ure 6 shows the results of running the circuit gen-

erated from each algorithm that obtains |D

≤

i =

∑

x∈{0,1}

,wt(x)≤2

|xi for 10

shots. Unlike the

noiseless case, quantum states with incorrect weights

≥ k were observed with a small probability. For the

desired states, Algorithm 3 generated |00000i and

states near |00000i with a higher probability than

other states. Therefore, the result is somewhat dif-

ferent from the uniform superposition. Algorithm 2

seems to generate correct weight states more uni-

formly than Algorithm 3. We transpile two cir-

cuits generated from Algorithm 2 and Algorithm 3.

The backend of the transpiler is ibm nairobi from

IBMQ, which is a 7-qubits quantum machine we will

use in our experiments in the next subsection. Ba-

sic gates implemented in ibm nairobi are [’id’,

’rz’, ’sx’, ’x’, ’cx’, ’reset’]. The depth of

the transpiled circuit of Algorithm 3 is 382, which is

twice as large as 189 of Algorithm 2. The result is

reasonable given that the depth complexity of Algo-

rithm 3 is O(k log k) larger than that of Algorithm 2.

5.3 Real Quantum Machine

We deployed Algorithm 2 and Algorithm 3 on a

real quantum machine produced by IBM Quan-

tum for toy examples. The 7-qubits quantum

Concrete Quantum Circuits to Prepare Generalized Dicke States on a Quantum Machine

335

Figure 7: Transpiled quantum circuits for |D

≤

i generated from Algorithm 2 (top) and Algorithm 3 (bottom).

Figure 8: Probability distribution obtained from Algorithm 2 and Algorithm 3 on ibm nairobi quantum machine for |D

≤

(top) and |D

≤

i (bottom).

Table 1: Gate summary of quantum circuits for Algorithm 2

and Algorithm 3 transpiled in ibm nairobi quantum ma-

chine.

Target State |D

≤

i |D

≤

Algorithm Alg. 2 Alg. 3 Alg. 2 Alg. 3

X gates 0 4 0 6

√

X gates 10 10 28 37

Rz gates 10 10 52 141

CNOT gates 8 16 79 234

Circuit Width 3 4 4 6

Circuit Depth 29 39 133 298

machine ibm nairobi was utilized in the exper-

iment. Our target generalized Dicke states are

≤

i =

(|000i + |001i + |010i + |100i) and

≤

i =

√

(|0000i + |0001i + |0010i + |0100i +

|1000i + |0011i + |0110i + |1100i + |0101i +

|1010i + |1001i). Again, we compared the prob-

ability distribution achieved by Algorithm 2 and

Algorithm 3. Table 1 summarizes the number of

gates for each circuit transpiled in the ibm nairobi

quantum machine. Figure 7 displays the transpiled

circuits to construct |D

≤

i for both algorithms. You

can also see the circuits for |D

≤

i in Appendix 6.

The average CNOT error during the experiment was

0.01225. The number of shots is 20000. To eliminate

coincidence, 100 batch executions for the same

circuit were performed and we calculated the average

value. The result is displayed in Figure 8. For |D

≤

both algorithms can generate correct weight states

with high probability (94.2% for Algorithm 2 and

93.4% for Algorithm 3). For the standard deviation

in the correct states, Algorithm 3 achieves 0.118,

which is smaller than 0.145 for Algorithm 2. Thus,

Algorithm 3 obtained better uniformity than Algo-

rithm 2 for |D

≤

i. The success probability for |D

≤

is 84.6% for Algorithm 2 and

88.9% for Algorithm 3. The standard deviation

in the correct states is 0.0516 for Algorithm 2 and

0.0631 for Algorithm 3. In particular, it is surpris-

ing that the success probability of Algorithm 3, which

uses more than 200 CNOT gates, is higher than that

of Algorithm 2 even though the CNOT error rate ex-

ceeds 0.01. The presumed reason for this is that Al-

gorithm 3 has a circuit structure with no dependencies

among the qubits except for the ancilla. Since errors

are assumed to accumulate in the ancilla, quantum er-

ICISSP 2023 - 9th International Conference on Information Systems Security and Privacy

336

ror correction should be performed preferentially on

the ancilla qubits. To construct generalized Dicke

states for larger n,k precisely, a quantum computer

with error correction capability is essentially needed.

6 CONCLUSION

We generalized a quantum circuit to prepare a su-

perposition of Dicke states proposed by (Esser et al.,

2021) with the same circuit complexity. We imple-

mented and compared two generalized Dicke state

preparation circuits using the Qiskit library. In our

experiments, we validated that both circuits can con-

struct the generalized Dicke state in a noisy quantum

simulator and a real quantum device. Future work

includes developing end-to-end quantum circuits to

solve a combinatorial problem employing the super-

position constructed from a generalized Dicke state

preparation circuit.

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APPENDIX

We show the quantum circuits for |D

≤

iwith elemen-

tary gates transpiled by the ibm nairobi backend in

the IBM quantum experience service in Figure 9 and

Figure 10.

Figure 9: Transpiled quantum circuit for |D

≤

i generated

from Algorithm 2.

Figure 10: Transpiled quantum circuit for |D

≤

i generated

from Algorithm 3.

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