Excitations in Qubit Space using Single-Particle Dipole Operator,

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Excitations in Qubit Space using
Single-Particle Dipole Operator
Nifeeya Singh1,∗Abhishek2,†Pooja Siwach3,
Ashutosh Singh1, and P. Arumugam1
1Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
2Department of Physics, University of Surrey, Guildford, Surrey, GU2 7XH and
3Lawrence Livermore National Laboratory (LLNL), Livermore, CA 94551, United States
Introduction
Quantum computers have potential applica-
tions in many fields, such as quantum chem-
istry and condensed matter physics; quantum
computing has become an efficient method for
modeling complicated physical systems. In
this study, we present an algorithm for the
transition between the states in qubit space
using the single-particle dipole operator [1].
This transition between the qubit states is
essential in modeling the nuclear response
within the regime of linear response theory.
An N-state quantum system is modeled us-
ing an N-qubit quantum circuit, where the
occupied states are represented by the qubits
in state |1iand unoccupied ones with state
|0i. We have employed a time-dependent state
preparation [2] approach for preparing excited
states for nuclear dynamics on a quantum
computer. For that, the quantum register is
initially set in the state |ψ0i. The primary ob-
jective is to generate the excited state |φiby
applying the operator Donto the initial state
|ψ0i. The efficiency of implementing this pro-
cess holds significant importance for enhanc-
ing the performance of quantum algorithms
developed for the investigation of nuclear re-
sponse. To illustrate our approach, we will
use the example of the nucleus 120Sn, where
the nuclear structure is modeled using the har-
monic oscillator (H.O.) mean field. The initial
state is a many-body state of the four H.O.
states where two are below the Fermi level
∗Electronic address: n_singh@ph.iitr.ac.in
†Electronic address: a.abhishek@surrey.ac.uk
(occupied), and the other two are above the
Fermi level (unoccupied).
We will showcase the initial state and the
states resulting from the operation of the
single-particle dipole operator. In Sec. 1, we
explain the Jordan-Wigner (JW) transforma-
tion to map the fermionic operator to the
qubit operator. In Sec. 2, we discuss the oper-
ator’s details and the basis size relevant to our
specific example,120 Sn. Following this, we will
explain the time evolution method to generate
the excited state. Our findings from quantum
simulators, real quantum computers, and dis-
cussion are presented in Sec. 3.
1. Jordan-Wigner transformation
The Jordan-Wigner transformation [3] is
a mapping of fermionic operators (creation
and annihilation operators) to qubit opera-
tors. The operation 1
2(Xj±Yj) is used in this
transformation to update the qubit j based on
the occupation of the basis state |ji, and op-
eration Zkis used to incorporate the parity
for all the qubits with k < j. Hence, the
fermionic creation and annihilation operations
can be written as
a†
j=1
2(Xj−iYj)⊗Y
k<j
Zk,(1)
aj=1
2(Xj+iYj)⊗Y
k<j
Zk.(2)
2. Single-particle dipole operator
ˆ
Dαis the single-particle dipole operator
where αrepresents the three spatial direc-
tions. ˆ
Dαis defined as [1]
Dα=NZ
A(rN
com −rP
com),(3)
Proceedings of the DAE Symp. on Nucl. Phys. 67 (2023) 67
Available online at www.sympnp.org/proceedings

where rN
com and rP
com are the centers of mass of
neutrons and protons, respectively. Here, N,
Z, and Aare the neutron, proton, and atomic
mass numbers, respectively. We have consid-
ered the case of a spherical nucleus 120Sn in
the H.O. basis. The N= 4 state represents
the Fermi level for 120Sn as every state has a
degeneracy of (N+ 1)(N+ 2) and we consider
the states from N= 3 to 6. The single-particle
dipole operator matrix in the basis N= 3 to
6 is given by
D=


0 1.121 0 0
1.121 0 1.179 0
0 1.179 0 2.692
0 0 2.692 0


.
This dipole operator matrix is transformed
from fermonic to qubit space using JW trans-
formation as given in Sec. 1, the transformed
operator DJ W is given as
DJW = 0.560X0X1+ 0.560Y0Y1+ 0.897X1X2
+ 0.897Y1Y2+ 1.346X2X3+ 1.346Y2Y3.
Next, to perform the state preparation, we
start with a quantum register initialized in the
ground state, |ψ0i=|0011i, and we produce
the excited state |φi=O|ψ0iby using the
time-evolution operator connected to the ex-
citation operator DJ W .
August 24, 2022
H X H
|Ψ0⟩U(γ)U†(γ)
1
FIG. 1: Quantum circuit for state preparation.
U(γ) = exp(−iγDJ W ) (4)
= cos γDJ W −isin γDJ W .
The “time” argument in this case, is γ. We
perform this operation with the circuit shown
in Fig. 1. The final state created by preparing
the ancilla qubit in the state |0iand using the
circuit is
|Ω(γ)i=|0i ⊗ cos (γDJ W )|Ψ0i
−i|1i ⊗ sin (γDJ W )|Ψ0i(5)
If we measure the ancilla in |0i, we start the
process over from scratch and make another
attempt as we are interested only in the second
component of the state in Eq. 5.
3. Result and Discussion
0011
1010
0101
1100
1001
0110
States
0.0
0.2
0.4
0.6
0.8
1.0
Probabilities
Ground State
Excited State (QASM)
Excited State (IBMQ)
FIG. 2: Comparison of Initial and operator oper-
ated excited state using the actual quantum com-
puter (IBMQ) and QASM simulator.
In Fig. 2, we show our result for the ex-
cited state prepared with the single-particle
dipole operator calculated on the QASM sim-
ulator and compare it with the results ob-
tained on the actual quantum devices (IBMQ
Nairobi). The difference in the results is
due to the noise in the real quantum com-
puter. However, the state with the maximum
probability is the same for both cases where
particles from below the fermi level are pro-
moted above. In conclusion, using the single-
particle dipole operator, our algorithm har-
nesses quantum computing to facilitate tran-
sitions between qubit states efficiently. This
holds significant promise for accurately mod-
eling nuclear responses and showcasing the
potential of quantum computing in quantum
state manipulation.
References
[1] Abhishek Sharma et al. Phys. Scr. 98
035303 (2023 ).
[2] Alessandro Roggero et al. Phys. Rev. C
102, 064624 (2020).
[3] Pooja Siwach, P. Arumugam, Phys. Rev.
C104, 034301 (2021).
Proceedings of the DAE Symp. on Nucl. Phys. 67 (2023) 68
Available online at www.sympnp.org/proceedings