Brief Notes on The Possibility of Copying Qubits in Quantum Systems
ARTYOM GRIGORYAN1, SOS AGAIAN2
1Electrical and Engineering Department,
The University of Texas at San Antonio, U.S.A.
2Computer Science Department,
City University of New York / CSI, U.S.A.
Abstract: - Copying the quantum states is contradictory to classical information processing since the fundamental
difference between classical and quantum information is that while classical information can be copied perfectly,
quantum information cannot. However, this statement does not rule out the risk of building a device that can
reproduce a set of quantum states. This paper investigates the naturally arising question of how well or under
what conditions one can copy and measure an arbitrary quantum superposition of states. The CNOT and XOR
operation-based quantum circuit is presented that exhibits entanglement of states and allows for measuring the
doubled qubits.
Key-Words: - Quantum computing, Quantum bits, Entanglement state, Qubit duplication.
Received: April 21, 2021. Revised: January 8, 2022. Accepted: January 26, 2022. Published: February 10, 2022.
1 Introduction
As In 1982, Wootters and Zurek showed that no
unitary process could create exact copies of arbitrary
quantum states [1]. The statement that non-
orthogonal quantum states cannot be copied was also
made by Dieks [2], and new no-cloning theorems can
be found in [3,4]. Many publications exist devoted to
the no-cloning theorem that uses the linearity and
unitarity of the copying transformations [5-7], perfect
cloning of no commuting mixed states [8], perfect
cloning with assistance [9], multiple copying of
qubits [10,11], supplementary information [12], the
additional measurement for probabilistic quantum
cloning [13,14], cloning of identical mixed qubits
[15], cloning with the local operation and classical
communication [16], and unclonable encryption [17].
With such an assertion, quantum computing theory
has become particularly distinctive from computing
on traditional computers. In addition, such theorems
were used to justify the security of quantum
cryptography [18,19].
The main goal of this work is to show that copies
of qubits can be nested in other more complex states
of systems with a large number of qubits, from where
they can be measured and processed. We illustrated
it, by presenting a quantum circuit that exhibits
entanglement of states and allows for measuring the
doubled qubits.
The rest of the paper is organized in the following
way. Section 2 discusses the well-known statement
about the impossibility of copying qubits. Section 3
analyses a simple quantum circuit with two CNOT
and XOR operations for nesting the doubled qubits in
3- qubit. Such a circuit allows us to observe and
measure the duplicated states of a qubit.
2 Problem Formulation
In this section, we discuss the known statement
about copying qubits. The unitary operator for
performing such a copy of an individual qubit in a
superposition
is considered to be the 2-qubit operator
Here, the amplitudes are such that (or
in the complex case). Also, and
denote the quantum computational basis states
of the single qubit,
and is the operation of the tensor product, or
Kronecker product, of vectors.
When measuring the qubit, by using the Hermitian
projection on the basis and , the probability of
the outcome is and the probability of outcome
is . The axioms of quantum mechanics
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demonstrate that after the measurement, a qubit is
collapsed to the measured basis state so that a qubit
will be destroyed in this measurement. Thus, we may
perhaps state the well-known no-cloning theorem: no
quantum procedure exists that can reproduce
perfectly an arbitrary quantum state. Therefore, the
measurement is irreversible; in contrast, it is pretty
easy to copy information, even in a reversible manner
in classical computers. It means that using the well-
known teleportation protocol, we may create a
perfect replica of the original qubit, but this will be at
the cost of destroying information encoded in the
original qubit [10,18]. Thus, a) if we are only
interested in producing imperfect copies, then it is
possible to design machines (actually, to find unitary
transformations) that can copy quantum states, b) if
we do two identical copies, then the quality of these
copies depends on the input state; and c) we may
formulate the quantum copying problem goal is to
produce a copy of the initial qubit, which is as close
as possible to the original state, while the output state
of the original qubit is minimally disturbed [18].
However, this theorem does not rule out the
possibility of building a device that can copy a
particular set of orthonormal quantum states [20]. In
quantum computing, the CNOT operation does not
allow copying the qubits, as in traditional computing,
when copying the bits. In a digital computer, when
copying a bit, a new cell is allocated in the computer's
memory, the value of the bit is read, and then this
value is written to the cell. Such a read-write-out
procedure is likely to take place in quantum systems.
Undoubtedly, other operators and quantum circuits
are needed here. For each state of a qubit, probably
somewhere in space, an identical state is reproduced.
Maybe it is not in its pure form, but in some entangled
state with other qubits. In other words, it is possible
that such copies can be nested in other shells of
systems with a large number of qubits, from where
they can be measured and processed.
2.1 No Cloning Qubits
Let us consider the common calculations in the
statement of copying qubits. We consider the qubit
in the Hadamard superposition
If a unitary transform copies this qubit and it is a
linear operator, we obtain two qubits in the following
state:
The state of doubled qubits is
Thus, we obtain different 2-qubit quantum
superpositions and What we use
in the above calculations is the assumption that if
an unitary operator copies a single qubit in the
superposition , then it copies any other
superposition of the qubit, i.e.,
2.1.1 Matrix of the Transformation
We can consider the quantum states and
as 4-D vectors,
and
We are looking for a 4×4 unitary matrix , such
that
It is clear that the coefficients of such a matrix will
be defined by the values of inputs, and . Unitary
transformations for copying qubits are parameterized
by amplitudes of quantum superpositions of states of
the qubits. They are not universal, i.e., they cannot
copy any qubits. It is clear that additional operators
are needed to complete this task. The only operators
that can be used in additional to the unitary
transforms are operators of measurement, or
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projection operators. However, in order to use them,
we need to first include the required 2-qubit in a
system with a large number of qubits.
3 Quantum Circuit for Nesting
Doubled Qubits
This section presents a method of nesting the
doubled qubits in a larger state. In other words, we
discuss the circuit which might be used to calculate
the doubled qubits in arms of 3-qubit. The
measurement and separation of states of doubled
qubits are described.
Let us consider a qubit in the state
with the required condition that
These coefficients are considered real. The
duplicated copy of this state is the 2-qubit state
(6)
When applying the CNOT operator (X) with
control qubit and controlling (target) state
the result is the 2-qubit state
(7)
as it is illustrated in Fig. 1. This operation changes
the 2nd qubit state to , when the control qubit
is , i.e., Except for
the cases when and the states
and are different. Thus, the qubit in its general
state is not copying by this circuit with the CNOT
gate.
Figure 1 The circuit with the CNOT operation.
Now, we apply the second CNOT operator with
a new control qubit in the state
This qubit can be obtained, by using the
Walsh-Hadamard gate on the basis state ,
The target is the second qubit of the 2-qubit state
. The result of this operation is
the following superposition of 3-qubit (without
coefficient ):
We consider the 3-qubit permutation ,
for which we will use the gate shown in Fig. 2 in part
(a) and call it the 2-XOR operator. This permutation
is
The matrix of this permutation can be written as
The logic element in this figure is not a Toffoli
gate over 3-qubit state, which performs the
permutation (6,7). The circuit representation of the
Toffoli gate is shown in part (b).
(a) (b)
Figure 2 Circuit representation of the 3-qubit permutations (a)
and (b) (6,7).
Applying this operator on 3-qubit in superposition of
Eq. 8, we obtain the following state:
It is a superposition of the first four basis states ,
, and . Now, considering the normalization
coefficient , we obtain
Q2
Q1
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One can note that the second qubit is in state
The abstract circuit for calculating this
3-qubit is given in Fig. 3 (in Appendix). The
same diagram in compact form is shown in Fig.
4 (see Appendix).
Now, we consider the doubled qubits . When
measuring the first qubit of , we obtain the state
, ,
and the state with
probability As follows from Eq. 9, the 3-qubit
superposition can be written as
Measuring the first qubit of , we obtain with the
same probability 0.5 the following two states. If the
measured qubit state is 0, the state will be
If the measured first qubit state is 1, then the state will
be
Thus, after measuring the first qubit in the 3-qubit
state , in the first two qubits of the measurement
we obtain one of the states of the doubled qubits ,
namely
or
The last qubit of both measurements is 0. The full
circuit of processing the given qubit and
measuring the doubled qubits nested in the 3-qubit
state is shown in Fig. 5 (in Appendix). The parameter
of measurement or when the measured first
qubit is 0 or 1, respectively.
Thus, this circuit shows that the doubled qubits
can be nested in the 3-qubit state, namely in the first
two qubits of this state. The output of this scheme is
a kind of shell containing doubled qubits, from where
they can be measured.
Algorithm of nesting and measuring the doubled
qubits in the 3-qubit state:
1.
2.
3.
.
4.
.
5.
.
6.
The doubled qubits are described by the
first two qubits of the measured 3-qubit,
when or .
Each of these measured 2-qubit superpositions
carries information of the original qubit ;
and
Measuring any of these 2-qubits, we obtain the
original qubit
4 Conclusion
The main challenges in quantum computing are not
only in developing algorithms and quantum circuits.
The accurate measurement of the calculated multi-
qubit state is also a difficult task to be solved. Unlike
the calculation in the traditional computer, in
quantum computing, many measurements are
required, i.e., the circuit should be run repeatedly.
The quantum circuits to get only the copy of the
qubit, i.e., the doubled qubits, are unknown, namely,
such circuits do not exist, according to what is said in
the current literature. In this paper, we present a
quantum circuit with CNOT operations, Hadamard
gate, permutation and measurement, which shows the
doubled qubits in two qubits of the calculated 3-qubit
state. In other words, we have shown that there exist
quantum schemes that allow us to measure doubled
qubits. As stated in the introduction, in this work we
presented our vision of transforming and copying
qubits.
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5 Appendix
Figure 3 The 3-qubit circuit with two CNOT and 2-XOR operations.
Figure 4 The 3-qubit circuit with two CNOT operations and 2-XOR operations.
Figure 5 The 3-qubit circuit with measurement of the duplicated qubit.
Q1
Q3
Q2
Q1
Q3
Q2
Q1
Q2
Q3
easurement
Q1
Q3
Q2
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Contribution of individual authors to the creation
of a scientific article (ghostwriting policy)
Artyom Grigoryan: Conceptualization,
Methodology, Circuits and Algorithm, Original
draft preparation.
Sos Agaian: Problem formulation, Methodology,
Original draft preparation and structuring.
'Not applicable'
Conflicts of interest/Competing interests
The author declares no conflicts of interest regarding the
publication of this paper
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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