# Fault-tolerance of the [[8,1,4]] non-CSS code

Pranav Maheshwari1 and Ankur Raina2 1Department of Physics, Indian Institute
of Science Education and Research Pune, India
2Department of EECS, Indian Institute of Science Education and Research
Bhopal, India
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show the fault-tolerance of the not-so-well known $[[8,1,4]]$ non-CSS code
and study the logical error rates of the code. To do so, we adopt the
procedure of the bare ancilla method presented by Brown _et al._ We choose the
encoding procedure for stabilizer codes given by Gottesman and modify it to
suit the setting of a class of non-CSS codes. We consider two types of noise
models for this study, namely the depolarizing noise and anisotropic noise to
depict the logical error rates obtained in decoding.

##  I Introduction

While quantum computing boasts immense power, it contends with a relentless
adversary: environmental noise that disrupts quantum systems. This challenge
is particularly pronounced in NISQ (Noisy Intermediate-Scale Quantum)
computers, which must grapple with error correction to conduct meaningful
computations amidst this noise. Quantum Error Correcting Codes (QECC) are
specialized encoding schemes that distribute quantum information redundantly
across multiple qubits in such a way that errors can be detected and
corrected. QECC are organized into distinct families, a well studied family is
of Calderbank-Shor-Steane (CSS) Codes which treats $X$ and $Z$ Pauli errors
separately[1][2].

Integrating error correction introduces errors due to noisy gates, thereby
underscoring the significance of fault tolerance. Fault tolerance ensures that
the system remains resilient against these newly induced errors, providing an
additional layer of protection to maintain the integrity of the quantum
computation. The Threshold theorem in quantum computing asserts that if the
error rate in a quantum computation is below a certain threshold, then it’s
possible to perform arbitrarily long quantum computations reliably by using
quantum error correction codes [3][4]. Many fault-tolerant schemes have been
proposed which require two or more ancillary qubits for QECC. [5][6][7]. Brown
_et. al_ proposed a fault-tolerant seven-qubit code that requires a single
ancilla to correct any error propagated from the syndrome qubit[8].

We discovered an eight-qubit non-CSS code utilizing cluster states
measurements[9] and moulded it to obtain a bare ancillary fault-tolerant code.
We upgraded the bare ancillary method in [8] to correct all two-qubit errors
and discard the limitations poised by hook errors. We reveal a unitary encoder
for non-CSS codes and present an effective implementation of any quantum error
correction code using IBM’s qiskit[10]. We sketch the procedure to obtain
pseudo-threshold values and the leading order of corrected error rates.

This paper studies the fault-tolerant properties of the eight-qubit non-CSS
code under the standard depolarizing noise model and the anisotropic noise
model. We test the traditional simulation scheme against a physically suitable
error correction procedure and obtain pseudo-thresholds and the leading order
term in error rates after quantum error correction (QEC).

The paper is organized as follows: Section II introduces quantum error
correction and implementation procedure. Section III describes the unitary
encoding scheme and its enhanced version that is suitable for non-CSS codes.
Section IV introduces the eight-qubit code and details the single-qubit fault
tolerance method. It explains how the code can bypass any propagated error and
the corresponding encoder and detector circuits. Noise Models of our interest
are outlined in Section V followed by simulation schemes in Section VI. We
present one complete cycle of bare ancillary error correction and its
modification to obtain pseudo-threshold values. In Section VII, we discuss the
code performance and the reason why the standard depolarising model shows more
optimistic error rate values. We also comment on practical error correction
procedure and why modifications are required in simulations to obtain
benchmarks.

###  I-A Notation

We list the notations used in this work:

$n$ | Number of physical qubits.
---|---
$k$ | Number of logical qubits.
$d$ | Minimum Hamming distance of the code.
$\mathcal{S}$ | Stabilizer subgroup.
$g$ | Generators of the stabilizer group.
$M$ | Standard form generators
$\mathcal{C}$ | Code or Codespace.
$X_{i},Y_{j},Z_{k}$ | Pauli matrices on qubits $i,j,k$ respectively.
$H$ | Check matrix of the code.
$I$ | Identity operator
$\mathbf{H}$ | Hadamard gate.
$\mathbf{S}$ | Phase gate.
$\ket{\psi_{i}}$ |  $k$ qubit initial state
$\ket{\psi_{e}}$ |  $n$ qubit encoded state
$\ket{\psi_{d}}$ |  $n$ qubit decoded state
$\mathbb{F}_{2}$ | Binary field $\\{0,1\\}$

##  II Quantum error correction

First of the quantum error correcting codes were discovered independently by
Shor [11] and Steane [12] followed by extensive work to create quantum codes
from classical codes. This section provides insights in classical coding
theory and the structure of QECCs.

###  II-A Preliminaries

In classical error correction, a linear code, $\mathcal{C}$, encodes $k$ bits
of information into $n$ bits and is denoted as an $[n,k]$ code. The code is
specified by a matrix of size $n\times k$ called the generator matrix, $G$,
whose entries are elements of $\mathbb{F}_{2}$. To encode all messages
uniquely, we require that columns of $G$ be linearly dependent.

A codespace of an $[n,k]$ code is defined to consist of all vectors of length
$n$, $x\in\mathbb{F}_{2}^{n}$, such that $Hx=0$ where $H$ is a matrix of size
$n-k\times n$, known as Parity Check Matrix. We can also say that codespace is
the kernel space of $H$. By Rank-Nullity Theorem, $H$ should have linearly
independent rows. The matrices $G$ and $H$ are related as $HG=0$.

Hamming Distance, d(x,y) between two codewords $x$ and $y$ is defined to be
the number of places at which $x$ and $y$ differ. The weight of a codeword
(any vector in codespace) is defined to be the distance from the string of all
zeroes, i.e.,

$\text{wt}(x)=d(x,0_{n})\quad\forall x\in\mathcal{C}$ (1)

Distance of a code, $d$ is the minimum distance between any two codewords.

$d(C)=\min\\{d(x,y)\quad|\quad x,y\in C\;\;\text{and}\;\;x\neq y\\}$ (2)

which implies $d(\mathcal{C})=\min\\{\text{wt}(x)|x\in\mathcal{C}\\}$. This
defines a $[n,k,d]$ code.

Consider a codeword $y=Gx$ but an error is induced such that $y^{\prime}=y+e$.
Since $Hy=0$, we get $Hy^{\prime}=He=s$, called the error syndrome. If for all
possible errors $e_{j}$, we know corresponding $s_{j}$, we can match the error
syndrome obtained to identify the error and perform error correction
accordingly. For a code to correct errors up to weight $t$, the minimum
distance of the code should be $2t+1$ so that $y^{\prime}$ can be corrected to
$y$ satisfying $d(y,y^{\prime})\leq t$.

In quantum computation, $n$ length vectors are replaced by $n$ qubit state
vectors, and operators replace the matrices. Akin to the parity check matrix,
$H$, we create a $n-k\times 2n$ check matrix with independent rows. These rows
correspond to $n-k$ independent operators acting on $n$ qubits, called the
Stabilizer generators. The span of these operators generates a group called
the Stabilizer group, $\mathcal{S}$. QECCs harnessing stabilizers are called
as Stabilizer Codes and we refer to the corresponding error correction
mechanism as Stabilizer Formalism [13].

A single qubit can exist in a $2^{1}$ dimensional complex vector space,
denoted by $\mathcal{V}_{1}$. Correspondingly, the operators acting on a
single qubit state can be expressed as a linear combination of Pauli Matrices
forming a basis group, $\mathcal{G}_{1}$. For a single qubit, we can define
Pauli Basis Group as:

$\mathcal{G}_{1}\equiv\\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm
iZ\\}$ (3)

This group can be generated by operators $iI,X,Z$ represented as
$\mathcal{G}_{1}=\langle iI,X,Z\rangle$. For $n$ qubits, Pauli group
($\mathcal{G}_{n}$) contains $n$-fold tensor products of Pauli Matrices with
multiplicative factors $\\{\pm 1,\pm i\\}$.

Tensor products of Pauli Matrices follow the following two properties:

  1. 1.

For any two elements $g_{1},g_{2}\in\mathcal{G}_{n}$, either they commute or
anti-commute, i.e., either $[g_{1},g_{2}]=0$ or $\\{g_{1},g_{2}\\}=0$.

  2. 2.

Each element is hermitian and involutory, i.e., $\forall
g\in\mathcal{G}_{n},\;g^{\dagger}=g$ and $g^{2}=1$.

###### Theorem 1

Let $\mathcal{S}=\langle g_{1},\dots,g_{n-k}\rangle$ be generated by $n-k$
independent and commuting elements from $\mathcal{G}_{n}$, such that
$-I\notin\mathcal{S}$, then $\mathcal{V}_{\mathcal{S}}$ is a $2^{k}$
dimensional vector space.

This $\mathcal{V}_{\mathcal{S}}$ serves as the codespace for an $[[n,k,d]]$
quantum code containing $k$ qubits information in $n$ qubits. All the
stabilizers commute with each other and all the codewords are in +1 eigenspace
of the stabilizers.

###### Definition 1

Normalizer of a subgroup $\mathcal{S}$ in group $\mathcal{G}_{n}$ is the the
set of elements $\mathcal{N}$ given by

$\mathcal{N}_{\mathcal{G}_{n}}(\mathcal{S})=\\{g\in\mathcal{G}_{n}\quad|\quad
g\mathcal{S}g^{-1}=\mathcal{S}\\}$ (4)

Thus, the normalizer subgroup contains all elements that commute with the
stabilizer subgroup. The elements in
$\mathcal{N}_{\mathcal{G}_{n}}(\mathcal{S})$ are also called logical
operators. For an $[[n,k]]$ code, there exists $n-k$ stabilizer generators and
$2k$ normalizer generators. These generators correspond to the encoded
$\mathcal{G}_{k}$ group and are referred as $\bar{X}$ and $\bar{Z}$ operators.

###### Theorem 2

Let $\mathcal{S}$ be the stabilizer subgroup for a code $\mathcal{C}$. Suppose
$\\{E_{j}\\}$ is a set of operators in $\mathcal{G}_{n}$ such that
${E_{j}}^{\dagger}E_{k}\notin\mathcal{N}_{\mathcal{G}_{n}}(\mathcal{S})\setminus\mathcal{S}\;\forall
j,k$. Then $\\{E_{j}\\}$ is a correctable set of errors for code
$\mathcal{C}$.

There exists $n-k$ independent error operators, each of which anticommutes to
one of the stabilizer and commutes to all others. The check matrix provides a
non-zero value to syndrome vector of an error corresponding to the row which
anticommutes with the error.

$\mathcal{G}_{n}$ group has $2n$ generators. If we replace each by its encoded
version i.e., $P\rightarrow\bar{P}$ where $P$ is a generator of
$\mathcal{G}_{n}$, Fig. 1 provides a simple visualization to Stabilizers
($I$), Normalizers ($\mathcal{G}_{k}$), and Errors ($\mathcal{E}$).

$\displaystyle\overbrace{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bar{Z}_{1},\dots,\bar{Z}_{n-k}}}^{I}\;\;{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\bar{Z}_{n-k+1},\dots,\bar{Z}_{n}}$
$\displaystyle\underbrace{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bar{X}_{1},\dots,\bar{X}_{n-k}}}_{\mathcal{E}}\;\;\underbrace{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\bar{X}_{n-k+1},\dots,\bar{X}_{n}}}_{\mathcal{G}_{k}}$
Figure 1: Visualization of encoded operators in $\mathcal{G}_{n}$
corresponding to Stabilizer Formalism

A deeper understanding of the stabilizer formalism could be gained from [4]. A
mathematically extensive explanation is given in [14].

###  II-B Error Correction Protocol

Given the stabilizer generators, $\\{g_{1},\dots,g_{n-k}\\},$ of a QECC, we
encode $k$ logical qubit state information, $\ket{\psi_{i}}$, into an $n$
physical qubit state $\ket{\psi_{e}}$ that belongs to the codespace
$\mathcal{C}$. Such a code can tolerate certain errors during computation. The
entire process of QEC can be summarized as follows:

  1. 1.

Initialize an $n$ qubit state, $\ket{0}^{\otimes n-k}\ket{\psi_{i}}$ where
$\ket{\psi_{i}}$ is the $k$ qubit state to be encoded.

  2. 2.

Apply a unitary quantum circuit called Encoder which maps the above $n$ qubit
state to the simultaneous +1 eigenspace of all stabilizer generators, giving
the $\ket{\psi_{e}}$ state.

  3. 3.

Take an ancilla qubit in $\ket{+}$ state, and apply controlled $g_{1}$
operation on the encoded state, controlled on the ancilla.

  4. 4.

Measure the ancilla qubit in $X$ basis and store the outcome. Reset the
ancilla to $\ket{+}$ state and repeat step 3 for all $g_{i}$ where
$i\in\\{2,\dots,n-k\\}$.

  5. 5.

Combine the stored outcomes in order and call it the syndrome. Feed the
syndrome to a corrector function, which applies the appropriate recovery
operation.

  6. 6.

Since the action of the encoder is a unitary operation, we can apply inverse
of each operator in encoder, in reverse order, to obtain the decoded state,
$\ket{\psi_{d}}$.

A schematic of the steps involved in QEC is presented in Fig. 2. The details
of constructing the encoder is given in Section III. Circuits specific to
[[8,1,4]] code are given in Section IV-E.

[wire types = q,q,n] $\ket{0}$ & n-k [2]Encoder [3]Detector
[3,steps=4,style=dashed,rounded corners,fill=blue!20, inner
xsep=1pt,background,label style=label
position=below,anchor=north,yshift=-0.2cm]Decoder [2]Corrector [2]Encoder^†
[2]$\ket{\psi_{d}}$
$\ket{\psi_{i}}$ k
$\ket{+}$ q c [vertical wire=c]-1 n n n

Figure 2: Block diagram description of the $[[n,k]]$ quantum error correcting
code.

##  III Encoder Design

Encoding requires embedding a $2^{k}$ dimensional subspace inside a $2^{n}$
dimensional Hilbert space $\mathbb{C}^{2^{n}}$ of $n$ qubits. Step 2 to Step 4
given in Section II aim at applying projection operators which take an
$n$-qubit state to the simultaneous +1 eigenspace of the stabilizer
generators, commonly called the codespace, $\mathcal{C}$. The detector and
corrector combined act as an encoder, which we call the measurement based
encoder. Since it involves non-unitary measurement operations, we cannot use
its inverse to decode a codeword state back to $\ket{\psi_{i}}$. An efficient
encoding technique is presented in [15] which shuffles the stabilizer
generators in a certain form so as to obtain an encoding unitary circuit as
described next.

###  III-A Efficient encoding mechanism

The check matrix, $H$ corresponding to the stabilizer generators contains two
blocks $H^{x}$ and $H^{z}$ corresponding to qubit location of $X$ and $Z$
operators in the stabilizer generators as given in Equation 5:

$\mathcal{S}\longrightarrow[H^{x}_{n-k\times n}|H^{z}_{n-k\times n}].$ (5)

Adding any two rows in $H$ is equivalent to multiplying the respective
stabilizer generators, and swapping two columns is equal to relabeling the
qubits. Following these two actions, we can translate any check matrix to the
standard form [4] given in Equation 8, where $r$ is the rank of $H^{x}$ block
and $l=n-k-r$:

$\displaystyle\left[\begin{array}[]{ccc|ccc}I_{r\times r}&A_{r\times
l}&A^{\prime}_{r\times k}&B_{r\times r}&0_{r\times l}&C_{r\times k}\\\
0_{l\times r}&0_{l\times l}&0_{l\times k}&D_{l\times r}&I_{l\times
l}&E_{l\times k}\end{array}\right].$ (8)

We can also obtain logical operators ($\bar{X}$, $\bar{Z}$) from the standard
form using Equation 13:

$\displaystyle\left[\begin{array}[]{c}\bar{X}\\\
\bar{Z}\end{array}\right]=\left[\begin{array}[]{ccc|ccc}0_{k\times
r}&E^{T}_{k\times l}&I_{k\times k}&C^{T}_{k\times r}&0_{k\times l}&0_{k\times
k}\\\ 0_{k\times r}&0_{k\times l}&0_{k\times k}&A^{\prime T}_{k\times
r}&0_{k\times l}&I_{k\times k}\end{array}\right].$ (13)

We can retrieve a new set of stabilizer generators, $M_{1},\dots,M_{n-k}$,
from the standard form. To create the corresponding encoder circuit, we
initialize $n-k$ ancillae in $\ket{0}$ state and $k$ information qubits in
$\ket{\psi_{i}}$ state. The action of encoder gives the encoded state
$\ket{\psi_{e}}$ equivalent to

$\ket{\psi_{e}}=\alpha\times\prod\limits_{i=1}^{n-k}(I+M_{i})\ket{0}^{\otimes
n-k}\ket{\psi_{i}},$ (14)

where $\alpha$ is the normalization factor. Applying Hadamard gate
$\mathbf{H}$ on first ancillary qubit followed by controlled-$M_{1}$
operation, with control on teh ancilla, gives the state
$(I+M_{1})\ket{0}^{\otimes n-k}\ket{\psi_{i}}$. Proceeding similarly for all
ancilla up to $r$ gives the $\ket{\psi_{i}}$ state. To accommodate logical
operations within the encoder, we first need to apply controlled-$\bar{X}$
operations as depicted in Fig. 3 showing an illustration of the encoder [15].

[2]$\ket{0}^{\otimes r}$ & H 1 M_2 [4]$\ket{\psi_{e}}$
[3]M_1 H 1[u]q
$\ket{0}^{\otimes l}$ ¯X [2]M_2
$\ket{\psi_{i}}\;\;$ -1

Figure 3: An schematic of the efficient encoder example for 2 stabilizers
$(M_{1},M_{2})$ in standard form.

A modification given in [16] points to errors in [13] and provides suggestions
on removing non-essential gates. In the next section, we adapt the mechanism
to suit for encoding non-CSS codes.

###  III-B Enhanced Encoder for non-CSS Codes

The efficient encoding mechanism in [15] defines $Y:=XZ$ and omits any
associated phase. [14] We notice that $Y^{2}=I$ but $(XZ)^{2}=-I$, which might
change the syndrome values and the encoded state. Thus, the above method suits
perfectly to CSS codes but fails with certain non-CSS codes.

To incorporate the required sign, we add another column, phase column, in the
check matrix, $H$. Corresponding to $\\{+,i,-,-i\\}$, phase of the stabilizer,
we add values $\\{0,1,2,3\\}$ respectively in the last column. Similar row
operations (modulo 4 for phase columns) are applicable leading to the standard
phase column of the standard matrix.

Each Pauli $Y$ in the standard form stabilizers devours a multiple of $i$ from
the phase column. We can then obtain the standard stabilizer generators and
corresponding phases. While creating circuit, we add an $\mathbf{S}$ gate
after $\mathbf{H}$ gate to encode $(I+Y)\ket{0}$. And to account for
stabilizer phase, add an $\mathbf{S},Z$ or $\mathbf{S}Z$ gate respectively for
$i,-1$, and $-i$ in phase column. We check if the state has been mapped to +1
eigenspace of all stabilizer generators. Through additional $\mathbf{S}$ and
$Z$ operators, we can correct any phase difference in the encoded state
leading to unexpected syndrome values.

##  IV Single Qubit Fault Tolerance

Perfect codes such as the [[5,1,3]] code saturates the quantum Hamming bound,
i.e., utilizes all possible syndrome values to detect errors correctable by
the code. For an $[[n,k,d]]$ code, we expect $2^{n-k}$ distinct syndrome
values. A non-degenerate code of distance, $d=3$, requires $3\times$ $k\choose
1$ distinct syndrome values for error correction. The difference of total
possible values and the required syndrome values are the spare syndromes which
could be leveraged to detect faults that occur during the error correction
protocol. Utilizing those values of syndrome measurements sets the footing of
single-qubit fault tolerance method that requires a single ancillary qubit to
correct errors due to fault gates which otherwise would lead to higher weight
errors or logical errors.

Following the syndrome detection method as outlined above, while obtaining
syndrome value corresponding to a stabilizer, say, $M=X_{1}Y_{2}Z_{3}$, a
$P\otimes X$ error on the ancilla qubit after the three gates propagates to
the following errors: $P_{1}Y_{2}Z_{3}$, $P_{2}Z_{3}$, and $P_{3}$
respectively as shown in figure LABEL:fig:bare_normal_detector. Here
$P_{i}\in\\{I_{i},X_{i},Y_{i},Z_{i}\\}$. For fault tolerance, we ensure that a
single fault must lead only to an error correctable by the code. Thus, in
addition to errors correctable by the distance property of the code, we only
need to detect the data qubit errors arising due to faulty gates. This
requires each propagated error to have syndrome values distinct from the
correctable errors as well as other propagated errors.

###  IV-A Reordering Stabilizers

In certain cases, the syndrome values corresponding to faults on one
stabilizer may match the syndrome of fault on another stabilizer or a
correctable error. We use the reordering trick to eliminate such similarities.
We notice how the propagated errors change if we reorder $M$ to
$Y_{2}Z_{3}X_{1}$ as shown in figure LABEL:fig:bare_reordered_detector. We’ll
reorder the gates such that all propagated errors have distinct syndromes. A
similar technique is used in [8] to achieve fault tolerance under anisotropic
noise model.

Consider a stabilizer of form $S_{i}S_{j}S_{k}S_{l}$, if a $P\otimes X$ error
occurs on gate corresponding to $S_{i}$, the propagated data error will be
$P_{i}S_{j}S_{k}S_{l}\equiv P^{\prime}_{i}$ which is a single qubit pauli
error on $i^{\mathrm{th}}$ qubit. Similarly error on controlled $S_{j}$ gate
leads to $P_{j}S_{k}S_{l}\equiv S_{i}P^{\prime}_{j}$ error where
$P^{\prime}=(PS)$ is another Pauli operator. Considering a code that can
correct any single qubit error, we are concerned only about propagated errors
of weight two or more. We observe that a weight four stabilizer of form
$S_{i}S_{j}S_{k}S_{l}$ can only lead to $S_{i}P^{\prime}_{j}$ and
$P^{\prime}_{k}S_{l}$ kind of higher weight errors while a weight six
stabilizer $S_{i}S_{j}S_{k}S_{l}S_{m}S_{n}$ can only lead to
$S_{i}P^{\prime}_{j}$, $S_{i}S_{j}P^{\prime}_{k}$, $P^{\prime}_{l}S_{m}S_{n}$,
and $P^{\prime}_{m}S_{n}$ kind of higher weight errors where
$S,P,P^{\prime}\in\\{I,X,Y,Z\\}$.

###  IV-B Details of [[8,1,4]] Code

We use the non-CSS code construction method outlined in [9] to obtain
stabilizer generators for a $[[8,1,4]]$ code. The code is modified to reduce
the weight of the stabilizer generators such that it shows fault-tolerant
properties requiring a single ancillary qubit. Table I presents the stabilizer
generators and the logical $\bar{X}$ and $\bar{Z}$ operators of the code.

Stabilizer Generators | Logical Operators
---|---
$Z_{0}X_{1}Z_{2}Z_{4}$ |
$Y_{0}Y_{2}Z_{3}Z_{4}$ |
$Z_{0}Z_{1}X_{4}Z_{6}$ | $\bar{X}=Z_{0}Z_{1}X_{2}Z_{5}$
$X_{1}Z_{3}X_{5}X_{6}$ | $\bar{Z}=Z_{0}Z_{2}Z_{5}Z_{6}$
$Z_{3}Z_{5}Z_{6}X_{7}$ |
$Z_{0}X_{3}Z_{6}Z_{7}$ |
$Z_{1}X_{2}X_{3}Z_{4}X_{6}X_{7}$ |
TABLE I: Stabilizers and Logical operators for [[8,1,4]] code

We analysed the single ancilla fault-tolerant properties of the code and found
that certain propagated errors have similar syndrome values. For example,
while measuring the syndrome for $Z_{0}X_{3}Z_{6}Z_{7}$ stabilizer, an $X$
error on the ancilla after $X_{3}$ gate would propagate to $Z_{6}Z_{7}$ error
on the data qubits. Its syndrome is same as $Y_{5}$ error and after applying
correction, we will end up with $Y_{5}Z_{6}Z_{7}$ error which is the logical
$\bar{Y}$ error.

We propose to reorder certain stabilizers of [[8,1,4]] code in the following
manner:

$\displaystyle Z_{3}Z_{5}Z_{6}X_{7}$ $\displaystyle\longrightarrow
Z_{5}Z_{3}Z_{6}X_{7}$ $\displaystyle Z_{0}X_{3}Z_{6}Z_{7}$
$\displaystyle\longrightarrow Z_{7}X_{3}Z_{0}Z_{6}$ $\displaystyle
Z_{1}X_{2}X_{3}Z_{4}X_{6}X_{7}$ $\displaystyle\longrightarrow
Z_{1}X_{3}Z_{4}X_{2}X_{6}X_{7}$

Such a reordering scheme for syndrome measurement leads to distinct syndromes
for all possible propagated errors shown in column 2 of Table II. The exact
syndrome values corresponding to correctable errors and the errors propagated
from faults during detection using reordered scheme are given in Section IV-D.
We notice that all the errors have distinct syndrome values upto a stabilizer.
Thus, we can achieve fault-tolerance in the $[[8,1,4]]$ code without requiring
extra ancilla qubits.

Stabilizers | Errors
---|---
$Z_{0}X_{1}Z_{2}Z_{4}$ | $Z_{0}P_{1}$ | $P_{2}Z_{4}$
$Y_{0}Y_{2}Z_{3}Z_{4}$ | $Y_{0}P_{2}$ | $P_{3}Z_{4}$
$Z_{0}Z_{1}X_{4}Z_{6}$ | $Z_{0}P_{1}$ | $P_{4}Z_{6}$
$X_{1}Z_{3}X_{5}X_{6}$ | $X_{1}P_{3}$ | $P_{5}X_{6}$
$Z_{5}Z_{3}Z_{6}X_{7}$ | $Z_{5}P_{3}$ | $P_{6}X_{7}$
$Z_{7}X_{3}Z_{0}Z_{6}$ | $Z_{7}P_{3}$ | $P_{0}Z_{6}$
$Z_{1}X_{3}Z_{4}X_{2}X_{6}X_{7}$ | $Z_{1}P_{3}$ | $Z_{1}X_{3}P_{4}$
| $P_{2}X_{6}X_{7}$ | $P_{6}X_{7}$
TABLE II: Propagated Errors where $P\in\\{X,Y,Z\\}$

###  IV-C Merit of Fault Tolerance

The [[8,1,4]] code presented is capable of correcting a single qubit error and
additionally detect a second error. The above reordering scheme makes it
fault-tolerant without requiring extra ancilla qubits other than the one used
while syndrome extraction. We expect a single fault in the procedure should
not lead to a logical error. The code achieves the same as follows:

  * •

All $P\otimes X$ errors are distinguishable by syndromes, upto a stabilizer
and a phase, and thus our reordering scheme tackles all hook errors,
considering the lookup table decoder.

  * •

Any $P\otimes Z$ kind of error leads to a single qubit error in data while
also flipping the syndrome value. We perform minimum two rounds of syndrome
extraction and compare the measurement results. Same results ensure no $Z$
error on ancilla while different outcomes require another round of syndrome
extraction. When two consecutive syndrome extraction rounds give same
outcomes, we correct the errors accordingly.

  * •

Errors of kind $P\otimes Y$ are equal to $(P\otimes X)(I\otimes Z)$. Repeated
application of detector circuit gets rid of $Z$ on ancilla and the syndrome
outcome tells the propagated error to be corrected.

###  IV-D Errors and Syndrome Values

This section provides the syndrome values corresponding to the single qubit
errors correctable by the code and the propagated errors due to faults in
error correction procedure.

All single qubit errors are correctable by the [[8,1,4]] code with each error
having distinct syndromes given in table III. Errors arising due to faults in
detection mainly accumulates through error propagation. Table IV presents
syndrome values corresponding to higher weight errors. We observe that the
syndrome corresponding to $Z_{1}X_{3}X_{4}$ is same as $Z_{7}$ error as
$Z_{1}X_{3}X_{4}Z_{7}$ is an stabilizer.

Single Qubit Errors
---
$X_{0}:1110010$ | $Y_{0}:1010010$ | $Z_{0}:0100000$
$X_{1}:0010001$ | $Y_{1}:1011001$ | $Z_{1}:1001000$
$X_{2}:1100000$ | $Y_{2}:1000001$ | $Z_{2}:0100001$
$X_{3}:0101100$ | $Y_{3}:0101111$ | $Z_{3}:0000011$
$X_{4}:1100001$ | $Y_{4}:1110001$ | $Z_{4}:0010000$
$X_{5}:0000100$ | $Y_{5}:0001100$ | $Z_{5}:0001000$
$X_{6}:0010110$ | $Y_{6}:0011111$ | $Z_{6}:0001001$
$X_{7}:0000010$ | $Y_{7}:0000111$ | $Z_{7}:0000101$
TABLE III: Syndromes of single-qubit errors Errors due to Faults
---
$Z_{0}X_{1}Z_{2}Z_{4}$
$Z_{0}X_{1}$ | $0110001$ | $X_{2}Z_{4}$ | $1110000$
$Z_{0}Y_{1}$ | $1111001$ | $Y_{2}Z_{4}$ | $1010001$
$Z_{0}Z_{1}$ | $1101000$ | $Z_{2}Z_{4}$ | $0110001$
$Y_{0}Y_{2}Z_{3}Z_{4}$
$Y_{0}X_{2}$ | $0110010$ | $X_{3}Z_{4}$ | $0111100$
$Y_{0}Y_{2}$ | $0010011$ | $Y_{3}Z_{4}$ | $0111111$
$Y_{0}Z_{2}$ | $1110011$ | $Z_{3}Z_{4}$ | $0010011$
$Z_{0}Z_{1}X_{4}Z_{6}$
$Z_{0}X_{1}$ | $0110001$ | $X_{4}Z_{6}$ | $1101000$
$Z_{0}Y_{1}$ | $1111001$ | $Y_{4}Z_{6}$ | $1111000$
$Z_{0}Z_{1}$ | $1101000$ | $Z_{4}Z_{6}$ | $0011001$
$X_{1}Z_{3}X_{5}X_{6}$
$X_{1}X_{3}$ | $0111101$ | $X_{5}X_{6}$ | $0010010$
$X_{1}Y_{3}$ | $0111110$ | $Y_{5}X_{6}$ | $0011010$
$X_{1}Z_{3}$ | $0010010$ | $Z_{5}X_{6}$ | $0011110$
$Z_{5}Z_{3}Z_{6}X_{7}$
$Z_{5}X_{3}$ | $0100100$ | $X_{6}X_{7}$ | $0010100$
$Z_{5}Y_{3}$ | $0100111$ | $Y_{6}X_{7}$ | $0011101$
$Z_{5}Z_{3}$ | $0001011$ | $Z_{6}X_{7}$ | $0001011$
$Z_{7}X_{3}Z_{0}Z_{6}$
$Z_{7}X_{3}$ | $0101001$ | $X_{0}Z_{6}$ | $1111011$
$Z_{7}Y_{3}$ | $0101010$ | $Y_{0}Z_{6}$ | $1011011$
$Z_{7}Z_{3}$ | $0000110$ | $Z_{0}Z_{6}$ | $0101001$
$Z_{1}X_{3}Z_{4}X_{2}X_{6}X_{7}$
$Z_{1}X_{3}$ | $1100100$ | $X_{6}X_{7}$ | $0010100$
$Z_{1}Y_{3}$ | $1100111$ | $Y_{6}X_{7}$ | $0011101$
$Z_{1}Z_{3}$ | $1001011$ | $Z_{6}X_{7}$ | $0001011$
$Z_{1}X_{3}X_{4}$ | $0000101$ | $X_{2}X_{6}X_{7}$ | $1110100$
$Z_{1}X_{3}Y_{4}$ | $0010101$ | $Y_{2}X_{6}X_{7}$ | $1010101$
$Z_{1}X_{3}Z_{4}$ | $1110100$ | $Z_{2}X_{6}X_{7}$ | $0110101$
TABLE IV: Syndromes of Propagated Errors

###  IV-E Quantum Circuits

This section presents the quantum circuits used in simulating the $[[8,1,4]]$
code with the noise models as described in section V.

We use the methods presented in Section III to create an efficient unitary
encoder for the chosen stabilizers. The encoder circuit given in Fig. 5
utilizes the enhanced standard form mechanism for non-CSS codes without any
non-essential gates. We notice the last step involves certain SWAP gates whose
purpose is to map relabeled qubits to the original order of qubits.

Figure 5: Encoder Circuit for $[[8,1,4]]$ code. The last $k=1$ qubit is
initialized in $\ket{\psi_{i}}$ state.

The syndrome measurement circuit, here referred to as Detector, is built
following the reordering scheme presented in Section IV-B. We utilize fast
reset of qubits to use a single ancilla for syndrome measurements. It enhances
the simulation speed exponentially as fewer space resources are required to
store quantum states in classical systems. The complete circuit is shown in
Fig. 6.

Figure 6: Detector circuit for syndrome measurements, incorporating reordering
scheme for [[8,1,4]] code. All measurements are in $X$ basis followed by
resetting the qubit to $\ket{+}$ state.

The corrector circuit is a classical-quantum hybrid circuit that applies
certain fixed operations on the data qubits given the syndrome values. We note
that we are not using a standard lookup table decoder that contains the most
likely error configuration for every possible syndrome value. Instead, our
lookup table only consists of syndrome and error pairs as given in Section
IV-D.

One can test the efficiency of the code under different decoders for
correcting errors. Details of the error rate calculation and simulation runs
are given in Section VI.

##  V Noise Models

To study the performance of the proposed $[[8,1,4]]$ code, we test it on two
noise models: the standard depolarising noise model and the anisotropic noise
model.

###  V-A Standard Depolarising Noise Model

It is the most applicable noise model to test physical systems, in which each
gate is followed by a symmetric depolarization with probability $p$. Thus,
with probability $p$, the state is replaced by $I/2$ and is left the same by
probability $1-p$. The simulation model of same is as follows:

  1. 1.

With probability $p_{s}$, each single qubit gate is followed by a single qubit
Pauli error drawn uniformly and independently from $\\{X,Y,Z\\}$. Thus either
of the three error occurs with probability $p_{s}/3$.

  2. 2.

With probability $p_{t}$, each two qubit gate is followed by a two qubit Pauli
error drawn uniformly and independently from
$\\{I,X,Y,Z\\}\otimes\\{I,X,Y,Z\\}\setminus\\{I\otimes I\\}$. Each of 15
errors occurs with probability $p_{t}/15$.

  3. 3.

With probability $p_{meas}$, each single qubit measurement has its outcome
flipped.

  4. 4.

With probability $p_{prep}$, each preparation of $\ket{0}$ is flipped to
$\ket{1}$ and each preparation of $\ket{+}$ is flipped to $\ket{-}$.

###  V-B Anisotropic Noise Model

A suitable noise model that mimics over or under-rotation of gates in ion-trap
qubits is the anisotropic noise model. It considers noise due to gate coupling
and thus two qubit errors are only the ones aligned with gates [8].
Anisotropic noise model is simulated as follows:

  1. 1.

Single qubit error, readout error, and preparation error are applied with
probability $p_{s}$, $p_{meas}$, and $p_{prep}$ respectively as defined above
for Standard Depolarising Noise model.

  2. 2.

With probability $p_{t}$, every two qubit $CP$ (controlled-$P$) gate is first
followed by $Z\otimes P$ error. Each of the two qubits involved are then acted
upon by a single qubit error with probability $p_{s}$.

For our purpose, we have taken all probabilities to be same, i.e.,
$p_{s}=p_{t}=p_{meas}=p_{prep}$, in each of the noise model. A further robust
noise model may consider single qubit errors on idle locations too.

##  VI Simulation Scheme

We designed the complete error correction simulator using qiskit following the
scheme described in Fig. 2[10]. We consider $\ket{\psi_{i}}=\ket{0}$ and
initialize state $\ket{0}^{\otimes 8}$, we apply the encoder circuit which
creates the logical $\ket{\bar{0}}$ state for the given code. The detector
circuit is then applied twice. The classical outputs of the two detector
circuits are compared, if they are same we proceed with corrector, else we
apply detector circuit again and feed its output to the corrector. Ideally, we
should keep applying the detector until two consecutive outcomes are
identical. Still, due to resource constraints, we limit our simulation to a
maximum of three detector applications. The error-corrected state is fed into
a circuit which is the adjoint of the encoder, and the data qubits are
measured. For our purpose, the encoder, the corrector, and the adjoint of the
encoder are kept ideal, i.e., noise-free. The detector circuit can observe
errors which follow from the noise models as described in Sections V-A and
V-B.

###  VI-A Error Rate Calculation

We analyze three metrics for the code on each noise model.

  1. 1.

Fidelity: Probability of the decoded state, $\ket{\psi_{d}}$ being same as the
encoded state, $\ket{\psi_{i}}$. For practical quantum computation, we require
as high fidelity as possible.

  2. 2.

Logical error rate: Probability of the decoded state to remain in the
codespace but different from the encoded state i.e.,
$\ket{\psi_{i}}\neq\ket{\psi_{d}}\in\mathcal{C}$. A logical error is an
unwanted operation which will always go undetected, altering the state of
interest and hence giving wrong computational results. The lower the logical
error rate, the better the code.

  3. 3.

Total error rate: Probability of the decoded state being different from the
encoded state. It encompasses logical and other errors that brings the state
out of codespace. We can trivially say total error rate = 1-fidelity.

There are three possible measurement outcomes of data qubits. If the first
$n-k$ qubits are in state $\ket{0}$, the decoded state,
$\ket{\psi_{d}}\in\mathcal{C}$. Otherwise, the error-corrected state is
outside the codespace. In reverse order of qubit measurements,
$\ket{0}\ket{0}^{\otimes 7}$ contributes to fidelity, while
$\ket{1}\ket{0}^{\otimes 7}$ contributes to logical error. The complete
simulation requires the following steps:

  1. 1.

For a given value of physical error rate, $p$, run the simulation circuit $N$
(atleast $100\times p$) times and note down the outcomes.

  2. 2.

Define logical error rate = (no. of logical errors / no. of times the circuit
was run) and fidelity = (no. of times same as initial state occurs / no. of
times the circuit was run)

  3. 3.

Repeat step 1 and 2 for $m$ rounds to obtain a distribution of error rate
values at a given $p$. The mean of $m$ rounds along with minimum and maximum
value is plotted against the physical error rate.

  4. 4.

Repeat step 1, 2, and 3 for several different values of $p$.

Once we have obtained the metrics for a range of values of $p$, we fit a
polynomial curve to judge the pseudo-thresholds. The error correcting scheme
is expected to reduce any linear dependence of logical or total error rates on
the physical error rate. It is encouraged to fit a curve of form
$a_{0}x^{2}+a_{1}x^{3}+\dots+a_{n-2}x^{n}$ where $a_{0}$ is the leading order
coefficient. Since we initialize the state in logical $\ket{\bar{0}}$ state,
the effect of logical $Z$ operator will not be noticed and only logical $X$
and logical $Y$ will cause an observable logical error. Thus we compare the
obtained error rates to $\frac{2}{3}p$ instead of $p$.

The pseudo-threshold is the value at which the error rate after error
correction crosses the unencoded error rate ($\frac{2}{3}p$). It denotes the
value of physical error rate below which the code in test would be more
suitable for performing quantum computation than without any encoding.
Comparing two codes, a code with higher pseudo-threshold will be preferable.
The leading coefficient defined above is another metric to compare codes. The
lower the coefficient, the better performing the code.

###  VI-B Modified Simulation

The above method is more realistic to physical devices. Nonetheless, it does
not account for all correctable errors. QEC simulations often involves a
noise-free round of projecting the error-corrected state to the codespace and
comparing it to the encoded state. This involves additional noise-free error
detection and correction circuit before $\text{Encoder}^{\dagger}$ operation.

The modified approach leads to a state containing only those errors not
correctable by the code. This leads to an increase in fidelity and an increase
in logical error rate as errors of comprising a logical operator times a
correctable error leads to logical error in measurement output.

##  VII Discussion of the results

This section presents results obtained from simulation runs and the properties
of the code obtained from analysing results. It also includes the authors’
perspective on the results and comparison. We call the simulation run without
the noise-free projection step described in Section VI-A as the Practical
Method. And the simulation run with the noise-free detection-correction step
described in Section VI-B as the Modified Method. We obtain pseudo-threshold
for the code in both the noise models. We also obtain the leading order terms
from the best-fit curve.

We first analyze the performance of the code in a standard depolarizing noise
model. Fig. LABEL:fig:std_dep_plots presents the descent of logical and total
error rates compared among the two simulation methods. We obtain leading order
terms for the practical and modified simulation runs with respective logical
error threshold values of 0.3% and 0.11%. The practical method could not
provide a pseudo-threshold for total error rates, thus showing a linear
relation to unencoded error rates. The modified approach nonetheless provides
a threshold of 0.034%.

Error Rates | pseudo-threshold | Leading Order
---|---|---
Logical Error Rates | Practical | 0.003041 | 260
Modified | 0.001181 | 564
Total Error Rates | Practical | - | -
Modified | 0.0003498 | 1980
TABLE V: Metrics for Standard depolarizing noise model

For the anisotropic noise model, the descent of logical and total error rates
is presented in Fig. LABEL:fig:anisotropic_plots. The threshold values for the
practical and modified methods are found to be 0.041% and 0.018% respectively
for logical error rates. The practical method fails again with total error
rates, but the modified method provides a threshold of $4.77\times 10^{-3}\%$
on extrapolating the best-fit curve.

Error Rates | pseudo-threshold | Leading Order
---|---|---
Logical Error Rates | practical | 0.0004148 | 1757
Modified | 0.0001768 | 3883
Total Error Rates | practical | - | -
Modified | $4.77\times 10^{-5}$ | 14090
TABLE VI: Anisotropic noise model

Fig. LABEL:fig:leading_orders presents the error rates divided by $p^{2}$ to
reveal the leading order of the best fit curve. Logical error rates crosses
the unencoded error rates for both noise models and for both simulation
methods. Total error rates shows the dependence of practical method results on
physical error rates while the modified method leads to a distinctive
convergence for the leading coefficient of $p^{2}$.

Evidently, the code performance is superior in the standard depolarizing model
than the anisotropic model. This can be attributed to the error correction
capabilities of the code as well the noise in the model. The anisotropic noise
has a two qubit error followed by single qubit errors increasing the chance of
$Z$ error on the syndrome which can only be corrected by consecutive rounds of
error detection. The constraint of three rounds allow many $Z$ errors to go
uncorrected. On the other hand, the code is well structured to identify any
$P\otimes X$ error, getting rid of hook errors and increasing the pseudo-
threshold.

While the practical method of simulation is more appropriate to realistic
settings, the modified method is more suitable to characterize code
performance and provide benchmarks. Modified method leaves the code with only
uncorrectable errors, either logical errors or higher weight errors, thus it
puts a bar on the efficiency of the code. We further observe that the leading
order of $p^{2}$ is also an appropriate metric to compare performance of a
code. The lower the leading order coefficient, the higher is the pseudo-
threshold value.

##  VIII Conclusions

We used the $[[8,1,4]]$ non-CSS code and showed Fault-tolerance using the
modified Bare ancillary method of [8]. We obtained pseudo-thresholds of the
code under the depolarizing and anisotropic noise models. Our fault tolerant
scheme is much suitable to standard depolarising noise and provides higher
pseudo-threshold values than widely studied fault tolerant techniques
implemented on various codes [17]. We compared the practical error correction
procedure to the traditional simulation run and compared the merits of both,
presenting the requirement of noise-free projection to encoded state for
reliable simulation results.

A promising future direction would be studying the code performance under
various noise models and to generalize the single qubit fault tolerance method
to other codes.

## Acknowledgment

The authors thank Harsh Gupta for implementing methods given in [9] and
providing us with the stabilizers of a non-CSS [[8,1,4]] code. We thank
Krishnakumar Sabapathy for encouraging us to study Bare ancillary fault
tolerance method [8] and providing fruitful comments in the manuscript
preparation.

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#  pre-saddle neutron multiplicity for fission reactions induced by heavy ions
and light particles

S. Soheyli111Corresponding author<EMAIL_ADDRESS>M. K. Khalili  Bu-Ali
Sina University, Department of Physics, Hamedan, Iran

###### Abstract

Pre-saddle neutron multiplicity has been calculated for several fission
reactions induced by heavy ions and light particles. Experimentally, it is
impossible to determine the contribution of neutrons being emitted before the
saddle point and those emitted between the saddle and the scission points.
Determination of the pre-saddle neutron multiplicity in our research is based
on the comparison between the experimental anisotropies and those predicted by
the standard saddle-point statistical model. Analysis of the results shows
that the pre-saddle neutron multiplicity depends on the fission barrier height
and stability of the compound nucleus. In heavy ion induced fission, the
number of pre-saddle neutrons decreases with increasing the excitation energy
of the compound nucleus. A main cause of this behavior is due to a reduction
in the ground state-to-saddle point transition time with increasing the
excitation energy of the compound nucleus. Whereas in induced fission by light
particles, the number of pre-saddle neutrons increases with increasing the
excitation energy of the compound nucleus.

###### pacs:

25.70.Jj, 25.85.Ge

##  I Introduction

In the last two decades, much theoretical attention has directed towards
understanding the dynamics of fission. According to reports, measuring the
number of neutrons emitted during fission most likely gives information on the
timescale of fission as well as on the nuclear dynamics. The transition state
model of fission, based on appropriate level densities, predicts the widths
(and thus lifetimes) of fission and neutron emission. This model is also
suitable for determining the pre-fission neutron multiplicity if the
calculated lifetimes are long compared to the dynamically constrained fission
lifetime. Several groups have invested an extensive effort in measuring the
number of emitted neutrons associated with fission reactions induced by heavy
ions [1-17]. The measurement of emitted neutrons is usually limited to the
measurement of pre-scission neutron multiplicity, post-scission neutron
multiplicity, and therefore total neutron multiplicity. These measurements
show that the transition state model of fission leads to an underestimation of
the number of measured pre-scission neutrons emitted in heavy ion induced
fission at high excitation energies. This discrepancy can be related to the
viscosity of the hot nucleus [18]. Hence, the fission lifetime of a hot
nucleus is substantially longer than that determined by statistical model of
Bohr and Wheeler [19]. As a result, it is natural to expect that a dissipative
dynamical model would provide an appropriate description of nuclear fission at
high excitation energies [20].

Pre-scission neutrons $\nu_{pre}$, can be emitted between the ground state of
the compound nucleus and the saddle point (pre-saddle neutrons) $\nu_{gs}$, or
between the saddle and the scission points (saddle-to-scission neutrons)
$\nu_{ss}$. The number of pre-saddle neutrons as well as the number of saddle-
to-scission neutrons can be determined by a combined dynamical statistical
model ( CDSM ) [21-23]. The contributions $\nu_{gs}$ and $\nu_{ss}$ to the
pre-scission neutron multiplicity are also estimated by a stochastic approach
based on three-dimensional Langevin equation [24]. Recently, a more accurate
four-dimensional Langevin model as an extension of the three-dimensional
Langevin model by adding the fourth collective coordinate ( the projection of
the total spin about the symmetry axis of the fissioning nucleus ) is used to
calculate the pre-scission neutron multiplicity [25].

A common assumption in the calculation of the angular anisotropy of fission
fragments using by the transition state model is that all pre-scission
neutrons are emitted prior to reaching the saddle-point, since it is not
straightforward to separate experimentally the contribution of neutrons being
emitted before the saddle-point and those emitted between the saddle and the
scission points [26-35]. It is well known that the standard saddle-point
statistical model (SSPSM) has become the standard theory of fission fragment
angular distributions and received great success since it was proposed. The
effect of neutron evaporation prior to reaching the saddle-point is to reduce
the temperature of the fissioning nucleus which in turn increases the fission
fragment anisotropy prediction by using this model. Only $\nu_{gs}$ has an
influence over the prediction of angular anisotropy by using SSPSM. The upper
limit of the angular anisotropy of fission fragments, based on the prediction
of SSPSM is determined by assuming that all the pre-scission neutrons are
emitted before the saddle-point.

The pre-saddle neutrons as a crucial quantity in determining the angular
anisotropy of fission fragments by using SSPSM plays a main role, although any
precise method to determine it has not been introduced. In this article, we
calculate the number of pre-saddle neutrons by a novel method. In this method,
the values of $\nu_{gs}$ for several induced fission reactions by light
particles and heavy ions are determined by the fission fragments angular
distribution method. This method is based on comparison between the
experimental anisotropies and those predicted by the standard saddle-point
statistical model. This method is limited to the calculation of pre-saddle
neutrons in induced fission in which the angular anisotropy of fission
fragments has a normal behavior, i.e., it is observed a good agreement between
the angular anisotropy of fission fragments and that predicted by the SSPSM.

In order to make the present paper self-contained, we present in Sec. II, an
brief description of the standard saddle-point statistical model as well as
the calculating method of the pre-saddle neutron multiplicity on the basis of
the SSPSM in detail. Section III is devoted to the results obtained in this
study. Finally, the concluding remarks are given in Sec. IV.

##  II Method of Calculations

###  II.1 Standard saddle-point Statistical Model

The standard transition-state model has been used to analyze the angular
anisotropy of fission fragments in fission. In the transition-state model, the
equilibrium distribution over the K degree of freedom (the projection of total
angular momentum of the compound nucleus (I) on the symmetry axis of the
fissioning nucleus) is assumed to be established at the transition state. Two
versions of the transition-state models based on assumptions on the position
of the transition state: standard saddle-point statistical model (SSPSM), and
scission-point statistical model (SPSM) can be used for the prediction of
fission fragment angular distributions. The basic assumption of the SSPSM is
that fission proceeds along the symmetry axis of a deformed compound nucleus,
and that the distribution of K is frozen from the saddle point to the scission
point. In this model, the fission fragment angular distribution $W(\theta)$
for the fission of spin zero nuclei is given by the following expression
Vandenbosch1:1973

$W(\theta)\propto\sum_{I=0}^{\infty}\frac{(2I+1)^{2}T_{I}\exp[-p\sin^{2}{\theta}]J_{0}[-\emph{i}p\sin^{2}{\theta}]}{\textrm{erf}[\sqrt{2p}]}.$
(1)

Where ${T_{I}}$, and ${J_{0}}$ are the transmission coefficient for fission,
and the zeroth-order Bessel function,
$p=(I+\frac{1}{2})^{2}/(4K_{\circ}^{2})$, and the variance of the equilibrium
K distribution ($K_{\circ}$) is

$K_{\circ}^{2}=\frac{\Im_{eff}T}{\hbar^{2}},$ (2)

here ${\Im_{eff}}$ and $T$ are the effective moment of inertia and the nuclear
temperature of the compound nucleus at the saddle point, respectively.

The angular anisotropy of fission fragments is defined as

$A=\frac{W(0^{\circ})}{W(90^{\circ})}.$ (3)

The nuclear temperature of the compound nucleus at the saddle point is given
by

$T=\sqrt{\frac{E_{ex}}{\emph{a}}},$ (4)

where $E_{ex}$ is the excitation energy of the fissioning system and a is the
nuclear level density parameter at the saddle point. $E_{ex}$ can be expressed
by the following relation

$E_{ex}=E_{c.m.}+Q-B_{f}(I)-E_{R}(I)-{\nu_{gs}}E_{n}.$ (5)

In this equation, $E_{c.m.}$, $Q$, $B_{f}(I)$, $E_{R}(I)$, $\nu_{gs}$, and
$E_{n}$ represent the center-of-mass energy of the projectile, the $Q$ value,
the spin dependent fission barrier height, the spin dependent rotational
energy of the compound nucleus, the number of pre-saddle neutrons, and the
average excitation energy lost due to evaporation of one neutron from the
compound nucleus prior to the system reaching to the saddle point,
respectively. In the case of $p\gg 1$, the angular anisotropy of fission
fragments by using Eq. (1) is given by the following approximate relation

$A\approx 1+\frac{<I^{2}>}{4K_{\circ}^{2}}.$ (6)

The prediction of angular anisotropy of fission fragments by using the SSPSM
is valid only under restrictive assumptions. At high angular momentum, or at
high fissility, the rotating liquid drop model (RLDM) predicts that the
fission barrier height($B_{f}(I)$) vanishes even for a spherical nucleus,
which leads to $K_{\circ}^{2}\rightarrow\infty$. Subsequently, the
distribution of K is uniform and hence the prediction of the SSPSM for the
fission fragments angular anisotropy is nearly uniform by using Eq. (1). This
predicted tendency toward isotropy for fission fragments at high angular
momentum is not seen in the experiments. This discrepancy is taken as a clear
indication that the width of the K distribution is not determined at the
predicted spherical saddle point shape, but at a point where nucleus is more
deformed. Therefore, it has been proposed that the standard saddle-point
statistical model breaks down at high spin and/or large values of
$\frac{Z^{2}}{A}$ of the compound nucleus (CN), and the angular distribution
of fission fragments is governed by an effective transition state different
from saddle point transition state.

###  II.2 Pre-saddle Neutron multiplicity

It is clear that because of the hindrance to fission, a large number of
particles more that those predicted by the statistical model are emitted from
the fissioning system. In heavy ion fusion reactions, due to the formation of
a heavy compound nucleus, the competition between neutron emission and fission
describes the decay possibilities rather well. During the collective motion to
the scission point, neutrons will be evaporated if energetically possible, and
would be experimentally as pre-fission, or more correctly, pre-scission
neutrons. A longer saddle to scission time due to the viscosity effect, will
result in a higher pre-scission neutron multiplicity Hinde3:1989 . The
calculation of pre-saddle neutrons in heavy ion induced reactions based on the
comparison between the experimental data of angular anisotropy and those
predicted by the SSPSM depends on the kinetic energy and the binding energy of
evaporated neutron from the compound nucleus prior to the system reaching to
the saddle point. The energy spectrum of evaporated neutrons is usually given
by the following form (an evaporation spectrum) Tsang:1983

$\frac{dN}{dE}=CE\exp(-\frac{E}{T}).$ (7)

Hence, the average kinetic energy of the emitted neutron, $\overline{E}_{K}$
is given by

$\overline{E}_{K}=2T.$ (8)

The average excitation energy lost due to evaporation of one neutron from the
compound nucleus prior to the system reaching to the saddle point is given by

$E_{n}=B_{n}+2T,$ (9)

where, $B_{n}$ denotes the average neutron separation energy.

In this work, the average energy lost by an emitted neutron over the energy
range of the projectile is calculated by Eq. (9), for heavy ion induced
fission reactions, as well as for induced fissions by light projectiles. The
level density parameter, a is taken $\frac{A_{C.N.}}{8}$ ( Considering the
level density parameter as $\frac{A_{C.N.}}{10}$, rather than
$\frac{A_{C.N.}}{8}$, the number of pre-saddle neutrons varies at most by
10$\%$). Hence, number of pre-saddle neutrons is not sensitive to the level
density parameter selected in the calculation. $\Im_{eff}$, $B_{f}(I)$, and
$E_{R}(I)$ are accounted by the use of rotating finite range model (RFRM)
Sierk:1986 , while $<I^{2}>$ quantities are calculated by several models
[39-44]. In the following sections, the determination of the number of pre-
saddle neutrons, $\nu_{gs}$ for these systems is based on the comparison
between the experimental data of angular anisotropies and those predicted by
the SSPSM. In the present work, it is determined pre-saddle neutron
multiplicities for several systems undergoing heavy ion induced fission in
which fission fragments angular anisotropies have a normal behavior as well as
those systems undergoing light particle induced fission. In order to determine
number of pre-saddle neutrons in heavy ion reactions with anomalous angular
anisotropies, it is necessary to predict the average contribution of non
compound nucleus fission events Soheyli:2012 .

##  III Results and discussion

The calculated multiplicities of pre-saddle neutrons as a function of $E_{ex}$
for the two ${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems leading to Protactinium isotopes, are shown in Fig. 1(a). For above
studied systems, the experimental data of angular anisotropy are taken from
literature [46, 47]. As illustrated in the figure, the number of pre-saddle
neutrons decreases with increasing the excitation energy of the compound
nucleus. This behavior is due to the fact that the fission barrier height (and
thus the ground state-to-saddle point transition time ) decreases with
increasing the excitation energy of the compound nucleus, which can be lead to
that $\nu_{gs}$ decreases with $E_{ex}$. In this figure, the general trend of
the number of pre-saddle neutrons as a function of the excitation energy of
the compound nucleus is represented by a line using the method of least
squares. Fig. 1(b), shows a similar case for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system. For this system, the experimental data of angular anisotropy are taken
from literature [8]. Multiplicities of pre-saddle neutrons for the two
${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, both populating the
same compound nucleus ${}^{248}\textrm{Cf}$ are also shown in Fig. 1(c). For
these two systems, the experimental data of $<\textrm{A}>$ are taken from
literature [56-58]. It is interesting to note that for these two systems, as
well as for the two ${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa
and ${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems as shown in Fig. 1(a), the number of pre-saddle neutrons at any given
excitation energy appears to be nearly equal. As a result, the multiplicities
of pre-saddle neutrons for heavy ion fusion reactions populating the same
compound nucleus are nearly independent of the entrance channel asymmetry and
depend on the mass number of the compound nucleus.

Figure 1: Calculated multiplicities of pre-saddle neutrons (a) for the two
${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems. Thick and dotted lines represent the general trends of $\nu_{gs}$
against the excitation energy of the compound nucleus for the two
${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems, respectively. (b) For the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}{\textrm{Th}}$ reaction
system. Thick line represents the general trend of $\nu_{gs}$ against the
excitation energy of the compound nucleus, and (c) for the two
${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, both populating the
same compound nucleus ${}^{248}\textrm{Cf}$. Thick and thin lines represent
the general trends of $\nu_{gs}$ against the excitation energy of the compound
nucleus for the two ${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, respectively.

The ratio of the calculated pre-saddle neutron multiplicity , $\nu^{gs}_{cal}$
to experimental pre-scission neutron multiplicity $\nu^{pre}_{exp}$
Rossner:1992 , and also the ratio of theoretical pre-saddle neutron
multiplicity to theoretical pre-scission neutron multiplicity Frobrich1: 1994
for the ${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$
reaction system, are given in Table I.

$E_{c.m.}(MeV)$ |  $E_{ex}(MeV)$ |  $\nu^{pre}_{exp}$ |  $\nu^{gs}_{cal}$ |  $\nu^{gs}_{cal}/{\nu^{pre}_{exp}}$ |  $\nu^{gs}_{th}/{\nu^{pre}_{th}}$
---|---|---|---|---|---
76.9 | 22.7 | 1.50 |  1.81 |  1.21 |  0.96
82.6 | 27.6 | 1.90 |  1.60 |  0.84 |  0.91
92.0 | 32.0 | 2.40 |  1.30 |  0.54 |  0.78
105.9 | 42.5 | 2.80 |  0.52 |  0.18 |  0.64
119.0 | 55.0 | 3.40 |  0.00 |  0.00 |  0.56
Table 1:  Comparison between the calculated $\nu^{gs}_{cal}$,
$\nu^{gs}_{cal}/{\nu^{pre}_{exp}}$ and $\nu^{gs}_{th}/\nu^{pre}_{th}$
Frobrich1: 1994  for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system.

As can be seen from Table I, the calculated number of pre-saddle neutrons for
the ${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system is greater than ${\nu^{pre}_{exp}}$ at
$\textrm{E}~{}_{\textrm{c.m.}}=76.9~{}\textrm{MeV}$. This unexpected result
can be related to the measured value of fission fragment angular anisotropy at
low energy . It seems that the measured value of the angular anisotropy at
$\textrm{E}~{}_{\textrm{c.m.}}=76.9~{}\textrm{MeV}$ is reported more than its
actual value.

As the nucleus is heated, the excitation energy of the compound nucleus,
$E_{ex}$ exceeds the fission barrier height, $B_{f}$. Hence, it becomes
possible for the nucleus to fission after passing through excited states above
the fission barrier ( transient state ) Hinde5: 1993 . In this transient state
picture, the fission width, $\Gamma_{f}$ depends on the level density above
the fission barrier. The fission width and the neutron width can be shown to
be approximately given by $\Gamma_{f}\propto\exp(-\frac{B_{f}}{T})$ and
$\Gamma_{n}\propto\exp(-\frac{B_{n}}{T})$ ( $B_{n}$ is the neutron binding
energy ), respectively. Therefore, the energy dependence of the ratio
$\frac{\Gamma_{n}}{\Gamma_{f}}$ is expected to be dominated by the ratio of
appropriate Boltzmann factors, i.e., $\frac{\Gamma_{n}}{\Gamma_{f}}\approx
exp[(B_{f}-B_{n})/T]$.

In general, in heavy ion induced fission, $B_{f}$ will be relatively high at
low excitation energy or at low angular momentum, $I$, however as $I$, as well
as $E_{ex}$ is increased, the larger moment of inertia of the elongated saddle
point configuration causes its energy to increase less rapidly than that of
the compact equilibrium deformation, so the barrier height falls to zero at
some $I$. The ratio $\frac{\Gamma_{n}}{\Gamma_{f}}$ is known to decrease
sharply as $E_{ex}$ increases in nuclei of the Lead-Bismuth region, and it is
expected to do just the opposite for nuclei with the largest known atomic
numbers Bishop: 1972 . For the lighter group of fissioning elements
$B_{f}\gg{B_{n}}$, and for the very heavy ones, it is expected that
$B_{n}\gg{B_{f}}$. For nuclei of intermediate mass like the Neptunium, $B_{n}$
and $B_{f}$ are nearly equal and one expects only a slow variation of
$\frac{\Gamma_{n}}{\Gamma_{f}}$ with $E_{ex}$. In a heavy ion reaction, there
is sufficient excitation energy to emit several neutrons, and fission can
compete at each stage ( if the excitation energy is greater than the fission
barrier height ), thus the fission probability and neutron evaporation
probability at stage i, are given by
$p_{{}_{f,i}}=(\frac{\Gamma_{{}_{f}}}{\Gamma_{{}_{tot}}})_{{}_{i}}$ and
$p_{{}_{n,i}}=(\frac{\Gamma_{{}_{n}}}{\Gamma_{{}_{tot}}})_{{}_{i}}=1-(\frac{\Gamma_{{}_{f}}}{\Gamma_{{}_{tot}}})_{{}_{i}}$,
respectively. As a result, the total fission probability, $P_{{}_{f}}$ is
given by

$P_{{}_{f}}=\sum_{k=1}^{\nu}\prod_{i=1}^{k}(p_{{}_{f,i}})(p_{{}_{n,i-1}}),$
(10)

where $\Gamma_{tot}=\Gamma_{f}+\Gamma_{n}$. The mean number of neutrons
emitted before fission, $\nu_{pre}$ can be derived by the following expression

$\nu_{pre}=(\frac{1}{P_{f}})\sum_{k=1}^{\nu}(k-1)\prod_{i=1}^{k}(p_{{}_{f,i}})(p_{{}_{n,i-1}}).$
(11)

As $I$ increases, the fission barrier height decreases, then $p_{{}_{f,1}}$
along the decay chain approaches unity, and steps with $k>1$ become
insignificant, and $\nu_{pre}\longrightarrow 0$; thus fission is predicted to
occur at the first step in the decay chain. It is obvious that as the
projectile energy rises, $\nu_{pre}$ will initially rises, due to more chances
for fission, but should subsequently falls as the angular momentum reaches the
value at which $P_{f}$ nears unity. It is shown that, the transient time at
the scission point, $\tau_{sci}$ by using a diffusion model for the fission
process is given by Hassani: 1984

$\tau_{sci}\simeq\tau_{sad}+\overline{\tau}=\beta^{-1}\ln(10B_{f}/T)+\overline{\tau},$
(12)

where, $\tau_{sad}$, $\overline{\tau}$ and $\beta$ are the transient time at
the saddle point, the average traveling time between the saddle and scission
points, and the nuclear friction, respectively. The time $\overline{\tau}$ is
a function of the value of the nuclear friction, of the shape of potential and
of the excitation energy. The above equation shows that $\tau_{sad}$ depends
sensibily on the nuclear friction $\beta$ and on the excitation energy of the
compound nucleus.

Earlier calculations of fission fragment anisotropies based on SSPSM have been
corrected to include the effect of pre scission neutron emission. The
calculation of fission fragment anisotropies with taking into account the
effect of pre-scission neutron emission better compares with the SSPSM
predictions with the experimental results. However, there is a small
discrepancy between model predictions and the data at high excitation
energies. A fraction of pre-scission neutrons is expected to be emitted
between saddle to scission. These latter neutrons do not longer influence the
prediction of angular anisotropy by SSPSM, since it is assumed that the SSPSM
is decided at the saddle point. In Fig. 2, the effect of pre-saddle neutrons
in the prediction of angular anisotropy by SSPSM is demonstrated for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system Frobrich1: 1994 . As it is shown in the figure, the discrepancy between
the experimental data of angular anisotropies and the prediction of the SSPSM
can be removed to a large extent by taking into account the pre-saddle neutron
emission correction.

Figure 2:  Experimental and calculated anisotropies in
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system [8, 54]. Thick, dashed, dotted curves are theoretical analysis in the
framework of the SSPSM without neutron emission correction, with pre-saddle
neutrons [${\nu^{pre}_{exp.}}(\nu^{gs}_{th.}/\nu^{pre}_{th.})$] correction,
and pre-scission neutrons correction [$\nu^{pre}_{exp.}$], respectively.

We observe that for the above studied system, the ratio of the calculated pre-
saddle neutron multiplicity to experimental pre-scission neutron multiplicity,
$\nu^{gs}_{cal}/{\nu^{pre}_{exp}}\approx\frac{1}{4.1}$ at $\frac{B_{f}}{T}=1$
is in agreement with
$\frac{\tau_{gs}}{\tau_{gs}+\tau_{ss}}\approx\frac{1}{3.7}$( where,
$\tau_{gs}$ and $\tau_{ss}$ are ground-to-saddle and saddle-to-scission
transition times, respectively) Saxena1:1994 . Hence, the neutron emission
rate by the compound nucleus in the transition from the ground state to the
saddle point and then in the transition from saddle to the scission points are
approximately uniform.

The calculated multiplicities of pre-saddle neutrons as a function of $E_{ex}$
for the
${}^{11}\textrm{B}+^{197}\textrm{Au},^{209}\textrm{Bi},^{235}\textrm{U},^{237}\textrm{Np}$
reaction systems are shown in Fig. 3(a). For these studied systems, the
experimental data of angular anisotropies are taken from literature [46,
48-50, 54-56]. The values of $\nu_{gs}$ as a function of $E_{ex}$ for the
${}^{14}\textrm{N},^{16}\textrm{O}+^{197}\textrm{Au}$ and
${}^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$ reaction systems are
also shown in Fig. 3(b). For these systems, the experimental data of angular
anisotropies are taken from literature [46, 47]. The calculated multiplicities
of pre-saddle neutrons as a function of the excitation energy of the compound
nucleus for induced fission of the ${}^{209}\textrm{Bi}$ target by using
different projectiles are shown in Fig. 3(c).

Figure 3: Calculated pre-saddle neutron multiplicities, (a) for the
${}^{11}\textrm{B}+^{197}\textrm{Au},^{209}\textrm{Bi},^{235}\textrm{U},^{237}\textrm{Np}$
reaction systems. Thin, dashed, thick, and dashed-dotted lines represent the
general trends of the number of pre-saddle neutrons against the excitation
energy of the compound nucleus, respectively. (b) for the
${}^{14}\textrm{N},^{16}\textrm{O}+^{197}\textrm{Au}$ and
${}^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$ reaction systems. Thin,
thick, dashed, and dashed-dotted lines represent the general trends of the
number of pre-saddle neutrons against the excitation energy of the compound
nucleus for these systems, respectively., and (c) for the
${}^{11}\textrm{B},^{12}\textrm{C},^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$
reaction systems. Thick, dashed-dotted, thin, and dashed lines represent the
general trends of the number of pre-saddle neutrons against the excitation
energy of the compound nucleus for these systems, respectively.

The average values of $\nu_{gs}$, as well as ranges of pre-saddle neutron
multiplicities for the fission reactions of different targets induced by the
same projectile over the same projectile energy range are shown in Table II.
In this Table, the quantity $V_{b}$ denotes the Coulomb barrier height. It can
be observed that $\overline{\nu}_{gs}$ decreases with increasing the mass
number of the target.

reaction systems |  Projectile energy (in $\frac{E_{c.m.}}{V_{b}}$) | $\nu_{gs}$ |  $\overline{\nu}_{gs}$
---|---|---|---
${}^{11}\textrm{B}+^{197}\textrm{Au}$ | 1.4-1.9 | 3.1-1.4 | 2.0
${}^{11}\textrm{B}+^{209}\textrm{Bi}$ | 1.4-1.9 | 1.8-0.8 | 1.6
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{12}\textrm{C}+^{197}\textrm{Au}$ | 1.3-1.8 | 2.4-1.6 | 2.1
${}^{12}\textrm{C}+^{209}\textrm{Bi}$ | 1.3-1.8 | 1.5-0.4 | 1.0
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{14}\textrm{N}+^{197}\textrm{Au}$ | 1.2-1.7 | 3.0-0.5 | 1.9
${}^{14}\textrm{N}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.6-0.1 | 0.9
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{16}\textrm{O}+^{197}\textrm{Au}$ | 1.0-1.6 | 3.3-0.7 | 2.0
${}^{16}\textrm{O}+^{208}\textrm{Pb}$ | 1.0-1.6 | 1.9-0.1 | 1.5
${}^{16}\textrm{O}+^{209}\textrm{Bi}$ | 1.0-1.6 | 1.7-0.9 | 1.4
Table 2:  Comparison between the calculated pre-saddle neutron multiplicity in
the form of a range, as well as $\overline{\nu}_{gs}$ for fission reactions of
the different targets induced by the same projectile.

The average values of $\nu_{gs}$, as well as the pre-saddle neutron
multiplicity in the form of a range for the induced fission of the same target
by different projectiles over the same projectile energy are also given in
Table III. As can be seen in Table III, the quantity $\overline{\nu}_{gs}$
decreases with increasing the mass number of projectile. All heavy ion induced
reactions show that $\nu_{gs}$ falls quite rapidly with increasing the mass
asymmetry, since it is partly due to a reduction of the dynamical fission time
scale with the mass asymmetry.

reaction systems |  Projectile energy (in $\frac{E_{c.m.}}{V_{b}}$) | $\nu_{gs}$ |  $\overline{\nu}_{gs}$
---|---|---|---
${}^{11}\textrm{B}+^{209}\textrm{Bi}$ | 1.2-1.7 | 2.2-1.2 | 1.9
${}^{12}\textrm{C}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.8-0.8 | 1.2
${}^{14}\textrm{N}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.6-0.2 | 0.9
${}^{16}\textrm{O}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.4-0.6 | 0.8
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{12}\textrm{C}+^{197}\textrm{Au}$ | 1.2-1.6 | 2.8-1.8 | 2.4
${}^{14}\textrm{N}+^{197}\textrm{Au}$ | 1.2-1.6 | 3.0-1.0 | 2.3
${}^{16}\textrm{O}+^{197}\textrm{Au}$ | 1.2-1.6 | 2.5-0.7 | 2.0
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{16}\textrm{O}+^{208}\textrm{Pb}$ | 1.1-1.6 | 1.9-1.0 | 1.9
${}^{19}\textrm{F}+^{208}\textrm{Pb}$ | 1.1-1.6 | 1.4-0.4 | 1.4
Table 3:  Comparison between the calculated pre-saddle neutron multiplicity in
the form of a range, as well as $\overline{\nu}_{gs}$ for fission reactions of
the same target induced by different heavy ions.

We now attempt to estimate the pre-saddle neutron multiplicities in several
fission reactions induced by light projectiles. We must pay attention to some
important points expressing the difference between fission induced by light
projectiles and heavy ions. In the fission induced by light projectiles, the
energy in the center-of-mass framework, $E_{c.m.}$ is roughly the same as that
in the laboratory framework, as well as due to the low weight of projectile,
rotational energy, $E_{R}$ can be neglected. Fig. 4, shows calculated pre-
saddle neutron multiplicities for the two $\alpha+^{182}\textrm{W}$, and
$\textrm{p}+^{185}\textrm{Re}$ reaction systems which are leading to similar
${}^{186}\textrm{Os}$ compound nucleus, as well as for the two
$\textrm{p}+^{209}\textrm{Bi}$, and $\alpha+^{206}\textrm{Pb}$ that formed the
same ${}^{210}\textrm{Po}$ compound nucleus. For these systems, the
experimental data of angular anisotropies are taken from literature [57-59].
The values of $<I^{2}>$ for these systems are given by Ignatyuk1: 1984 :

$<I^{2}>=\frac{\sum(2I+1)T_{I}I(I+1)}{\sum(2I+1)T_{I}}$ (13)

where $T_{I}$ is the entrance channel transmission coefficients and satisfy
$T_{I}=1$ for $I\leq{I_{max}}$ and $T_{I}=0$ for $I>I_{max}$. If the maximum
angular momentum is determined by the relation $<I^{2}>=1/2I_{max}^{2}$, the
following relations give the values of the mean square angular momentum of the
compound nucleus for the fission of pre-actinide nuclei induced by proton and
$\alpha$ particle, respectively:

$<I^{2}>=2.08E_{p}(MeV)-15,$ (14) $<I^{2}>=10.2E_{\alpha}(MeV)-199.$ (15)

In heavy ion induced fission at low bombarding energies, several neutrons are
evaporated prior to the reaching to the saddle point, and at the highest
bombarding energy essentially all the neutrons are evaporated by the fission
fragments, i.e., the fission process is rapid compared to the time scale for
neutron evaporation. However, the number of pre-saddle neutrons, $\nu_{gs}$
increases with increasing the excitation energy of the compound nucleus in
fission induced by light projectiles. This behavior is mainly due to that in
the induced fission by light projectile, the fission barrier height is higher
than the neutron binding energy, as well as $B_{f}$ is approximately
independent of the excitation energy of the compound nucleus. Therefore, the
fission probability, $P_{f}=\frac{\Gamma_{f}}{\Gamma_{tot}}$ is negligible at
low energies. When $E_{ex}<{B_{f}}$, it is impossible that the compound
nucleus undergoes fission, but there is sufficient excitation energy to emit
several neutrons. It is clear that the fission becomes significant if
$E_{ex}>{B_{f}}$.

Figure 4: The values of $\nu_{gs}$ for the two $\alpha+^{206}\textrm{Pb}$ ,
and $\textrm{p}+^{209}\textrm{Bi}$ reaction systems which are leading to the
similar ${}^{210}\textrm{Po}$ compound nucleus, as well as for the two
$\alpha+^{182}\textrm{W}$, and $\textrm{p}+^{185}\textrm{Re}$ reaction systems
that formed the same ${}^{186}\textrm{Os}$ compound nucleus. Thick, thin,
dashed, and dashed-dotted lines represent the general trends of the pre-saddle
neutrons against the excitation energy of the compound nucleus for these
systems, respectively.

##  IV Summary and Conclusions

We have presented in this paper the calculated pre-saddle neutron
multiplicities for several heavy ion induced fission reactions, as well as for
several fission reactions induced by light projectiles. The calculation by
using the experimental data of fission fragment angular anisotropies, as well
as the prediction of the SSPSM is a novel method, which has been carried out
in this work for the first time. We have also considered the behavior of pre-
saddle neutron multiplicities in fission reactions induced by heavy ions and
light projectiles. In heavy ion induced fission, the number of pre-saddle
neutrons decreases with increasing the excitation energy of the compound
nucleus. Whereas in fission induced by light particles, the number of pre-
saddle neutrons increases with increasing the excitation energy of the
compound nucleus. The fission barrier height in heavy ion fission reaction
depends on the excitation energy of the compound nucleus. On the other hand,
the fission barrier height ( and thus ground-to-saddle transition time ;
$\tau_{gs}\propto\ln(10B_{f}/T)$ ) decreases with increasing the excitation
energy of the compound nucleus. As a result, in heavy ion induced fission the
number of pre-saddle neutrons decreases with increasing the excitation energy
of the compound nucleus. Our results also shows that the emission rate of
neutrons is approximately constant in transition from the ground state to the
saddle point and then from the saddle to the scission points. On the contrary,
in fission induced by light projectiles, the fission barrier height is greater
than the neutron binding energy, and the fission barrier is approximately
independent of the excitation energy. Hence, the compound nucleus does not
undergo fission, unless the excitation energy of the compound nucleus exceeds
the fission barrier. As a result, in fission induced by light projectiles, the
number of pre-saddle neutrons exhibits an increasing function against the
excitation energy of the compound nucleus as shown our calculations. The
number of pre-saddle neutrons for reactions lead to the same compound nucleus
at any given excitation energy appears to be nearly equal, since the number of
pre-saddle neutrons depends only on the mass of the compound nucleus and it is
independent of the entrance mass asymmetry parameter. This behavior of the
number of the pre-saddle neutrons as a function of the projectile mass and/or
of the target mass may also be related to the size of compound nucleus. We
observe that the average number of pre-saddle neutrons decreases with
increasing the mass number of projectile in fission reactions of the same
target induced by different projectiles. A similar behavior in the
multiplicities of pre-saddle neutrons is also observed in fission reactions of
different targets induced by the same projectile. At the end, our results may
provide useful information on the ground state-to-saddle and saddle-to-
scission transition times.

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$\displaystyle\geq\left[2\gamma(1-4\gamma^{2})\right]^{2S}\left(1-4\gamma^{2}\right)^{S}\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{L-1}}\left(\frac{\partial}{\partial\theta_{q,n}}{\rm
Tr}\left[\sigma_{\bm{3|i}}\rho_{2L-2}\right]\right)^{2}$ (129)
$\displaystyle\geq\left[2\gamma(1-4\gamma^{2})\right]^{2S}\left(1-4\gamma^{2}\right)^{(L-q)S}\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q}}\left(\frac{\partial}{\partial\theta_{q,n}}{\rm
Tr}\left[\sigma_{\bm{3|i}}\rho_{2q}\right]\right)^{2}.$ (130)

Eq. (127) follows from the definition of $\rho_{2L}$ (122). Eq. (128) is
obtained since

${\rm CZ}(\sigma_{j}\otimes\sigma_{k}){\rm
CZ}^{{\dagger}}=\sigma_{j}\otimes\sigma_{k},\forall j,k\in\\{0,3\\}.$

InEq. (129) is derived by using the Lemma B.1. We repeat the derivation in
Eqs. (127-129) inductively for parameters
$(\bm{\theta}_{L},\cdots,\bm{\theta}_{q+1})$, which yields InEq. (130).

Next, we consider the expectation with respect to $\bm{\theta}_{q}$. We have

$\displaystyle\text{Eq.~{}(\ref{tqnn_partial_derivative_2_6})}=\left[2\gamma(1-4\gamma^{2})\right]^{2S}\left(1-4\gamma^{2}\right)^{(L-q)S}\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q}}\left(\frac{\partial}{\partial\theta_{q,n}}{\rm
Tr}\left[\sigma_{\bm{3|i}}\rho_{2q-1}\right]\right)^{2}$ $\displaystyle\geq{}$
$\displaystyle\left[2\gamma(1-4\gamma^{2})\right]^{2S}\left(1-4\gamma^{2}\right)^{(L-q)S}\left(1-4\gamma^{2}\right)^{S-1}\left[4\gamma^{2}(1-4\gamma^{2})\right]4\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q-1}}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{2q-2}\big{]}^{2},$ (131)

where expectations with respect to parameters $\\{\theta_{q,j}\\}_{j\in
I_{S},j\neq n}$ are calculated via Lemma B.1 and the expectation with respect
to $\theta_{q,n}$ is calculated via Lemma B.2.

Finally we proceed from Eq. (131) and take the expectation for parameters in
$(\bm{\theta}_{q-1},\cdots,\bm{\theta}_{1})$. We have

$\displaystyle\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q-1}}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{2q-2}\big{]}^{2}$
$\displaystyle=\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q-1}}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\text{CZ}_{q-1}R_{q-1}(\bm{\theta}_{q-1})\rho_{2q-4}R_{q-1}(\bm{\theta}_{q-1})^{\dagger}\text{CZ}_{q-1}^{\dagger}\big{]}^{2}$
(132)
$\displaystyle=\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q-1}}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}R_{q-1}(\bm{\theta}_{q-1})\rho_{2q-4}R_{q-1}(\bm{\theta}_{q-1})^{\dagger}\big{]}^{2}$
(133)
$\displaystyle\geq\left(1-4\gamma^{2}\right)^{S}\mathop{\mathbb{E}}\limits_{\bm{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{\bm{\theta}_{q-2}}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{2q-4}\big{]}^{2}$ (134)
$\displaystyle\geq\left(1-4\gamma^{2}\right)^{(q-1)S}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{0}\big{]}^{2}.$ (135)

Eq. (132) is derived by using the definition of $\rho_{2q-2}$. Eq. (133) is
obtained since

${\rm CZ}(\sigma_{j}\otimes\sigma_{k}){\rm
CZ}^{{\dagger}}=\sigma_{j}\otimes\sigma_{k},\forall j,k\in\\{0,3\\}.$

InEq. (134) is derived by using Lemma B.1. We repeat the derivation in Eqs.
(132-134) inductively for parameters
$(\bm{\theta}_{q-1},\cdots,\bm{\theta}_{1})$, which yields InEq. (135).
Employing Eq. (135) to Eq. (131) yields

Eq. (130) $\displaystyle\geq
4\left(4\gamma^{2}\right)^{S+1}\left(1-4\gamma^{2}\right)^{S(L+2)}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{0}\big{]}^{2}$ (136)
$\displaystyle=4\left(\frac{1}{S(L+2)}\right)^{S+1}\left(1-\frac{1}{S(L+2)}\right)^{S(L+2)}\text{Tr}\big{[}\sigma_{\bm{3|i}}\rho_{0}\big{]}^{2}$
(137) $\displaystyle\geq
4\left(\frac{1}{S(L+2)}\right)^{S+1}\left(1-\frac{1}{2}\right)^{2}{\rm
Tr}\big{[}\sigma_{\bm{3|i}}\rho_{0}\big{]}^{2}$ (138)
$\displaystyle=\frac{1}{S^{S+1}(L+2)^{S+1}}{\rm
Tr}\left[\sigma_{\bm{3|i}}\rho_{0}\right]^{2}.$ (139)

Eq. (137) is derived by using the condition $\gamma^{2}=\frac{1}{4S(L+2)}$.
Eq. (138) is obtained by noticing that function $g(x)=(1-\frac{1}{x})^{x}$ is
monotonically increasing when $x\geq 2$. Thus, we have proved Eq. (120).

∎

##  Appendix D Proof of Theorem 4.2

###### Proof.

To begin with, we define several notations for convenience. Denote by
$\rho_{j}$ the state after the $j$-th parameterized operator, i.e.

$\rho_{j}(\theta_{1},\cdots,\theta_{j})=\left(\prod_{i=j}^{1}V_{i}(\theta_{i})\right)\rho_{\rm
in}\left(\prod_{i=1}^{j}V_{i}(\theta_{i})^{{\dagger}}\right).$ (140)

Denote by $O_{j}$ the observable, i.e.

$O_{j}=V_{j}(0)^{\dagger}\cdots V_{L}(0)^{\dagger}OV_{L}(0)\cdots V_{j}(0),\
\forall j\in\\{1,\cdots,L\\}.$ (141)

Now we begin to prove the Theorem. First, we remark that $\forall j\in[L]$,
the $a_{j}\neq 1$ case can be converted to the $a_{j}=1$ case by using the
transformation

$\theta_{j}^{\prime}=\frac{\theta_{j}}{a_{j}},$

where the variance of the new and the old parameter satisfies

${\rm Var}[\theta_{j}^{\prime}]=\frac{1}{a_{j}^{2}}{\rm Var}[\theta_{j}].$

In the following proof, we assume that $a_{j}=1$, $\forall j\in[L]$. By using
the parameter-shift rule, $\frac{\partial f}{\partial\theta_{\ell}}$ could be
written as the linear sum of $2h$ expectations on the observable $O$ with
coefficients $\pm 1$. Then for the case $\ell\leq L-1$, we have

$\displaystyle{}\mathop{\mathbb{E}}\limits_{\bm{\theta}}\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}={}\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{L}}\left(\frac{\partial}{\partial\theta_{\ell}}\text{Tr}\left[OV_{L}(\theta_{L})\rho_{L-1}V_{L}(\theta_{L})^{\dagger}\right]\right)^{2}$
(142) $\displaystyle\geq{}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{L-1}}\left(\frac{\partial}{\partial\theta_{\ell}}\text{Tr}\left[OV_{L}(0)\rho_{L-1}V_{L}(0)^{\dagger}\right]\right)^{2}-\left[12h_{L}(h_{L}-1)+4\right]4h_{L}^{2}\gamma_{L}^{2}\|O\|_{2}^{2}$
(143) $\displaystyle={}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{L-1}}\left(\frac{\partial}{\partial\theta_{\ell}}\text{Tr}\left[O_{L}\rho_{L-1}\right]\right)^{2}-\left[12h_{L}(h_{L}-1)+4\right]4h_{L}^{2}\gamma_{L}^{2}\|O\|_{2}^{2}$
(144) $\displaystyle={}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{L-1}}\left(\frac{\partial
f}{\partial\theta_{\ell}}(\theta_{1},\cdots,\theta_{L-1},0)\right)^{2}-\left[12h_{L}(h_{L}-1)+4\right]4h_{L}^{2}\gamma_{L}^{2}\|O\|_{2}^{2},$
(145)

where Eq. (142) follows from the definition of $\rho_{j}$ in Eq. (140). InEq.
(143) is obtained by using Lemma B.3, where $\|\bm{c}\|_{1}=2h$. Eq. (144)
follows from the definition of $O_{j}$ in Eq. (141). Eq. (145) follows from
the formulation $f(\bm{\theta})=\text{Tr}[O\rho(\bm{\theta})]$. By proceeding
the derivation (142-145) for $L-\ell$ times, we have

$\displaystyle{}\mathop{\mathbb{E}}\limits_{\bm{\theta}}\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}\geq{}\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{\ell}}\left(\frac{\partial
f}{\partial\theta_{\ell}}(\theta_{1},\cdots,\theta_{\ell},0,\cdots,0)\right)^{2}-\sum_{j=\ell+1}^{L}\left[12h_{j}(h_{j}-1)+4\right]4h_{j}^{2}\gamma_{j}^{2}\|O\|_{2}^{2}$
$\displaystyle={}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{\ell}}\left(\frac{\partial}{\partial\theta_{\ell}}\text{Tr}\left[O_{\ell+1}V_{\ell}(\theta_{\ell})\rho_{\ell-1}V_{\ell}(\theta_{\ell})^{\dagger}\right]\right)^{2}-\sum_{j=\ell+1}^{L}\left[12h_{j}(h_{j}-1)+4\right]4h_{j}^{2}\gamma_{j}^{2}\|O\|_{2}^{2}$
(146) $\displaystyle\geq{}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{\ell-1}}(1-4\gamma_{\ell}^{2})\left(\frac{\partial}{\partial\theta_{\ell}}\text{Tr}\left[O_{\ell+1}V_{\ell}(\theta_{\ell})\rho_{\ell-1}V_{\ell}(\theta_{\ell})^{\dagger}\right]\right)^{2}\bigg{|}_{\theta_{\ell}=0}-96h_{\ell}^{2}(h_{\ell}-1)\gamma_{\ell}^{2}\|O\|_{2}^{2}$
$\displaystyle{}-20h_{\ell}^{2}(h_{\ell}-1)(h_{\ell}-2)\gamma_{\ell}^{2}\|O\|_{2}^{2}-\sum_{j=\ell+1}^{L}\left[12h_{j}(h_{j}-1)+4\right]4h_{j}^{2}\gamma_{j}^{2}\|O\|_{2}^{2}$
(147) $\displaystyle\geq{}$
$\displaystyle\mathop{\mathbb{E}}\limits_{{\theta}_{1}}\cdots\mathop{\mathbb{E}}\limits_{{\theta}_{\ell-1}}\left(\frac{\partial
f}{\partial\theta_{\ell}}(\theta_{1},\cdots,\theta_{\ell-1},0,0,\cdots,0)\right)^{2}-4\gamma_{\ell}^{2}(2h_{\ell})^{2}\|O\|_{2}^{2}-96h_{\ell}^{2}(h_{\ell}-1)\gamma_{\ell}^{2}\|O\|_{2}^{2}$
$\displaystyle{}-20h_{\ell}^{2}(h_{\ell}-1)(h_{\ell}-2)\gamma_{\ell}^{2}\|O\|_{2}^{2}-\sum_{j=\ell+1}^{L}\left[12h_{j}(h_{j}-1)+4\right]4h_{j}^{2}\gamma_{j}^{2}\|O\|_{2}^{2},$
(148)

where Eq. (146) follows from definitions $\rho_{j}$ (140) and $O_{j}$ (141).
InEq. (147) is derived by using Lemma B.4. InEq. (148) follows from the
parameter-shift rule. We proceed from InEq. (148) by employing the derivation
(142-145) for parameters $(\theta_{\ell-1},\cdots,\theta_{1})$, which yields

$\displaystyle{}\mathop{\mathbb{E}}\limits_{\bm{\theta}}\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}$ $\displaystyle\geq{}$
$\displaystyle\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}\bigg{|}_{\bm{0}}-\sum_{j=1}^{\ell-1}16h_{j}^{2}\left[3h_{j}(h_{j}-1)+1\right]\gamma_{j}^{2}\|O\|_{2}^{2}-\sum_{j=\ell+1}^{L}16h_{j}^{2}\left[3h_{j}(h_{j}-1)+1\right]\gamma_{j}^{2}\|O\|_{2}^{2}$
$\displaystyle{}-16h_{\ell}^{2}\gamma_{\ell}^{2}\|O\|_{2}^{2}-96h_{\ell}^{2}(h_{\ell}-1)\gamma_{\ell}^{2}\|O\|_{2}^{2}-20h_{\ell}^{2}(h_{\ell}-1)(h_{\ell}-2)\gamma_{\ell}^{2}\|O\|_{2}^{2}$
(149) $\displaystyle\geq{}$ $\displaystyle\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}\bigg{|}_{\bm{0}}-\sum_{j=1}^{L}16h_{j}^{2}\left[3h_{j}(h_{j}-1)+1\right]\gamma_{j}^{2}\|O\|_{2}^{2}$
$\displaystyle\geq{}$ $\displaystyle(1-\epsilon)\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}\bigg{|}_{\bm{0}}.$ (150)

InEq. (149) is obtained by using Lemma B.3, where $\|\bm{c}\|_{1}=2h$. InEq.
(150) follows from the condition
$\gamma_{j}^{2}\leq\frac{a_{j}^{2}\epsilon}{16h_{j}^{2}(3h_{j}(h_{j}-1)+1)L\|O\|_{2}^{2}}\left.\left(\frac{\partial
f}{\partial\theta_{\ell}}\right)^{2}\right|_{\bm{\theta}=\bm{0}}$ and
$a_{j}=1$, $\forall j\in[L]$. Thus, we have proved the theorem.∎