# Making the zeroth-order process fidelity independent of state preparation
and measurement errors

Yu-Hao Chen<EMAIL_ADDRESS>Department of Physics, National Taiwan
University, Taipei, Taiwan  Renata Wong<EMAIL_ADDRESS>Department of
Artificial Intelligence, Chang Gung University, Taoyuan, Taiwan  Hsi-Sheng
Goan<EMAIL_ADDRESS>Department of Physics, National Taiwan University,
Taipei, Taiwan

###### Abstract

A protocol called zero-fidelity was recently proposed as an approximation to
the process fidelity. It can be understood as the zeroth-order process
fidelity. It remedies problems of the process fidelity such as complex
experimental settings and long running times. However, like the process
fidelity, zero-fidelity is also not robust to state preparation and
measurement (SPAM) errors. This limits both protocols’ usefulness for real
devices. Here, we investigate how SPAM errors affect the zero-fidelity for two
operation channels: one channel containing controlled-NOT gates and the other
containing controlled-Z gates. Then, we compare these two channels by checking
the difference in the zero-fidelities between them. Finally, to make the zero-
fidelity protocol independent of SPAM errors, we combine it with the
randomized benchmarking (RB) protocol.

##  I Introduction

Quantum computation may one day replace classical computation for some
difficult problems as quantum computers are expected to become more powerful
for performing calculations than classical computers. In order to evaluate the
performance of real quantum devices, as opposed to classical simulators of
quantum processes, we can use the widely used protocol called process
fidelity[1]. This protocol is known to require a large number of experimental
settings and frequent changes of the initial states and the measurement basis
on a shot-by-shot case. Recently, a protocol called zero-fidelity[2] was
proposed, which can be seen as the zeroth-order approximation to the process
fidelity and which addresses these issues. However, it has not yet been shown
how the zero-fidelity protocol can be quantified to make it independent of
state preparation and measurement (SPAM) errors, which limit the usefulness of
the zero-fidelity for real devices. In the present work, we will combine the
zero-fidelity with the randomized benchmarking protocol [3, 4, 5] to make it
robust to SPAM errors.

It has been demonstrated in [1] that in general, instead of a full tomography,
one can use a constant number of randomly chosen Pauli measurements to
evaluate the process fidelity to a low variance figure as follows

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[\Lambda(P_{j})P_{i}]\mathbf{Tr}[\Gamma(P_{j})P_{i}],$
(1)

where $\Lambda$, $\Gamma$ are an ideal channel and an actual channel,
respectively, and $P_{i}$ are Pauli operators. However, one cannot directly
use Pauli operators to be initial states in experimental setting because Pauli
operators are not quantum states. For the sake of satisfying the experimental
requirement we could prepare the eigenstates of Pauli operators to be our
initial states. An issue with this is that one needs to prepare eight
different initial states for a single qubit (two eigenstates per each Pauli
operator). Furthermore, it is also necessary to frequently change the initial
states and measurement bases shot-by-shot, and the efficiency of this change
relies on the speed of the device used. For that reason, the zero-fidelity
protocol uses symmetric, informationally complete (SIC) states instead of
Pauli eigenstates. The four SIC-states can be used directly as initial states
because they are quantum states and we just need to prepare four different
initial states for a single qubit. Therefore we can use fewer experimental
settings and do not need to change initial states and measurement bases shot-
by-shot in order to estimate the process fidelity [2].

The main reason why it is important for protocols to be robust to SPAM errors
is that it is not easy to characterize SPAM errors experimentally. If we are
unable to characterize SPAM errors, and the effect of SPAM errors happens to
be large, we may obtain a fidelity evaluation that does not correspond to the
actual state of the channel in question. SPAM errors refer to state
preparation and measurement errors. For example, when preparing an initial
state it may get over-rotated (state preparation error). And when measuring
the state it may flip (e.g., $|0\rangle\rightarrow|1\rangle$) with a certain
probability (measurement error). Furthermore, the higher the number of qubits
in the system the more significant the impact of SPAM errors becomes because
more qubits can over-rotate and more outcomes can become flipped.

The paper is structured as follows. In Section II, we introduce the zero-
fidelity protocol, including SIC-states, which constitute the input states of
zero-fidelity. In Section III, we talk about the randomized benchmarking
protocol and combine it with the zero-fidelity protocol in such a way that
zero-fidelity becomes robust to SPAM errors. Section IV presents our results.
Section V concludes.

##  II Zero-fidelity as approximation to process fidelity

In this section we will briefly introduce symmetric, informationally complete
(SIC) states [6], the process fidelity and a hierarchy of its approximations
called $k$-fidelities [2]. The zero-fidelity is the zeroth-order approximation
to the process fidelity among the $k$-fidelities, with the accuracy level
increasing with the increasing order. After that, we will use the SIC states
(a total of four) for our state preparation in the zero-fidelity. This is
different from the process fidelity which uses the density matrices prepared
by the eigenstates of Pauli operators for state preparation [7].

The process fidelity can be written as

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i}\mathbf{Tr}[\Lambda(P_{i})\Gamma(P_{i})],$
(2)

where $P_{i}$ ranges over a set of Pauli operators and $\Lambda$ and $\Gamma$
are an ideal channel and an actual channel, respectively. The zero-fidelity
protocol proposed in [2] approximates the process fidelity, albeit without
considering SPAM errors. Here, we will augment the zero-fidelity to account
for SPAM errors in its estimation. To verify the accuracy of our augmentation
procedure, we compare the obtained results with those in which SPAM errors
were not considered.

###  II.1 Symmetric, informationally complete states

According to quantum theory, positive operator valued measures (POVM) must be
used to perform measurements on quantum systems. A POVM is a set of $m$
positive semidefinite operators $\\{E_{i}\\}^{m}_{i=1}$ that span Hilbert
space $\mathcal{H}$ of dimension $d$ ($2^{n}$). Those operators must satisfy
the requirement of their sum adding up to identity:

$\sum^{m}_{i=1}E_{i}=I.$ (3)

Therefore, for a single qubit, one can use the Pauli group as a set of POVMs
$P_{i}\in\\{I,X,Y,Z\\}$ while for $n$ qubits, the set of POVMs will be the
tensor product of the Pauli operators $P_{i}\in\\{I,X,Y,Z\\}^{\otimes n}.$ For
$n=1$ there is also another set of operators that can span a $d$-dimensional
Hilbert space $\mathcal{H}$ with a minimal number of elements $d^{2}=4$. And
these are the four SIC states. (Note that for $n=1$ we need $d^{2}=4$
operators to describe the entire Hilbert space $\mathcal{H}$). However these
states are not like Pauli operators which are mutually orthogonal, i.e.

$\begin{cases}\mathbf{Tr}[P_{i}P_{j}]=0,&\mbox{if }i\neq j\\\
\mathbf{Tr}[P_{i}P_{j}]=1,&\mbox{if }i=j\end{cases}$

where $P_{i}\in\\{X,Y,Z\\}.$

The inner products of the four SIC states are equal to [6]

$|\langle\psi_{i}|\psi_{j}\rangle|^{2}=\frac{1}{d+1},\quad i\neq j.$ (4)

That is to say that no matter what the initial state is we can project it onto
the remaining three states with the same probability. This set of POVMs is
called symmetric, informationally complete states POVM (SIC-POVM). For a
single qubit, the four SIC states can be defined as

$|\psi\rangle_{1}=|0\rangle$ (5)
$|\psi\rangle_{2}=\frac{1}{\sqrt{3}}|0\rangle+\sqrt{\frac{2}{3}}|1\rangle$ (6)
$|\psi\rangle_{3}=\frac{1}{\sqrt{3}}|0\rangle+\sqrt{\frac{2}{3}}e^{i\frac{2\pi}{3}}|1\rangle$
(7)
$|\psi\rangle_{4}=\frac{1}{\sqrt{3}}|0\rangle+\sqrt{\frac{2}{3}}e^{i\frac{4\pi}{3}}|1\rangle$
(8) $|\langle\psi_{i}|\psi_{j}\rangle|^{2}=\frac{1}{3},\quad i\neq j$ (9)

On the Bloch sphere, the four SIC states above form a regular tetrahedron.

###  II.2 Process fidelity and k-fidelity

The process fidelity, which is also known as entanglement fidelity, is defined
as the overlap between ideal channel (quantum operation) and actual channel.
In other words, the process fidelity checks how close the ideal quantum
operation and the actual quantum operation are. Consider an ideal channel
$\Lambda$ (here we use the same notation as [2]) which maps the initial state
$\rho_{I}$ to the desired final state $\rho_{f}=\Lambda(\rho_{I})$, and an
actual channel $\Gamma$ which will not perfectly match the ideal channel
$\Lambda$. We can represent the similarity between the ideal channel $\Lambda$
and the actual channel $\Gamma$ with the following equation [2]

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})],$
(10)

where $\sigma_{i}\in P_{i}$ is Pauli operator (here, we do not normalize Pauli
operators so the maximum fidelity is $2^{n}$ instead of $1$). For $n=1$,

$\sigma_{1}=I,\quad\sigma_{2}=X,\quad\sigma_{3}=Y,\quad\sigma_{4}=Z.$

However, the process fidelity cannot be evaluated experimentally because in
order to be able to do so our input state must be a quantum state, which is
Hermitian, positive-semidefinite (for non-negative probabilities) and trace
preserving (the total probability of all outcomes of a state is
$1=\mathbf{Tr}[\rho]$). Hence, $\sigma_{i}$ is not a quantum state since Pauli
operators are traceless. In order to estimate the process fidelity we need to
rewrite equation (10) to satisfy this experimental requirement (input state
must be a quantum state). The process fidelity is defined as

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}[B^{-1}]_{ij}\mathbf{Tr}[\Lambda(\rho_{i})\Gamma(\rho_{j})],$
(11)

where $B_{ij}=\mathbf{Tr}[\rho^{\dagger}_{i}\rho_{j}]$ checks the overlap
between the state $\rho_{i}$ and the state $\rho_{j}$ and performs
normalization. It is worth noticing that if the density matrices (quantum
states) $\rho_{i}$ are prepared by the eigenstates of Pauli operators then the
coefficient

$\begin{cases}B_{ij}=2^{n},&\mbox{if }i=j\\\ B_{ij}=0,&\mbox{if }i\neq
j.\end{cases}$

Equation (11) will then reduce to [7]

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}2^{-n}\mathbf{Tr}[\Lambda(\rho_{i})\Gamma(\rho_{i})].$
(12)

From equation (11) and in agreement with Born’s rule we can consider the term
$\mathbf{Tr}[\Lambda(\rho_{i})\Gamma(\rho_{j})]$ to be the expectation value
of the observable $\Lambda(\rho_{i})$ with respect to the final state whose
initial state $\rho_{j}$ is evolved by the actual channel $\Gamma$. In the
theoretical view, equation (11) seems reasonable because the inputs are
quantum states and the goal of estimating the process fidelity is to know how
close the ideal quantum operations and the actual quantum operations are.
Nevertheless, equation (11) is not practical for measuring the expectation
values of observables $\Lambda(\rho_{i})$ experimentally because for most
channels $\Lambda$ and initial states $\rho_{i}$, the observables
$\Lambda(\rho_{i})$ have entangled eigenstates [2]. The reason why we don’t
want observables with entangled eigenstates is that entangled eigenstates
cannot be separated. In other words, they cannot be described by a single
vector (individual) and as such cannot therefore be measured separately.
Instead we would be forced to measure the entire system, which is often
difficult or impossible to do in practice.

On the other hand, let us take two qubits for example. Assume the initial
state $\rho_{i}$ evolves through the ideal channel $\Lambda$ so the final
state $\Lambda(\rho_{i})$ becomes the density matrix of a Bell state
$|\psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).$ It is not
appropriate to consider $\Lambda(\rho_{i})$ to be an observable because the
density matrix of a Bell state is not an observable but a quantum state. A
quantum state can be measured by an observable, such as a Pauli operator
$X,Y,Z$, or it can be used to prepare other observables [8]. Therefore in
order to estimate process fidelity experimentally equation (11) should be
rewritten as

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,l}C_{il}\mathbf{Tr}[\Gamma(\rho_{i})W_{l}],$
(13)

with

$C_{il}=\sum^{4^{n}}_{j=1}[B^{-1}]_{ji}\mathbf{Tr}[\Lambda(\rho_{j})W_{l}],$
(14)

where $W_{l}$ is a set of Pauli operators $\\{I,X,Y,Z\\}^{\otimes n}.$

###  II.3 Zero-fidelity with SPAM errors

Equation (13) is not scalable because we need to prepare $4^{n}$ initial
states and $4^{n}$ observables for measurements. That is, we have to use a
total of $16^{n}$ experimental settings to estimate the process fidelity
experimentally. Thankfully, in [1] the authors show that one can use joint
probability distribution to sample a small subset of initial states and
observables instead, which means that the experimental resources needed for
estimating the process fidelity can be significantly reduced.

In order to use the protocol in [1] and not to be forced to frequently change
initial state and measurement basis one can use the zeroth-order approximation
to process fidelity called the zero-fidelity and given as

$F_{0}(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[\Lambda(\rho_{i})W_{j}]\mathbf{Tr}[\Gamma(\rho_{i})W_{j}],$
(15)

where $\rho_{i}$ are prepared by the four SIC-states and $W_{j}$ are Pauli
operators. Nevertheless, the zero-fidelity doesn’t take the effects of state
preparation and measurement errors into account. Here, we will remedy that
while using equation (15) to estimate the zero-fidelity. We will also estimate
the zero-fidelity of the target channel with controlled-NOT (CNOT, CX) and
controlled-Z (CZ) gates in order to compare the difference between the two
channels 111All simulations in this paper were performed using IBM Qiskit.

##  III Zero-fidelity with randomized benchmarking

In this section we will use the randomized benchmarking protocol [3, 4, 5]
together with the zero-fidelity to make the zero-fidelity robust to SPAM
errors.

Based on [10], the process fidelity can be made robust to SPAM errors by
applying certain quantum operations $m$ times and gradually increasing the
number of measurements. Therefore the process fidelity equation can be written
in the following form

$F^{m}(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda^{m}(\sigma^{\dagger}_{i})\Gamma^{m}(\sigma_{i})],$
(16)

where $\sigma_{i}$ is a tensor product over single-qubit Pauli operators.
However, as mentioned in Section II, equation (16) cannot be estimated
experimentally because Pauli operators are traceless which means that these
operators are not quantum states. Instead, we need to prepare the eigenstates
of Pauli operators to be the quantum states and also avoid using observables
$\Lambda(\rho_{i})$ with entangled eigenstates. Due to the above reasons,
equation (16) should be rewritten as

$F^{m}(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}2^{-n}\mathbf{Tr}[\Lambda^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\Gamma^{m}(\rho_{i})W_{j}].$
(17)

Based on the argumentation in [10], one can also decouple the SPAM errors from
the zero-fidelity and the equation then becomes

$F_{0}^{m}(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}\mathbf{Tr}[\Lambda^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\Gamma^{m}(\rho_{i})W_{j}].$
(18)

The combination of randomized benchmarking with the zero-fidelity may be more
useful and scalable experimentally as it has the potential to be robust to
errors and requires fewer experimental settings.

###  III.1 Randomized benchmarking and the process fidelity

Here, we will demonstrate how to combine the process fidelity and randomized
benchmarking to obtain the process fidelity for different Clifford lengths
$m$, i.e., a channel is applied $m$ times

$F^{m}(\mathcal{U},\tilde{\mathcal{U}})=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i}),\tilde{\mathcal{U}}^{m}(\rho_{i})]$
(19)

where $\mathcal{U},\tilde{\mathcal{U}}$ are ideal channel and noisy channel,
respectively, and $\rho_{i}$ are quantum states prepared by the eigenstates of
Pauli operators.

A quantum state $\rho$ that evolves through an ideal channel $\mathcal{U}$ can
be described as

$\rho\rightarrow\mathcal{U}(\rho)=U\rho U^{\dagger},$ (20)

Similarly, a quantum state that evolves through a noisy channel can be
described as

$\rho\rightarrow\tilde{\mathcal{U}}(\rho)=\tilde{U}\rho\tilde{U}^{\dagger}.$
(21)

Assume that the noisy channel $\tilde{\mathcal{U}}$ can be decomposed into
$\Lambda\mathcal{U}$ so that equation (21) is of the form

$\rho\rightarrow\tilde{\mathcal{U}}(\rho)=\Lambda(U\rho U^{\dagger}).$ (22)

Therefore, from the process fidelity we have

$F(\mathcal{U},\tilde{\mathcal{U}})=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\rho_{i}U^{\dagger}\Lambda(U\rho_{i}U^{\dagger})U].$
(23)

Now, consider ideal channels and noisy channels applied to a quantum state $m$
times, which is defined as

$\rho^{\prime}_{i}=\tilde{\mathcal{C}}^{m}\rho_{i}=\Lambda\mathcal{U}^{-1}\Lambda\mathcal{U}_{m}...\Lambda\mathcal{U}_{2}\Lambda\mathcal{U}_{1}\rho_{i}$
(24)

and

$\rho^{\prime}_{i}=\mathcal{C}^{m}\rho_{i}=\mathcal{U}^{-1}\mathcal{U}_{m}...\mathcal{U}_{2}\mathcal{U}_{1}\rho_{i},$
(25)

where
$\mathcal{U}^{-1}=(\mathcal{U}_{m}...\mathcal{U}_{2}\mathcal{U}_{1})^{-1}.$
Using the superoperator representation
$\Lambda\rightarrow\hat{\Lambda}=\sum_{k}A_{k}\otimes A^{*}_{k}$ and
$U\rightarrow\hat{U}=U\otimes U^{*}$ one obtains the process fidelity as

$\displaystyle F(\mathcal{C}^{m},\tilde{\mathcal{C}}^{m})$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\mathcal{C}^{m}(\rho_{i})\tilde{\mathcal{C}}^{m}(\rho_{i})]$
(26)
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\rho_{i}(\hat{U}^{-1}_{m}\hat{\Lambda}_{m}\hat{U}_{m}\cdots\hat{U}^{-1}_{1}\hat{\Lambda}_{1}\hat{U}_{1})\rho_{i}].$
(27)

From
$\mathbb{E}_{U}(F_{g})=\mathbf{Tr}\big{[}\rho(\int_{U(D)}dU\hat{U}\hat{\Lambda}\hat{U}^{-1})\rho\big{]}$
we obtain the formula for averaging over the Haar measure for the unitary
operators

$\displaystyle F_{avg}(\mathcal{C}^{m},\tilde{\mathcal{C}}^{m})$
$\displaystyle=\mathbb{E}_{U}(F(\mathcal{C}^{m},\tilde{\mathcal{C}}^{m}))$
(28) $\displaystyle=\int_{U(D^{\otimes
n})}(\Pi^{m}_{j=1}dU_{j})(F(\mathcal{C}^{m},\tilde{\mathcal{C}}^{m}))$ (29)
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}(\rho_{i}\Lambda[\Pi^{m}_{j=1}\hat{\Lambda}^{avg}_{j}]\rho_{i}),$
(30)

where $dU_{j}$ denotes the Haar measure and

$\hat{\Lambda}^{avg}_{j}\equiv\mathbb{E}_{U_{j}}(\hat{\Lambda}_{j})\equiv\int_{U(D)}dU\hat{U}^{-1}_{1}\hat{\Lambda}_{1}\hat{U}_{1}.$
(31)

As given in [11],

$\hat{\Lambda}^{avg}_{j}\rho=p_{j}\rho+(1-p_{j})\frac{\mathbb{I}}{D}$ (32)

with the

$p_{j}=\frac{\mathbf{Tr}(\hat{\Lambda}_{j})-1}{D^{2}-1}.$ (33)

With the above, we can finally obtain the relation

$\displaystyle F^{m}(\mathcal{U},\tilde{\mathcal{U}})$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i})\tilde{\mathcal{U}}^{m}(\rho_{i})]$
(34) $\displaystyle=F^{m}=A_{0}p^{m}+B_{0},$ (35)

where the coefficients $A_{0}$ and $B_{0}$ absorb state preparation and
measurement errors and $p$ can be considered the parameter for the average
gate fidelity $F_{ave}=p+\frac{1-p}{2^{n}}$ (with the assumptions of same
strength $p_{j}$, gate-independent and time-independent [10, 12]). Now, from
[1] we can also estimate the process fidelity by the following equation

$F^{m}(\mathcal{U},\tilde{\mathcal{U}})=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}2^{-n}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\tilde{\mathcal{U}}^{m}(\rho_{i})W_{j}],$
(36)

where the quantum states $\rho_{i}$ are prepared by the eigenstates of Pauli
operators.

###  III.2 Randomized benchmarking and the zero-fidelity

Since the zero-fidelity is not robust to SPAM errors, we will combine it with
randomized benchmarking in order to make it more useful and scalable
experimentally. From equation (36) we can write down the equation for the
zero-fidelity as

$F^{m}(\mathcal{U},\tilde{\mathcal{U}})=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\tilde{\mathcal{U}}^{m}(\rho_{i})W_{j}],$
(37)

where the quantum states $\rho_{i}$ are prepared by the SIC-states instead of
the eigenstates of Pauli operators. From the discussion above, the form of
zero-fidelity in (37) is robust to SPAM errors (see Appendix B for the
derivation).

We consider two-qubit randomized benchmarking. We use Gottesman–Knill theorem
[13], which is an efficient method for obtaining the inverse of the first $m$
Clifford operators to generate the randomized benchmarking circuits [14]. In
our circuit, we use Qiskit’s noisy model to add depolarizing noise on every
gate and also readout error on the measurement gates. Note that one could just
use $m$ Clifford operators to represent Clifford length $m$. However, in the
present work we use a set of Clifford operators to represent a single Clifford
operator. We do that in order to reduce the number of gate types, namely we
restrict the gates to the Hadamard, phase gate and CNOT gates, plus the gates
of the Pauli group. Such a reduction has the double benefit of being easily
implementable on real devices and of memory optimization. It has been shown in
[15] that the 92897280-element 3-qubit Clifford group is the semidirect
product of the 1451520-element 3-qubit symplectic group with the Pauli group.
This means that one can reconstruct the entire Clifford group by storing just
the symplectic group in memory. We will present the results of our numerical
experiments in Section IV.

##  IV Numerical results

In this section we will show our simulation results, which include the
estimated zero-fidelities with SPAM errors. We will also present the
simulation results for our combination of the zero-fidelity with randomized
benchmarking.

###  IV.1 Three-qubit zero-fidelity

Here, we will present our numerical results for the zero-fidelity with SPAM
errors. We will also compare the performance of the zero-fidelity with
different channels.

When estimating the zero-fidelity for the channel with CZ gates (see figure 1
for the channel and figure 2 for the numerical results), we consider our SPAM
errors to be over-rotation for state preparation and readout error for
measurement. Likewise, for the channel with the CX gate we consider the same
errors to be our SPAM errors (figure 3). Note that the channel with CX gates
is essentially the same as the channel in figure 1 but instead of CZ gates we
have CX gates. In figure 4 and 5 we show the fidelities of the channels we
mentioned above that are without(with) SPAM errors and have fixed gate error.
We use Qiskit’s noisy model to create a depolarizing channel whose
mathematical form can be written as

$E(\rho)=(1-\lambda)\rho+\lambda\mathbf{Tr}[\rho]\frac{\mathbf{I}}{2^{n}},$
(38)

where $\rho$ are the SIC-states, $n$ is the number of qubits and $\lambda$ is
fixed to $0.01$ in our experiments (4, 5). Figure 6 shows the comparison
between the two types of channels mentioned above. The fidelities of both
types are almost the same. That is to say that SPAM errors and gate errors
will not be affected even when the initial states are the four SIC-states and
despite the fact that these states have a larger coefficient on the
$|1\rangle$ state which may skew the calculation of expectation values. We
found that the state preparation error can cause different effects on
different kinds of channels and so the zero-fidelities shows a difference
between this two kinds of channels in our experiments (for convenience, we
adjusted the scale to compare the effect of errors). The zero-fidelity for the
perfect channel including CX gates is slightly higher than $1$ while the zero-
fidelity for the perfect channel including CZ gates is $1$. Therefore we can
notice that no matter whether there is state preparation error or not the
zero-fidelities of the channel with CX gate are a bit lower than the zero-
fidelities of the channel with CZ gate. We think it is because the
coefficients of the zero state and the one state in the four SIC-states are
different. CX gate flips the bit while CZ gate changes the phase. This
indicates that the expectation values of Pauli operators will be affected
differently by projective error depending in which gate is used. In order to
increase accuracy we can increase the number of shots for each circuit and the
number of qubits.

Figure 1: An example three-qubit circuit. The first U gate on each qubit is
for state preparation, where we rotate $|0\rangle$ to one of the SIC-states.
The second U gate on each qubit is state preparation error. The actual target
channel contains CZ gates. The gates behind the CZ gates are for performing
measurement basis changes. We also add readout error on each measurement gate.
The ideal target channel contains no SPAM and gate errors. Figure 2: The zero-
fidelity for three qubits. We consider the effect on fidelity with increasing
gate error. The error on CZ gate is depolarizing error. The blue circle
indicates only gate error without readout error. The orange circle indicates
gate error and readout error which affects the measurement. The green circle
indicates over-rotation which affects state preparation and gate error. The
degrees of over-rotation are sampling from Gaussian distribution (with zero
mean and $\sqrt{5}$ standard deviation) so each circuit has different degree
of over-rotation. The red circle indicates both gate error and SPAM errors.
Figure 3: The zero-fidelity for three qubits. We consider the effect that
increasing gate error has on fidelity. The error on CX gate is depolarizing
error. The blue circle indicates only gate error. The orange circle indicates
gate error and readout error which affects the measurement, while the green
circle indicates over-rotation which affects state preparation and gate error.
The degrees of over-rotation are sampling from Gaussian distribution (with
zero mean and $\sqrt{5}$ standard deviation) so each circuit has different
degree of over-rotation. The red circle indicates both gate error and SPAM
errors. Figure 4: The zero-fidelity for three qubits with 10 runs. The degrees
of over-rotation are sampled from the Gaussian distribution (with zero mean
and $\sqrt{5}$ standard deviation) and the readout error is described by the
matrix (39). The orange circle indicates that depolarizing error was added on
CZ gates but no SPAM errors are added to the circuit. The average of zero-
fidelity without SPAM is $0.968$. The blue circuit indicates that SPAM errors
were added to the circuit and depolarizing error was added on CZ gates. The
average of zero-fidelity with SPAM is $0.943$. We fixed the parameter
$\lambda=0.01$ in equation (38). Figure 5: The zero-fidelity for three qubits
with 10 runs. The degrees of over-rotation are sampled from the Gaussian
distribution (with zero mean and $\sqrt{5}$ standard deviation) and the
readout error is described by the matrix (39). The orange circle indicates
that we do not add SPAM errors to the circuit but add depolarizing error on CX
gates. The average of zero-fidelity without SPAM is $0.968$. The blue circuit
indicates that we add SPAM errors and add depolarizing error on CX gates. The
average of zero-fidelity with SPAM is $0.941$. We fixed the parameter
$\lambda=0.01$ in equation (38). Figure 6: The zero-fidelity for three qubits.
We compare the difference kinds of channels that are affected by increasing
gate error. The triangle shows that the circuits have not over-rotation error
and the circle shows that circuits have over-rotation sampling from Gaussian
distribution (with zero mean and $\sqrt{5}$ standard deviation). The blue and
green indicate that the depolarizing error is added on CX gates. The orange
and red indicate that the depolarizing error is added on CZ gates. The readout
error can be described by the matrix (40).

###  IV.2 Combining zero-fidelity with randomized benchmarking

Now, we present the numerical results obtained from combining the zero-
fidelity and randomized benchmarking (RB).

We consider the circuit without over-rotation gates and without readout error
and the circuit with over-rotation gates and a readout error with values as
below:

$\begin{pmatrix}0.997&0.003\\\ 0.005&0.995\end{pmatrix}$ (39)
$\begin{pmatrix}0.97&0.03\\\ 0.05&0.95\end{pmatrix}.$ (40)

The following procedures show how we use RB:

$\mathbf{1.}$ Apply a sequence of $m+1$ operators that are chosen uniformly
from the Clifford group. The first $m$ operators can be chosen randomly,
however the choice of the $m+1$-th operator depends on the first $m$ operators
because the $m+1$-th operator is the inverse of all the other $m$ operators.
For example, let $C$ be an operator and let
$S=C_{m+1}C_{m}...C_{3}C_{2}C_{1}$, where
$C_{m+1}=(C_{m}...C_{3}C_{2}C_{1})^{-1}$. If the operators are ideal (i.e., no
noise) the sequence applied onto the quantum state is $S(\rho)=\rho.$ For a
noisy quantum operator $\tilde{C}$ and a sequence $\tilde{S}$, we can describe
them as $\tilde{S}_{m}=\bigcirc^{m+1}_{k=1}(\Lambda\circ C_{j}).$ Here, we
generate more than one sequence
$\tilde{S}_{l_{m}}=\bigcirc^{m+1}_{k=1}(\Lambda\circ C_{l_{j}})$ and then
apply all of the generated sequences on the initial states.

$\mathbf{2.}$ For each sequence we calculate the expectation values of Pauli
operators $W_{j}$ with respect to the final states $S_{l_{m}}(\rho_{i})$ and
$\tilde{S}_{l_{m}}(\rho_{i})$ respectively, where $\rho_{i}$ are initial
states prepared using SIC-states. And then we calculate the overlap

$\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[S_{l_{m}}(\rho_{i})W_{j}]\mathbf{Tr}[\tilde{S}_{l_{m}}(\rho_{i})W_{j}]$
(41)

$\mathbf{3.}$ Estimate equation (41) for different sequence $l$ and different
Clifford length $m$ and then average the random sequence to find the averaged
sequence fidelity

$F^{m}_{seq}(\mathcal{S}^{m},\tilde{\mathcal{S}}^{m})=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[\mathcal{S}^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\tilde{\mathcal{S}}^{m}(\rho_{i})W_{j}],$
(42)

where

$\mathcal{S}^{m}=\frac{1}{L}\sum^{L}_{l=1}S^{m}_{l}.$

$\mathbf{4.}$ Finally, use the results estimated using equation (42) to fit
the model mentioned above

$F^{m}_{seq}(\mathcal{S}^{m},\tilde{\mathcal{S}}^{m})=A_{0}p^{m}+B_{0}.$ (43)

$\mathbf{5.}$ With the result from Step 4 plot the exponential decay curve of
the process fidelity and Clifford length $m$.

Figure 7 shows the decay of zero-fidelity as a function of Clifford number $m$
without SPAM errors for blue line, with SPAM (weak readout error) for orange
line and with SPAM (strong readout error) for green line, respectively. We can
notice that the zero-fidelity is affected by SPAM errors, however with the
increasing $m$ the effect of those errors decreases. The values of the zero-
fidelity almost converge to the same value when $m$ is large. For the cases
without(with) SPAM errors, we use the values of the zero-fidelity to fit the
curve by using equation $F^{m}=A_{0}p^{m}+B_{0}$. For blue fitted curve we
obtain the decay rate $p\simeq 0.977$ and the coefficients $A_{0}=0.744\quad
B_{0}=0.243$. The average gate fidelity $F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq
0.983$. We also obtain the average error rate $r\equiv
1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. And for the orange fitted curve,
we obtain the decay rate $p\simeq 0.977$ and the coefficients
$A_{0}=0.718\quad B_{0}=0.244$. The average gate fidelity
$F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq 0.983$. We also obtain the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. For the last fitted
curve, we obtain the decay rate $p\simeq 0.977$ and the coefficients
$A_{0}=0.623\quad B_{0}=0.218$. The average gate fidelity
$F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq 0.983$. We also obtain the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. Therefore, no
matter whether there are SPAM errors or not, the decay rate of the zero-
fidelity will not be affected. Only the coefficients $A_{0}$ and $B_{0}$ will
be different. This shows that the combination of the zero-fidelity with
randomized benchmarking is indeed robust to SPAM errors.

Above we have shown the results of two-qubit randomized benchmarking. Here, we
will further show the results of three-qubit randomized benchmarking. Figure 8
shows the fidelity-decay of zero fidelity as a function of Clifford number
$m$. Like in figure 7 the values of the zero-fidelity almost converge to the
same value when $m$ is large. For the cases without(with) SPAM errors, we use
the values of the zero-fidelity to fit the curve by using equation
$F^{m}=A_{0}p^{m}+B_{0}$. For blue fitted curve we obtain the decay rate
$p\simeq 0.952$ and the coefficients $A_{0}=0.819\quad B_{0}=0.125$. The
average gate fidelity $F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq 0.958$. We also
obtain the average error rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.20\cdot
10^{-2}$. And for the orange fitted curve, we obtain the decay rate $p\simeq
0.952$ and the coefficients $A_{0}=0.817\quad B_{0}=0.122$. The average gate
fidelity $F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq 0.958$. We also obtain the
average error rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.20\cdot 10^{-2}$. For
the last fitted curve, we obtain the decay rate $p\simeq 0.952$ and the
coefficients $A_{0}=0.670\quad B_{0}=0.103$. The average gate fidelity
$F_{\textbf{ave}}=p+(1-p)/2^{n}\simeq 0.958$. We also obtain the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.23\cdot 10^{-2}$. Therefore, like two
qubit RB, no matter whether there are SPAM errors or not, the decay rate of
the zero-fidelity will not be affected. Only the coefficients $A_{0}$ and
$B_{0}$ will be different. This shows that, for the three qubit case as well,
the combination of the zero-fidelity with randomized benchmarking is indeed
robust to SPAM errors.

Now, we interleave the two kinds of channels that we compared between the two
cases (with SPAM and without SPAM) into RB (i.e., we use interleaved
randomized benchmarking [16] to obtain the fidelity of a particular gate).
Figure 9 shows the decay of zero-fidelity as a function of Clifford number $m$
with SPAM errors. The readout error is described by the matrix 39. The blue
line indicates three-qubit RB, the orange line indicates that we interleave
the channel (figure 1), and the green line indicates that we interleave the
channel 1 but with CX gates instead. By calculating the decay rate from each
fitted curve for the channel with CZ, we obtain the zero-fidelity $F_{0}\simeq
0.967$, which is nearly equal to the fidelity shown in figure 4. For channel
with CX we obtain the zero-fidelity $F_{0}\simeq 0.968$, which is nearly equal
to the fidelity shown in figure 5. Figure 10 shows the decay of zero-fidelity
as a function of Clifford number $m$ with SPAM errors. The readout error is
described by the matrix 40 (here, the parameter $\lambda$ of depolarizing
error (38) on each gates is $0.01$ ). The blue line indicates three-qubit RB,
the orange line indicates that we interleave the channel (figure 1), and the
green line indicates that we interleave the channel (figure 1) but with CX
gates instead. Like in figure 9, we can obtain the decay rate from each fitted
curve. For the channel with CZ, we obtain the zero-fidelity $F_{0}\simeq
0.963$, which is nearly equal to the fidelity shown in figure 4. For the
channel with CX we obtain the zero-fidelity $F_{0}\simeq 0.967$, which is
nearly equal to the fidelity shown in figure 5. We can observe that even when
the strength of readout error is increased, we can also obtain the same zero-
fidelities for these two kinds of channels. Therefore, the results show that
with this method one can calculate zero-fidelities of any channel without the
effects caused by SPAM errors.

Figure 7: Two-qubit RB. The zero-fidelity as the function of the number of
Clifford gates without SPAM errors. The horizontal axis is the number of
Clifford gates while the vertical axis is the zero-fidelity. The blue line
indicates that our circuits only have depolarizing error on gates. The orange
line means that our circuits have SPAM errors and the readout error is
described by the matrix (39). The green line means that our circuits also have
SPAM errors but the readout error is described by the matrix (40) instead. The
decay rate for blue line $p\simeq 0.977$ and error per Clifford (EPC) is
$1.71\mathrm{e}{-02}$. The decay rate for orange line $p\simeq 0.977$ and EPC
is $1.71\mathrm{e}{-02}$. The decay rate for green line $p\simeq 0.977$ and
EPC is $1.71\mathrm{e}{-02}$. Figure 8: Three-qubit RB. The zero-fidelity as
the function of the number of Clifford gates without SPAM errors. The
horizontal axis is the number of Clifford gates while the vertical axis is the
zero-fidelity. The blue line indicates that our circuits only have
depolarizing error on gates. The orange line means that our circuits have SPAM
errors and the readout error is described by the matrix (39). The green line
means that our circuits also have SPAM errors but the readout error is
described by the matrix (40) instead. The decay rate for blue line $p\simeq
0.952$ and error per Clifford (EPC) is $4.20\mathrm{e}{-02}$. The decay rate
for orange line $p\simeq 0.952$ and EPC is $4.20\mathrm{e}{-02}$. The decay
rate for green line $p\simeq 0.952$ and EPC is $4.20\mathrm{e}{-02}$. Figure
9: The zero-fidelity as the function of the number of Clifford gates with SPAM
errors for three qubits. Here, the readout error is described by the matrix
(39). As shown by orange fitted curve in figure 8 the blue line is indicating
three-qubit RB decay rate. The orange line indicates that we interleave the
channel shown in figure 1. The green line is for the channel like 1 but with
CX gates instead. The orange and green fitted curves are indeed lower than the
blue line because we interleave extra channels. However they all converge to
the same values when the number of Cliffords is large. Figure 10: The zero-
fidelity as a function of the number of Clifford gates with SPAM errors for
three qubits. Here, the readout error is described by the matrix (40). As
shown by green fitted curve in figure 8 the blue line is indicating three-
qubit RB decay rate. The orange line indicates that we interleave the channel
shown in figure 1. The green line is for the channel like 1 but CX gates
instead. The orange and green fitted curves are indeed lower than the blue
line because we interleave extra channels. However, they all converge to the
same values when the number of Cliffords is large.

##  V Conclusion

In Section II we introduced the process fidelity and the difficulties in
estimating it experimentally. In order to satisfy the experimental
requirements and use much fewer experimental settings we can use the
eigenstates of Pauli operators to be our initial states and then estimate the
expectation values of Pauli observables. However, in this scenario we need to
frequently change the initial states on a shot-by-shot basis, which is not
desirable in experiments due to the hardship of state preparation. In order to
deal with this problem, we considered using SIC-states as initial states.
However, these states do not possess an estimator with statistical properties
similarly favourable to that of Pauli eigenstates [1]. It is because, unlike
Pauli eigenstates, the SIC-states are not mutually orthogonal. This means that
they are not independent. However, the SIC states are a set of states closest
to being mutually orthogonal. And this makes them still useful and makes the
zero-fidelity feasible in the sense that it can be shown experimentally that
it indeed approximates the process fidelity well.

Similarly to the process fidelity, the zero-fidelity as given in [2] is not
robust to SPAM errors. In order to make it robust to SPAM errors we combined
it with randomized benchmarking. The respective equation that makes such a
combination possible is derived in Section III.1.

In Section IV we showed the results of this combination of zero-fidelity with
randomized benchmarking. We tested two cases - two qubits and three qubits -
for robustness to SPAM errors. For two qubits and the case without SPAM
errors, we obtained the decay rate $p\simeq 0.977$ and the coefficients
$A_{0}=0.744\quad B_{0}=0.243$ by using equation $F^{m}=A_{0}p^{m}+B_{0}$ to
fit the results of zero-fidelity. The average gate fidelity
$F_{\textbf{ave}}=p+(1-p)/2^{n}=0.983$. We also obtained the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. For the cases with
SPAM errors (weak and strong readout error cases) the results were as follows.
For weak readout error cases, we obtained the decay rate $p=0.977$ and the
coefficients $A_{0}=0.718\quad B_{0}=0.244$. The average gate fidelity was
$F_{\textbf{ave}}=p+(1-p)/2^{n}=0.983$. We also obtained the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. And for strong
readout error case, we obtained the decay rate $p=0.977$ and the coefficients
$A_{0}=0.623\quad B_{0}=0.218$. The average gate fidelity was
$F_{\textbf{ave}}=p+(1-p)/2^{n}=0.983$. We also obtained the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 1.71\cdot 10^{-2}$. For three qubits
and the case without SPAM errors, we obtained the decay rate $p\simeq 0.952$
and the coefficients $A_{0}=0.819\quad B_{0}=0.125$ by using equation
$F^{m}=A_{0}p^{m}+B_{0}$ to fit the results of zero-fidelity. The average gate
fidelity $F_{\textbf{ave}}=p+(1-p)/2^{n}=0.958$. We also obtained the average
error rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.20\cdot 10^{-2}$. For the cases
with SPAM errors (weak and strong readout error cases) the results were as
follows. For weak readout error cases, we obtained the decay rate $p=0.952$
and the coefficients $A_{0}=0.817\quad B_{0}=0.122$. The average gate fidelity
was $F_{\textbf{ave}}=p+(1-p)/2^{n}=0.983$. We also obtained the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.20\cdot 10^{-2}$. And for strong
readout error case, we obtained the decay rate $p=0.952$ and the coefficients
$A_{0}=0.670\quad B_{0}=0.103$. The average gate fidelity was
$F_{\textbf{ave}}=p+(1-p)/2^{n}=0.983$. We also obtained the average error
rate $r\equiv 1-F_{\textbf{ave}}\simeq 4.20\cdot 10^{-2}$. Hence, no matter
whether there are SPAM errors or not, the decay rate of zero fidelity will not
be affected, which means that the combination is robust to SPAM errors. We
also interleave the two different channels mentioned above to obtain their
zero-fidelities, which turned out to be the same as the results without SPAM
errors in figure 4 and in figure 5. Note that in our experiments, we assumed
gate-independence and time-independence so the decay rate $p$ is the parameter
of the equation $F_{ave}=p+\frac{1-p}{2^{n}}$ [10, 12]. We could relax the
assumption of gate-independence but then we get a different interpretation of
the decay rate $p$ [17]. We leave this work for the future.

By combining zero-fidelity with randomized benchmarking we may use much fewer
experimental settings and not only do not need to frequently change the
initial states shot-by-shot [2] but also do not need to consider the effects
caused by SPAM errors.

It is known that the randomized benchmarking protocol is not as scalable as
the cycle benchmarking protocol. Therefore, in a future work, we shall
consider combining the zero-fidelity with cycle benchmarking [7] to make the
zero-fidelity even more scalable.

###### Acknowledgements.

We wish to acknowledge the support of the National Taiwan University’s High
Performance Computing facilities, on which our numerical results were
obtained.

##  Appendix A The other form of the process fidelity

In appendix A we will derive the equation

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{{4^{n}}}_{{i,j=1}}C_{ij}\mathbf{Tr}[\Gamma(\rho_{i})W_{j}]$
(44)

with
$C_{ij}=\sum^{{4^{n}}}_{{l=1}}[B^{-1}]_{li}\mathbf{Tr}[\Lambda(\rho_{l})W_{j}].$

From the equation

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})],$
(45)

where $W_{j}$ is a set of Pauli operators, $\rho_{i}$ are the quantum states
which are prepared by the eigenstates of Pauli operators, and $\Lambda,\Gamma$
are the ideal channel and the actual channel respectively. Before the
derivation we will introduce some notation and representation.

###  A.1 Preliminary notation

In this section we will define the vectorization of a matrix, which is a
linear transformation converting a matrix into a vector form.

Let $A$ be a $n\times n$ matrix and let $\mathbf{Vec}(A)$ denote a
vectorization of the matrix $A$. So

$\mathbf{Vec}(A)=[a_{1,1},...a_{m,1},a_{1,2},...,a_{m,2},...,a_{m,n}]^{\mathbf{T}}$
(46)

, where ${\mathbf{T}}$ stands for transposition. For example, consider a
matrix

$M=\begin{pmatrix}1&2\\\ 3&4\end{pmatrix},$ (47)

Its vectorization form can be written as

$\mathbf{Vec}(M)=\begin{pmatrix}1\\\ 3\\\ 2\\\ 4\end{pmatrix}.$ (48)

Consider the product of three matrices $ABC$. The vectorization is

$\mathbf{Vec}(ABC)=(C^{\mathbf{T}}\otimes A)\mathbf{Vec}(B)$ (49)

and the inner product (Hilbert–Schmidt inner product) between the vectorized
matrices denotes

$\mathbf{Vec}(A)^{\dagger}\mathbf{Vec}(B)=\mathbf{Tr}(A^{\dagger}B).$ (50)

For convenience let us define $\mathbf{Vec}(A)=|A\rangle\rangle$ and
$\mathbf{Vec}(A)^{\dagger}=\langle\langle A|.$ Now assume a density matrix
$\rho$ which is evolved by a channel $\mathcal{U}$. It can be represented by
this form

$\rho^{\prime}\rightarrow\mathcal{U}(\rho)=U\rho U^{\dagger},$ (51)

where $U$ is a unitary matrix. Because of equation (49) we can rewrite
$\rho^{\prime}$ as

$\displaystyle\mathbf{Vec}(\rho^{\prime})$
$\displaystyle=\mathbf{Vec}(\mathcal{U}(\rho))$ (52)
$\displaystyle=\mathbf{Vec}(U\rho U^{\dagger})$ (53)
$\displaystyle=(U^{\dagger}\otimes U)\mathbf{Vec}(\rho)$ (54)
$\displaystyle=\mathcal{U}|\rho\rangle\rangle,$ (55)

Here we use the superoperator representation to describe the channel
$\mathcal{U}=(U^{\dagger}\otimes U).$ It can be understood that the
superoperator $\mathcal{U}$ in the term (55) is an operator and the vectorized
$\rho$ is a vector.

###  A.2 Derivation

With the introduction in A.1 we can vectorize the Pauli operator

$P_{i}\rightarrow|P_{i}\rangle\rangle/\sqrt{2^{n}},$ (56)

where $n$ is the number of qubits. Take $n=1$ for example. The vectorized
Pauli operators can be represented as

$|P_{i}\rangle\rangle\in\\{|I\rangle\rangle\quad|X\rangle\rangle\quad|Y\rangle\rangle\quad|Z\rangle\rangle\\}$
(57)

, where

$|I\rangle\rangle=\begin{pmatrix}1\\\ 0\\\ 0\\\
1\end{pmatrix}|X\rangle\rangle=\begin{pmatrix}0\\\ 1\\\ 1\\\
0\end{pmatrix}|Y\rangle\rangle=\begin{pmatrix}0\\\ i\\\ -i\\\
0\end{pmatrix}|Z\rangle\rangle=\begin{pmatrix}1\\\ 0\\\ 0\\\ -1\end{pmatrix}$
(58)

Since the density matrix can be represented by the Pauli operator we represent
a quantum state as this form

$|\rho\rangle\rangle=\sum_{i}c_{i}|P_{i}\rangle\rangle,$ (59)

where $c_{i}=\langle\langle P_{i}|\rho\rangle\rangle$ (we have normalized
Pauli operators in order to avoid the factor). Note that the coefficients
$c_{i}$ can be considered to be the expectation values of Pauli operators with
respect to the quantum state $\rho$. Putting the coefficients $c_{i}$ back
into equation (59), the quantum state becomes

$|\rho\rangle\rangle=\sum_{i}|P_{i}\rangle\rangle\langle\langle
P_{i}|\rho\rangle\rangle.$ (60)

Consider another quantum state

$|\sigma\rangle\rangle=\sum_{j}|P_{j}\rangle\rangle\langle\langle
P_{j}|\sigma\rangle\rangle.$ (61)

We can perform the Hilbert–Schmidt inner product to obtain

$\displaystyle\mathbf{Tr}[\sigma^{\dagger}\rho]$
$\displaystyle=\sum_{i}\sum_{j}\langle\langle\sigma|\Large[P_{j}\rangle\rangle(\langle\langle
P_{j}|P_{i}\rangle\rangle)\langle\langle P_{i}\Large]|\rho\rangle\rangle$ (62)
$\displaystyle=\sum_{i}\langle\langle\sigma|P_{i}\rangle\rangle\cdot\langle\langle
P_{i}|\rho\rangle\rangle$ (63) $\displaystyle=\sum_{i=1}\mathbf{Tr}[\sigma
P_{i}]\mathbf{Tr}[\rho P_{i}].$ (64)

because Pauli operators are mutually orthogonal, which means that

$\begin{cases}\mathbf{Tr}[P_{i}P_{j}]=0,&\mbox{if }i\neq j\\\
\mathbf{Tr}[P_{i}P_{j}]=1,&\mbox{if }i=j\end{cases}$

and recall that $\langle\langle P_{i}|\rho\rangle\rangle=\mathbf{Tr}[\rho
P_{i}].$ Now, back to the equation

$F(\Lambda,\Gamma)=\frac{1}{4^{2}}\sum^{4^{2}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})],$
(65)

where $\\{\sigma_{i}\\}^{4}_{i=1}$ is a set of Pauli operators, for n qubits
$\sigma_{i}\in\\{I,X,Y,Z\\}^{\otimes n}.$

We can rewrite

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})]$
(66)

as

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}[B^{-1}]_{ij}\mathbf{Tr}[\Lambda(\rho_{i})\Gamma(\rho_{j})],$
(67)

where $B_{ij}=\mathbf{Tr}[\rho^{\dagger}_{i}\rho_{j}]$ and $\rho_{i},\rho_{j}$
are density matrices (quantum states of a system) which can be prepared from
the eigenstates of Pauli operators. Let $\rho^{\prime}=\Lambda(\rho)$ and
$\sigma=\Gamma(\rho)$ so the form (67) can be

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}[B^{-1}]_{ij}\mathbf{Tr}[\rho^{\prime}_{i}\sigma_{j}].$
(68)

From equation (62) to equation (64) we obtain

$\displaystyle F(\Lambda,\Gamma)$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}[B^{-1}]_{ij}\mathbf{Tr}[\rho^{\prime}_{i}\sigma_{j}]$
(69)
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}\sum^{4^{n}}_{k=1}[B^{-1}]_{ij}\mathbf{Tr}[\sigma_{j}P_{k}]\mathbf{Tr}[\rho^{\prime}_{i}P_{k}]$
(70)
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}\sum^{4^{n}}_{k=1}[B^{-1}]_{ij}\mathbf{Tr}[\Gamma(\rho_{j})P_{k}]\mathbf{Tr}[\Lambda(\rho_{i})P_{k}].$
(71)

As mentioned in the begin of this appendix we finally get

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{{4^{n}}}_{{i,j=1}}C_{ij}\mathbf{Tr}[\Gamma(\rho_{i})W_{j}]$
(72)

with
$C_{ij}=\sum^{{4^{n}}}_{{l=1}}[B^{-1}]_{li}\mathbf{Tr}[\Lambda(\rho_{l})W_{j}],$
where $W_{j}$ is a set of Pauli operators.

##  Appendix B Decay rate of zero-fidelity

In this appendix we will show that the equation

$F^{m}(\mathcal{U},\tilde{\mathcal{U}})=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\tilde{\mathcal{U}}^{m}(\rho_{i})W_{j}],$
(73)

where $\rho_{i}$ are the quantum state prepared by the SIC-states and $W_{j}$
is a set of Pauli operators, is equal to

$A_{0}p^{m}+B_{0},$ (74)

where the coefficients $A_{0}$ and $B_{0}$ absorb state preparation and
measurement errors, and $m$ is the number of Clifford operators.

Because the process fidelity

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})]$
(75)

can be written as the form (the derivation is shown in Appendix A)

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\rho_{i})W_{j}]\mathbf{Tr}[\Gamma(\rho_{i})W_{j}]$
(76)

we only need to prove that the process fidelity with the channel applied $m$
times

$F^{m}(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda^{m}(\sigma^{\dagger}_{i})\Gamma^{m}(\sigma_{i})]$
(77)

is equal to eq. (74).

As we have shown (Section III)

$\displaystyle F^{m}(\mathcal{U},\tilde{\mathcal{U}})$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\mathcal{U}^{m}(\rho_{i})\tilde{\mathcal{U}}^{m}(\rho_{i})]$
(78) $\displaystyle=A_{0}p^{m}+B_{0}$ (79)

and as we have also shown

$\displaystyle F(\Lambda,\Gamma)$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i=1}\mathbf{Tr}[\Lambda(\sigma^{\dagger}_{i})\Gamma(\sigma_{i})]$
(80)
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j=1}C_{ij}\mathbf{Tr}[\Lambda(\rho_{i})W_{j}],$
(81)

where

$C_{ij}=\sum^{4^{n}}_{l=1}[B^{-1}]_{li}\mathbf{Tr}[\Gamma(\rho_{i})W_{j}],$
(82)

and if the density matrix $\rho_{i}$ is prepared by eigenstates of Pauli
operators we can write the process fidelity as [7]

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}2^{-n}\mathbf{Tr}[\Gamma(\rho_{i})W_{j}]\mathbf{Tr}[\Lambda(\rho_{i})W_{j}].$
(83)

Now let the quantum states be prepared by the SIC-states instead of the
eigenstates of Pauli operators. In this case, the coefficient $2^{-n}$ in
equation (83) is $1$ and then eq. (83) becomes

$F(\Lambda,\Gamma)=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[\Gamma(\rho_{i})W_{j}]\mathbf{Tr}[\Lambda(\rho_{i})W_{j}]$
(84)

Finally, we can conclude that

$\displaystyle F^{m}(\Lambda,\Gamma)$
$\displaystyle=\frac{1}{4^{n}}\sum^{4^{n}}_{i,j}\mathbf{Tr}[\Gamma^{m}(\rho_{i})W_{j}]\mathbf{Tr}[\Lambda^{m}(\rho_{i})W_{j}]$
(85) $\displaystyle=A_{0}p^{m}+B_{0}$ (86)

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#  Confinement effect enhanced Stoner ferromagnetic instability in monolayer
1T-VSe2

Junyi He  Q. Xie  Wuhan National High Magnetic Field Center and School of
Physics, Huazhong University of Science and Technology, Wuhan 430074, China
Gang Xu<EMAIL_ADDRESS>Wuhan National High Magnetic Field Center and
School of Physics, Huazhong University of Science and Technology, Wuhan
430074, China

###### Abstract

Monolayer 1T-VSe2 has been reported as a room-temperature ferromagnet. In this
work, by using first-principles calculations, we unveil that the
ferromagnetism in monolayer 1T-VSe2 is originated from its intrinsic huge
Stoner instability enhanced by the confinement effect, which can eliminate the
interlayer coupling, and lead to a drastic increase of the density of states
at the Fermi level due to the presence of Van Hove singularity. Our
calculations also demonstrate that the Stoner instability is very sensitive to
the interlayer distance. These results provide a useful route to modulate the
nonmagnetic to ferromagnetic transition in few-layers or bulk 1T-VSe2, which
also shed light on the enhancement of its Curie temperature by enlarging the
interlayer distance.

##  I Introduction

Ferromagnetic order in two-dimensional (2D) materials is a highly desirable
property that provides a new physical degree of freedom to manipulate spin
behaviors in spintronic devices Burch  _et al._ (2018); Gibertini  _et al._
(2019). Previously, magnetism in 2D was mainly realized through depositing
films onto magnetic substrates, magnetic atoms adsorption, or doping González-
Herrero  _et al._ (2016); Nair  _et al._ (2012); Avsar  _et al._ (2014). The
shortcomings of these methods are obvious: (i) one does not have an ideal 2D
system from depositing and it is impractical to integrate with spintronic
devices, and (ii) disorder effects make the electronic properties hard to
design. Due to these drawbacks, 2D materials with intrinsic magnetic order
have been actively pursued.

The CrI3 Huang  _et al._ (2017) and Cr2Ge2Te6 Gong  _et al._ (2017) are the
first two experimentally reported 2D materials exhibiting long-range
ferromagnetic (FM) order with the Curie temperatures $T_{c}$ $\sim$ 45 K and
30 K, respectively Huang  _et al._ (2017); Gong  _et al._ (2017). These
discoveries have stimulated numerous research interests on 2D magnetic
materials. Very recently, several materials with higher $T_{c}$ have been
experimentally and theoretically explored, including MnSex O Hara  _et al._
(2018), Fe3GeTe2 Deng  _et al._ (2018), and 1T-VSe2 Bonilla  _et al._ (2018);
Gao  _et al._ (2013); Yu  _et al._ (2019); Zhang  _et al._ (2019); Vinai  _et
al._ (2020). Among them, the 1T-VSe2 is of particular interest since the bulk
1T-VSe2 has a van der Waals (vdW) nature, which can be easily exfoliated to
few-layers thickness. This gives 1T-VSe2 the advantage to be tailored and
manipulated for nano spintronic devices at low cost.

However, the nature of the ground phase of 1T-VSe2 is still under hot debate.
Two groups have observed charge-density-wave (CDW) ground states and concluded
that magnetic order is absent in the monolayer limit due to the CDW
suppression Fumega  _et al._ (2019); Coelho  _et al._ (2019). Wong _et al._
claimed that a spin frustrated phase was observed and the FM phase must be
attributed to extrinsic factors Wong  _et al._ (2019). Chua _et al._ Chua  _et
al._ and Yu _et al._ Yu  _et al._ (2019) suggested that the observed FM is
_not_ intrinsic, but caused by defects. Nevertheless, Bonilla _et al_. Bonilla
_et al._ (2018) and many others Gao  _et al._ (2013); Yu  _et al._ (2019);
Zhang  _et al._ (2019); Vinai  _et al._ (2020) have presented strong
experimental evidences for intrinsic 2D magnetism in monolayer 1T-VSe2, which
also reported a NM to FM phase transition from bulk to the monolayer limit van
Bruggen and Haas (1976). This is in contrast with other 2D magnetic materials,
where the FM phase is more stable in the bulk system.

Figure 1:  (a) The crystal structure of bulk 1T-VSe2. $d$ denotes the
interlayer distance. (b) and (c) are the top view and side view of the VSe2
monolayer, respectively. (d) The total and projected DOS of bulk 1T-VSe2 in
the NM phase. (e) NM band structures of bulk VSe2 with V-3$d$ orbitals
projections.

In this paper, we study the ground-state properties of 1T-VSe2 by first-
principles calculations. Through a comprehensive study of the density of
states (DOS) and band structures of the bulk and few-layers 1T-VSe2, we reveal
that the monolayer system has the strongest FM instability due to the presence
of Van Hove singularity (VHS) originated from saddle points at the Fermi
level. We also find that in the few-layers case, the couplings of $d_{z^{2}}$
orbitals between interlayer V atoms split the saddle points away from the
Fermi level and weaken the FM instability. The strongest FM instability in the
monolayer is confirmed by the largest energy difference between the NM and FM
phases and also verified by using the phenomenological Stoner theory Stoner
(1938); Janak (1977); Stollhoff  _et al._ (1990); Zhuang  _et al._ (2016). We
thus conclude that the room-temperature FM order in the monolayer 1T-VSe2 is
intrinsic due to its unique electronic structures. Finally, we study the FM
instability with respect to the interlayer distance $d$ [see Fig. 1(a)] and
predict that it is possible to tune the NM to FM phase transition in few-
layers 1T-VSe2 by enlarging the interlayer distance $d$. Our study provides an
explanation to the origin of FM order in monolayer 1T-VSe2 and also proposes a
mechanism to tune the NM to FM phase transition in few-layers 1T-VSe2.

Figure 2:  Results of the monolayer 1T-VSe2 in the NM phase. (a) The total DOS
and projected DOS of the V-$3d$ orbitals. (b) The corresponding band
structures along the high symmetry $k$-path. (c) Three dimensional plot of the
$d_{z^{2}}$ band in the first BZ. (d) Zoom-in band dispersion near the saddle
point $S$.

##  II Computational details

VSe2 usually adopts the 2H and 1T structures. While the 2H-VSe2 shows
semiconducting behavior, the 1T-VSe2 is a metal Ma  _et al._ (2012); Fuh  _et
al._ (2016); Lebègue  _et al._ (2013) and shows strong experimental evidence
for FM ordering in the few-layers limit Bonilla  _et al._ (2018). Different
from the triangular prismatic crystal field in the 2H structure, 1T-VSe2 has
an octahedral crystal structure and belongs to $P\bar{3}m1$ space group, where
V atoms form a triangular lattice and each V atom occupies the center of the
octahedron surrounded by six Se atoms, as shown in Figs. 1(a), 1(b) and 1(c).
As a result, each layer of VSe2 is stoichiometric Zhuang and Hennig (2016).
The bulk crystal is composed of an AA stacking of VSe2 sandwiches.

First-principles calculations based on density functional theory are carried
out by using the Vienna _ab initio_ simulation package (VASP) Kresse and
Furthmüller (1996). The Perdew-Burke-Ernzerhof functional Perdew  _et al._
(1996) is employed to treat the exchange-correlation interactions. The cutoff
energy for wave function expansion is set to 500 eV. We use
19$\times$19$\times$9 and $21$$\times$21$\times$1 $\Gamma$-centered k meshes
to sample the Brillouin zone (BZ) in the bulk and slab calculations,
respectively. Structures are optimized until the force on each atom is less
than 0.001 eV/Å. A vacuum layer of 15 Å is set to minimize artificial
interactions between layers in the slab calculations. For the bulk
calculations, the lattice constants $a=b=3.356$ Å, and $c=6.105$ Å are used
Bayard and Sienko (1976).

##  III Results And Discussion

We first calculate and plot the total and projected DOS of the NM bulk 1T-VSe2
in Fig. 1(a), which are in good agreement with previous results Reshak and
Auluck (2004); Li  _et al._ (2014); Feng  _et al._ (2018); Yang  _et al._
(2018). The density at the Fermi level is about 2.9 states/eV, confirming its
metallic nature. The projected DOS demonstrates that the states between $-0.9$
eV and $3.5$ eV are mainly contributed by the V-$3d$ orbitals. In an
octahedron crystal field, the five 3$d$ orbitals split into the lower $t_{2g}$
and the upper $e_{g}$ manifolds, which mainly contribute to the DOS around
$-0.9\sim 1.4$ eV and $1.9\sim 3.5$ eV, respectively. Furthermore, due to the
presence of a triangular field, the $t_{2g}$ manifold splits into the lower
$a_{1g}$ ($d_{z^{2}}$) and upper $e_{g}^{\prime}$ ($d_{xy}$ and
$d_{x^{2}-y^{2}}$) orbitals. This picture is also verified from the projected
band structures in Fig. 1(e), in which the $d_{z^{2}}$ is the lowest 3$d$
orbital that crosses the Fermi level and dominates the low-energy physics of
the bulk 1T-VSe2.

We further plot the total and projected DOS of the NM monolayer 1T-VSe2 in
Fig. 2(a). Compared with the bulk DOS in Fig. 1(d), we notice that these two
DOS plots share many similarities. For instance, they both have high densities
around 1 eV and they both possess energy gaps at $\sim$1.4 eV. This is
reasonable due to that 1T-VSe2 is a layered vdW material. The interlayer
coupling does not significantly alter the electronic structures. Nevertheless,
a shape peak appears at the Fermi level $E_{F}$ in the monolayer case as shown
in Fig. 2(a). The DOS at $E_{F}$ is about $N(E_{F})\approx$ 6.4 state$/$eV,
much higher than the bulk value. Such high DOS suggests the presence of VHS on
the band structures. We thus plot the band structures along high symmetry
$k$-path in Fig. 2(b) and observe that only the $d_{z^{2}}$ band crosses the
Fermi level. On the $\Gamma-M$ path there is a maximum at $\Gamma$ and on the
$\Gamma-K$ path a minimum at $S$ appears. These characteristics indicate the
presence of a saddle point. To show more details, we plot this band on the
whole BZ in Fig. 2(c), where there are six saddle points on the $\Gamma-K$ and
$\Gamma-K^{\prime}$ paths. Figure 2(d) zooms in the band structures around the
saddle point $S$. The red and blue solid lines are the band dispersions along
$\Gamma-K$ and its orthogonal directions, which represent hole-like and
electron-like dispersions, respectively. These dispersions evidently show the
topology of a saddle point Ziletti  _et al._ (2015). Therefore, we have
demonstrated that the high DOS and its divergent behavior are due to the
presence of saddle-points VHS on the band structures.

Figure 3:  Results of the monolayer 1T-VSe2 in the FM phase. (a) The spin
polarized DOS. $\tilde{E}_{\uparrow}$ and $\tilde{E}_{\downarrow}$ denote the
energy shift of the VHS peaks, which are used to evaluate the energy
difference between the spin-up and spin-down states. (b) The evolution of the
Stoner criterion with respect to the number of layers $N$. (c) The evolutions
of DOS at the Fermi level $N(E_{F})$ and the Stoner parameter $I$ with respect
to $N$. (d) The energy difference $\Delta E=E_{FM}-E_{NM}$ as a function of
$N$.

We suggest that the VHS in monolayer 1T-VSe2 may cause FM instability
according to the phenomenological Stoner theory Stoner (1938), which states
that the FM phase is favored when the Stoner criteria $N(\epsilon_{f})\cdot
I>1$ is satisfied. Here $N(\epsilon_{f})$ is the DOS at the Fermi level in the
NM state, and $I$ is the Stoner parameter that measures the strength of the
magnetic exchange interaction, which is related to the energy splitting
between the spin-up and spin-down states in the FM phase via the following
formulas Blundell (2003)

$\displaystyle\begin{split}E_{\uparrow}(k)&=E_{0}(k)-I\frac{n_{\uparrow}}{n},\\\
E_{\downarrow}(k)&=E_{0}(k)+I\frac{n_{\downarrow}}{n}.\\\ \end{split}$ (1)

Here $E_{0}(k)$ is the energy of the NM phase, $E_{\sigma}(k)$ and
$n_{\sigma}$ are the energy and number of electrons with spin $\sigma$
$(\sigma=\uparrow,\downarrow)$ in the FM phase, respectively. The total number
of electrons is $n=n_{\uparrow}+n_{\downarrow}$. Since only the $d_{z^{2}}$
band is responsible for the Stoner instability in monolayer 1T-VSe2,
$n_{\uparrow}$ and $n_{\downarrow}$ can be estimated as 1. Therefore we have
$n=2$. Finally, the Stoner parameter $I$ can be estimated as
$E_{\downarrow}(k)-E_{\uparrow}(k)$.

Figure 3(a) presents the DOS of the spin-up and spin-down states in the FM
phase of the monolayer 1T-VSe2. By comparing with the NM results in Fig. 2(a),
we observe that the sharp VHS peak splits into two peaks, which is a typical
signature of the FM exchange interaction. We assume the exchange interaction
is $k$-independent and use the energy difference of the VHS peaks to evaluate
its average magnitude Blundell (2003), which gives $I=0.68$ eV. Together with
$N(E_{F})=6.4$ state/eV in the NM phase, we obtain a Stoner criterion
$N(E_{F})\cdot I=4.3$. This large value indicates a strong FM instability in
the monolayer 1T-VSe2.

Recent experiments have shown that the monolayer 1T-VSe2 exhibits FM order,
while the bulk 1T-VSe2 displays a NM property Bonilla  _et al._ (2018); Gao
_et al._ (2013); Yu  _et al._ (2019); Zhang  _et al._ (2019); Vinai  _et al._
(2020). To study this transition, we show the evolution of the Stoner
criterion $N(E_{F})\cdot I$ with respect to the number of layers $N$ in Fig.
3(b), from which a drastic decrease of $N(E_{F})\cdot I$ from the mono- to the
bilayer is observed, indicating the decrease of FM instability in the bilayer
1T-VSe2. We further plot the evolutions of $N(E_{F})$ and $I$ with respect to
$N$ in Fig. 3(c), which clearly shows that the drastic decrease is due to the
decrease of $N(E_{F})$ since $I$ is insensitive to $N$ [see the red curve in
Fig. 3(c)]. We notice that when $N\geq 2$, the Stoner criterion $N(E_{F})\cdot
I$, density $N(E_{F})$, and Stoner parameter $I$ fluctuate slightly around
their saturated values. These results prove that when the system transforms
from the monolayer to bulk, only the monolayer exhibits a strong FM
instability. Our result well explains the recent experiment by Bonilla  _et
al._ , where a strong FM signal has been detected in the monolayer while the
bilayer has a significantly weak FM signal comparable with the bulk Bonilla
_et al._ (2018). This trend is also manifested by the energy difference
$\Delta E=E_{FM}-E_{NM}$ between the FM and NM phases as depicted in Fig.
3(d), from which we observe that the maximal energy difference occurs in the
monolayer case, and a drastic decrease of $\Delta E$ takes place from the
monolayer to the bilayer.

Figure 4:  Results of the bilayer 1T-VSe2. (a) and (b) are the NM DOS and band
structures, respectively. The inset of (a) shows the detail of the two peaks.
(c) The blue curve shows the evolution of the two peaks’ energy splitting
$\Delta E_{d}$ with the ratio of interlayer distance $\varepsilon$. The red
curve shows the $N(E_{F})$ as a function of $\varepsilon$. (d) shows the
energy difference $\Delta E=E_{FM}-E_{NM}$ as a function of $\varepsilon$.

To understand the drastic decrease of $N(E_{F})$ from the mono to the bilayer,
we plot the NM total DOS of bilayer 1T-VSe2 in Fig. 4(a). Due to the vdW
nature, the bilayer DOS is very similar with that of the monolayer, except
that two peaks emerge near the Fermi level at $E_{1}=-0.03$ and $E_{2}=0.08$
eV [see the inset in Fig. 4(a)]. These two peaks also originate from the
saddle points on the band structures as shown in Fig. 4(b). We notice that two
bands are crossing the Fermi level. They are contributed by the $d_{z^{2}}$
orbitals of the two V atoms in the bilayer unit cell. The upper and lower
$d_{z^{2}}$ bands are anti-bonding and bonding states, respectively. The
energy difference of these two $d_{z^{2}}$ bands at the $\Gamma$ is about 0.44
eV, which gives an estimation of the interlayer coupling strength. The VHS
splitting at the $S$ point is determined by the two minima on the $\Gamma-K$
path, which is about 0.11 eV, consistent with the two peaks on the DOS in Fig.
4(a). Such splitting is larger than that in typical vdW materials Ersan  _et
al._ (2019); Jung  _et al._ (2018); McGuire  _et al._ (2015). This is because
that the VHS peak in 1T-VSe2 is mainly contributed by the $d_{z^{2}}$
orbitals, whose lobes from interlayer V atoms are head-to-head aligned along
the $z$-direction and form relatively strong $dd\sigma$ bonds. As a result,
the VHSs no longer present at the Fermi level, and the $N(E_{F})$ is
significantly reduced. Finally, the Stoner criterion $N(E_{F})\cdot I$
decreases severely, and the FM instability is weakened.

To summarize, the transition from the bulk NM phase to the monolayer FM phase
in 1T-VSe2 can be understood as follows. In the bulk system, the coupling of
$d_{z^{2}}$ orbital between interlayer V atoms splits the VHSs away from the
Fermi level. Especially from the bilayer to the monolayer, the enhanced
confinement effect eliminates the interlayer $d_{z^{2}}$ coupling. Thus the
VHSs are pushed to the Fermi level, which leads to a drastic enhancement of
the $N(E_{F})$ and the Stoner criterion $N(E_{F})\cdot I$. Eventually, this
enhanced $N(E_{F})$ causes a strong FM instability in the monolayer. In other
words, the confinement effect in the monolayer 1T-VSe2 prevents the interlayer
coupling between the $d_{z^{2}}$ orbitals of V atom and pushes the saddle-
point VHS at the Fermi level, which results in a large Stoner criterion and
leads to a stable FM ground state. Our numerical results can well explain
recent experiments Bonilla  _et al._ (2018); Gao  _et al._ (2013); Yu  _et
al._ (2019); Zhang  _et al._ (2019); Vinai  _et al._ (2020).

Based on this understanding, we expect that the magnetic property of a few-
layers 1T-VSe2 can be tuned by the interlayer distance $d$ Zhou  _et al._
(2012) [see Fig. 1(a)]. The evolution of the VHS splitting $\Delta
E_{d}=E_{2}-E_{1}$ between the two peaks in the bilayer 1T-VSe2 with the ratio
$\varepsilon=(d-d_{0})/d_{0}$ is shown in Fig. 4(c). Here $d_{0}=3.067$ Å is
the interlayer distance of the bulk. The corresponding evolution of $N(E_{F})$
with $\varepsilon$ is also shown in this figure. We observe that $\Delta
E_{d}$ monotonically decreases with the increase of $d$. This means the
$dd\sigma$ bond is weakened when the lobes of $d_{z^{2}}$ orbitals in adjacent
layers are moving apart. During this process, the $N(E_{F})$ monotonically
goes up, reflecting an enhanced Stoner FM instability. The enhancement of the
Stoner instability is also confirmed by the energy differences $\Delta E$
between the FM and NM phases in Fig. 4(d). We find that $\Delta E$ also
monotonically decreases with $\varepsilon$, which indicates that the FM phase
becomes more and more stable when $d$ increases. This effect provides a useful
route to control the NM to FM transition in few-layers 1T-VSe2 through
enlarging the interlayer distance $d$. It also sheds light on tuning the Curie
temperature of 1T-VSe2 through nanoengineering. Further experimental studies
are highly desirable to verify these conjectures.

##  IV  Acknowlegments

This work is supported by the Ministry of Science and Technology of China (No.
2018YFA0307000) and the National Natural Science Foundation of China (No.
11874022).

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