# Quantum Simulation of the N flavor Gross-Neveu Model

Muhammad Asaduzzaman  Department of Physics, Syracuse University, Syracuse, NY
13244, USA  Simon Catterall  Department of Physics, Syracuse University,
Syracuse, NY 13244, USA  Goksu Can Toga  Department of Physics, Syracuse
University, Syracuse, NY 13244, USA  Yannick Meurice  Department of Physics
and Astronomy, University of Iowa, Iowa City, IA 52242, USA  Ryo Sakai
Department of Physics, Syracuse University, Syracuse, NY 13244, USA

###### Abstract

We discuss the use of quantum simulation to study an $N$ flavor theory of
interacting relativistic fermions in (1+1) dimensions on NISQ era machines.
The case of two flavors is particularly interesting as it can be mapped to the
Hubbard model. We derive the appropriate qubit Hamiltonians and associated
quantum circuits. We compare classical simulation and DMRG/TEBD calculations
with the results of quantum simulation on various platforms for $N$=2 and 4.
We demonstrate that the four steps of the calculations of real-time scattering
can actually be implemented using current NISQ devices.

##  I Introduction

Doing ab-initio lattice QCD calculations in real-time or at finite density
would have a significant impact on our interpretation of hadron collider data
and our understanding of nuclear matter. However, because of sign problems,
such calculations cannot be handled efficiently with importance sampling
(Monte Carlo) methods that have been very successful in the study of masses
and form factors. In contrast, these calculations could be handled efficiently
by using quantum devices capable of manipulating large enough Hilbert spaces
that can be mapped into those relevant for the QCD Hamiltonian. The
possibility of using universal quantum computers Jordan et al. (2014, 2012);
Lloyd (1996); Mezzacapo et al. (2015); Cervera-Lierta (2018); Yeter-Aydeniz et
al. (2019); Raychowdhury and Stryker (2020); Lamm and Lawrence (2018);
Macridin et al. (2018); Klco et al. (2018); Bauer et al. (2021); Lamm et al.
(2020, 2019); Gustafson et al. (2019); Gustafson et al. (2021a, b); Kharzeev
and Kikuchi (2020); Honda et al. (2022); Bhattacharya et al. (2021); Ji et al.
(2022), or analog quantum simulations with cold atoms Bloch et al. (2012);
Lewenstein et al. (2007); Cirac et al. (2010); Kapit and Mueller (2011); Kuno
et al. (2017); Martinez et al. (2016); Danshita et al. (2017); Zhang et al.
(2018); Davoudi et al. (2021, 2020); Monroe et al. (2021); González-Cuadra et
al. (2017); Nguyen et al. (2022); Aidelsburger et al. (2021); Schweizer et al.
(2019) has motivated road maps Bauer et al. (2022); Bañuls et al. (2020);
Kasper et al. (2020); Dalmonte and Montangero (2016); Meurice et al. (2022) to
implement sequences of models of increasing complexity and dimension using the
rapidly evolving NISQ technology Kogut (1979).

In this context, the Schwinger model has often been the first model to try
Hauke et al. (2013); Kühn et al. (2014); Martinez et al. (2016); Klco et al.
(2018); Thompson and Siopsis (2022); Nguyen et al. (2022); Shaw et al. (2020);
Kharzeev and Kikuchi (2020). The Gross Neveu (GN) model with $N$ species of
fermions in 1+1 dimensions is also a particularly important step in the study
of relativistic fermions. As for QCD, it is asymptotically free and capable of
dynamical mass generation and has a rich phase structure at finite temperature
and finite density Gross and Neveu (1974). An efficient initial state
preparation for this model has been developed by Moosavian and Jordan Hamed
Moosavian and Jordan (2018). Because of the limited entanglement entropy in
one spatial dimension, this model can also be handled efficiently with
classical computers using the density matrix renormalization group (DMRG) and
the time-evolving block decimation (TEBD) methods based on the matrix product
states (MPS) Moosavian et al. (2019); Roose et al. (2021, 2022) in order to
explore and validate quantum simulations.

In this article we show how to map the continuum Hamiltonian for the GN model
to a lattice qubit system and derive quantum circuits which can be used for
its time evolution and to find the ground state wavefunction for a range of
values of the mass and four fermion coupling. We then compare quantum
simulations on two platforms - the IBM-Q Guadalupe and Honeywell Quantinuum
machines - with the results of exact diagonalization and DMRG/TEBD
calculations. We demonstrate that the four steps of the calculations of real-
time scattering outlined by Jordan, Lee and Preskill (JLP) Jordan et al.
(2014, 2012), namely, 1) vacuum preparation, 2) excitation of single-particle
wavepackets, 3) unitary time evolution, and 4) measurements for the final
state, can all be achieved with the current NISQs technology for small
systems.

##  II From continuum Hamiltonian to quantum circuit

We start from the general form of the continuum Hamiltonian for one Dirac
fermion in one spatial dimension:

$H=\int
dx\;i{\psi}^{\dagger}\alpha\partial_{x}\psi+m{\psi}^{\dagger}\beta\psi.$ (1)

To discretize we place the theory on a lattice and replace the continuum
derivative by the symmetric difference operator
$\partial\to\Delta_{n^{\prime}\;n}=\frac{1}{2}\left(\delta_{n^{\prime}\;n+1}-\delta_{n^{\prime}\;n-1}\right)$.
In addition, we employ a staggered fermion construction in which the original
field $\psi$ is replaced by a new field $\lambda$ via the unitary
transformation $\psi(n)=\alpha^{n}\lambda(n)$ where $n$ labels the lattice
site. This yields the lattice Hamiltonian

$H=\sum_{n}\frac{i}{2}\,{\lambda}^{\dagger}(n)\left[\lambda(n+1)-\lambda(n-1)\right]+m{\lambda}^{\dagger}(n)\beta\lambda(n).$
(2)

The sum is over all lattice sites and depending on the boundary conditions
some kinetic terms at the lattice edges need to be omitted or modified. Notice
that in one dimension there is no staggered fermion phase. In the usual
Euclidean path integral the staggering transformation has the effect of
reducing the spinor structure of the fermion operator to the unit matrix and
in consequence all but one of the spinor components can be discarded. In a
Hamiltonian formulation one can only do this for the derivative but not the
mass term. Instead we allocate the 2 spinor components to even and odd lattice
sites corresponding to the choice $\alpha=\sigma^{x}$. Choosing
$\beta=\sigma^{z}$ then generates the staggered mass term
$m\left(-1\right)^{n}\lambda^{\dagger}(n)\lambda(n)$. If we denote $\chi_{\rm
even}=\lambda^{1}$ and $\chi_{\rm odd}=\lambda^{2}$ this can be trivially
rewritten:

$\displaystyle H$
$\displaystyle=\frac{i}{2}\sum_{n}\chi^{\dagger}(n)\left[\chi(n+1)-\chi(n-1)\right]$
$\displaystyle+m\sum_{n}\left(-1\right)^{n}\chi^{\dagger}(n)\chi(n).$ (3)

For $N$ flavors of Dirac fermion we can then add a four fermion term to
generate a Gross-Neveu model Gross and Neveu (1974) and rescaling of $m$ and
$G^{2}$ allows to omit the factor in front of the kinetic term:

$\displaystyle H^{(N)}$
$\displaystyle=i\sum_{n}\sum_{a=1}^{N}\chi^{a\dagger}(n)\left[\chi^{a}(n+1)-\chi^{a}(n-1)\right]$
$\displaystyle+m\left(-1\right)^{n}\chi^{a\dagger}(n)\chi^{a}(n)+G^{2}\left(\chi^{a\dagger}(n)\chi^{a}(n)\right)^{2}.$
(4)

The resultant Hamiltonian has a manifest $SU(N)$ symmetry. To simulate this
system on a quantum computer we first need to rewrite the theory in terms of
Pauli matrix or qubit operators. We use the Jordan-Wigner transformation
Jordan and Wigner (1928); Dargis and Maassarani (1998)

$\displaystyle\chi^{a}(n)$
$\displaystyle=\prod_{b<a}P^{\left(\sigma^{b}\right)}(L)\prod_{a}P^{\left(\sigma^{a}\right)}(n-1)\sigma_{+}^{a}(n)$
(5)

where

$P^{\left(\sigma^{a}\right)}(n)=\prod_{y=1}^{n}\sigma_{3}^{a}(y),$ (6)

and $\sigma_{\pm}=\frac{1}{2}\left(\sigma_{1}\pm i\sigma_{2}\right)$. It is
straightforward, to show that this representation respects the fundamental
anticommutator required for fermion operators

$[\chi^{a\dagger}(x),\chi^{b}(y)]_{+}=\delta_{xy}\delta^{ab}.$ (7)

We have used open boundary conditions in our work. In this representation the
free massive Hamiltonian becomes

$\displaystyle H^{(N)}_{0,m}$
$\displaystyle=\sum_{a=1}^{N}\bigg{[}i\sum_{n=1}^{L-1}\Big{(}\sigma^{a}_{+}(n)\sigma^{a}_{-}(n+1)-\sigma^{a}_{-}(n)\sigma^{a}_{+}(n+1)\Big{)}$
$\displaystyle+m\sum_{n=1}^{L}\left(-1\right)^{n}\left(\sigma^{a}_{-}(n)\sigma^{a}_{+}(n)\right)\bigg{]}+{\rm
h.c}$
$\displaystyle=\sum_{a=1}^{N}\Bigg{[}\sum_{n=1}^{L-1}\Big{(}-\sigma^{a}_{1}(n)\sigma^{a}_{2}(n+1)+\sigma^{a}_{2}(n)\sigma^{a}_{1}(n-1)\Big{)}$
$\displaystyle+m\sum_{n=1}^{L}\left(-1\right)^{n}\left(1-\sigma_{3}^{a}\right)\Bigg{]}$
(8)

while the four fermi term is

$H^{(N)}_{G}=\frac{1}{2}G^{2}\sum_{n=1}^{L}\,\,\sum_{a=1}^{N}\,\,\sum_{b,b>a}(I-\sigma^{a}_{3}(n))(I-\sigma^{b}_{3}(n)).$
(9)

Notice that one can think of flavor as another lattice dimension. That is one
can imagine that the problem maps to a ladder geometry where each leg of an
$N$ leg ladder corresponds to the spatial lattice while the vertical rungs
correspond to four fermion interactions between pairs of flavors.
Alternatively we can map all $N$ legs into a single one dimensional lattice
with the different flavors mapping into consecutive lattice sites in such a
way that the unit cell of the lattice is of length $N$ Roose et al. (2021).

We consider Trotter evolution of the Hamiltonian with zero staggered mass.
This reduces $H^{(N)}_{0,m}$ to $H^{(N)}_{0,0}$ and can be obtained from Eqn.
II omitting the last term. Thus the Hamiltonian for the $N$-flavor massless
staggered fermions can be written as

$H^{(N)}_{m=0}=H^{(N)}_{0,0}+H^{(N)}_{G}.$ (10)

To evolve the system in time requires exponentiation of the Hamiltonian. Since
it is composed of non-commuting pieces we have employed the first order
Suzuki-Trotter approximation for a small time step $\Delta t=t/n$ Trotter
(1959); Suzuki (1992, 1990, 1993)

$e^{-i(H^{(N)}_{0,0}+H^{(N)}_{G})t}\approx\left(e^{-i\,H^{(N)}_{m=0}t/n}\,\,e^{-i\,H^{(N)}_{G}\frac{t}{n}}\right)^{n}+\mathcal{O}(t\Delta
t),\\\ $ (11)

where the contribution of the kinetic term $H_{k}$ for a fixed flavor $a$ can
be decomposed into an elementary 2-qubit operation $Q_{1}=\exp{\left[i\Delta
t\,\sigma^{a}_{1}(n)\otimes\sigma^{a}_{2}(n+1)\right]}$ and
$Q_{2}=\exp{\left[-i\Delta
t\,\sigma^{a}_{2}(n)\otimes\sigma^{a}_{1}(n+1)\right]}$. The four fermion
interaction term $H^{(N)}_{G}$ couples two such flavors $a$ and $b$ at the
same physical site introducing additional qubit operators of the form
$Q_{3}=\exp{\left[i\frac{\Delta
t}{2}G^{2}\sigma_{3}^{a}(n)\otimes\sigma_{3}^{b}(n)\right]}$ and
$R_{z}=\exp{\left[-i\frac{\Delta t}{2}G^{2}\sigma_{3}^{a}(n)\right]}$. A
schematic diagram showing how these operations are combined to generate a
single time step is shown in Fig. 1. The individual gates needed to generate
$Q_{1}$, $Q_{2}$, and $Q_{3}$ appear in Figs. 15, 16, and 14 in appendix B,
which gives a detailed description of how these elementary circuit blocks
involving CNOT, Hadamard, and rotation gates implement the elementary qubit
operations needed for the Hamiltonian.

Figure 1: Schematic diagram of the quantum operations in the circuit form is
shown for a single step of Trotter evolution for two flavors. Here,
$\phi_{1}=2\Delta t$ and $\phi_{2}=G^{2}\Delta t$.

##  III Two flavor results

Interestingly the model at $m=0$ can be mapped into the Hubbard model Hubbard
(1963) at a particular value of the chemical potential Melzer (1995). The four
fermi interaction clearly corresponds to a Hubbard term after identifying
$n^{\uparrow}=\chi^{1\dagger}\chi^{1}$ and
$n^{\downarrow}=\chi^{2\dagger}\chi^{2}$. In addition, the kinetic operator
can be mapped to the usual Hubbard hopping term after performing an additional
unitary transformation $\chi^{a}(n)\to i^{n}\chi(n)$. In this case the
manifest $SU(2)$ symmetry of the two flavor theory is enhanced to $SO(4)$
which is most easily seen by decomposing each staggered field in terms of real
(or reduced staggered) fields via the mapping

$\displaystyle\chi^{1\dagger}$ $\displaystyle=\xi_{1}+i\xi_{2},$ (12)
$\displaystyle\chi^{1}$ $\displaystyle=\xi_{1}-i\xi_{2},$ (13)
$\displaystyle\chi^{2\dagger}$ $\displaystyle=\xi_{3}-i\xi_{4},$ (14)
$\displaystyle\chi^{2}$ $\displaystyle=\xi_{3}+i\xi_{4}.$ (15)

The Hamiltonian including the four fermi term can then be written

$H_{m=0}^{(2)}=\sum_{x}\xi^{i}(x)\xi^{i}(x+1)+\frac{G^{2}}{12}\epsilon_{ijkl}\xi^{i}(x)\xi^{j}(x)\xi^{k}(x)\xi^{l}(x).$
(16)

In this form it can be identified with recent path integral studies of reduced
staggered fermions capable of symmetric mass generation in (spacetime)
dimension $D\geq 2$ Ayyar and Chandrasekharan (2015); Catterall (2016); Ayyar
and Chandrasekharan (2017, 2016); Butt et al. (2018).

###  III.1 Time evolution

Figure 2: Comparison of 1st order Trotter-bound with the practical bound:
differences in the norm of the unitaries $||\Delta U||$ computed using matrix
exponentiations of our model.  Figure 3: Trotter evolution for the $N=2$
flavor model with $L=2$ lattice sites and $G^{2}=2.0,m=0.0$ and time step
$\delta t=0.6$ from initial state $|\psi\rangle=|0010\rangle$. Number of shots
used for Guadalupe and Quantinuum simulation is 4000 and 300 respectively.

We have simulated the model using a first order Trotter update on both the
IBM-Q Guadalupe quantum processing unit (QPU) and the Honeywell Quantinuum
platforms. Quantinuum provider gives access to two H1 generation of QPU: H1-1
and H1-2. Results of the Trotter evolution are shown in Fig. 3 for
$G^{2}=2.0$, $m=0.0$ and Trotter step $\delta t=0.6$ on a lattice with two
sites. Initial wavefunction considered can be written in the computational
basis $\ket{\psi}=\ket{0100}$. From the classical exact diagonalization
analysis, it is found out that to capture the characteristics of the time-
evolved wavefunction at $G^{2}=2.0$ we need to consider computing Trotter
evolution upto a time $t\sim 3.0$. Hence, we used a large Trotter step due to
practical limitations of computing Trotter evolution for large number of step
with NISQ-era machines. It has been demonstrated previously for the quantum
Ising model that the Trotter step can be taken 20 or 30 times larger compared
to the theoretical bounds of the first order Trotter step $\Delta t^{2}$
before showing large discretization error Gustafson et al. (2021a); Meurice
(2021). We performed identical analysis with our model and found the
conclusion to be true for our model too. We compute the norm of the following
operator

$\displaystyle\Delta U$ $\displaystyle=\mathrm{e}^{-iH^{(2)}_{m=0}\delta
t}-e^{-iH^{(2)}_{0,0}\delta t}e^{-iG^{2}H^{(2)}_{G}\delta t}.$ (17)

We use the following definition of the norm — it is the largest singular value
of the corresponding operator. Theoretical bound up-to the second order in
$\delta t$ of the norm of the above quantity is
$B=(G^{2}/2)||[H^{(2)}_{0,0},H^{(2)}_{G}]||(\delta t)^{2}$. However actual
bound $||\Delta U||$ is less strict in the region $0.55<t<0.95$, and we
numerically find that the actual-bound is consistent with linear approximation
in $\delta t$

$||\Delta U||\sim 1.49(1)\delta t-0.304(7).$ (18)

Figure 2 shows the comparison of the second order Trotter bound ($B$) from 1st
order Trotterization with the actual value of the difference in the norm
($||\Delta U||$). Choice of the Trotter step $\delta t=0.6$ is justfied where
$||\Delta U||\sim 0.6$. In Fig 2, the quantum simulations are compared with
exact diagonalization, first order Trotterization code, and the TEBD algorithm
written using ITensor library Fishman et al. (2020).

Before implementing the circuit on quantum hardware we also simulated the
circuit using the device noise model. For the IBMQ QPU, we used the aer-
simulator using a basic device noise model derived from the backend
properties. The noise model incorporates a simplified model for the gate error
probability of each basis gate on each qubit taking into account the
relaxation time and readout probability of each qubit. Figure 3 shows that for
a small number of qubits $Q=N\times L=4$, the error model predicts the results
from the QPU quite well out to four Trotter steps. We also performed analysis
with the noise model of the Quantinuum machine in the native simulator of
Quantinuum provider. The noise model simulator was seen to predict the Trotter
evolution from the Quantinuum machine extremely accurately upto five Trotter
step.

| IBMQ | Quantinuum
---|---|---
Trotter step, $n$ |  $d$ |  CX |  $d$ |  CX
1 |  24 |  4 |  10 |  4
2 |  47 |  18 |  22 |  12
3 |  70 |  32 |  34 |  20
4 |  93 |  46 |  46 |  28
5 |  116 |  60 |  58 |  36
Table 1: Circuit depth $d$ and the number of CNOT (CX) gates required for the
implementation of the Trotter evolution for different numbers of Trotter steps
$n$.

Table 1 shows the circuit depth of the implemented circuits 111 The original
circuit is transpiled before submitting the circuit to the QPU in order to
express the circuit in terms of the native gates and optimize the mapping of
qubits to the QPU. Depth noted here is the depth for the transpiled circuit..
We also note the number of CNOT gates for each case. Notice that twice as many
CNOT gates are required in the case of the IBM-Q relative to the Honeywell
machine. In addition the latter possesses all-to-all qubit connectivity which
eliminates the need for additional SWAP operations - which require three CNOT
gates. These facts account for the observed difference in the two platforms.
Our overall conclusion is that the layout of the physical qubits plays a very
important role in the efficiency with which quantum simulation can be
accomplished on NISQ era hardware.

Figure 4: Trotter evolution for $N=4$ flavor model with $L=2$ lattice sites
$G^{2}=-2.0,m=0.0$ and time step $\delta t=0.6$ from the initial state
$|\psi\rangle=|0010\rangle$. Number of shots used for Guadalupe and Quantinuum
simulation is 4000 and 300 respectively.  Figure 5: Quantum circuit used in
estimating the ground state wavefunction for 2 flavors

##  IV Four flavor results

In this brief section, we present the results of the Trotter evolution of the
$N=4$ flavor Gross Neveu model. Quantum simulation with QISKIT demonstrates
that the formulation of the $N$ flavor model is straightforward. Results
obtained from the quantum circuit formulation match with exact diagonalization
and TEBD calculations while simulated using the Aer simulator. However, it is
evident from Fig. 4 that the results obtained from the QPUs deviate from the
exact results. Quantinuum’s H1 QPU demonstrates superior behavior to the
IBMQ’s Guadalupe machine. The superiority of the H1 machine can mainly be
attributed to its all-to-all connectivity of physical qubits. Due to the
nature of the interaction terms, it is evident that more flavors translate to
the requirement of more SWAP gates. For both cases, Trotter evolution results
with QPUs deviate from the exact results more for the $N=4$ flavor case than
for the $N=2$ flavor model. For the Quantinuum provider, the deviation from
the exact result is $\sim 19\%$ for the two flavor, whereas $\sim 49\%$ for
the four flavor GN model. Whereas for the Guadalupe, the difference is $\sim
71\%$ and $\sim 98\%$ for the $N=2$ and $N=4$ flavor cases, respectively.
Furthermore, we see that the native simulator of the Quantinuum provider
predicts the result quite well. However, the Aer simulator with the device
noise model of the Guadalupe machine does not provide an accurate description
of the quantum processing unit of the Guadalupe QPU.

Figure 6: Projection of the ground state $\ket{\psi_{g}}$ on the computational
basis $\ket{n}$ derived from the VQE. VQE result is compared with the exact
diagonalization and results obtained from DMRG. Figure 7: Energy minimization
using VQE alogorithm with two different classical optimizers at $G^{2}=1.0$.

##  V Computation of the ground state

One of the fundamental goals of investigating lattice fermionic model is to
probe the phase structure of the model. The first step of doing that is to
prepare the ground state. In our work, we designed a quantum circuit that is
suitable for use with the Variational Quantum Eigensolver (VQE) algorithm
McClean et al. (2016) to determine the ground state wavefunction of the system
as a function of the four fermi coupling. The inputs to the algorithm are the
qubit Hamiltonian and a parametrized quantum circuit whose function is to
evaluate the expectation value of the Hamiltonian on a trial ground state
wavefunction. The algorithm uses a classical computer to minimize the energy
of the state with the quantum circuit being used to evaluate the expectation
value of the Hamiltonian in the trial wavefunction at each stage of the
iteration. The wavefunction ansatz for the ground state for two flavors is the
well-known Hardware Efficient Approximation (HEA) Kandala et al. (2017) shown
in Fig. 5, where a set of rotation angles $\theta_{a},a=0,\,1\ldots 15$ are
used as variational parameters. For a $Q$ qubit lattice model, expression for
the HEA wavefunction ansatz in terms of Rotation gates $R$ and 2-qubit
entangling operators $U^{mn}$ entangling mth qubit and nth qubit can be
written as

$\displaystyle\ket{\psi}$
$\displaystyle=\sum_{i=1}^{Q}\Bigg{[}(\prod_{a}{R^{i}_{a}})(\prod_{b}U^{i,i+1}_{b})(\prod_{c}{R^{i}_{c}})\Bigg{]}\ket{\psi_{0}}$
$\displaystyle\equiv M_{\alpha}(\\{\theta\\})\ket{\psi_{0}}.$ (19)

Here, the subscript in $R^{i}_{a}$ denotes different rotation operators along
different axis with $a,c=\\{x,y,z\\}$, and the subscript in $U_{b}$ denotes
different entangling operators
$U_{b}\in\\{CX,\,CY,\,CZ,\,CH,\,CRZ,\,CU,\cdots\\}$. The number of product
terms in each part can be varied and a suitable number of terms can be chosen
for approximating the ground state. In principle, the operator
$M_{\alpha}(\\{\theta\\})$ can be repeated as many times as needed with a new
set of parameters for each $M_{\alpha}$ block. Thus, in general, the structure
of the HEA ansatz can be written as

$\displaystyle\ket{\psi}=\prod_{\alpha(\\{\theta\\})}M_{\alpha}(\\{\theta\\})\ket{\psi_{0}}.$
(20)

Repeating the block ($M_{\alpha}$) $N$-times increases the number of
parameters $N$ times. If the number of layers in the rotation blocks
$\ell_{1}$ and $\ell_{3}$ in the first and the last stage respectively, and
$\ell_{2}$ layers entangling gates then the total circuit depth is then
$N\sum_{i}\ell_{i}$, number of parameters $p$ needed is then
$p<QN\sum_{i}\ell_{i}$. VQE uses Ritz’s variational principle to update the
parameters $\theta$ of the operator that approximates the ground state
MacDonald (1933). We used Constrained Optimization BY Linear Approximation
(COBYLA) optimizer Powell (1994, 1998, 2007) with the statevector simulator of
Qiskit to do the determine the change of the parameters at each stage of the
iteration. The COBYLA optimizer is based on the linear approximation of the
objective functions and the constraints. To verify whether the ‘true’ ground
state is reached, we compared the results obtained from the COBYLA optimizer
with the SLSQP optimizer Kraft (1988). SLSQP uses Sequential Least SQuares
Programming to minimize a function of several variables. Any combination of
bounds, equality and inequality constraints can be incorporated in the SLSQP
optimization routine. Figure 7 shows a typical relaxation of the energy to the
ground state at $G^{2}=1.0$ for both optimizers. We assumed the algorithm
reaches ground state when two successive iteration matches upto the fourth
order after the decimal point. For the same $G^{2}$, projection of the
obtained ground state $\ket{\psi_{g}}$ on computational basis $\\{\ket{n}\\}$
are shown and compared with results from the exact diagonalization and the
DMRG result in Fig. 6. Similar analysis were performed at different values of
$G^{2}$ and computed ground state energy from Variational Quantum Eigensolver
is compared with the exact diagonalized result in Fig. 8. For the computation
of the ground state, we choose an arbitrary wavefunction by choosing the
parameters $\theta_{i}$ of the ‘ansatz wavefunction’ from random distribution
of floating point numbers $-2\pi\leq\theta_{i}<2\pi$. Different set of
parameters are randomly chosen for wavefunction ansatz and error-bars are
computed from the standard deviation of the different results obtained.

Figure 8: Ground state energy computed from the VQE compared with the results
of the exact diagonalization. DMRG results are not shown here as they match
exactly with the exact diagonalization result.

##  VI Wave Packet preparation and measurement

In the previous three sections, we showed results of the Trotter evolution and
the ground state preparation procedure for the Gross-Neveu (GN) model. In this
section, using the DMRG algorithm we demonstrate that Jordan, Lee, Presskill
(JLP) prescription for real-time scattering can be implemented in the context
of the two flavor GN model. For the DMRG computation, we used a different
labeling of the qubits. We used the concept of unit cell, each unit cell
contains $N$ virtual sites corresponding to the $N$ different flavors of the
model. Unit cells are considered as the physical sites. Thus for a GN model
with $L$ sites, there would $L$ unit cells each with $N$ virtual sites
totalling to a $Q=N\times L$ effective sites in this language. Reformulated
Hamiltonian is written in Appendix A, see Eqn. A.

Preparing ground state is an intermediate step for the construction of the
scattering states. We obtain the ground state of the system by running the
DMRG algorithm at $G^{2}=-3.0$, $m=0.5$ and the ground state is expressed an
MPS. The prepared ground state is compared with the results obtained using
variational quantum eigensolver method described in the previous section. The
ground state for any large negative $G^{2}$ value consists of all down states
which makes it very suitable to be a starting point for wave packet creation.
After we get the ground state in terms of a MPS we can feed it into the TEBD
algorithm as an initial state to start our time evolution for the wave
packets.

Since the ground state is all down states we need to add a new term in our
Hamiltonian to excite particles in precise locations on the chain which is the
second step in JLP prescription,

$H_{e}=\sigma^{+}(x).$ (21)

Note that there is no implicit sum over the site index $x$. Applying this term
will change the $\ket{0}$ to a $\ket{1}$ at that location. Then we can give it
some finite momentum so that this excitation can move around the spin chain:

$H_{e}(t)=e^{ik\sigma^{+}(x)}.$ (22)

To mimic single particle physics we choose to excite only one site within the
unit cell this results in an excitation that moves within the lattice on only
even or odd sites depending on the original excited site. Since we do not want
to generate particles indefinitely we will only include this term in the time
evolution for one Trotter step and the remaining time evolution then will be
carried out by the original Hamiltonian.

Now we give simple examples for this procedure. In Fig. 10 we have excited the
first site with $k=0.5$ by including $H_{e}$ in the time evolution for $1<t<2$
with a Trotter step size $\delta t=0.5$ and then let the system evolve with
the original Hamiltonian up to $t=20$ as can be seen from the plot this
resulted in a right-moving wave packet. Notice that we have only shown the odd
sites in Figs. 10 and 10 this is due to the fact that we have excited the
$1st$ site which resulted in an excitation that moves only on odds sites while
even sites remain dark. By omitting the even sites from the plots the
propagation becomes more visible. Also this effectively reduces our model to
be a single flavor $L=8$ model which might be accessible to NISQ era machines.

Figure 9: Time evolution for a Right Moving Wave Packet.

Figure 10: Time evolution for $\ket{100\dots 0}$.

One can also excite both ends of the spin chain to create scattering states.
As can be seen from Fig. LABEL:scattered. Due to the way we have constructed
our spin chain the ends of the spin chain belongs to different flavors of
fermions hence we see excitations all over the lattice.

Alternatively we can just pick our initial state by hand to simulate the wave
propagation for example starting with a $\ket{100\dots 0}$ and then time
evolving this state under the original Hamiltonian with $G^{2}=0,m=0$ would
result in a right moving wave packet which can be seen from Fig. 10.
Furthermore one can start with an initial state of form $\ket{100\dots 1}$ to
obtain a scattering like state in Fig. LABEL:101. The advantage of choosing
the initial state by hand and using only the kinetic term for the time
evolution is that it will allow us to easily test our results with quantum
hardware in the future.

Concerning the measurements of the final state, general methods to map the
position space basis into a momentum space space basis for fermions have been
developed in Refs. Ferris (2014); Kivlichan et al. (2020). Practical
implementations with four qubits have been used to measure phase shifts using
IBMQ and trapped ions Gustafson et al. (2021b). Implementations with eight
qubits are under active investigation.

##  VII Conclusions

In this paper we have described a mapping of the N flavor Gross Neveu model
into a suitable qubit Hamiltonian and have benchmarked quantum simulations of
the latter by comparing time evolution and determinations of the ground state
wavefunction of the model with results from classical simulations, exact
diagonalization and DMRG/TEBD calculations. We have concentrated on the case
of two flavors (which can be mapped to the spinful 1+1 Hubbard model) and four
flavors on small lattices for a range of four fermion couplings.

## Acknowledgements

We acknowledge useful discussions with the members of QuLat collaboration,
Erik Gustafson and Bharath Sambasivam. We acknowledge support from Microsoft’s
Azure Quantum team for providing credits and access to the Quantinuum
hardwares and Quantinuum simulators. We thank the IBM-Q hub at Brookhaven
National Laboratory for providing access to the IBMQ quantum computers. S.C
has been supported under U.S. Department of Energy grants DE-SC0009998 and DE-
SC0019139.

##  Appendix A Details of the DMRG Calculation

Figure 13: Comparing the ground state energy obtained via DMRG with exact
diagonalization for different numbers of lattice sites $L$

To get a version of Eqn. II suitable for DMRG simulations White (1992, 1993);
Schollwöck (2005); Fishman et al. (2020) for any number of flavors $N$ we need
to make a few adjustments. Firstly we need to map all degrees of freedom to
individual lattice sites on the one dimensional lattice. This means the unit
cell of the lattice is now of length $N$. After this transformation the
lattice action can be written as follows,

$\displaystyle
H^{(N)}=i\sum_{x=1}^{L-N}-\sigma^{+}(x)\sigma^{-}(x+N)+\sigma^{-}(x)\sigma^{+}(x+N)$
$\displaystyle+\frac{1}{2}G^{2}\sum_{x=1,1+N,..}^{L-N}\sum_{a=0}^{N-1}\sum_{\begin{subarray}{c}b=0\\\
b>a\end{subarray}}^{N-1}\sigma^{+}(x+a)\sigma^{-}(x+a)$
$\displaystyle\times\sigma^{+}(x+b)\sigma^{-}(x+b).$ (23)

A mass term can be introduced using a more general staggered phase $\eta(x)$
that changes it’s sign between the unit cells rather than between each site:

$M_{s}=m\sum_{x=1}^{L}\eta(x)\sigma^{+}(x)\sigma^{-}(x).$ (24)

To verify our DMRG code is correct we have calculated the ground state energy
using exact diagonalization of the original Hamiltonian and compared it with
the energies obtained from DMRG. As can be seen in Fig. 13 there is very good
agreement between the two calculations.

##  Appendix B Circuit Blocks

ABCD Figure 14: Circuit block for the implementation of
$Q_{3}(\phi_{2})=\exp(-i\,(\phi_{2}/2)\sigma_{3}\otimes\sigma_{3})$.

For the computation of the Trotter evolution, each term in the Hamiltonian is
exponentiated. This amounts to creating circuit blocks of the exponential of
the tensor product of the $\sigma$ operators
$\exp{(\alpha\prod_{n}^{\otimes}\sigma_{n})}$. As an example we describe first
the creation of the operator
$\exp(-i\,(\phi_{2}/2)\sigma_{3}\otimes\sigma_{3})$ in terms of elementary
unitary gates 222This manuscript follows the physics textbook convention,
where the qubits are ordered from left to right. Thus a 2 qubit state is
represented as $\ket{q_{0}q_{1}}$, where the 0th qubit is represented as the
most significant bit in the bit string.. To understand the construction let us
first explain the construction of the CNOT gate

$\mathrm{CNOT}_{10}=\ket{1}\bra{1}\otimes\sigma_{1}+\ket{0}\bra{0}\otimes\mathbf{I}_{2}.$
(25)

Here the subscript of the $\mathrm{CNOT}$ gate, denotes that it is applied on
qubit 1 and qubit 0 with qubit 0 as a control bit. We identify the basis of
the one qubit states as column vectors

$\ket{0}=\begin{bmatrix}1\\\ 0\end{bmatrix},\qquad\ket{1}=\begin{bmatrix}0\\\
1\end{bmatrix}.$ (26)

and $\sigma_{i}$, $i=1,2,3$, denotes the usual Pauli matrices. Thus, if the
initial state at position A is $\ket{\psi}$, the state at position B is
$\mathrm{CNOT}_{10}\ket{\psi}$. With these definitions, it is easy to verify
that the CNOT gate applied on 2 qubits where 0th qubit works as a control bit
satisfy these identities

$\displaystyle\mathrm{CNOT}_{10}\ket{00}$ $\displaystyle=\ket{00},$
$\displaystyle\mathrm{CNOT}_{10}\ket{01}$ $\displaystyle=\ket{01},$
$\displaystyle\mathrm{CNOT}_{10}\ket{10}$ $\displaystyle=\ket{11},$
$\displaystyle\mathrm{CNOT}_{10}\ket{11}$ $\displaystyle=\ket{10}.$ (27)

Rotation by an angle $\phi_{2}$ around z-axis can be represented by the
rotation operator $\mathrm{R}_{z}$, described by

$\mathrm{R}_{z}(\phi_{2})=\exp\left(-i\frac{\phi_{2}}{2}\sigma_{3}\right)=\cos\frac{\phi_{2}}{2}\,\mathrm{I}_{2}-i\,\sin\frac{\phi_{2}}{2}\,\sigma_{3}.$
(28)

This implies that up to the point ‘C’ in Fig. 14 the operator that is applied
on a 2 qubit initial state $\ket{\psi}$ is

$\displaystyle(\mathrm{I}_{2}\otimes\mathrm{R}_{z})\mathrm{CNOT}_{10}$
$\displaystyle=\cos\frac{\phi_{2}}{2}\,\mathrm{CNOT}_{10}$
$\displaystyle+\,\sin\frac{\phi_{2}}{2}(\ket{1}\bra{1}\otimes\sigma_{2}+\ket{0}\bra{0}\otimes\sigma_{3}).$
(29)

At the final stage at position ‘D’, the operator takes the form

$\displaystyle\mathrm{CNOT}_{10}(\mathrm{I}_{2}\otimes\mathrm{R}_{z})\mathrm{CNOT}_{10}$
$\displaystyle=\cos\frac{\phi_{2}}{2}\,\mathrm{I}_{4}-i\,\sin\frac{\phi_{2}}{2}\big{(}\ket{0}\bra{0}-\ket{1}\bra{1}\big{)}\otimes\sigma_{3}$
$\displaystyle=\exp(-i\,\frac{\phi_{2}}{2}\,\sigma_{3}\otimes\sigma_{3}).$
(30)

Thus, after the application of the second CNOT gate, the state obtained is
$\exp(-i\,(\phi_{2}/2)\,\sigma_{3}\otimes\sigma_{3})\ket{\psi}$.

The circuit needed for $Q_{1}(\phi_{1})$ is similar but requires Hadamard
gates to rotate the $\sigma_{z}$’s to $\sigma_{x}$ and rotation gates about
the x-axis $R_{x}(\pi/2)$ to rotate $\sigma_{z}$ to $\sigma_{y}$, see Fig. 15.
Likewise, $Q_{2}(-\phi_{1})$ can be described by the circuit in Fig. 16.

Figure 15: Circuit block for the implementation of
$Q_{1}(-\phi_{1})=\exp(i\,(\phi_{1}/2)\sigma_{1}\otimes\sigma_{2})$.  Figure
16: Circuit block for the implementation of
$Q_{2}(\phi_{1})=\exp(-i\,(\phi_{1}/2)\sigma_{2}\otimes\sigma_{1})$.

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# Bogomolov multiplier of Lie Algebras

Pradeep K. Rai  Mahindra University, Hyderabad, Telangana,
India<EMAIL_ADDRESS>
###### Abstract.

In the work of Rostami et al., the Bogomolov multiplier of a Lie algebra $L$
over a field $\Omega$ is defined as a particular factor of a subalgebra of the
exterior product $L\wedge L$. If $L$ is finite dimensional, we identify this
object as a certain subgroup of the second cohomology group $H^{2}(L,\Omega)$
by deriving a Hopf-Type formula. As an application, we affirmatively answer
two questions posed by Kunyavskiĭ regarding the invariance of the Bogomolov
multiplier under isoclinism of Lie algebras and the existence of a family of
Lie algebras with Bogomolov multipliers of unbounded dimension.

###### Key words and phrases:

Bogomolov multiplier, Lie algebras, Second cohomology group

###### 2010 Mathematics Subject Classification:

17B56, 14E08

##  1\. Introduction

The Bogomolov multiplier of a finite group $G$ is a cohomological invariant of
$G$ that has been studied in connection with Noether’s problem, which asks
whether the fixed subfield $k(G)$ of the function field $k(x_{g}:g\in G)$ is
purely transcendental over an algebraically closed field $k$ of characteristic
zero. The Bogomolov multiplier has played a key role in the discovery of
counter examples to Noether’s problem over the complex numbers $\mathbb{C}$.
Saltman [7] was the first to provide such counter examples, showing that if
the unramified cohomology group $H_{nr}^{2}(k(G),\mathbb{C})$ is nonzero, then
$G$ has a negative solution to Noether’s problem over $\mathbb{C}$. Bogomolov
later showed that $H_{nr}^{2}(k(G),\mathbb{C})$ is naturally isomorphic to the
subgroup $B_{0}(G)$ of the second cohomology group $H^{2}(G,\mathbb{C})$
consisting of classes that vanish when restricted to the abelian subgroups of
$G$ [2]. This has led to the discovery of numerous other counter examples to
Noether’s problem. Kunyavskiĭ later gave the name “Bogomolov multiplier" to
$B_{0}(G)$ [3], and Moravec provided a homological description of $B_{0}(G)$
as a quotient of $H^{2}(G,\mathbb{C})$ [5]. Following Moravec’s construction,
Rostami et. al [6] extended the notion of Bogomolov multiplier to Lie
algebras. For the convenience of the reader we define it here.

Let $L$ be a Lie algebra over $\Omega$. The exterior square of $L$ is defined
to be the Lie algebra $L\mbox{\Large$\wedge$}L$ generated by the symbols
$m\wedge n$, where $m,n\in L$, subject to the following relations:

  * (i)

$\alpha(m\wedge n)=\alpha m\wedge n=m\wedge\alpha n$,

  * (ii)

$(m+m^{\prime})\wedge n=m\wedge n+m^{\prime}\wedge n$,

  * (iii)

$m\wedge(n+n^{\prime})=m\wedge n+m\wedge n^{\prime}$,

  * (iv)

$[m,m^{\prime}]\wedge n=m\wedge[m^{\prime},n]-m^{\prime}\wedge[m,n]$,

  * (v)

$m\wedge[n,n^{\prime}]=[n^{\prime},m]\wedge n-[n,m]\wedge n^{\prime}$,

  * (vi)

$[(m\wedge n),(m^{\prime}\wedge
n^{\prime})]=-[n,m]\wedge[m^{\prime},n^{\prime}]$,

  * (vii)

$m\wedge n=0$ whenever $m=n$,

for all $\alpha\in\Omega,m,m^{\prime},n,n^{\prime}\in L$.

It is easy to see that $\mathcal{K}:L\times L\mapsto[L,L]$ given by
$(m,n)\mapsto[m,n]$ induces a homomorphism
$\bar{\mathcal{K}}:L\mbox{\Large$\wedge$}L\mapsto[L,L]$, such that
$\bar{\mathcal{K}}(m\wedge n)=[m,n]$, for all $m,n\in L$. It is known that the
kernel of $\bar{\mathcal{K}}$ is isomorphic to the Schur Multiplier
$H^{2}(L,\Omega)$ (defined below). We denote it by $\mathcal{M}(L)$. Define
$\mathcal{M}_{0}(L)$ to be the group $\big{\langle}m\wedge n\ \mid\ m,n\in
L,[m,n]=0\big{\rangle}$. The factor group
$\frac{\mathcal{M}(L)}{\mathcal{M}_{0}(L)}$ is defined to be the Bogomolov
multiplier $B(L)$ of the Lie algebra $L$.

In this article, we define a cohomological object $B_{0}(L)$ for a finite
dimensional Lie algebra $L$ over a field $\Omega$, and show that it is
isomorphic to the Bogomolov multiplier $B(L)$. Before that, we recall the
definition of the Schur multiplier of a finite dimensional Lie algebra. Let
$L$ be a finite dimensional Lie algebra and $A$ be a trivial $L$-module. A map
$f:L\times L\mapsto A$ said to be a 2-cocycle if it is bilinear, alternating
and satisfies

$f([x_{1},x_{2}],x_{3})+f([x_{2},x_{3}],x_{1})+f([x_{3},x_{1}],x_{2})=0.$

And $f$ is said to be a 2-coboundry if there exists a linear $\sigma:L\mapsto
A$ such that

$f(x_{1},x_{2})=-\sigma([x_{1},x_{2}]).$

The sets of 2-cocycles and 2-coboundries are denoted by $Z^{2}(L,A)$ and
$B^{2}(L,A)$, respectively and form abelian groups with respect to usual
addition. The group $Z^{2}(L,A)/B^{2}(L,A)$ is said to be the second
cohomology group with coefficients in $A$, and is denoted by $H^{2}(L,A)$.

Schur multiplier of the Lie algebra $L$ is defined as the abelian Lie algebra
$H^{2}(L,\Omega)$, considering $\Omega$ as a central $L$-module. We are now
ready to define $B_{0}(L).$

######  Definition.

For a finite dimensional Lie algebra $L$ over $\Omega$, we define $B_{0}(L)$
as follows:

$B_{0}(L)=\\{\overline{f}\in H^{2}(L,\Omega)\ \ |\ \ f(x_{1},x_{2})=0\ \
\text{whenever}\ \ [x_{1},x_{2}]=0\\}.$

Batten [1, Theorem 3.6] established the following Hopf Formula for the Schur
multiplier of the Lie algebra $L$:

$H^{2}(L,\Omega)\cong\frac{F^{\prime}\cap R}{[F,R]},$

where $1\mapsto R\mapsto F\mapsto L\mapsto 1$ is a free a presentation of $L$
and $F^{\prime}$ is the derived subalgebra of $F$.

Let $K(L)$ denote the set $\\{[x,y]\ \ |\ \ x,y\in L\\}$. In the following
theorem we derive a Hopf-type formula for $B_{0}(L)$.

######  Theorem 1.1.

Let $L$ be a finite dimensional Lie algebra with a free presentation $L\cong
F/R$. Then $B_{0}(L)\cong\frac{F^{\prime}\cap R}{\big{\langle}K(F)\cap
R\big{\rangle}}$.

The following corollary follows from [6, Proposition 4.1] and the Theorem 1.1.

######  Corollary 1.2.

Let $L$ be a finite dimensional Lie algebra. Then $B(L)\cong B_{0}(L).$

As an application we answer a couple of questions of Kunyavskiĭ [4]. He asked
the following questions for finite dimensional Lie algebras $L$:

######  Question 1.3.

[4, Question 7.1] Can the dimension of $B(L)$ be as large as possible?

######  Question 1.4.

[4, Question 7.2] Is $B(L)$ invariant under isoclinism of Lie algebras?

Two Lie algebras $L$ and $K$ are said to be isoclinic if there exist
isomorphisms $\alpha:L/Z(L)\mapsto K/Z(K)$ and $\beta:L^{\prime}\mapsto
K^{\prime}$ such that the following diagram commutes.

$\begin{CD}\frac{L}{Z(L)}\times\frac{L}{Z(L)}@>{\phi}>{}>L^{\prime}\\\
@V{\alpha\times\alpha}V{}V@V{\beta}V{}V\\\
\frac{K}{Z(K)}\times\frac{K}{Z(K)}@>{\theta}>{}>K^{\prime}\\\ \end{CD}$

where

$\phi\Big{(}l_{1}+Z(L),l_{2}+Z(L)\Big{)}=[l_{1},l_{2}]\ \ \text{for}\ \
l_{1},l_{2}\in L$

and

$\theta\Big{(}k_{1}+Z(K),k_{2}+Z(K)\Big{)}=[k_{1},k_{2}]\ \ \text{for}\ \
k_{1},k_{2}\in K.$

The pair $(\alpha,\beta)$ is called an isoclinism between $L$ and $K$.

In the following theorems we give an affirmative answer to Questions 1.3 and
1.4.

######  Theorem 1.5.

Let $n\geq 1$ be a natural number. There exists a finite dimensional nilpotent
Lie algebra $L$ of nilpotency class 2 such that dimension of $B(L)$ is greater
than or equal to $n$.

######  Theorem 1.6.

Let $L$ and $M$ be two isoclinic finite dimensional Lie Algebras over the
field $\Omega$. Then $B(L)\cong B(M)$.

##  2\. Hopf-Type Formula

Consider a finite dimensional Lie algebra $L$ over a field $\Omega$, a central
ideal $H$ of $L$, and a trivial $L$-module $A$. The restriction map
$\operatorname{Res}:\operatorname{Hom}(L,A)\rightarrow\operatorname{Hom}(H,A)$
is defined as follows: for a homomorphism $f:L\rightarrow A$,
$\operatorname{Res}(f)$ is the restriction of $f$ to $H$. There is also an
inflation map
$\operatorname{Inf}:\operatorname{Hom}(L/H,A)\rightarrow\operatorname{Hom}(L,A)$
defined by sending a homomorphism $\alpha\in\operatorname{Hom}(L/H,A)$ to the
homomorphism $\alpha^{\prime}\in\operatorname{Hom}(L,A)$ defined by
$\alpha^{\prime}(x,y)=\alpha(\beta(x),\beta(y))$ for all $x,y\in L$, where
$\beta:L\rightarrow L/H$ is the natural group homomorphism. Another inflation
map $\operatorname{Inf}:H^{2}(L/H,A)\rightarrow H^{2}(L,A)$ is defined
similarly by sending $[\alpha]\in H^{2}(L/H,A)$ to $[\alpha^{\prime}]\in
H^{2}(L,A)$ where $\alpha^{\prime}(x,y)=\alpha(\beta(x),\beta(y))$ for all
$x,y\in L$. Next, we define a transgression map
$\operatorname{Tra}:\operatorname{Hom}(H,A)\rightarrow H^{2}(L/H,A)$ as
follows: for a fixed section $\mu$ of $\beta$, define a map $f:L/H\times
L/H\rightarrow L$ by $f(x,y)=[\mu(x),\mu(y)]-\mu([x,y])$ for all $x,y\in L/H$.
Given a homomorphism $\chi\in\operatorname{Hom}(H,A)$, we can verify that
$\chi f\in Z^{2}(L/H,A)$ and that the cohomology class of $\chi f$ does not
depend on the choice of $\mu$. The transgression map $\operatorname{Tra}$ is
then defined as the map that sends a homomorphism $\chi$ to the cohomology
class of $\chi f$

We are now ready to quote some results required for our subsequent
investigations. The following 5-term exact sequence was established by P.
Batten in her Ph.D. Thesis [1, Theorem 3.1]

######  Theorem 2.1.

Let $L$ be a Lie algebra, $H$ be central ideal of $L$, $1\mapsto H\mapsto
L\mapsto L/H\mapsto 1$ be the natural exact sequence and $A$ be a trivial
$L$-module. Then the induced sequence

(2.1)
$1\xrightarrow{}\operatorname{Hom}(L/H,A)\xrightarrow{\operatorname{Inf}}\operatorname{Hom}(L,A)\xrightarrow{\operatorname{Res}}\operatorname{Hom}(H,A)\xrightarrow{\operatorname{Tra}}H^{2}(L/H,A)\\\
\xrightarrow{\operatorname{Inf}}H^{2}(L,A)$

is exact.

Let $L$ be a lie algebra and $T$ be a subset of $L$. By
$\big{\langle}T\big{\rangle}$ we denote the subspace of $L$ generated by $T$
and by $\operatorname{Hom}_{T}(L,A)$ we denote the set of those homomorphisms
which maps $T$ to 0. The following lemma is instrumental in our subsequent
investigations.

######  Lemma 2.2.

Let $L$ be a finite dimensional Lie algebra over the field $\Omega$ and $H$ be
a central ideal of $L$. Then $\operatorname{Tra}(\lambda)\in B_{0}(L/H)$ if,
and only if $\lambda\in\operatorname{Hom}_{T}(H,\Omega)$ where
$T=\big{\langle}K(L)\cap H\big{\rangle}$.

###### Proof.

Let $\mu:L/H\rightarrow L$ be a section such that $\mu(0)=0$ and let
$\operatorname{Tra}$ be defined using $\mu$ as
$\operatorname{Tra}(\lambda)=[\lambda f]$, where

$f(x,y)=[\mu(x),\mu(y)]-\mu([x,y])\ \ \forall\ x,y\in L/H.$

Let $\lambda\in\operatorname{Hom}_{T}(H,\Omega)$ and $x,y\in L/H$ be such that
$[x,y]=0.$ Then $\lambda f(x,y)=\lambda([\mu(x),\mu(y)]).$ But
$[\mu(x),\mu(y)]\in T.$ Therefore $\lambda f(x,y)=0$. This proves that
$[\lambda f]\in B_{0}(L/H).$ Thus $\operatorname{Tra}(\lambda)\in B_{0}(L/H)$.
Conversly, suppose that $\operatorname{Tra}(\lambda)\in B_{0}(L/H)$. Let
$l_{1},l_{2}\in L$ such that $[l_{1},l_{2}]\in H.$ Notice that
$[l_{1},l_{2}]=[\mu(l_{1}+H),\mu(l_{2}+H)]$ because $H$ is central. Also
$\mu([l_{1}+H,l_{2}+H])=\mu([l_{1},l_{2}]+H)=0$ because $[l_{1},l_{2}]\in H$
and $\mu(0)=0.$ Therefore,

$\lambda([l_{1},l_{2}]=\lambda([\mu(l_{1}+H),\mu(l_{2}+H)])-\lambda\mu([l_{1}+H,l_{2}+H])=\lambda
f(l_{1}+H,l_{2}+H).$

Since $\operatorname{Tra}(\lambda)=[\lambda f]\in B_{0}(L/H)$ and
$[l_{1}+H,l_{2}+H]=0$ in $L/H$ we have that $\lambda f(l_{1}+H,l_{2}+H)=0$.
Thus $\lambda([l_{1},l_{2}])=0$ whenever $[l_{1},l_{2}]\in H.$ This proves
that $\lambda\in\operatorname{Hom}_{T}(H,\Omega).$ ∎

######  Theorem 2.3.

Let $L$ be a finite dimensional Lie algebra over the field $\Omega$, $H$ be
central ideal of $L$, and $T=\big{\langle}K(L)\cap H\big{\rangle}$. Then the
induced sequence

(2.2)
$1\xrightarrow{}\operatorname{Hom}(L/H,\Omega)\xrightarrow{\operatorname{Inf}}\operatorname{Hom}(L,\Omega)\xrightarrow{\operatorname{Res}}\operatorname{Hom}_{T}(H,\Omega)\xrightarrow{\operatorname{tra}}B_{0}(L/H)\\\
\xrightarrow{\operatorname{inff}}B_{0}(L)$

is exact, where $\operatorname{tra}$ and $\operatorname{inff}$ are the
restrictions of $\operatorname{Tra}$ and $\operatorname{Inf}$.

###### Proof.

The theorem follows from the exactness of the Sequence 2.1, Lemma 2.2 and the
following straight forward observations:

  * (i)

$\operatorname{Res}(\operatorname{Hom}(L,\Omega)\leq\operatorname{Hom}_{T}(H,\Omega)$.

  * (ii)

$\operatorname{Inf}(B_{0}(L/H)\leq B_{0}(L).$

∎

The next theorem follows from Lemma 2.2 and [1, Corollary 3.7].

######  Theorem 2.4.

Let $L$ be a finite dimensional Lie algebra, $L^{*}$ be its cover with $A\leq
L^{*}$ satisfying the following three conditions

  1. (1)

$A\leq Z(L^{*})\cap[L^{*},L^{*}]$;

  2. (2)

$A\cong M(L)$;

  3. (3)

$L\cong L^{*}/A,$

and $T=\big{\langle}K(L^{*})\cap A\big{\rangle}$. Then the map
$\operatorname{tra}:\operatorname{Hom}_{T}(A,\Omega)\mapsto B_{0}(L)$ is
bijective.

######  Corollary 2.5.

Let $L$ be a finite dimensional Lie algebra and $L^{*}$ be its cover with
$A\leq L^{*}$ satisfying the following three conditions

  1. (1)

$A\leq Z(L^{*})\cap[L^{*},L^{*}]$;

  2. (2)

$A\cong M(L)$;

  3. (3)

$L\cong L^{*}/A$.

Then $B_{0}(L)\cong\frac{A}{\big{\langle}K(L^{*})\cap A\big{\rangle}}$.

###### Proof.

Let $T=\big{\langle}K(L^{*})\cap A\big{\rangle}$. Since $A$ is an abelian lie
algebra, homomorphisms are nothing but linear transformations. It follows that
$\operatorname{Hom}_{T}(A,\Omega)\cong\operatorname{Hom}\Big{(}\frac{A}{T},\Omega\Big{)}\cong\frac{A}{T}$.
The result follows from Theorem 2.4. ∎

######  Theorem 2.6.

Let $H$ be a central ideal in $L$ and $T=\big{\langle}K(L)\cap
H\big{\rangle}$. Then $\frac{L^{\prime}\cap H}{\big{\langle}K(L)\cap
H\big{\rangle}}$ is isomorphic to the image of the map
$\operatorname{tra}:\operatorname{Hom}_{T}(H,\Omega)\mapsto B_{0}(L/H)$. In
particular, $\frac{L^{\prime}\cap H}{\big{\langle}K(L)\cap
H\big{\rangle}}\cong B_{0}(L/H)$ if the map $\operatorname{tra}$ is
surjective.

###### Proof.

By Theorem 2.3, we have the following exact sequence

$\operatorname{Hom}(L,\Omega)\xrightarrow{\operatorname{Res}}\operatorname{Hom}_{T}(H,\Omega)\xrightarrow{\operatorname{tra}}B_{0}(G/H).$

It follows that
$\frac{\operatorname{Hom}_{T}(H,\Omega)}{\operatorname{Res}(\operatorname{Hom}(L,\Omega))}$
is isomorphic to the image of $\operatorname{tra}$. Thus, to prove the
theorem, we only need to show that

$\frac{L^{\prime}\cap H}{\big{\langle}K(L)\cap
H\big{\rangle}}\cong\frac{\operatorname{Hom}_{T}(H,\Omega)}{\operatorname{Res}(\operatorname{Hom}(L,\Omega))}.$

Since $H$ is abelian it follows that the natural restriction map
$\operatorname{Res}_{1}:\operatorname{Hom}_{T}(H,\Omega)\rightarrow\operatorname{Hom}_{T}(L^{\prime}\cap
Z,\Omega)$ is surjetive. Therefore

$\frac{\operatorname{Hom}_{T}(H,\Omega)}{\ker\operatorname{Res}_{1}}\cong\operatorname{Hom}_{T}(L^{\prime}\cap
H,\Omega).$

If we consider the natural restriction map
$\operatorname{Res}_{2}:\operatorname{Hom}(H,\Omega)\mapsto\operatorname{Hom}(L^{\prime}\cap
H,\Omega)$, it is straight forward to note that
$\ker\operatorname{Res}_{1}=\ker\operatorname{Res}_{2}.$ Let $J$ be the subset
of $\operatorname{Hom}(H,\Omega)$ consisting of all $\chi$ which can be
extended to a homomorphism $L\rightarrow\Omega$. Invoking the proof of [1,
Theorem 3.2] we have $J=\ker\operatorname{Res}_{2}.$ But it is obvious that
$J=\operatorname{Res}(\operatorname{Hom}(L,\Omega))$. Hence, it follows that

$\frac{\operatorname{Hom}_{T}(H,\Omega)}{\operatorname{Res}(\operatorname{Hom}(L,\Omega))}\cong\operatorname{Hom}_{T}(L^{\prime}\cap
H,\Omega).$

But

$\operatorname{Hom}_{T}(L^{\prime}\cap
H,\Omega)\cong\operatorname{Hom}\bigg{(}\frac{L^{\prime}\cap
H}{T},\Omega\bigg{)}\cong\frac{L^{\prime}\cap H}{T}$

because $L^{\prime}\cap H$ is abelian. This completes the proof. ∎

Proof of Theorem 1.1: Let $\bar{R}=\frac{R}{[F,R]}$ and
$\bar{F}=\frac{F}{[F,R]}$. Then $\bar{R}$ is a central ideal of $\bar{F}$ and
$L\cong\frac{\bar{F}}{\bar{R}}$. By [1, Lemma 3.4] the transgression map
$\operatorname{Tra}:Hom(\bar{R},\Omega)\mapsto H^{2}(L,\Omega)$ is surjective.
Therefore by Lemma 2.2 the map
$\operatorname{tra}:\operatorname{Hom}_{\big{\langle}K(\bar{F})\cap\bar{R}\big{\rangle}}(\bar{R},\Omega)\mapsto
B_{0}(L)$ is also surjective. It therefore follows, from Theorem 2.6, that
$B_{0}(L)\cong\frac{\bar{R}\cap[\bar{F},\bar{F}]}{\big{\langle}K(\bar{F})\cap\bar{R}\big{\rangle}}$.
But $\bar{R}\cap[\bar{F},\bar{F}]=\frac{F^{\prime}\cap R}{[F,R]}$ and
$\big{\langle}K(\bar{F})\cap\bar{R}\big{\rangle}=\frac{\big{\langle}K(F)\cap
R\big{\rangle}}{[F,R]}$. Hence $B_{0}(L)\cong\frac{F^{\prime}\cap
R}{\big{\langle}K(F)\cap R\big{\rangle}}$.

##  3\. Applications

The proof of the following proposition is exactly the same as the proof of [6,
Proposition 4.3].

######  Theorem 3.1.

Let $L$ be a Lie algebra with a free presentation $L\cong F/R$, and $M$ be an
ideal of $L$, such that $T=\ker(F\mapsto L/M)$. Then the sequence

$0\mapsto\frac{R\cap\big{\langle}K(F)\cap
T\big{\rangle}}{\big{\langle}K(F)\cap R\big{\rangle}}\mapsto B_{0}(L)\mapsto
B_{0}(L/M)\mapsto\frac{M\cap L^{\prime}}{\big{\langle}K(L)\cap
M\big{\rangle}}\mapsto 0,$

is exact.

######  Definition.

A Lie algebra $L$ is called generalized Heisenberg of rank $n$ if
$L^{\prime}=Z(L)$ and $\dim L^{\prime}=n$.

A freest generalized Heisenberg Lie algebra is a $d$-generated (minimally
generated by $d$ elements) generalized Heisenberg Lie algebra of rank
$\frac{1}{2}d(d-1)$ for some $d\geq 2$. We shall denote it by $L_{d}$. Notice
that $L_{d}$ has the following presentation:

$\big{\langle}x_{1},\ldots,x_{d},\ y_{ij}\mid\ [x_{i},x_{j}]=y_{ij},1\leq
i<j\leq d,\ \text{class}\ 2\big{\rangle},$

and $\dim L_{d}=\frac{1}{2}d(d+1).$

######  Theorem 3.2.

Let $L_{d}$ be the freest generalized Heisenberg Lie algebra of rank
$\frac{1}{2}d(d-1)$. Then $B_{0}(L_{d})=0.$

###### Proof.

Let $f:L_{d}\times L_{d}\mapsto\Omega$ be a 2-cocycle such that $\bar{f}\in
B_{0}(L_{d})$. Let $B=\\{x_{1},\ldots,x_{d},[x_{i},x_{j}]\mid\ 1\leq i<j\leq
d\\}$. Define a map $\mu:B\mapsto\Omega$ as follows:

$\mu(x_{i})=0\ \text{for}\ 1\leq i\leq d,$
$\mu([x_{i},x_{j}])=-f(x_{i},x_{j})\ \text{for}\ 1\leq i<j\leq d.$

Since $B$ is basis for $L_{d}$ we can extend this map linearly to $L_{d}$ and
call it $\sigma$ so that $\sigma$ is a linear transformation. Let $x,y\in
L_{d}$. Since $B$ is a basis we can write $x$ and $y$ as

$x=\sum_{i=1}^{d}\alpha_{i}x_{i}+\sum_{1\leq i<j\leq
d}\alpha_{ij}[x_{i},x_{j}],$ $y=\sum_{k=1}^{d}\beta_{k}x_{k}+\sum_{1\leq
k<l\leq d}\beta_{kl}[x_{k},x_{l}],$

for some $\alpha_{i},\beta_{i},\alpha_{ij},\beta_{ij}\in\Omega.$ Now using the
bilinearity of $f$ and the fact that $f(a,b)=0$ whenever $[a,b]=0$, we get
that

$f(x,y)=\sum_{i=1}^{d}\sum_{k=1}^{d}\alpha_{i}\beta_{k}f(x_{i},x_{k}).$

Also, using the bilinearity of Lie bracket, linearity of $\sigma$ and the fact
that $f(l_{1},[l_{2},l_{3}])=0$ for all $l_{1},l_{2},l_{3}\in L_{d}$, we get
that

$\sigma([x,y])=-\sum_{i=1}^{d}\sum_{k=1}^{d}\alpha_{i}\beta_{k}f(x_{i},x_{k}).$

Therefore $f(x,y)=-\sigma([x,y]).$ Thus $f$ is a coboundry and $\bar{f}=0$.
This completes the proof.

∎

Rostami et al. [6] proved that the Bogomolov multiplier of a Heisenberg Lie
algebra is trivial. We prove this fact as a Corollary of Theorem 3.2.

A Heisenberg Lie algebra of dimension $2n+1,n>0,$ is given by the following
presentation:

$\big{\langle}x_{1},\ldots,x_{2n},\ v\mid\
[x_{2i-1},x_{2i}]=v,[x_{j},x_{k}]=0,1\leq i\leq
n,(j,k)\neq(2i-1,2i)\big{\rangle}.$

######  Corollary 3.3.

Let $H_{2n+1}$ be the Heisenberg Lie algebra of dimension $2n+1.$ Then
$B_{0}(H_{2n+1})=0.$

###### Proof.

Let $L_{2n}$ be the generalized Heisenberg Lie algebra of rank $n(2n-1)$ and
$M$ be its ideal generated by $[x_{2r-1},x_{2r}]-[x_{2s-1},x_{2s}],1\leq
r<s\leq n$ and $[x_{t},x_{u}],1\leq t<u\leq 2n,(t,u)\neq(2i-1,2i)$ for any
$i\leq n.$ Then it is easy to see that $L_{2n}/M$ is isomorphic to $H_{2n+1}.$
Since $B_{0}(L_{2n})=0$ by Theorem 3.2, it follows from Theorem 3.1 that

$B_{0}(L_{2n}/M)\cong\frac{M\cap L_{2n}^{\prime}}{\big{\langle}K(L_{2n})\cap
M\big{\rangle}}.$

Since for $1\leq r<s\leq n$,

$[x_{2r-1}-x_{2s-1},x_{2r}+x_{2s}]=[x_{2r-1},x_{2r}]-[x_{2s-1},x_{2s}]+[x_{2r},x_{2s-1}]+[x_{2r-1},x_{2s}],$

it follows that $M\cap L_{2n}^{\prime}=\big{\langle}K(L_{2n})\cap
M\big{\rangle}.$ Thus $B_{0}(L_{2n}/M)=0$ so that $B_{0}(H_{2n+1})=0.$

∎

We now proceed to prove Theorem 1.5.

Proof of Theorem 1.5: Let $d$ be a natural number greater than $4n$ and
$L_{d}$ be the $d$-generated freest generalized Heisenberg Lie algebra
generated by $x_{1},x_{2},\ldots x_{d}$. Let $M$ be the ideal generated by
$[x_{1},x_{2}]+[x_{3},x_{4}],[x_{5},x_{6}]+[x_{7},x_{8}],\ldots[x_{4n-3},x_{4n-2}]+[x_{4n-1},x_{4n}].$
Since $B_{0}(L_{d})=0$, it follows from Theorem 3.1 that

$\dim B_{0}(L_{d}/M)=\dim\frac{L_{d}^{\prime}\cap M}{\big{\langle}K(L_{d})\cap
M\big{\rangle}}.$

Next we prove that $K(L_{d})\cap M=\\{0\\}$. For this let $l\in K(L_{d})\cap
M.$ Since $l\in M$ there exists $\gamma_{k}^{\prime}s$ such that

$l=\sum_{k=0}^{n-1}\gamma_{k+1}\Big{(}[x_{4k+1},x_{4k+2}]+[x_{4k+3},x_{4k+4}]\Big{)}.$

Also there exist $\alpha_{i}$’s and $\beta_{j}$’s such that

$l=\bigg{[}\sum_{i=1}^{d}\alpha_{i}x_{i},\sum_{j=1}^{d}\beta_{j}x_{j}\bigg{]}$

because $l\in K(L_{d}).$ Hence

$\sum_{k=0}^{n-1}\gamma_{k+1}\Big{(}[x_{4k+1},x_{4k+2}]+[x_{4k+3},x_{4k+4}]\Big{)}=\sum_{i=1}^{d-1}\sum_{j=i+1}^{d}(\alpha_{i}\beta_{j}-\alpha_{j}\beta_{i})[x_{i},x_{j}].$

It follows that

(3.1) $\alpha_{i}\beta_{j}-\alpha_{j}\beta_{i}=0\ \ \text{when}\ \
(i,j)\neq(2k+1,2k+2)\ \text{for any}\ k=0,1,\ldots n-1,$

and

$\gamma_{k+1}=\alpha_{4k+1}\beta_{4k+2}-\alpha_{4k+2}\beta_{4k+1}=\alpha_{4k+3}\beta_{4k+4}-\alpha_{4k+4}\beta_{4k+3}$

for any $k=0,1,\ldots n-1$. From Equation 3.1 we have

(3.2) $\alpha_{4k+1}\beta_{4k+3}-\alpha_{4k+3}\beta_{4k+1}=0$ (3.3)
$\alpha_{4k+1}\beta_{4k+4}-\alpha_{4k+4}\beta_{4k+1}=0$ (3.4)
$\alpha_{4k+2}\beta_{4k+3}-\alpha_{4k+3}\beta_{4k+2}=0$ (3.5)
$\alpha_{4k+2}\beta_{4k+4}-\alpha_{4k+4}\beta_{4k+2}=0,$

for $k=0,1,\ldots n-1.$

Suppose $\alpha_{4k+1}=\beta_{4k+1}=0.$ Then $\gamma_{k+1}=0.$ Assume then
that $\alpha_{4k+1}=0$ but $\beta_{4k+1}\neq 0.$ From Equations 3.2 and 3.3,
$\alpha_{4k+3}=\alpha_{4k+1}=0$. As a result, $\gamma_{k+1}=0$ in this case as
well. Thus we have shown that if $\alpha_{4k+1}=0$, then $\gamma_{k+1}=0$.
Similarly, if either of
$\alpha_{4k+2},\alpha_{4k+3},\alpha_{4k+4},\beta_{4k+1},\beta_{4k+2},\beta_{4k+3},\beta_{4k+4}$
is zero, then $\gamma_{k+1}=0.$ Hence, we can now assume that neither of
$\alpha_{4k+i},\beta_{4k+i}$ is zero for $i=1,2,3,4.$ From Equations 3.2 and
3.3 we can deduce that
$\beta_{4k+3}/\beta_{4k+4}=\alpha_{4k+3}/\alpha_{4k+4}.$ Hence
$\gamma_{k+1}=0$ so that $l=0$. It follows that $K(L_{d})\cap M=\\{0\\}$.

Also, $M\leq L_{d}^{\prime}$. Hence $\dim B_{0}(L_{d}/M)=\dim(M)=n.$ By
Corollary 1.2 $\dim B(L_{d}/M)=n.$ Taking $L$ to be $L_{d}/M$ completes the
proof.

Proof of Theorem 1.6: Let $(\theta,\phi)$ be an isoclinism between $L$ and
$M$, i.e., $\theta:\frac{L}{Z(L)}\mapsto\frac{M}{Z(M)}$ and
$\phi:\gamma_{2}(L)\mapsto\gamma_{2}(M)$ be isomorphisms and whenever
$\theta(l_{i}Z(L))=m_{i}Z(M)$ for $i=1,2$, we have
$\phi([l_{1},l_{2}])=[m_{1},m_{2}].$ Let $\bar{f}\in B_{0}(L)$ where
$f:L\times L\mapsto\Omega$ be a cocycle. Define $c_{f}:M\times M\mapsto\Omega$
by $c_{f}(m_{1},m_{2})=f(l_{1},l_{2})$, where $l_{1}$ and $l_{2}$ are given by
$\theta^{-1}(m_{i}+Z(M))=l_{i}+Z(L)$ for $i=1,2.$ The rest of the proof
follows from the following lemmas:

######  Lemma 3.4.

Let $c_{f}$ be the map defined above. Then

  * (i)

$c_{f}$ is well defined.

  * (ii)

$c_{f}$ is a 2 cocycle.

  * (iii)

$\overline{c_{f}}\in B_{0}(M).$

###### Proof.

Since $f$ is bilinear and $f(k,l)=0$ whenever $[k,l]=0$. It follows that
$f(l_{1}+z_{1},l_{2}+z_{2})=f(l_{1},l_{2})$ for every $z_{1},z_{2}\in Z(L)$.
This shows that $c_{f}$ is well-defined.

To see that $c_{f}$ is a 2-cocycle, let $m_{1},m_{2},m_{3}\in M$ and let
$\theta^{-1}(m_{i}+Z(M))=l_{i}+Z(L)$ for $i=1,2,3.$ It is obvious that
$\theta^{-1}(m_{1}+m_{2}+Z(M))=l_{1}+l_{2}+Z(L).$ Therefore
$c_{f}(m_{1}+m_{2},m_{3})=f(l_{1}+l_{2},l_{3})$ which equals
$f(l_{1},l_{3})+f(l_{2},l_{3})$ that is equal to
$c_{f}(m_{1},m_{3})+c_{f}(m_{2},m_{3}).$ Similarly
$c_{f}(m_{1},m_{2}+m_{3})=c_{f}(m_{1},m_{2})+c_{f}(m_{1},m_{3})$. Thus $c_{f}$
is bilinear. Also, it is easy to see that $c_{f}$ is alternating because $f$
is alternating. Next, For $i,j,k\in\\{1,2,3\\}$ note that
$c_{f}([m_{i},m_{j}],m_{k})=f([l_{i},l_{j}],l_{k})$ because

$\begin{split}\theta^{-1}\Big{(}[m_{i},m_{j}]+Z(M)\Big{)}&=\theta^{-1}\Big{(}\Big{[}m_{i}+Z(M),m_{j}+Z(M)\Big{]}\Big{)}\\\
&=\Big{[}\theta^{-1}\Big{(}m_{i}+Z(M)\Big{)},\theta^{-1}\Big{(}m_{j}+Z(M)\Big{)}\Big{]}\\\
&=\Big{[}l_{i}+Z(L),l_{j}+Z(L)\Big{]}\\\ &=[l_{i},l_{j}]+Z(L).\end{split}$

t follows that $c_{f}$ is a 2-cocycle, since $f$ is a 2-cocycle.

To see that $\overline{c_{f}}\in B_{0}(M)$, suppose that $[m_{1},m_{2}]=0$.
But then $[l_{1},l_{2}]=0$ because $\phi([l_{1},l_{2}])=[m_{1},m_{2}].$ Since
$\overline{f}\in B_{0}(L)$ it follows that $f(l_{1},l_{2})=0.$ Hence
$c_{f}(m_{1},m_{2})=0.$

######  Lemma 3.5.

The map $\eta:B_{0}(L)\mapsto B_{0}(M)$ defined by
$\eta(\overline{f})=\overline{c_{f}}$ is an isomorphism.

###### Proof.

We begin by ensuring that the map is well-defined. To verify this consider
$\sigma:L\times L\mapsto\Omega$ to be a coboundary. Then

$c_{f+\sigma}(m_{1},m_{2})=(f+\sigma)(l_{1},l_{2})=f(l_{1},l_{2})+\sigma(l_{1},l_{2})=c_{f}(m_{1},m_{2})+c_{\sigma}(m_{1},m_{2}).$

Thus we have, $c_{f+\sigma}=c_{f}+c_{\sigma}.$ Notice that $c_{\sigma}$ is a
coboundary because $\sigma$ is coboundary. Therefore
$\overline{c_{f}}=\overline{c_{f+\sigma}}$, i.e.,
$\eta(\overline{f})=\eta(\overline{f+\sigma}).$ This proves that $\eta$ is
well-defined.

In a similar fashion one can see that $c_{f_{1}+f_{2}}=c_{f_{1}}+c_{f_{2}}$
and $c_{\alpha f_{1}}=\alpha c_{f_{1}}$ for each $\alpha\in\Omega$ and each
cocycles $f_{1}$, $f_{2}$ from $L\times L$ to $\Omega$. So that
$\eta(\overline{f_{1}+f_{2}})=\eta(\overline{f_{1}})+\eta(\overline{f_{2}})$
and $\eta(\alpha\overline{f_{1}})=\alpha\eta(\overline{f_{1}}).$ Thus $\eta$
is a linear map.

Finally, in order to see that $\eta$ is a bijection, we define another map
$\chi:B_{0}(M)\mapsto B_{0}(L)$ in the same way as $\eta$ is defined from
$B_{0}(L)$ to $B_{0}(M).$ Then it is easy to see that $\eta\chi$ and
$\chi\eta$ both are identity maps and thus $\eta$ is a bijection. This
completes the proof. ∎

∎

## References

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