#  Symbolic, numeric and quantum computation of Hartree-Fock equation

Ichio Kikuchi1, Akihito<EMAIL_ADDRESS>(The
corresponding author; a visiting researcher in IFQT)
1Internationales Forschungszentrum für Quantentechnik
2International Research Center for Quantum Technology, Tokyo

###### Abstract

In this article, we discuss how a kind of hybrid computation, which employs
symbolic, numeric, classic, and quantum algorithms, allows us to conduct
Hartree-Fock electronic structure computation of molecules. In the proposed
algorithm, we replace the Hartree-Fock equations with a set of equations
composed of multivariate polynomials. We transform those polynomials to the
corresponding Gröbner bases, and then we investigate the corresponding
quotient ring, wherein the orbital energies, the LCAO coefficients, or the
atomic coordinates are represented by the variables in the ring. In this
quotient ring, the variables generate the transformation matrices that
represent the multiplication with the monomial bases, and the eigenvalues of
those matrices compose the roots of the equation. The quantum phase estimation
(QPE) algorithm enables us to record those roots in the quantum states, which
would be used in the input data for more advanced and more accurate quantum
computations.

##  1  Introduction

There is a symbolic-numeric method of quantum chemistry [1], whereby the
computations are carried out in the following way:

  * •

The molecular integrals are represented by the polynomial approximation of
analytic formulas, which are computed symbolically if we use analytic atomic
bases, such as Gaussian-Type or Slater-Type orbitals (GTO or STO) [2]. Those
formulas are the analytic functions of several variables, namely, those of
orbital exponents and atomic coordinates. By Tailor expansion with respect to
those variables, the molecular integrals are approximated by polynomials.

  * •

The total energy is a polynomial composed of molecular integrals and the
undetermined coefficients of LCAO. The ortho-normalization conditions are
similarly treated.

  * •

We compose the objective function from the total energy and the ortho-
normalization condition with the Lagrange multipliers which represent the
orbital energies.

  * •

By symbolic differentiation, we obtain a system of polynomial equations that
gives the optima.

  * •

To get the roots of the system of polynomial equations, we apply several
methods of computer algebra, where Gröbner bases and the primary ideal
decomposition play central roles in getting the quantum eigenstates [3, 4, 5,
6, 7, 8]. Namely, we compose an ideal $I$ from the given polynomials and
transform them into another system that has a more suitable form for root-
finding [9, 10, 11, 12, 13, 11]. The ideal representing a Hartree-Fock
equation could be decomposed into several subsystems described by primary
ideals. Each primary ideal would represent one solution set, namely, one
quantum state if the decomposition is executed to the full.

  * •

Up to now, we have reported the results of several simple molecules, using STO
and n-GTO models ( [1] and [14]). In those works, the adopted algorithms are
classical, not quantum. One might use the term Molecular Algebraic Geometry to
refer to this algebraic computational scheme for molecular orbital theory.

The algebraic method described above could relate to the quantum algorithms,
and the theme of the present study is to demonstrate it. This article is
structured as follows. First, we show the computational step whereby the
classical symbolic computation prepares the eigenvalue problem that gives the
roots of the given equation. Second, we show how the quantum algorithm could
solve the problem. Then we discuss several points that should be treated with
care.

##  2 Computational process

In this section, we incarnate the algorithms for symbolic-numeric and
classical-quantum computation in quantum chemistry. The computational process
is composed of two phases. The first phase uses the symbolic-numeric classical
algorithm and converts the Hartree-Fock equations into a representation
suitable for quantum computation. The second phase uses the data generated in
the first phase and computes the roots of the Hartree-Fock equations. We
describe the algorithm in each phase, using two examples.

###  2.1 Phase 1: symbolic-numeric classical algorithm

####  2.1.1 Tools for symbolic, numeric, and classical computation

We solve the set of polynomial equations through the computational steps
explained in [15].

  * •

Let $I$ be an ideal made of multivariate polynomials $(f_{1},f_{2},...,f_{t})$
in $R[x_{1},x_{2},...,x_{n}]$. Once the Gröbner for the ideal $I$ is computed,
it is an easy task to represent any element in $R[x_{1},x_{2},...,x_{n}]/I$
uniquely as the linear combination of the monomial basis of the quotient ring.

  * •

Let $\bar{x}_{1},\bar{x}_{2},...,\bar{x}_{2}$ be the representatives of
$x_{1},x_{2},...,x_{n}$ in $R[x_{1},x_{2},...,x_{n}]/I$. Additionally, let $b$
be a vector that is composed of the representatives of the monomial basis of
the quotient ring.

  * •

For any $i$, the multiplication $\bar{x}_{i}\cdot b$ is represented by

$\displaystyle\bar{x}_{i}\cdot b=b\cdot M_{x_{i}}$ (1)

with a transformation matrix $M_{x_{i}}$. The entries of the matrix are
numbers, but not symbols.

  * •

As $M_{x_{i}}\cdot M_{x_{j}}=M_{x_{j}}\cdot M_{x_{i}}$, those transformation
matrices share common eigenvectors $\\{v_{j}|j=1,...,M\\}$, where $M$ is size
of the monomial basis $b$.

  * •

Let us consider the eigenvalue problems, defined as follows,

$\displaystyle\bar{\xi}_{i}^{(j)}\ v_{j}=v_{j}\cdot M_{x_{i}}$ (2)

for $i=1,...,n$ and $j=1,...,M$. Those equations are solved numerically, and
the eigenvalues give the common zeros of the polynomials included in the ideal
$I$. Namely, the eigenvalues give the roots of the set of polynomial equations
defined by

$\displaystyle
f_{1}(x_{1},x_{2},...,x_{n})=f_{2}(x_{1},x_{2},...,x_{n})=...=f_{t}(x_{1},x_{2},...,x_{n})=0$
(3)

in such a way that

$\displaystyle(x_{1},x_{2},...,x_{n})=(\bar{\xi}_{1}^{(j)},\bar{\xi}_{2}^{(j)},...,\bar{\xi}_{n}^{(j)})$
(4)

for $j=1,...,M$. Note that if eigenvectors $\\{v_{j}\\}_{j}$ for one $M_{i}$
is obtained, the other components of the roots are computed by

$\displaystyle\bar{\xi}_{i}^{(j)}=\frac{(v_{j}\cdot
M_{x_{i}},v_{j})}{(v_{j},v_{j})}.$ (5)

The root-finding of a system of polynomial equations is replaced by a set of
eigenvalue problems, which could be solved by quantum algorithms. We put the
eigenvectors $\\{v_{j}\\}_{j}$ into a set of quantum states
$\\{|v_{j}\rangle\\}$, and the computational steps are carried out by a
quantum circuit, which conducts the following transformation:

$\displaystyle|v_{j}\rangle|Ancilla_{1}\rangle|Ancilla_{2}\rangle\cdots|Ancilla_{n}\rangle\rightarrow|v_{j}\rangle|\bar{\xi}_{1}^{(j)}\rangle|\bar{\xi}_{2}^{(j)}\rangle\cdots|\bar{\xi}_{n}^{(j)}\rangle$
(6)

where the eigenvalues of $M_{x_{i}}$ $(i=1,...,n)$ for $v_{j}$ are recorded in
ancilla qubits through a successive application of quantum phase estimation.

####  2.1.2 Computation for a simple toy model

Let us compute a simple toy model, where the secular equation is given by

$\displaystyle\begin{pmatrix}V(x,y)&-1\\\
-1&V(x,y)\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix}=e\begin{pmatrix}x\\\
y\end{pmatrix}$ (7)

along with the normalization condition $x^{2}+y^{2}=1$. The variables $(x,y)$
are the amplitudes of the wavefunction and $e$ is the orbital energy. $V(x,y)$
is the on-site potential that is the function of the amplitude of the
wavefunction. We assume that the roots are real.

The polynomial ideal that represents the secular equation is given by

$\displaystyle I=(xV(x,y)-y-e\,x,yV(x,y)-x-e\,y,x^{2}+y^{2}-1)$ (8)

In the case of $V(x,y)=0$, the Gröbner basis is given by

$\displaystyle I_{std}=(e^{2}-1,2y^{2}-1,x+ye)$ (9)

The roots of the set of polynomial equations are given by

$\displaystyle(x,y,e)=\left(\pm\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}},-1\right),\left(\pm\frac{1}{\sqrt{2}},\mp\frac{1}{\sqrt{2}},1\right)$
(10)

The entries in the quotient ring $Q(x,y,e)/I$ are the linear combinations of
the monomial basis $b=(b[0],b[1],b[2],b[3])$:

$\displaystyle b[0]$ $\displaystyle=y\,e$ (11) $\displaystyle b[1]$
$\displaystyle=y$ (12) $\displaystyle b[2]$ $\displaystyle=e$ (13)
$\displaystyle b[3]$ $\displaystyle=1$ (14)

In the quotient ring, the multiplications of the entries of the basis $b$ by
$x$, $y$, and $e$ are represented by the transformation matrices: $b\cdot
p=b\cdot m_{p}$ for $p=x,y,e$.

$\displaystyle m_{x}=\begin{pmatrix}0&0&0&-1\\\ 0&0&-1&0\\\ 0&-0.5&0&0\\\
-0.5&0&0&0\\\ \end{pmatrix}$ (15)

$\displaystyle m_{y}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 0.5&0&0&0\\\
0&0.5&0&0\\\ \end{pmatrix}$ (16)

$\displaystyle m_{e}=\begin{pmatrix}0&1&&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{pmatrix}$ (17)

For the above example, the related properties are given as follows.

$\displaystyle\begin{pmatrix}v&(vm_{x},v)&(vm_{y},v)&(vm_{e},v)\\\
(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},-1,1)&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&-1\\\
(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},-1,1)&-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&-1\\\
(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},1,1)&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&1\\\
(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},1,1)&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&1\\\
\end{pmatrix}$ (18)

The data in Table 18 covers all the solutions of the secular equation.

###  2.2 Computation for Hartree-Fock model

In this section, the restricted Hartree-Fock computation of a realistic
molecule HeH+ is used as an example. This molecule is the simplest
heteronuclear molecule and is used as a benchmark problem for the solving of
the Hartree-Fock model [2].

At first, we compute the total energy functional of the RHF model of the
molecule through STO-3g basis set [16]. The analytic formulas of the molecular
integrals are computed and substituted into the formula of the energy, namely,
the objective function. The total energy functional is a function of the LCAO
coefficients $(x,y)$, the orbital energy $e$, and the interatomic distance
$R$, as defined in the following.

$\displaystyle E_{HF}=\sum_{i}\langle
i|h|i\rangle+\frac{1}{2}\sum_{ij}\left([ii|jj]-[ij|ji]\right)$ (19)

$\displaystyle\langle i|h|i\rangle=\int d{\bf x}_{1}\chi_{i}^{*}({\bf
x}_{1})h({\bf r}_{1})\chi_{j}({\bf x}_{1})$ (20)

$\displaystyle[ij|kl]=\int d{\bf x}_{1}d{\bf x}_{2}\chi_{i}^{*}({\bf
x}_{1})\chi_{j}({\bf x}_{1})\frac{1}{r_{12}}\chi_{i}^{*}({\bf
x}_{2})\chi_{j}({\bf x}_{2})$ (21)

$\displaystyle\chi_{i}({\bf
x})=\left(x\,\phi_{1s,He}(r-R_{He})+y\,\phi_{1s,H}(r-R_{H})\right)\sigma_{i}$
(22)

$\displaystyle\sigma_{i}:\text{spin function}$ (23)
$\displaystyle\phi^{STO-3G}_{1s}({\bf r})=\sum_{i=1}^{3}c(i)\exp(-z(i)r^{2})$
(24)

For the computation of $HeH^{+}$, we use two spin orbitals.

$\displaystyle i=\alpha,\beta$ (25)

The total energy in the restricted Hartree-Fock model is given by

$\displaystyle h({\bf r})=-\frac{1}{2}\nabla^{2}-\frac{Z_{H_{e}}}{|\bf
r-R_{H_{e}}|}--\frac{Z_{H}}{|\bf r-R_{H}|}$ (26) $\displaystyle R=|{\bf
R}_{He}-{\bf R}_{H}|,Z_{He}=2,Z_{H}=1$ (27)

$\displaystyle
E_{tot}(x,y,e,R)=E_{HF}(x,y,R)-e\sum_{i}(\langle\chi_{i}|\chi_{j}\rangle-1)+\frac{Z_{He}Z_{H}}{R}$
(28)

  * •

The total energy functional is converted to a polynomial through the Taylor
expansion with respect to the atomic distance $R$. The expansion is carried
out at the center $R0=1.5$.

  * •

The numerical coefficients in the objective function are approximated by
fractional numbers so that the objective function, multiplicated by the powers
of ten, is given by a polynomial with integer coefficients. To this end, we
simply approximate the numerical coefficient $C$ by rounding $10^{n}C$ to the
nearest integer $N_{c}$ and get $N_{c}/10^{n}$. We use the polynomial with
integer coefficients as the objective function, denoted by $\Omega$.

  * •

A set of polynomial equations is derived by the partial differentiation with
respect to $(x,y)$ and $e$ so that the roots of those equations give the
optima of the objective function. For the sake of simplicity, we do not carry
out the optimization for $R$. Instead, we replace
$\frac{\partial\Omega}{\partial R}$ with $100R-146$ so that the interatomic
distance is fixed at $R=1.46$.

  * •

We apply algebra. We use the ring $Q[x,y,e]$ (a ring over the field of
rational numbers $Q$) with the degree reverse lexicographic monomial ordering,
such that $x>y>e$. The generators of the set of polynomial equations form an
ideal $I$. We compute the Gröbner basis of $I$, by which the quotient ring
$Q[x,y,e]/I$ is defined. In this quotient ring, the monomial basis and the
transformation matrices representing the operation of $x$, $y$, and $e$ over
$b$ are computed.

  * •

As the transformation matrices are numerical data, we then use classical or
quantum methods to compute eigenvalues.

The objective function $f_{obj}$ is computed as

OBJ=281*R**5*x**3*y + 1119*R**5*x**2*y**2 + 164*R**5*x**2 + 533*R**5*x*y**3 -
901*R**5*x*y - 70*R**5*y**2 - 1756*R**5 - 2892*R**4*x**3*y -
9431*R**4*x**2*y**2 - 2273*R**4*x**2 - 5040*R**4*x*y**3 + 8552*R**4*x*y +
712*R**4*y**2 + 15802*R**4 + 11305*R**3*x**3*y + 29175*R**3*x**2*y**2 +
12477*R**3*x**2 + 18393*R**3*x*y**3 - 30849*R**3*x*y - 1877*R**3*y**2 -
59260*R**3 - 15964*R**2*x**3*y - 32038*R**2*x**2*y**2 - 35996*R**2*x**2 -
27890*R**2*x*y**3 + 37012*R**2*x*y - 3516*R**2*y**2 + 118518*R**2 -
12479*R*x**3*y - 18807*R*x**2*y**2 + 58692*R*x**2 + 1281*R*x*y**3 +
52833*R*x*y + 28135*R*y**2 - 133334*R - 2*e*(114*R**5*x*y - 1281*R**4*x*y +
5600*R**3*x*y - 10194*R**2*x*y + 115*R*x*y + 10000*x**2 + 18221*x*y +
10000*y**2 - 10000) + 13071*x**4 + 45874*x**3*y + 59634*x**2*y**2 - 91649*x**2
+ 32206*x*y**3 - 146963*x*y + 7746*y**4 - 65195*y**2 + 79999;

The ideal that gives the optima of the objective function is composed of the
following components:

$\displaystyle I=\left(\frac{\partial f_{obj}}{\partial x},\frac{\partial
f_{obj}}{\partial y},\frac{\partial f_{obj}}{\partial R}\right)$ (29)

To save the computational cost, the atomic distance $R$ is fixed, and $I$ is
modified as

$\displaystyle I=\left(\frac{\partial f_{obj}}{\partial x},\frac{\partial
f_{obj}}{\partial y},100R-146\right)$ (30)

The quotient ring $Q[x,y,e]/I$ has the monomial basis
$b=(y^{2},xe,ye,e2,x,y,e,1)$, and the transformation matrices ($m_{x}$,
$m_{y}$, and $m_{e}$) for three variables ($x$,$y$, $e$) are obtained.

Let us inspect the computed result.

As a reference, the result of the Hartree-Fock computation by the standard
self-consistent method is shown in Table 1.

| $x$ | $y$ | $e$
---|---|---|---
STO-3G | 0.801918 | 0.336800 | -1.597448
Table 1:  The result of the Hartree-Fock computation of HeH+ by the standard
self-consistent method with STO-3g basis set, at the interatomic distance
$R=1.4632$.

The solutions obtained from the symbolic-numeric method are shown in Table 2.
We use the normalized right eigenvectors $\phi)$ and compute the expectation
values for $m_{x}^{T}$ and $m_{e}^{T}$. The solutions at the third and fourth
rows correspond to the ground state in the reference data. Those results are
quantitatively satisfactory in giving the electronic structure of the
molecule, although there is a bit of deviation from the reference data. The
cause of the deviation is that we have approximated the objective function as
a polynomial with integer coefficients after the Taylor expansion, and as a
result, this rough approximation dropped the subtle features of the numeric
data used in the standard self-consistent method.

$i$ | $Eig$ | $(\phi_{i}|x|\phi_{i})$ | $(\phi_{i}|y|\phi_{i})$ | $(\phi_{i}|e|\phi_{i})$
---|---|---|---|---
1 | -1.114772 | 0.604062 | -1.114772 | -0.537546
2 | 1.114772 | -0.604062 | 1.114772 | -0.537546
3 | -0.337484 | -0.801308 | -0.337484 | -1.600455
4 | 0.337484 | 0.801308 | 0.337484 | -1.600455
Table 2:  The result of the Hartree-Fock computation of HeH+ by the symbolic
numeric method. We used four normalized right eigenvectors
$|\phi_{1}),...,|\phi_{4})$ of $m_{y}^{T}$ that have real eigenvalues (which
are given as $Eig$ in the table) and computed $(\phi_{i}|m_{j}^{T}|\phi_{i})$
for $j=x,y,e$. The third and the fourth solutions give the ground state.

###  2.3 Phase 2: quantum computation

####  2.3.1 Tools for quantum computation

Now we have restated the given question as an eigenvalue problem, and we
anticipate the application of quantum phase estimation to get the eigenvalues.
The remaining question is that the QPE is not applied directly, since the
transformation matrices $m_{p}$ are not Hermitian, and the time-evolution
operator $\exp(-iTm_{p})$ is not unitary. To settle this issue, we use the
block-encoding, by which any complex matrix can be embedded in the diagonal
part of certain unitary matrices. Several algorithms enable us to conduct the
block-encoding and design the quantum circuits [17, 18, 19].

The block encoding of an n-qubit operator $A$ is formally defined as follows:

$\displaystyle\tilde{A}=\left(\langle 0|^{\otimes a}\otimes
I_{n}\right)U\left(|0\rangle^{\otimes a}\otimes I_{n}\right)$ (31)

In the above, $\tilde{A}=\alpha A$, for which the factor $\alpha$ is chosen in
such a way that $|\tilde{A}_{ij}|\leq 1$ for all $i$ and $j$. $U$ is a unitary
matrix operating on $a+n$ qubits, and its action on the qubits is given by

$\displaystyle U\left(|0\rangle^{\otimes
a}\otimes|\phi\rangle\right)=|0\rangle^{\otimes
a}\otimes\tilde{A}|\phi\rangle+\sqrt{1-\|\tilde{A}|\phi\rangle\|^{2}}|\sigma^{\perp}\rangle$
(32)

with

$\displaystyle\left(\langle 0|^{\otimes a}\otimes
I_{n}\right)|\sigma^{\perp}\rangle=0$ (33)

and

$\displaystyle\||\sigma^{\perp}\rangle\|=1$ (34)

A repetition of partial measurements of the ancilla qubits yields
$|0\rangle^{\otimes a}$ with probability $\|\tilde{A}|\phi\rangle\\}^{2}$, and
the circuit gives rise to
$\frac{\|\tilde{A}|\phi\rangle}{\|\tilde{A}|\phi\rangle\|}$.

For simplicity, let us assume that $\alpha=1$ and $|{A}_{ij}|\leq 1$ for all
$i$ and $j$. In this case, the matrix query operation $O_{A}$ is defined by

$\displaystyle
O_{A}|0\rangle|i\rangle|j\rangle=\left(a_{ij}+\sqrt{1-|a_{ij}|^{2}}\right)|i\rangle|j\rangle$
(35)

where $|i\rangle$ and $|j\rangle$ are n-qubit computational basis states. The
unitary representation of $O_{A}$ is given by

$\displaystyle O_{A}=\begin{pmatrix}c_{00}&&&&-s_{00}&&&&\\\
&c_{01}&&&&-s_{01}&&&\\\ &&\ddots&&&&\ddots&&\\\
&&&c_{N-1,N-1}&&&&-s_{N-1,N-1}&\\\ s_{00}&&&&c_{00}&&&&\\\
&s_{01}&&&&c_{01}&&&\\\ &&\ddots&&&&\ddots&&\\\
&&&s_{N-1,N-1}&&&&c_{N-1,N-1}&\\\ \end{pmatrix}$ (36)

where $c_{ij}=\cos(\theta_{ij})$, $s_{ij}=\sin(\theta_{ij})$, and
$\theta_{ij}=\text{arccos}(a_{ij})$. Keep in mind that the indices of $c_{ij}$
and $S_{ij}$ are given by n-qubit computational basis states.

The quantum circuit that embodies the block encoding is defined by

$\displaystyle U_{A}=(I_{1}\otimes H^{\otimes n}\otimes
I_{n})(I_{1}\otimes\text{SWAP})O_{A}(I_{1}\otimes H^{\otimes n}\otimes I_{n})$
(37)

where $I_{1}$ and $I_{n}$ means the identity operations; SWAP is the swap
gate; $H$ is the Hadamard gate. After algebra, one obtains

$\displaystyle\langle 0|\langle 0|^{\otimes n}\langle
i|U_{A}|0\rangle|0\rangle^{\otimes n}|j\rangle=\frac{1}{2^{n}}a_{ij}$ (38)

This relation means that, if the $n+1$ ancilla qubits are measured as the
zero-state, the signal register, which is initialized by $|\phi\rangle$,
returns $\frac{A|\phi\rangle}{\|A\phi\rangle\|}$.

If $\|A_{ij}\|>1$ for some $i$ and $j$, we must replace $A$ with $\alpha A$,
using a scale factor $\alpha$ such that $|\alpha|<1$. It increases the
complexity of the quantum circuit for the QPE. If we use the block encoding of
$U^{2k}$ for a unitary $U$ with $\alpha<1$, during the QPE, the controlled
$U^{2k}$ yields

$\displaystyle\frac{1}{\sqrt{2}}\left(|0\rangle+{\alpha}e^{i2^{k}\lambda}|1\rangle\right)\otimes|\psi\rangle$
(39)

To record the eigenvalue $\lambda$ in the bit string, however, the state at
the left qubit should be given by

$\displaystyle|0\rangle+e^{i2^{k}\lambda}|1\rangle$ (40)

To get the latter state, we prepare $U_{\alpha\bf{I}}$, and we apply
$(X\otimes{\bf I})U_{\alpha\bf{I}}(X\otimes{\bf I})$ though the controlled
gate operation. Then we get

$\displaystyle\frac{\alpha}{\sqrt{2}}\left(|0\rangle+e^{i2^{k}\lambda}|1\rangle\right)$
(41)

In (41), $\alpha$ could be neglected on account of the normalization of the
output state.

The problem in the above construction is that the naive design of the quantum
circuit to conduct the operation $O_{A}$ requires too many numbers of $R_{y}$
gates, which causes worse complexity than the classical case. To avoid it, the
FABLE algorithm uses Gray codes to designate the operations on the ancilla
qubits so that this algorithm achieves improved scaling with respect to the
number of the $R_{y}$ gates [19].

####  2.3.2 The quantum steps for the simple toy model and the Hartree-Fock
computation

In this section, the accuracy of block encodings for the simple examples (the
simple toy model and the Hartree-Fock computation for HeH+ are investigated.
Those models are equally given in $Q[x,y,e]$ (in the ring with three
variables), and they are studied together.

Using the FABLE algorithm, we construct the block encoding of the unitary
operator $A$. In Tables 3 and 4, the expectation values of the block encoding
form of unitary operators $(\phi|A|\phi)$ and $(\phi|\exp(-iA)|\phi)$ for
$A=m_{x}^{T},m_{y}^{T}$, and $m_{e}^{T}$ are shown, respectively, for the two
examples. They are computed by numerical linear algebra with a suitable choice
of $\phi$. Furthermore, those values are compared to $(\phi|O_{\exp}|\phi)$,
where $O_{\exp}=O_{\exp(-iM)}$ is obtained by the FABLE algorithm. The block
encodings by the FABLE algorithm are quantitatively accurate for representing
the corresponding evolution of non-unitary $A$.

i | $M$ | $(\phi_{i}|M|\phi_{i})$ | $(\phi_{i}|\exp(-\sqrt{-1}M^{T})|\phi_{i})$ | $(\phi|O_{\exp}|\phi_{i})$
---|---|---|---|---
1 | $m_{x}$ | 0.707107 | 0.760245-0.649637j | 0.760245-0.649637j
1 | $m_{y}$ | 0.707107 | 0.760245-0.649637j | 0.760245-0.649637j
1 | $m_{e}$ | -1.000000 | 0.540302+0.841471j | 0.540302+0.841471j
2 | $m_{x}$ | 0.707107 | 0.760245-0.649637j | 0.760245-0.649637j
2 | $m_{y}$ | -0.707107 | 0.760245+0.649637j | 0.760245+0.649637j
2 | $m_{e}$ | 1.000000 | 0.540302-0.841471j | 0.540302-0.841471j
Table 3:  The expectation values of the unitary operators $(\phi_{i}|\exp(-\sqrt{-1}A)|\phi_{i})$ for $A=m_{x}^{T},m_{y}^{T}$, and $m_{e}^{T}$ in the simple toy model. The table contains the result for two different solutions, distinguished by two different eigenvectors ($\phi_{1}$ and $\phi_{2}$), which correspond to the solutions for $e=-1$ and $e=1$, respectively. The eigenvectors $|\phi_{i})$ are computed from the analytic formula given in the previous section. The symbol $j$ in the table means the imaginary unit number $\sqrt{-1}$,  i | $M$ | $(\phi_{i}|M^{T}|\phi_{i})$ | $(\phi_{i}|\exp(-\sqrt{-1}M^{T})|\phi_{i})$ | $(\phi_{i}|O_{\exp}|\phi_{i})$
---|---|---|---|---
1 | $m_{x}$ | 0.604062 | 0.823035-0.567990j | 0.823035-0.567990j
1 | $m_{y}$ | -1.114772 | 0.440383+0.897810j | 0.440383+0.897810j
1 | $m_{e}$ | -0.537546 | 0.858968+0.512030j | 0.858968+0.512030j
2 | $m_{x}$ | -0.604062 | 0.823035+0.567990j | 0.823035+0.567990j
2 | $m_{y}$ | 1.114772 | 0.440383-0.897810j | 0.440383-0.897810j
2 | $m_{e}$ | -0.537546 | 0.858968+0.512030j | 0.858968+0.512030j
3 | $m_{x}$ | -0.801308 | 0.695768+0.718267j | 0.695768+0.718267j
3 | $m_{y}$ | -0.337484 | 0.943591+0.331114j | 0.943591+0.331114j
3 | $m_{e}$ | -1.600455 | -0.029654+0.999560j | -0.029654+0.999560j
4 | $m_{x}$ | 0.801308 | 0.695768-0.718267j | 0.695768-0.718267j
4 | $m_{y}$ | 0.337484 | 0.943591-0.331114j | 0.943591-0.331114j
4 | $m_{e}$ | -1.600455 | -0.029654+0.999560j | -0.029654+0.999560j
Table 4:  The expectation values of the unitary operators
$(\phi_{i}|\exp(-\sqrt{-1}A)|\phi_{i})$ for $A=m_{x}^{T},m_{y}^{T}$, and
$m_{e}^{T}$ in the Hartree-Fock computation. For the computation of
expectation values, we used four eigenvectors ($\\{\phi_{i}|i=1,...,4\\}$) of
$m_{y}^{T}$, which have real eigenvalues.

In the computations presented here, we used the eigenvectors that were
analytically derived or computed by the eigenvalue solver. In the Hartree-Fock
case, we cast off the eigenvectors with complex eigenvalues since those
useless vectors are easily detected by classical computations. However, in
quantum computations, it is not so easy to examine the state vectors in the
quantum circuit. In the next section, we discuss how to carry out the state
preparation properly.

##  3 Discussion

In this section, we discuss several points that should be treated with care.

###  3.1 The difficulty of quantum algorithm concerning complex-valued
solutions

The existence of complex roots of the given system of polynomial equations is
an obstacle to full-fledged quantum computation in the current problem
setting. The standard quantum phase estimation is applied to the Hermitian
operators which have real eigenvalues only. Let $\lambda$ be an eigenvalue,
which is represented in the following way:

$\displaystyle\frac{\lambda}{2\pi}=\frac{j_{1}}{2^{1}}+\frac{j_{2}}{2^{2}}+\cdots+\frac{j_{n}}{2^{n}}$
(42)

In the intermediate stage of the computation by the QPE, the quantum state
vectors are generated and transformed as follows.

$\displaystyle|0\rangle|\phi\rangle\xrightarrow{H}\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)|\phi\rangle$
(43)
$\displaystyle\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)|\phi\rangle\xrightarrow{\Lambda(U^{2^{n-1}})}\frac{1}{\sqrt{2}}\left(|0\rangle+e^{i(2\pi)j_{n}}|1\rangle\right)|\phi\rangle$
(44)
$\displaystyle\frac{1}{\sqrt{2}}\left(|0\rangle+e^{i(2\pi)j_{n}}|1\rangle\right)|\phi\rangle\xrightarrow{H}\
|j_{n}\rangle|\phi\rangle$ (45)

However, if the eigenvalue is given by $\lambda+\sqrt{-1}\nu$, the quantum
circuit yields

$\displaystyle\frac{1}{\sqrt{2}}\left(|0\rangle+\exp(-2^{k}\cdot
2\pi\nu)e^{i(2\pi)j_{n}}|1\rangle\right)|\phi\rangle,$ (46)

from which, the integer $j_{n}$ cannot be extracted at
$|j_{n}\rangle|\phi\rangle$ by the Hadamard transformation.

Several approaches tackle this problem [20, 17, 21].

  * •

The algorithm in [20] generates the state of the form of (46), estimates the
factor $|\exp(-2^{k}(2\pi)\nu)|$ by projective measurements on the index qubit
in the basis $|1\rangle\langle 1|$, rotates the quantum state to cancel that
factor, and obtains the wanted form of (45).

  * •

The algorithm in [17] similarly estimates $|\exp(-2^{k}(2\pi)\nu)|$ by
measurements and then obtains the phase part of the eigenvalue.

  * •

The algorithm in [21] prepares the initial state vector in such a way that

$\displaystyle|\psi_{init}\rangle=\sum\beta_{j}|E_{j}\rangle|\bar{E}_{j}\rangle$
(47)

where $|E_{j}\rangle$ and $|\bar{E}_{j}\rangle$ are the eigenvectors of a
matrix $M$, and they have the conjugated eigenvalues $\lambda_{i}+i\mu_{j}$
and $\lambda_{i}-i\mu_{j}$, respectively. The time evolution using $M\otimes
I+I\otimes M$ yields

$\displaystyle e^{2\pi i\lambda_{j}\Delta T}|E_{j}\rangle|\bar{E}_{j}\rangle$
(48)

In addition, the time evolution using $i(M\otimes I+I\otimes M)$ yields

$\displaystyle e^{2\pi i\mu_{j}\Delta T}|E_{j}\rangle|\bar{E}_{j}\rangle$ (49)

Thus, the real and imaginary parts of the eigenvalues are recorded separately
in two ancillae:

$\displaystyle|\lambda_{j}\rangle|\mu_{j}\rangle|E_{j}\rangle|\bar{E}_{j}\rangle$
(50)

Any of those approaches increases the complexity of the quantum circuits. The
former two approaches require additional measurements to determine the complex
amplitude Indeed, before applying those methods, we should prepare a
particular eigenvector that has a complex eigenvalue $\lambda+i\mu$. If not,
the measurements do not report correctly $|\exp(-2^{k}\mu)|$. The third
approach needs a special preparation of the initial state in which conjugate
states are paired.

The occurrence of complex eigenvalues is related to the question of how to
prepare good initial states for the QPE. If we could use classical algorithms,
it would be easy to get rid of complex eigenvalues. We prepare the randomized
initial state vector and project out the components that give rise to complex
eigenvalues. On the other hand, it is laborious to detect complex solutions
only by quantum algorithms. Regarding this issue, there are several ways of
filtering out eigenvalues before applying the quantum phase estimations for
Hermitian operators [22, 23, 24]. In pity, to the best of our knowledge, those
methods are not applied in the removal of complex eigenvalues, since the
existing filtering methods make use of convenient properties of Hermitian
matrices which always have real eigenvalues. Moreover, those methods are
composed to prepare the ground state, namely, the lowest eigenvalue.
Meanwhile, the request of the present study is to obtain all real eigenvalues.
However, the following measures would do that task.

  * •

The inverse power iteration yields the eigenvectors that have the eigenvalues
closest to the given $\lambda$. In the present work, we choose real $\lambda$,
and we use the parallel character of quantum computation.

  * •

To this end, we apply the method with $(A-\lambda I)^{-1}$, solving

$\displaystyle\begin{pmatrix}0&(A-\lambda I)\\\ (A^{T}-\lambda
I)&0\end{pmatrix}\begin{pmatrix}0\\\
x_{k}\end{pmatrix}=\begin{pmatrix}x_{k-1}\\\ 0\end{pmatrix}$ (51)

by some quantum linear system solver. We start from
$|x_{0}\rangle=|\beta_{init}\rangle$, repeat the computation, and after a
suitable number of iterations, obtain the desired quantum states
$|x_{k}\rangle$.

The initial state preparation goes as follows:

  * •

Prepare the initial state:
$\sum_{s}|\beta\rangle\to\sum_{s}|\beta\rangle|e_{s}\rangle$. This state is
implicitly given by $\sum_{s}\sum_{j}C_{j}|v_{j}\rangle|e_{s}\rangle$, where
$\\{v_{j}\\}_{j}$ are the eigenvectors of $A$, and $e_{s}$ is the index to the
sampling points for $\lambda$ in the inverse power method.

  * •

Apply the inverse power method:

$\displaystyle\sum_{s}\sum_{j}C_{j}(A-e_{s})^{-N}|v_{j}\rangle|e_{s}\rangle$
(52)

with a sufficiently large $N$. Then we get

$\displaystyle\sum_{s}\sum_{l}D_{l}|v_{e_{s}}(l)\rangle|e_{s}\rangle$ (53)

where $\\{v_{e_{s}}(l)\\}_{l}$ are the eigenvectors that have the eigenvalue
closest to $e_{s}$.

  * •

Similarly, doubly applying the inverse power method to
$|\beta\rangle|\beta^{\prime}\rangle$ by
$(A-(\lambda+i\mu)I)^{-1}\otimes(A-(\lambda-i\mu)I)^{-1}$, we get the state
$\sum C_{j}|E_{j}\rangle|\hat{E}_{j}\rangle$.

By the measures prescribed above, in general cases, we record the real
eigenvalues in the state vector as follows:

$\displaystyle|\hat{\lambda}(e_{s})\rangle|e_{s}\rangle$ (54)

where $\hat{\lambda}(e_{s})$ is the bit string representation of the
eigenvalue closest to $e_{s}$. Note that one label $|e_{s}\rangle$ shall catch
exactly one real eigenvalue. In exceptional cases, however, we would prepare
an initial state composed of two conjugated eigenvectors that have the
eigenvalues $\lambda\pm i\mu$. The initial state vector is given by

$\displaystyle|\psi\rangle=\left(p|\lambda+i\mu\rangle+q|\lambda-i\mu\rangle\right)|e_{s}\rangle.$
(55)

For such state vectors, the QPE cannot obtain the eigenvalues as in (54).
Instead, the result of the QPE is given by

$\displaystyle\sum_{s}\sum_{k}C_{k}|k\rangle|e_{s}\rangle$ (56)

where $|k\rangle=|k_{0}k_{1}\cdots k_{N}\rangle$. In this case, the label
$|e_{s}\rangle$ is connected to the noisy superposition of the states
$\\{|k\rangle\\}_{k}$. If such a result is measured for the label
$|e_{s}\rangle$, it means that the corresponding eigenvalues are complex. We
should discard them since complex eigenvalues are meaningless in our problem
setting. In the inverse power iteration, we could use
$\lambda-\sqrt{-1}\delta$ (with a shift by small positive $\delta$), instead
of genuine real $\lambda$, so that we shake off the eigenstate that has the
eigenvalue with the positive imaginary part, which causes the unbounded growth
of the amplitude in the time evolution. There is another rare exceptional case
where two different state vectors are involved. This case shall happen that
the sampling point $e_{s}$ is located exactly in the middle of two adjacent
real eigenvalues $lambda_{1}$ and $\lambda_{2}$. The measurement also gives
the noisy superposition of $\\{|k\rangle\\}_{k}$. However, such a circumstance
almost surely does not take place, and a small shift of the sampling point
$e_{s}\rightarrow e_{s}+\delta$ shall avoid it.

If the quantum algorithms for the state preparation do not work well, we are
obliged to use one diagonalization of one of the transformation matrices by
classical algorithms so that we get rid of the right eigenvectors with
complex-valued solutions. The initial state vector could be chosen as an
arbitrary linear combination of the right eigenvectors with real eigenvalues.
It is an expediency, but it has its merit, as we shall discuss later in
Section 3.3.

Note that if we calculate the eigenenergy of QUBO models by the present
method, the set of polynomial equations is given by

$\displaystyle
C_{1}\sum_{i}x_{i}+C_{2}\sum_{i_{1},i_{2}}x_{i_{1}}x_{i_{2}}+\cdots+C_{n}\sum_{i_{1},...,i_{n}}x_{i_{1}}\cdots
x_{i_{n}}-e=0$ (57)

and

$\displaystyle x_{i}^{2}-x_{i}=0\ \text{for}\ i=1,...,n.$ (58)

This kind of equation is without complex solutions, thanks to the restriction
of the ranges of $\\{x_{i}\\}_{i}$, which is explicitly written by the
polynomials. It follows that, if we construct a set of polynomial equations of
the Hartree-Fock model as a QUBO one, there is no problem concerning the
complex eigenvalues, although this construction increases the number of qubits
and the cost of symbolic computations.

###  3.2 The choice of basis vectors in the eigenvalue problems

Note that there is an ambiguity in the choice of the basis vectors. In the toy
model case, the matrix $m_{e}$ has two eigenvectors for eigenvalue 1, which
are given by

$\displaystyle v_{1}$
$\displaystyle=\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},-1,1\right)$ (59)
$\displaystyle v_{2}$
$\displaystyle=\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},-1,1\right)$ (60)

We could choose the basis vectors differently, such as

$\displaystyle w_{1}=\frac{1}{2}\left(v_{1}+v_{2}\right)$
$\displaystyle=(0,0,-1,1)$ (61) $\displaystyle
w_{2}=\frac{1}{\sqrt{2}}\left(v_{1}-v_{2}\right)$ $\displaystyle=(-1,1,0,0)$
(62)

However, $w_{1}$ and $w_{2}$ are not suitable choices in the present problem,
for they are not represented by the monomial basis vector $b=(ye,y,e,1)$.
Indeed, they are not the eigenvectors of $m_{x}$ or $m_{y}$.

$\displaystyle w_{1}\,m_{x}=(-1/2,1/2,0,0)$ (63) $\displaystyle
w_{2}\,m_{x}=(0,0,1,-1)$ (64) $\displaystyle w_{1}\,m_{y}=(-1/2,1/2,0,0)$ (65)
$\displaystyle w_{2}\,m_{y}=(0,0,-1,1)$ (66)

Therefore, if there is a degeneracy of the eigenvalues, we should make the
basis vectors for the corresponding subspace in such a way that all the basis
vectors are potentially given by the monomial basis vector $b$.

###  3.3 The merit of the quantum algorithm

To see the superiority of the quantum algorithm over the classical one, let us
consider the following circumstances. Let $\\{m_{i}\\}_{i}$ be the list of
transformation matrices, and assume that $m_{1}^{T}$ has two eigenvectors
($v_{1}$ and $v_{2}$) with a common eigenvalue $E_{v}$:

$\displaystyle m_{1}^{T}\,v_{1}=E_{v}v_{1},m_{1}^{T}\,v_{2}=E_{v}v_{2}.$ (67)

These two vectors are not necessarily the eigenvectors of the other $m_{i}$ if
they are not suitably prepared, as pointed out in Section 3.2.

In the classical algorithm, we prepare the eigenvectors of the other
$m_{i}^{T}$, say $m_{2}^{T}$, by the generalized eigenvalue problem:

$\displaystyle\left[\begin{pmatrix}(v_{1}|m_{2}^{T}|v_{1})&(v_{1}|m_{2}^{T}|v_{2})\\\
(v_{2}|m_{2}^{T}|v_{1})&(v_{2}|m_{2}^{T}|v_{2})\end{pmatrix}-E_{j}\begin{pmatrix}(v_{1}|v_{1})&(v_{1}|v_{2})\\\
(v_{2}|v_{1})&(v_{2}|v_{2})\end{pmatrix}\right]\begin{pmatrix}c_{1}^{j}\\\
c_{2}^{j}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}$ (68)

From this equation, we get two eigenvectors

$\displaystyle w_{j}=c_{1}^{j}v_{1}+c_{2}^{j}v_{2}\ \text{for}\ j=1,2$ (69)

and two corresponding eigenvalues $E_{w_{1}}$ and $E_{w_{2}}$.

On the other hand, this task of solving the eigenvalue problem can be skipped
in quantum algorithms. To see this, let us use the initial state vector
(combined with ancilla qubits) defined by

$\displaystyle|\psi\rangle$
$\displaystyle=(p|v_{1}\rangle+q|v_{2}\rangle)|Ancilla_{1}\rangle|Ancilla_{2}\rangle\cdots|Ancilla_{N}\rangle$
(70)
$\displaystyle=(s|w_{1}\rangle+t|w_{2}\rangle)|Ancilla_{1}\rangle|Ancilla_{2}\rangle\cdots|Ancilla_{N}\rangle.$

In the above, $p$ and $q$ would randomly be chosen. Consequently, $s$ and $t$
are determined. Then let us apply the QPE by $m_{1}^{T}$ and record the phase
at $|Ancilla_{1}\rangle$. We get

$\displaystyle|\psi\rangle=(s|w_{1}\rangle+t|w_{2}\rangle)|E_{v}\rangle|Ancilla_{2}\rangle\cdots|Ancilla_{N}\rangle$
(71)

Next, apply the QPE by $m_{2}^{T}$ and record the phase at
$|Ancilla_{2}\rangle$. We record the corresponding phases at
$|Ancilla_{2}\rangle$ and get

$\displaystyle|\psi\rangle$
$\displaystyle=s|w_{1}\rangle|E_{v}\rangle|E_{w_{1}}\rangle\cdots|Ancilla_{N}\rangle$
(72)
$\displaystyle+t|w_{2}\rangle|E_{v}\rangle|E_{w_{2}}\rangle\cdots|Ancilla_{N}\rangle$

If $E_{w_{1}}\neq E_{w_{2}}$, the two data $(E_{v},E_{w_{1}})$ and
$(E_{v},E_{w_{2}})$ in $|\psi\rangle$ are distinguished, and they are the
parts of two distinct roots of the given set of polynomial equations. If
$E_{w_{1}}=E_{w_{2}}$, we successively apply the QPE using
$m_{3}^{T}$,…,$m_{N}^{T}$. Then we finally get the distinct roots of the form
$(E_{v},E_{w},E_{w^{{}^{\prime}}},...,E_{w^{(^{\prime}\cdots^{\prime})}})$.
Each of the roots is recorded in one of the orthonormalized bases in the
output state and measured distinctly one from the other since the
orthogonality is guaranteed by the bit string representation of the ancilla.

###  3.4 On the enormous complexity concerning symbolic computation

Another obstacle is the enormous complexity in the computation of Gröbner
basis, which scales with the number of variables ($n$) and the maximal degree
of the input polynomials ($d$). If the primitive algorithm (as was initially
proposed) is applied, the complexity is doubly exponential in $n$ for the
worst case. However, detailed inspections have revealed the following fact
[25].

  * •

Let $(f_{1},...,f_{m})$ be a system of homogeneous polynomials in
$k[x_{1},...,x_{n}]$ where $k$ is an arbitrary field. (A homogeneous
polynomial is composed of nonzero monomials, all of which have the same
degree.)

  * •

The number of operations to compute a Gröbner basis of the ideal
$I=(f_{1},...,f_{m})$ for a graded monomial ordering up to degree $D$ scales
with

$\displaystyle O\left(m\,D\,\begin{pmatrix}n+D-1\\\
D\end{pmatrix}^{\omega}\right)\ \text{as}\ D\rightarrow\infty$ (73)

where $\omega$ is the exponent of matrix multiplication over $k$. Namely,
$\omega$ is the smallest constant such that two $N\times N$ matrices could be
multiplied by performing $O(N\omega+\epsilon)$ arithmetic operations for every
$\epsilon>0$.

In this estimation, the bound $D$ for a full Gröbner basis is not yet given.
However, it could be estimated under a certain assumption, and the conclusion
is that the complexity is simply exponential in $n$, thanks to the assumption
that the polynomials are homogeneous [25]. As any system of polynomials can be
transformed into this form by adding a variable and homogenizing, it means
that the doubly exponential complexity could be avoided.

Note that the estimation of the complexity is carried out for the worst cases,
meanwhile, the actual computations often finish with much lower computational
costs. Moreover, the algorithmic improvements are successful in facilitating
the computation. Currently, the F5 algorithm is regarded as the most effective
one [26]. The complexity of this algorithm was studied in [25]. The formula of
the complexity is given in a refined style that reflects the special feature
of the algorithm, although it is still exponential in $n$.

The complexity in computing Gröbner basis, however, would be mitigated by
quantum algorithms. The computational steps of the Gröbner basis are as
follows [3, 4, 5, 6, 8, 7]

  * •

Input: $F=(f_{1},...,f_{m})$; Output: the Gröbner basis $G$ for $F$

  * •

$G:=F$

  * •

For every pair of polynomials $(f_{i},f_{j})$ in $F$, compute the
s-polynomial, which is defined by

$\displaystyle
S(f_{i},f_{j})=\frac{a_{ij}}{g_{j}}f_{i}-\frac{a_{ij}}{g_{i}}f_{j}.$ (74)

In the above, $g_{i}$ (resp. $g_{j})$ the leading term of $f_{i}$ (resp.
$f_{j}$) in the given monomial ordering, and $a_{ij}$ is the least common
multiple of $g_{i}$ and $g_{j}$.

  * •

Reduce $S(f_{i},f_{j})$. By the division algorithm, it is represented as
$S(f_{i},f_{j})=\sum_{l}c_{l}f_{l}+r$, and the residual $r$ is the result. If
$r\neq 0$, add $r$ to $G$.

  * •

Repeat the computation of s-polynomials and the reduction until the extension
of $G$ terminates.

  * •

Return $G$.

This algorithm is essentially a Gaussian elimination that carries out the
reduction of rows in a matrix that holds the coefficients in the system of
polynomials [27, 26]. The difficulties in conducting that task are as follows.

  * •

In the reduction of $S(f_{i},f_{j})$, it happens that many pairs of the
polynomials reduce to zero, being completely useless to the construction of
the Gröbner basis. It is necessary to detect the unnecessary pairs beforehand,
and the trials to improve the efficiency of the algorithm are intensively
carried out regarding this issue.

  * •

The size of the Gaussian elimination would vary, indeed increase, during the
computation. The computation of s-polynomials shall increase the total number
of the maximum degree of the polynomials in $G$. The expanding matrix requires
ever-increasing usage of a vast amount of memory.

To reduce the computational costs enumerated above, the quantum algorithms
would be hopeful choices. First, the qubits could encompass a vast set of
quantum states that describe a set of data with enormous size. They could
embrace the incessant increase of polynomial data during the computation of
Gröbner basis. Second, the quantum algorithms are efficient in linear
computation and searching for unconstructed data. The HHL algorithm would
facilitate the computation of Gaussian elimination. Moreover, the Grover
database search algorithm would be useful in detecting the terms in
polynomials that should be eliminated in the reduction.

##  4 Conclusion

The main aim of this paper is to illustrate a computational scheme of quantum
computation that enables us to carry out the Hartree-Fock computation, in the
sense that the computation shall realize the optimization of molecular
orbitals composed of atomic bases. The proposed computational scheme uses
algebraic techniques to reform the Hartree-Fock equations into a set of
eigenvalue problems, wherein the eigenvalues give the LCAO coefficients, the
orbital energy, and if necessary, the optimized atomic coordinates. The
eigenvalue problems could be solved by the quantum phase estimation through
the block-encoding technique for non-Hermitian or non-unitary operators. The
computed results are recorded in quantum states, which shall be used for more
complicated quantum computations with the aid of quantum RAM. There are
several unsettled points in the present work. The first is the occurrence of
complex-valued eigenvalues in the eigenvalue problem, which is caused by the
potential occurrence of complex-valued solutions of the Hartree-Fock equation
which is treated as a system of polynomial equations. For the sound
application of the QPE, this sort of eigenstates should be removed by any
means – if it is algorithmically difficult, the quantum device should remove
them. The second is the possibly enormous complexity of symbolic computation.
However, the required symbolic computations are Gauss eliminations, which
would be facilitated by quantum algorithms ever proposed.

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Further author information: (Send correspondence to Y.T.)
Y.T.: E-mail<EMAIL_ADDRESS>
# FINER: Far-Infrared Nebular Emission Receiver for the Large Millimeter
Telescope

Yoichi Tamura  Department of Physics, Nagoya University, Nagoya, Aichi
464-8602, Japan  Takeshi Sakai  University of Electro-Communications, Chofu,
Tokyo 182-8585, Japan  Ryohei Kawabe  National Astronomical Observatory of
Japan, Mitaka, Tokyo 181-8588, Japan  Takafumi Kojima  National Astronomical
Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  Akio Taniguchi
Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan
Tatsuya Takekoshi  Kitami Institute of Technology, Kitami, Hokkaido 090-8507,
Japan  Haoran Kang  National Astronomical Observatory of Japan, Mitaka, Tokyo
181-8588, Japan  Wenlei Shan  National Astronomical Observatory of Japan,
Mitaka, Tokyo 181-8588, Japan  Masato Hagimoto  Department of Physics, Nagoya
University, Nagoya, Aichi 464-8602, Japan  Norika Okauchi  Department of
Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan  Airi Tetsuka
University of Electro-Communications, Chofu, Tokyo 182-8585, Japan  Akio K.
Inoue  Waseda University, Shinjuku, Tokyo 169-8555, Japan  Kotaro Kohno
Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015, Japan
Kunihiko Tanaka  Keio University, Yokohama, Kanagawa 223-8522, Japan  Tom
J.L.C. Bakx  Chalmers University of Technology, Onsala Observatory, SE-439 94
Onsala, Sweden  Yoshinobu Fudamoto  Chiba University, Chiba 263-8522, Japan
Kazuyuki Fujita  Institute of Low Temperature Science, Hokkaido University,
Sapporo, Hokkaido 060-0819, Japan  Yuichi Harikane  Institute of Cosmic Ray
Research, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan  Takuya
Hashimoto  Tsukuba University, Tsukuba, Ibaraki 305-8751, Japan  Bunyo
Hatsukade  National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588,
Japan  David H. Hughes  Instituto Nacional de Astrofísica, Óptica y
Electrónica (INAOE), Tonantzintla, Puebla 72480, Mexico  Takahiro Iino
Information Technology Center, The University of Tokyo, Kashiwa, Chiba
277-0882, Japan  Yuki Kimura  Institute of Low Temperature Science, Hokkaido
University, Sapporo, Hokkaido 060-0819, Japan  Hiroyuki Maezawa  Osaka
Metropolitan University, Sakai, Osaka 599-8531, Japan  Yuichi Matsuda
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  Ken
Mawatari  Waseda University, Shinjuku, Tokyo 169-8555, Japan  Taku Nakajima
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya,
Aichi 464-8601, Japan  Shunichi Nakatsubo  Institute of Space and
Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara,
Kanagawa 252-5210, Japan  Tai Oshima  National Astronomical Observatory of
Japan, Mitaka, Tokyo 181-8588, Japan  Hideo Sagawa  Kyoto Sangyo University,
Kyoto 603-8555, Japan  F. Peter Schloerb  University of Massachusetts,
Amherst, MA 01003, USA  Shigeru Takahashi  Information Technology Center, The
University of Tokyo, Kashiwa, Chiba 277-0882, Japan  Kotomi Taniguchi
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Akiyoshi Tsujita  Institute of Astronomy, The University of Tokyo, Mitaka,
Tokyo 181-0015, Japan  Hideki Umehata  Department of Physics, Nagoya
University, Nagoya, Aichi 464-8602, Japan  Teppei Yonetsu  Osaka Metropolitan
University, Sakai, Osaka 599-8531, Japan  Min S. Yun  University of
Massachusetts, Amherst, MA 01003, USA

###### Abstract

Unveiling the emergence and prevalence of massive/bright galaxies during the
epoch of reionization and beyond, within the first 600 million years of the
Universe, stands as a pivotal pursuit in astronomy. Remarkable progress has
been made by JWST in identifying an immense population of bright galaxies,
which hints at exceptionally efficient galaxy assembly processes. However, the
underlying physical mechanisms propelling their rapid growth remain unclear.
With this in mind, millimeter and submillimeter-wave spectroscopic
observations of redshifted far-infrared spectral lines, particularly the [O
iii] 88 $\mu$m and [C ii] 158 $\mu$m lines, offers a crucial pathway to
address this fundamental query.

To this end, we develop a dual-polarization sideband-separating
superconductor-insulator-superconductor (SIS) mixer receiver, FINER, for the
Large Millimeter Telescope (LMT) situated in Mexico. Harnessing advancements
from ALMA’s wideband sensitivity upgrade (WSU) technology, FINER covers radio
frequencies spanning 120–360 GHz, delivering an instantaneous intermediate
frequency (IF) of 3–21 GHz per sideband per polarization, which is followed by
a set of 10.24 GHz-wide digital spectrometers. At 40% of ALMA’s light-
collecting area, the LMT’s similar atmospheric transmittance and FINER’s 5
times wider bandwidth compared to ALMA culminate in an unparalleled spectral
scanning capability in the northern hemisphere, paving the way for finer
spectral-resolution detection of distant galaxies.

###### keywords:

Millimeter and submillimeter instrumentation, Heterodyne receiver, SIS
receiver, LMT, ALMA, High-redshift universe, Galaxy formation, Interstellar
medium

##  1 INTRODUCTION

###  1.1 Scientific Background and Key Science

How and when first stars and galaxies form is one of the most fundamental
questions in modern astronomy. The pristine materials in the early universe
cooled down to form neutral hydrogen at redshift $z\sim 1000$ or 380k years
after the Big Bang. Once first stars and galaxies are born, however, the
hydrogen gas in the intergalactic space is ‘reionized’ by hydrogen-ionizing
photons from the hot, massive stars. This so-called cosmic reionization is
indirect evidence for the birth of the first luminous objects. In recent
years, Planck satellite has reported that cosmic reionization occurred in the
period centered at redshift $z=7.8$ (660 million years after the Big Bang),[1]
whereas hundreds of galaxy candidates have been identified with Hubble and
James Webb Space Telescopes (HST and JWST) even before the period[2, 3, 4, 5,
6, 7, 8, 9, 10], meaning that we are likely witnessing the birth of the
earliest galaxies in the universe (Figure 1).

In particular, finding massive galaxies in the era is difficult but is
expected to offer unique information on galaxy formation, because the
probability of finding them (i.e., volume density or luminosity function)
depends on cosmic structure formation models. Recent discoveries of the
excessive massive galaxy candidates at $z>8$ made by HST and JWST suggest an
extremely high efficiency of galaxy assembly in the earliest Universe, while
it is still unclear what drives the rapid growth of the massive population.
Then, the key questions we are trying to address in this research include (i)
when massive galaxies emerge, (ii) how common they are in the pre-reionization
era, and (iii) what controls their growth. It is, however, difficult to
confirm galaxy ‘candidates’ in the era; in general, identifying an object as a
galaxy and determining its physical properties begin with spectroscopy of
atomic or molecular spectral lines. But, the combination of bright spectral
lines and a sensitive instrument that are sufficient for detection of a galaxy
even in the furthest universe was very limited.

In this study, we aim for a sensitive spectroscopic survey of galaxies in the
pre-reionization era using the far-infrared fine-structure lines, the
${}^{3}\\!P_{1}\rightarrow\,^{3}\\!P_{0}$ transition of doubly-ionized oxygen
[O iii] 88.3 $\mu$m (3393.01 GHz) and the
${}^{2}\\!P_{1/2}\rightarrow\,^{2}\\!P_{3/2}$ transition of ionized carbon [C
ii] 157.7 $\mu$m (1900.54 GHz), based on our novel mm and submm-wave receiver,
FINER, for the Large Millimeter Telescope (LMT).[11, 12] The [O iii] line is
demonstrated to be bright in ultraviolet-selected galaxies[13] and now becomes
a useful tracer in identifying their spectroscopic redshifts[14, 15] (Figure
2) even compared with JWST. We also aim to elucidate their physical
properties, such as gas-phase metallicity, ionizing photon and gas densities,
and multi-phase gas structure, such as a covering fraction of neutral gas in
photo-dissociation regions. Those physics are the key to unveil the origins of
the extraordinary efficiency of star formation in the earliest massive
galaxies JWST has found.

Figure 1: Galaxy formation at the epoch of reionization and beyond (pre-EoR)
in the history of the Universe. [Image credit: NASA, ESA, P. Oesch and B.
Robertson (University of California, Santa Cruz), and A. Feild (STScI)]

###  1.2 Broad Impact and Complementary Role of FINER

With these capabilities, FINER offers broader science cases in addition to the
EoR galaxy science (see Table 1), such as solar system, interstellar medium
and astrochemistry, star formation, nearby and distant submillimeter-bright
galaxies. Figure 3 shows how FINER complements the capabilities of LMT’s
existing receivers. FINER fully covers the same frequency bands as TolTEC but
with broad, high-dispersion spectroscopic capabilities. Although FINER is a
single-beam receiver, it offers the imaging spectroscopic capability, which
complements SEQUOIA and allows one to seamlessly cover the mm to submm bands.
As shown in the right panel of Figure 3, FINER is the unique instrument
offering the highest spectral-resolving power $R\equiv f/\Delta f$ over the
frequency range from 125 to 210 GHz, which is useful for planetary, Galactic,
and extragalactic spectroscopic studies and thus will benefit the LMT user
community. It also offers the widest instantaneous frequency coverage with a
modest spectral resolving power $R=10,500$–17,500 among the LMT receivers
operating at 120–360 GHz (the bottom edge of the ‘FINER’ rectangle in Figure 3
left).111New capabilities will be delivered by the tri-band receiver for Event
Horizon Telescope and a 1-mm array receiver OMAyA, which are being
developed.[12]

In addition, the improved continuum sensitivity offered by FINER’s wide
frequency coverage will benefit telescope calibration measurements, such as
relative pointing, focusing, astigmatism and beam efficiency measurements. The
instantaneous IF bandwidth covered by the DRS4 spectrometers is 33 GHz per
polarization, which is approximately $20\times$ wider than the current per-
polarization bandwidth of MSIP1mm. This wideband capability will increase the
continuum sensitivity by a factor of $\approx 4.5$, contributing not only to
the improvement in the calibration accuracy but also to a substantial decrease
in the overhead time for telescope calibration. Accurate measurements of
telescope performances with a single receiver system will reduce potential
systematics of the measurements and will allow us to more precisely understand
the thermal, mechanical, and optical performances of the telescope across a
wide frequency range from 120 to 360 GHz. This can also be verified and cross-
checked by TolTEC, providing reliable results for the telescope parameters
that benefit every type of science observation with the LMT.

Figure 2: The [O iii] 88 $\mu$m line detected a $z=8.31$ Lyman break galaxy, MACS0416_Y1[15]. The spectrum was obtained by four tunings of ALMA. Redshift uncertainty of a typical galaxy candidate at $z\sim 10$ is $\Delta z\sim 1$, requiring a $>30$ GHz spectral coverage. ALMA requires 4 to 6 tunings, whereas FINER only needs a single tuning to cover $>30$ GHz. Table 1: Summary of potential science cases afforded by LMT-FINER. Scientific field | Use cases
---|---
Solar system |  $\bullet$ High resolution spectral-line mapping of planetary atmosphere
|  $\bullet$ Molecular line survey
|  $\bullet$ ToO/monitoring observations
Star-formation and astrochemistry |  $\bullet$ Molecular line survey
|  $\bullet$ High angular resolution spectral-line mapping
Dusty star forming galaxies |  $\bullet$ Redshift identification in CO, [CI], and [CII]
|  $\bullet$ CO spectral energy distribution
|  $\bullet$ Physical and chemical properties of ISM
Galaxy formation |  $\bullet$ Redshift identification in [OIII] and [CII]
|  $\bullet$ Physical and chemical diagnostics of ISM
Figure 3: Instantaneous on-sky frequency coverage and resolving power of LMT
receivers as a function of frequency. The gray and blue rectangles represent
the capabilities of existing single and multi-beam receivers/camera,
respectively. FINER and its potential 3-mm extension (denoted as FINER+, see
also Section 3) are represented by the orange rectangles.

##  2 INSTRUMENT

FINER is a dual-polarization sideband-separating superconductor-insulator-
superconductor (SIS) heterodyne receiver system which comprises two frequency
bands covering 120–210 GHz (Band 4+5) and 210–360 GHz (Band 6+7), each of
which has an intermediate frequency (IF) bandwidth of 3–21 GHz per sideband.
The feeds of the two frequency bands are placed on the focal plane separately,
allowing simultaneous dual-band observations with an on-sky beam separation of
60 arcsec. FINER employs novel SIS mixers with a high critical current density
($J_{\rm c}$) circuit to increase the IF bandwidth[16, 17]. It is also
equipped with an array of FPGA-based digital spectrometers with the total
bandwidth of 163.84 GHz ($16\times 10.24$ GHz).

This offers an instantaneous bandwidth $5\times$ wider than ALMA, while
achieving the same sensitivity as ALMA (Figure 2). LMT-FINER will provide 40%
of ALMA’s light-collecting area, a similar atmospheric transmittance to ALMA,
and a $5\times$ wider bandwidth than ALMA. This offers the most sensitive
spectral-scanning capability among mm/submm telescopes in the northern
hemisphere (Figure 4).

Figure 4: The expected intensities of the [O iii] and [C ii] lines expected
for a bright Lyman break galaxy with an apparent rest-UV magnitude of $m_{\rm
AB}=25$ (blue and yellow curves) and the $5\sigma$ limiting sensitivity of
LMT-FINER (gray shade) as a function of observing frequency. The sensitivity
is estimated for a 10-hour on-source observation.

###  2.1 Science Requirements and Specifications

The level-1 science requirements are defined as follows.

  * •

A single Lyman break galaxy with $H=25.0$ AB and $\Delta z/(1+z)=0.1$ needs to
be detected in [O iii] 88 $\mu$m at a S/N = 5 or higher within a single night
(8 hr) observation.

  * •

The redshifted [O iii] and [C ii]at $z=9$ to 15 need to be observed by a
tunable receiver.

  * •

The expected peak flux density of [O iii] 88 $\mu$m is 5 mJy at $z=10$.

The level-1 requirements are broken down to the technical specifications as
listed in the science traceability matrix (Table 2). Table 3 summarizes the
specifications of FINER. Figure 4 shows how the specified capabilities meet
the science requirements.

Table 2: Science traceability matrix. Level | Requirements
---|---
Level 1: Science objectives |  |  $\bullet$ To measure the bright-end UVLF ($M_{\rm UV}<-22$) of galaxies at $z=8.7$ to 11.
---
$\bullet$ To constrain the bright-end UVLF ($M_{\rm UV}<-22.5$) of galaxies at
$z=12$ to 15.
$\bullet$ To constrain a covering fraction of neutral medium surrounding the
ionized regions
of bright-end galaxies.
Level 2: Science requirements |  Science observables |  |  $\bullet$ Spectroscopic redshift of a candidate galaxy selected by the
---
Lyman break technique.
$\bullet$ [C ii] 158 $\mu$m to [O iii] 88 $\mu$m luminosity ratio.
$\bullet$ The number of spectroscopically-identified galaxies.
Measurement requirements |  |  $\bullet$ To detect [O iii] 88 $\mu$m of a candidate $z\sim 10$ galaxy in 24 hours,
---
with a peak flux density $S_{\rm peak}=4$ mJy, a velocity width of
150 km/s, and a prior photometric-redshift uncertainty of $\Delta z=1$.
$\bullet$ To detect [C ii] 158 $\mu$m or put a meaningful constraint on its
flux in 24 hours.
$\bullet$ Forty (20) spectroscopically-identified galaxies for goal
(baseline).
Level 3: Instrument requirements |  Technical parameters |  |  $\bullet$ Tunable frequency range.
---
$\bullet$ Instantaneous on-sky frequency coverage.
$\bullet$ Spectral resolution.
$\bullet$ Sensitivity.
Technical requirements |  |  $\bullet$ Tunable frequency range: 120–360 GHz with no gap (except for
---
the telluric water vapor lines at 183 and 325 GHz).
$\bullet$ Instantaneous on-sky frequency coverage: $>30$ GHz ($\Delta z>1$ for
[O iii] at $z=10$) with as small a gap as possible.
$\bullet$ Spectral resolution: 20 MHz ($\Delta V=50$ km/s at 120 GHz,
17 km/s at 350 GHz).
$\bullet$ Sensitivity: Receiver noise temperature (single sideband) of 45 K
for 120–210 GHz and 75 K for 210–350 GHz, yielding system noise
temperature of $T_{\rm sys}\leq 100$ K at 120 GHz, $\leq 200$ K at 300 GHz,
$\leq 300$ K at 350 GHz under a precipitable water vapor (PWV) of
2 mm.
Level 4: Mission requirements |  |  $\bullet$ PWV $\leq$ 2 mm (270–350 GHz), $\leq$ 3 mm (120–270 GHz).
---
$\bullet$ Surface error $\leq 75~{}\mu$m RMS ($>270$ GHz), $\leq 90~{}\mu$m
RMS ($<270$ GHz).
$\bullet$ 10 hours per source (including overheads) times 40 (20) nights.
Table 3: FINER specifications. Item | Baseline | Goal
---|---|---
Frequency (Band 4+5) | 125–210 GHz | 120–210 GHz
Frequency (Band 6+7) | 210–350 GHz | 210–360 GHz
Beam size |  $7\times({\rm Frequency/200~{}GHz})^{-1}$ arcsec
No. of polarization | 2
$T_{\rm RX}$ (Band 4+5) | 45 K ($>95\%$ of IF band)
$T_{\rm RX}$ (Band 6+7) | 75 K ($>95\%$ of IF band)
IF | 4–20.5 GHz | 3–20.5 GHz
Sideband separation | 2SB, $>10$ dB | 2SB, $>25$ dB
No. of basebands | 8 | 16
Bandwidth (wide) | 81.92 GHz ($8\times 10.24$ GHz) | 163.84 GHz ($16\times 10.24$ GHz)
Spectral resolution (wide) | 20 MHz
Bandwidth (fine) | 15.0 GHz ($6\times 2.5$ GHz) | 20.0 GHz ($8\times 2.5$ GHz)
Spectral resolution (fine) | 88 kHz
Data dumping rate | 100 ms

###  2.2 System

The system block diagram is shown in Figure 5. The system comprises the
following subsystems; optics, frontend receiver, backend spectrometer,
computers and software, which are described in the following sub-sections.

Figure 5: The system block diagram of FINER.

####  2.2.1 Optics subsystem

Warm optics include a pickup mirror (M5) and a chain of focusing mirror(s), a
beam switching mechanism called the fast sky chopper (optional), and a load
calibration device. The detailed optical parameters and layout are to be fixed
(as of the time of writing). The pickup mirror is placed on a motorized linear
slider and can be controlled remotely. The fast sky chopper is a set of
removable mirrors to change the optical path slightly and to throw the on-sky
beam position by $120^{\prime\prime}$ at a frequency of $\sim 1$–2 Hz, which
allows fast beam-switching observations. The number of available bands is
limited to 1 when using the fast sky chopper due to the limitation of optical
configuration.

####  2.2.2 Front-end subsystem

The receiver is coupled with the optics through corrugated feed horns. FINER
uses the SIS receiver technology with wide RF and IF[16, 17], which has been
developed for ALMA Wideband Sensitivity Upgrade (WSU)[18]. Band 4+5 and 6+7
components are accommodated by a single cryostat, which is cooled with a two-
staged GM 4-K cryocooler (RDK-415DP, Sumitomo Heavy Industry Inc.). Both bands
are available simultaneously with the on-sky beam separation of
$60^{\prime\prime}$. The observed frequencies of both bands are set
independently. The receiver detects two polarizations in both upper and lower
sidebands (USB and LSB) separately, yielding 4 IF outputs in each band. Note
that it does not offer Stokes parameters other than intensity (i.e.,
Stokes-$I$).

####  2.2.3 Back-end subsystem

The IF signals are sent to the spectrometers. FINER offers two types of
spectrometers; wideband and high-resolution modes realized with DRS4 (Elecs
Inc., Figure 6, Hagimoto et al., 2024 in this volume of Proc. SPIE) and XFFTS
(RPG Inc.)[19], respectively. Users can select one of the spectrometer modes.

DRS4 directly samples the signal at 20.48 GSps with a good analog transmission
up to 20.48 GHz. This offers the capability of spectroscopy in the 10.24–20.48
GHz band with no secondary down-converters, in addition to the baseband (DC to
10.24 GHz). DRS4 is an FX-type spectrometer with an optional output of cross-
correlation for detailed calibration such as digital sideband separation
correction. The input signal is digitized with a 3 bit 20.48 GSps analog-to-
digital converter and is Fourier-transformed with one of window functions,
rectangular, Hann, and Hamming, and then is squared and integrated to produce
a 512 channel power spectrum at a dumping rate of 100 ms, while it does not
offer a polyphase filterbank capability due to a limitation of the
computational resource. It also offers a digital sideband separation
capability [20, 21], to compensate for the imbalance of complex gain (i.e.,
amplitude and phase) between the sidebands and to remove the image sideband
signal digitally (under development).

XFFTS is an off-the-shelf spectrometer with high spectral resolution (88 kHz,
corresponding to $\approx 0.13$ km s-1 at 200 GHz) and a 2.5-GHz
bandwidth.[19] Six boards of XFFTS are operational for B4R, and we will use
them for FINER.

Figure 6: The digital spectrometer DRS4 with four 20.48 GSps samplers (bottom)
and the 20.48 GHz reference signal generator (top).

####  2.2.4 Computer and software

A control computer at the LMT backend cabin controls receiver tuning including
local oscillator (LO) frequency setup and LO power optimization, spectrometer
mode selection, pickup mirror drive, beam switching, and load calibration. The
spectrometer outputs are recorded with NTP-synced timestamps on a data
acquisition computer in the LMT backend cabin at a dumping rate of 100 ms.
FINER is synchronized with LMT’s telescope control system (TCS) through TCP/IP
or UDP connection and does not require any hardware reference clocks from the
TCS.

###  2.3 Hardware Configuration

The proposed optics layout is shown in Figure 7. As we intend to replace the
existing B4R receiver222The B4R (band 4 receiver) is the 2-mm receiver system
which consists of a single-beam two-polarization SIS receiver operating at
125–163 GHz and four XFFTS spectrometers. B4R is based on the ALMA band-4 2SB
mixer technology [22]. See
http://lmtgtm.org/telescope/instrumentation/instruments/b4r/?lang=en for
details. (PI: Kawabe, Figure 8) with FINER, the layout is almost the same as
B4R; The ray from M4 to M5 (yellow beam toward SEQUOIA) is bent by a removable
pickup mirror. The bent ray is directly led into the FINER cryostat and is
coupled with the two feed horns of Band 4+5 and 6+7. FINER has a beam-
switching mechanism in between the pickup mirror and the cryostat, as well as
a chopper wheel for the standard single-load calibration.

The cryostat is placed at the position of B4R in the LMT receiver cabin. We
will replace B4R’s cryostat and cryocooler with new ones, while the helium
hoses and chiller of B4R will be used for FINER. The LO system and digitizers
are placed alongside. The IF signal is sent to the spectrometers in the
backend cabin downstairs.

Figure 7: The configuration of the FINER receiver and warm optics in the LMT
receiver cabin. The left and right panels show the CAD layout in bird’s eye
view and from the direction parallel to the elevation axis, respectively.
FINER is indicated by a green box. The beam ray from the fourth mirror (M4) to
the fifth mirror (M5) is represented as the yellow bars, part of which is
picked off by the FINER warm coupling optics. The location of FINER is similar
to that of B4R. Figure 8: The B4R receiver system.

###  2.4 Data Analysis Software

The observed data will be provided in the MeasurementSet (MS) format of
casa[23], in which time series spectra taken on- and off-source and for load
calibration are contained along with telescope boresight positions, ancillary
environmental records and metadata.

Users will receive MSs as initial datasets and will use casa for fundamental
data reduction including intensity and bandpass calibration, flagging,
velocity correction, spectrum making, and map/cube making. Further analysis
such as spectral fit can also be made with the single-dish tasks of casa. A
new data-scientific noise-removal method demonstrated using the B4R
receiver[24], which reduces the baseline ripples and improves the observed
sensitivity by a factor of $\sqrt{2}$, will also be provided by the
development team.

##  3 DEVELOPMENT PLAN

The development is led mainly by Nagoya University, the University of Electro-
Communications (UEC), Advanced Technology Center (ATC) at NAOJ, Kitami
Institute of Technology, the University of Tokyo, and Keio University in
collaboration with the LMT project. Development, commissioning, and early
science operations of FINER are planed in 2022–2027. We plan to deliver the
instrument to LMT and to start commissioning and science operations in
2025/26.

At the initial phase FINER will begin with the ‘Baseline’ specifications
(Table 3) and will aim for the ‘Goal’ specifications when technologies and
resources are ready. As the SIS mixer technology is likely being improved
after FINER gets operational, we may upgrade the SIS mixer device if this is
the case. The back-end spectrometers are also added to enhance the capability.
The data reduction pipeline will be improved by implementing a new algorithm
for noise removal.[24] The FINER cryostat and warm optics are designed so that
another additional single-beam receiver, such as a 3–4 mm band receiver, may
be accommodated in the future, without any conflict with the 120–360 GHz band
(see the FINER+ parameter space in Figure 3).

###### Acknowledgements.

This study is financially supported by JSPS KAKENHI (No. 22H04939, 23K20035),
NAOJ Joint Development Research Programs (No. 1901-0101, 2001-0104), and the
Grant for Joint Research Program of the Institute of Low Temperature Science,
Hokkaido University (23G037 and 24G036)

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