1]University of North Texas, Denton, TX, 76203, USA 2]Santa Fe Institute,
Santa Fe, NM, 87501, USA

# Orthogonal-Ansatz VQE: Locating excited states without modifying a cost-
function

Kyle Sherbert  Marco Buongiorno Nardelli  [  [<EMAIL_ADDRESS>
###### Abstract

Most literature in the Variational Quantum Eigensolver (VQE) algorithm focuses
on finding the ground state of a physical system, by minimizing a quantum-
computed cost-function. When excited states are required, the cost-function is
usually modified to include additional terms ensuring orthogonality with the
ground state. This generally requires additional quantum circuit executions
and measurements, increasing algorithmic complexity. Here we present a design
strategy for the variational ansatz which enforces orthogonality in candidate
excited states while still fully exploring the remaining subset of Hilbert
space. The result is an excited-state VQE solver which trades increasing
measurement complexity for increasing circuit complexity. The latter is
anticipated to become preferable as quantum error mitigation and correction
become more refined. We demonstrate our approach with three distinct ansatze,
beginning with a simple single-body example, before generalizing to
accommodate the full Hilbert space spanned by all qubits, and a constrained
Hilbert space obeying particle number conservation.

[wires=4]$\ket{z_{0}}$ & [wires=4]V_0 [wires=4]$\ket{\psi_{0}}$

(a) Ground-state circuit.

[wires=4]$\ket{z_{1}}$ & [wires=4]V_1(θ) [wires=4]V_0
[wires=4]$\ket{\psi_{1}}$

(b) First-excited state circuit.

Assumptions:

$\displaystyle\matrixelement{z_{0}}{\hat{V}_{1}}{z_{0}}=1$
$\displaystyle\matrixelement{z_{0}}{\hat{V}_{1}}{z_{1}}=0$

Guarantees:

$\displaystyle\innerproduct{\psi_{0}}{\psi_{1}}=0$

Figure 1: Given a quantum circuit $V_{0}$ to transform a particular reference
state $\ket{z_{0}}$ into a particular solution state $\ket{\psi_{0}}$ (Fig.
1(a)), design a quantum circuit $V_{1}({\bf\it\theta})$ which acts trivially
on the original reference state $\ket{z_{0}}$ for all ${\bf\it\theta}$.
Applying $V_{1}$ and then the original circuit $V_{0}$ to an orthogonal
reference state $\ket{z_{1}}$ (Fig. 1(b)) prepares an ansatz $\ket{\psi_{1}}$
guaranteed to be orthogonal to the original solution $\ket{\psi_{0}}$. Our
paper offers several circuit families with this property and applies them to
locate excited electronic states.

##  1 Introduction

Quantum chemistry and materials science are anticipated to be among the first
of applications transformed by quantum computing. Several promising algorithms
have been developed over the past two decades for characterizing electronic
states, including the Variational Quantum Eigensolver (VQE) [1, 2, 3], Quantum
Phase Estimation [4, 5, 6], Quantum Subspace Expansion [7, 8, 9], Quantum
Imaginary Time Evolution [10, 11], and many others. See [12] for a thorough
review.

Variational algorithms like VQE are particularly popular today because their
hybrid quantum-classical approach renders them compatible with the available
qubit count, connectivity, decoherence, and gate fidelity, which pose severe
engineering constraints on many other algorithms’ efficacy. In VQE, a
parameterized quantum circuit is used to prepare an arbitrary trial state,
whose energy is measured by estimating the expectation value of a molecule or
material’s Hamiltonian operator $\hat{H}$. The parameters are then updated
according to a classical optimization routine in an attempt to minimize the
energy. Despite theoretical challenges in minimizing the potential energy
surface of a chemical system [13, 14], many recent experiments have reported
that VQE implemented on actual quantum hardware is capable of producing a
high-quality estimate of the ground state.[15, 16, 17]

Many properties of interest require knowledge not only of the ground-state,
but of excited states as well. Most strategies for locating excited states
involve a modification of the system Hamiltonian, and therefore the cost-
function to be measured on the quantum computer. For example, if one is
interested in the lowest excited state above an energy $E_{0}$, one may
minimize the energy of a “folded” Hamiltonian $(\hat{H}-E_{0})^{2}$, which
will tend to quadratically increase the complexity of the cost-function.[18]
More commonly, one attempts to find excited states iteratively, beginning from
the ground state and working up one energy level at a time. One strategy is to
directly subtract from the Hamiltonian the subspace spanned by the ground
state [19], but we find this approach quickly leads to intractably complex
cost-functions. This method is refined in Orthogonally-Constrained VQE (OC-
VQE): one measures the energy of a trial state as usual, but adds to the cost-
function an additional term obtained from a subsequent circuit measuring
orthogonality with the ground state, which ensures the trial state will find
its optimal value in the excited subspace.[20, 21, 18] As each excited state
is located, every cost-function evaluation requires running additional
circuits to guarantee each trial state optimizes to the correct subspace.

In this paper, we present Orthogonal-Ansatz VQE (OA-VQE), an alternative
iterative eigensolver comparable to OC-VQE. In OA-VQE, trial-states are
directly constrained to explore the excited subspace by modification of the
parameterized quantum circuit rather than the cost-function. This has the
immediate benefit of keeping the cost-function consistent for each excited
state optimization - measurement complexity is determined only by the
complexity of the Hamiltonian itself. Additionally, because the circuit must
explore a smaller subspace each time a new excited state is located, the
optimization step in OA-VQE becomes easier as the algorithm proceeds (albeit
with a possibly longer circuit).

In Section 2, we describe the properties a parameterized quantum circuit
family must have to be compatible with OA-VQE, and we summarize the algorithm.
In Sections 3, 4, and 5, we present circuit families suitable for performing
OA-VQE for various chemical systems and qubit mappings. In Section 6, we
conclude by commenting on the limitations and potential of our algorithm.

##  2 Orthogonal-Ansatz VQE

Suppose the active space of our system of interest comprises the $N$
computational basis vectors $\ket{z_{i}}$, where $i\in\quantity{0,1...N-1}$.
Our objective is to iteratively construct an alternate basis (such as an
eigenbasis for some observable) with vectors
$\ket{\psi_{l}}\in\mathrm{Span}\quantity{\ket{z_{i}},i<N}$ guaranteeing
orthogonality: $\innerproduct{\psi_{k}}{\psi_{l}}=\delta_{kl}$.

Define the family of unitary operators $\hat{\Omega}_{l}$ with the following
conditions:

$\displaystyle\hat{\Omega}_{l}\hat{\Omega}_{l}^{\dagger}=\hat{\Omega}_{l}^{\dagger}\hat{\Omega}_{l}=\hat{I}$
(1) $\displaystyle\matrixelement{z_{i<l}}{\hat{\Omega}_{l}}{z_{i}}=1$ (2)
$\displaystyle\matrixelement{z_{i<l}}{\hat{\Omega}_{l}}{z_{l}}=0$ (3)

Eq. 1 is the unitary constraint. Eq. 2 ensures that the $l$th $\hat{\Omega}$
acts trivially on all basis vectors indexed before $l$. Note that Eq. 2
implies $\hat{\Omega}^{\dagger}$ must act trivially on the same vectors. Eq. 3
is redundant (it can be inferred from Eqs. 1 and 2) but instructive: it
ensures that the $l$th $\hat{\Omega}$ operating on the $l$th basis vector
results in a vector which spans only those basis vectors indexed on or after
$l$.

We now construct $\ket{\psi_{l}}$ as:

$\displaystyle\ket{\psi_{l}}=\prod_{i=0}^{l}\hat{\Omega}_{i}\ket{z_{l}}$ (4)

The product is ordered such that $\hat{\Omega}_{l}$ acts directly on
$\ket{z_{l}}$. In quantum computing parlance, the circuit implementing
$\hat{\Omega}_{l}$ is applied to the state $\ket{z_{l}}$, and then each
preceding circuit is applied, ending with the one implementing
$\hat{\Omega}_{0}$.

Consider the inner product $\innerproduct{\psi_{k}}{\psi_{l}}$ for $k<l$:

$\displaystyle\innerproduct{\psi_{k}}{\psi_{l}}$
$\displaystyle=\bra{z_{k}}\quantity(\hat{\Omega}_{k}^{\dagger}...\hat{\Omega}_{0}^{\dagger})\quantity(\hat{\Omega}_{0}...\hat{\Omega}_{k}\hat{\Omega}_{k+1}...\hat{\Omega}_{l})\ket{z_{l}}$
$\displaystyle=\bra{z_{k}}\hat{\Omega}_{k+1}...\hat{\Omega}_{l}\ket{z_{l}}$
(5)

where equality is due to Eq. 1. Because of Eq. 2, the bra
$\bra{z_{k}}\hat{\Omega}_{k+1}$ can be contracted to $\bra{z_{k}}$. Repeating
this contraction, we are left with

$\displaystyle\innerproduct{\psi_{k}}{\psi_{l}}=\innerproduct{z_{k}}{z_{l}}=0$
(6)

Thus, the orthogonality of $\ket{\psi_{l}}$ is guaranteed.

Now we will describe the Orthoganal-Ansatz Variational Quantum Eigensolver
(OA-VQE) algorithm. We would like to find the eigenstates of a Hermitian
operator $\hat{H}$ acting on a given active space. We should construct a
parameterized quantum circuit $\hat{V}_{0}({\bf\it\theta})$ such that the
ansatz $\ket{\Psi({\bf\it\theta})}=\hat{V}_{0}({\bf\it\theta})\ket{z_{0}}$
thoroughly explores the entire active space, and variationally locate the
parameters ${\bf\it\theta}_{0}$ which minimize the expectation value
$\expectationvalue{\hat{H}}{\Psi}$. We identify the first of our
orthogonalization operators $\Omega_{0}\equiv\hat{V}({\bf\it\theta}_{0})$.
Next we should construct a new parameterized quantum circuit
$\hat{V}_{1}({\bf\it\theta})$ such that $\hat{V}_{1}({\bf\it\theta})$ acts
trivially on $\ket{z_{0}}$ (Eq. 2), and
$\ket{\Phi({\bf\it\theta})}=\hat{V}_{1}({\bf\it\theta})\ket{z_{1}}$ thoroughly
explores the span of every basis vector in the active space except
$\ket{z_{0}}$ (Eq. 3). A new ansatz
$\Psi({\bf\it\theta})=\hat{\Omega}_{0}\ket{\Phi({\bf\it\theta})}$ is
guaranteed from the arguments above to be orthogonal to the ground state;
minimizing its expectation value $\expectationvalue{\hat{H}}{\Psi}$ therefore
locates the first excited state. If circuits can be effectively parameterized
to iteratively omit each basis vector, this protocol may be repeated to obtain
as many eigenstates as desired, all without needing to alter the cost-
function. As an additional benefit, one may expect each optimization to
require fewer and fewer resources because subsequent eigenstates are located
in a lower-dimensional space.

##  3 Single-Body Circuits

The eigendecomposition protocol described in Section 2 requires designing
quantum circuits which thoroughly explore the Hilbert space spanned by certain
basis vectors, while acting trivially on others. The next three sections
present examples to illustrate how such circuits may be constructed for a
specific problem. In the first example, we will briefly re-present a family of
circuits used to solve the band-structure of a periodic system.

(a) Toy model band structure

(b) Worst-case optimization error.

Figure 2: Fig. 2(a) is the band structure and quantum variational error for a
toy model of Polonium. The black curves give the analytically calculated band
structure, while diamonds mark the values obtained using OA-VQE on a noiseless
quantum simulator using the COBYLA optimization routine. The central inset is
a schematic of the $s$ and three $p$ orbitals in a simple cubic lattice. The
left inset labels the high-symmetry points in the corresponding Brillouin
zone. Fig. 2(b) marks the worst-case error for each band. The single-body
circuit from Section 3 is approximation-free, and optimization error is
negligible. The compact circuit from Section 4 introduces approximation error
via Trotterization, but this quickly falls off by Trotter-order $r=3$.

Band-structures are a tool for characterizing periodic materials under the
single-body approximation, presuming all electrons in the material behave
independently. Here we will use the reciprocal-orbital qubit mapping from
[22], which assigns one qubit to each orbital included in the material’s unit
cell. Since we consider only one electron, the active space consists only of
those basis vectors with a Hamming weight of one. Considering a toy model for
elemental polonium in a cubic lattice (Fig. 2(a)), we require four qubits; the
basis vectors $\ket{1000},\ket{0100},\ket{0010},\ket{0001}$ represent the
reciprocal $s$, $p_{x}$, $p_{y}$, and $p_{z}$ orbitals.

$\ket{1}$ & [wires=2]A_1 [wires=4,steps=3,background,style= ,label style=
label position=below, anchor=north, yshift=-0.2cm ]$V_{0}$
$\ket{0}$ [wires=2]A_2
$\ket{0}$ [wires=2]A_3
$\ket{0}$

(a) Ground-state circuit.

$\ket{0}$ & [wires=2]A_1 [wires=4,steps=3,background,style= ,label style=
label position=below, anchor=north, yshift=-0.2cm ]$V_{0}$
$\ket{1}$ [wires=2]A_4 [wires=3,steps=2,background,style= ,label style= label
position=below, anchor=north, yshift=-0.2cm ]$V_{1}$ [wires=2]A_2
$\ket{0}$ [wires=2]A_5 [wires=2]A_3
$\ket{0}$

(b) First-excited state circuit.

Figure 3: Variational circuits suitable for band-structure calculations using
the reciprocal-orbital qubit mapping in Section 3. In Fig. 3(a), all $A$ gates
have two independent parameters for a total of six. In Fig. 3(b), the $A$
gates comprising $V_{0}$ are fixed at their optimal values, so that only the
four parameters from $V_{1}$ must be varied.

In order to construct a quantum circuit, we use the two-qubit $A$ gates
presented in [23]. Each $A$ gate takes two independent angles $\theta$, $\phi$
and rotates between the $\ket{01}$ and $\ket{10}$, such that chaining each $A$
gate together as in Fig. 3(a) successfully spans the full active space. For
excited states, cutting out individual basis vectors is as simple as cutting
out qubits from the chain, as in Fig. 3(b). We have previously used this
variational circuit design for band-structures under a different qubit mapping
in [24].

##  4 Full Hilbert Space

In this section, we consider a case when the active space corresponds to the
full Hilbert space accessible by all qubits. To ease the transition, we will
continue to consider single-body band-structure calculations. Instead of the
reciprocal-orbital qubit mapping used above, we will adopt a compact mapping
which assigns each orbital to its own basis state, as was done for silicon in
[19]. Considering the same toy model for elemental polonium as above, we
require two qubits; the basis vectors $\ket{00},\ket{01},\ket{10},\ket{11}$
represent the reciprocal $s$, $p_{x}$, $p_{y}$, and $p_{z}$ orbitals.

To construct a suitable variational circuit, let us begin by writing down an
arbitrary wavefunction spanning the basis states $\ket{z\geq l}$:

$\displaystyle\ket{\phi_{l}}=\sum_{z\geq l}\phi_{z}\ket{z}$ (7)

subject to the normalization constraint $\innerproduct{\phi_{l}}{\phi_{l}}=1$.
Express the complex numbers $\phi_{z}$ in polar form as
$\phi_{z}=r_{z}e^{i\gamma}$. We facilitate parameterization of this wave-
function by adopting hyperspherical coordinates
$t_{z},\alpha_{z}\in\quantity[0,1]$ such that:

$\displaystyle r_{z}$
$\displaystyle=\sin(\frac{\pi}{2}t_{z})\prod_{i=0}^{z-1}\cos(\frac{\pi}{2}t_{i})$
(8) $\displaystyle\gamma_{z}$ $\displaystyle=2\pi\alpha_{z}$ (9)

Normalization is enforced by setting $t_{N-1}=1$, and an arbitrary global
phase is fixed by setting $\alpha_{l}=0$. This leaves $2N-2$ free parameters
for $\ket{\phi_{0}}$ to span the full Hilbert space. Eliminating successive
basis states for $\ket{\phi_{l>0}}$ is accomplished simply by setting
$t_{z<l}=\alpha_{z\leq l}=0$.

Now consider the following non-unitary operator:

$\displaystyle\hat{T}_{l}=\sum_{z<l}\outerproduct{z}{z}+\sum_{z\geq
l}\phi_{z}\outerproduct{z}{l}$ (10)

It is clear that $\hat{T}_{l}\ket{l}=\ket{\phi_{l}}$, and that $\hat{T}_{l}$
satisfies the constraints in Eqs. 2 and 3. In order to satisfy Eq. 1 we must
first construct the following Hermitian operator:

$\displaystyle\hat{H}_{l}=\frac{1}{2}\quantity(\hat{T}_{l}+\hat{T}_{l}^{\dagger})$
(11)

and unitary operator:

$\displaystyle\hat{\Omega}_{l}=e^{i\pi\hat{H}_{l}}$ (12)

This construction is similar in form to the popular Unitary Coupled Cluster
ansatz.[2, 21]

Implementing $\hat{\Omega}_{l}$ as a quantum circuit is equivalent to time-
evolving a Hamiltonian operator $\hat{H}$, a well-studied problem in the
quantum computing literature. The most straight-forward approach is Suzuki-
Trotter decomposition [25], in which $\hat{\Omega}_{l}$ is approximated as the
product
$\hat{\Omega}_{l}\approx\quantity[\prod_{k}\exp(i\frac{\pi}{r}\hat{H}_{lk})]^{r}$,
where $\hat{H}_{l}=\sum_{k}\hat{H}_{lk}$ and each factor
$\exp(i\frac{\pi}{r}\hat{H}_{lk})$ is straightforward to implement as a
quantum circuit (see Appendix A for a tutorial). Error in the approximation is
reduced by increasing $r$, leading to longer and longer circuits less suitable
for noise-prone devices.

Fig. 2 demonstrates the success of band-structure calculations with self-
orthogonalizing circuits on a noiseless quantum simulator. All optimizations
are carried out with the COBYLA algorithm implemented in the `scipy` Python
package. Results using the single-body circuits from Section 3 (four qubits)
are virtually indistinguishable from the analytically obtained band-structure.
The compact basis enables the same calculation with exponentially fewer qubits
(two qubits, in this case), but implementing the quantum circuit requires
Trotterization, which introduces noticeable approximation error. Nevertheless,
we observe that the worst-case error falls off rapidly even for very small
$r$. That said, implementing these circuits becomes much more difficult on a
real quantum device vulnerable to qubit decoherence and gate infidelity. One
possible alternative would be to exploit the Linear Combination of Unitaries
lemma [26, 27] for exponentially improved error scaling, at the cost of
additional ancillary qubits.

##  5 Many-body Circuits

In this section, we consider a standard formulation in which the active space
corresponds to all many-body states with fixed particle number. We again
associate one qubit with each orbital, but we now adopt the Jordan-Wigner
mapping [2, 28] to enforce fermionic anti-commutation relations. We shall
consider the hydrogen molecule in a minimal basis (two atoms each contributing
two spin-orbitals) with no symmetry reduction, so that we require four qubits.
Our active space consists of the basis vectors $\ket{1100}$, $\ket{1010}$,
$\ket{1001}$, $\ket{0110}$, $\ket{0101}$, $\ket{0011}$.

(a) Hydrogen dissociation curve

(b) Worst-case optimization error.

Figure 4: Fig. 4(a) is the dissociation curve for the hydrogen molecule. Solid
lines give the FCI energy while diamonds mark the values obtained from OA-VQE
on a noiseless quantum simulator using the COBYLA optimization routine. The
bottom curve gives the electronic ground-state energy under the Born-
Oppenheimer approximation over increasing atomic separation. Its minimum value
at approximately .74 Angstroms marks the bond of the hydrogen molecule. The
ground-state itself is multi-reference, consisting predominantly of the
Hartree-Fock bonding orbital, but increasingly large contributions from the
anti-bonding orbital as the molecule is stretched. The second-lowest curve is
triply-degenerate, including the spatially symmetric combination of bonding
and anti-bonding orbitals, and both states in which the electron spins are
parallel. It has no minimum value, illustrating that the triplet state of
molecular hydrogen is inherently unstable. Fig. 2(b) marks the worst-case
error for each energy level. The circuit used in Section 5 introduces
approximation error via Trotterization, but is practically negligible even for
$r=1$, on the same order as exponentiating the Hermitian operator
analytically, without Trotterization.

The self-orthogonalizing variational circuit and parameterization we adopt is
essentially the same as in Section 4. The crucial difference is that we
replace the ket $\ket{z}$ with the state $A^{\dagger}_{z}\ket{}$. Here
$\ket{}$ is the vacuum state, $z$ indexes the vector in our active space (note
that an order must be assigned to the vectors), and $A^{\dagger}_{z}$ is
shorthand for the normal-ordered product of creation operators which fill up
the orbitals corresponding to the bits indicated in the basis vector. For
example, if the basis vectors are ordered as in the previous paragraph, then
$A^{\dagger}_{0}=a^{\dagger}_{3}a^{\dagger}_{2}$,
$A^{\dagger}_{1}=a^{\dagger}_{3}a^{\dagger}_{1}$, and so on. These creation
operators are then mapped onto qubit operators via the Jordan-Wigner mapping,
and $\hat{\Omega}_{l}$ is implemented via Suzuki-Trotter decomposition or a
comparable method as described above.

Fig. 4 gives the ground-state dissociation curve of the hydrogen molecule, in
addition to the excited state energies for this basis. Orbital integral
calculations are made using the `pyscf` Python package [29] and the Jordan-
Wigner mapping is carried out with the `openfermion` Python package.[30] While
we should not expect the minimal basis to yield extremely accurate
calculations for excited states, we can observe the long-known fact that the
first-excited state (triply-degenerate) lacks a minimum, illustrating the fact
that electronic excitations in hydrogen gas tend to split the molecule
apart.[31] As observed in the compact single-body example in Fig. 2, the
circuits $\hat{\Omega}_{l}$ require only very small Trotterization constants
$r$ to be effective.

##  6 Conclusion

In this paper, we have formulated an Orthogonal-Ansatz Variational Quantum
Eigensolver (OA-VQE) algorithm for iteratively locating excited states, using
variational quantum circuits spanning only the subspace orthogonal to
previously-located states. The main distinction of this approach as compared
to Orthogonally-Constrained VQE (OC-VQE) is to leave the cost-function
unchanged for all excited states, simplifying measurement complexity.

We have offered several families of self-orthogonalizing variational circuits,
suitable for studying both single-body band structure calculations and many-
body molecular spectroscopy. However, we emphasize that the circuit families
presented in Sections 3-5 are by no means the only circuits consistent with
the self-orthogonality constraints presented in Section 2. If other
variational circuit families (eg. UCC, ADAPT, hardware-efficient ansatze,
etc.) can be adapted to be consistent with Eqs. 1-3, they will be compatible
with OA-VQE.

For example, the circuit families presented in Sections 4 and 5 have a similar
form to Unitary Coupled Cluster (UCC) circuits. The “cluster operators”
$\hat{T}_{l}$ used in this paper are designed to robustly explore the entire
Hilbert subspace orthogonal to previously located eigenstates. Meanwhile, the
standard UCC cluster operator is formulated to preferentially explore the
basis vectors most likely contributing to the ground state, for example by
considering only single- and double-excitations (UCCSD) from the Hartree-Fock
solution. Reconciling the UCCSD approximation with the self-orthogonality
constraints presented here could lead to fewer terms in the Hermitian operator
$\hat{H}_{l}$ and thus more efficient implementations of $\hat{\Omega}_{l}$ as
a quantum circuit.

The improved measurement complexity in OA-VQE is balanced by an increased
circuit complexity required to guarantee orthogonality. In the short-term,
this is a high price to pay due to relatively low coherence time and gate
fidelity in present-day quantum computers. However, the relatively high cost
of measuring and resetting qubits suggests that hybrid quantum-classical
algorithms such as VQE may soon transition to favoring fewer, longer circuits.
Moreover, not every circuit family will necessarily incur an asymptotic cost
in circuit complexity: for example, the single-body circuits presented in Fig.
3 can overlap such that the circuit length is consistently $\mathcal{O}(N)$.
This emphasizes the importance of problem-specific quantum circuit design,
rather than a one-size-fits-all approach, when developing algorithms for
quantum chemistry and materials science.

##  7 Acknowledgments

We thank Marco Fornari, Itay Hen, and Rosa Di Felice for useful discussions.
We acknowledge support from the US Department of Energy through the grant Q4Q:
Quantum Computation for Quantum Prediction of Materials and Molecular
Properties (DE-SC0019432).

##  Appendix A Suzuki-Trotter Circuits

This appendix is meant for readers unfamiliar with Hamiltonian simulation via
Suzuki-Trotter simulation. We remind the reader that other methods of
Hamiltonian simulation exist, such as in [26, 27], but we provide this
tutorial so that the industrious reader may reproduce our procedure.

Our objective is to implement the unitary operator
$\hat{\Omega}=\exp(i\pi\hat{H})$ as a quantum circuit. We first map the
Hamiltonian $\hat{H}$ onto a sum of qubit operators:

$\displaystyle\hat{H}\rightarrow\sum_{k=1}^{4^{n}}c_{k}\hat{P}_{k}$ (13)

Each $\hat{P}_{k}$ is a “Pauli word”, consisting of one of the Pauli-spin
operators $\hat{X}$, $\hat{Y}$, $\hat{Z}$, or the identity operator $\hat{I}$,
acting on each of $n$ qubits. The coefficients $c_{k}$ are determined by the
particular qubit mapping. In Section 4, we map projection operators
$\outerproduct{z}{z^{\prime}}$ directly onto Pauli words; we refer the reader
to [24] for details. In Section 5, we map a second-quantized Hamiltonian onto
Pauli words using the Jordan-Wigner mapping; see [28] or documentation for the
`openfermion` Python package [30] for details.

Next we take advantage of the second-order exponential product formula (also
known as the first-order symmetrized product formula):

$\displaystyle
e^{x(A+B+...Z)}=e^{xA/2}e^{xB/2}...e^{xZ}...e^{xB/2}e^{xA/2}+\mathcal{O}(x^{3})$
(14)

Higher-order formulae exist, but become needlessly complicated - see [25] for
an excellent tutorial. Therefore, the following decomposition is sufficient:

$\displaystyle\hat{\Omega}=e^{i\pi\hat{H}}\approx\prod_{k=1}^{4^{n}}e^{i\pi
c_{k}\hat{P}_{k}/2}\prod_{k=4^{n}}^{1}e^{i\pi c_{k}\hat{P}_{k}/2}$ (15)

If the coefficient $x$ in the exponential product formula is sufficiently
small, the single-operator exponentials on the right-hand side are an adequate
approximation of the complex operator exponential on the left. In order to
guarantee the operator $x$ is sufficiently small, we introduce Trotterization:

$\displaystyle\hat{\Omega}\approx\quantity[\prod_{k=1}^{4^{n}}e^{i\frac{\pi
c_{k}}{2r}\hat{P}_{k}}\prod_{k=4^{n}}^{1}e^{i\frac{\pi
c_{k}}{2r}\hat{P}_{k}}]^{r}$ (16)

Let $\Lambda=\max_{k}c_{k}$; then the error in the product formula scales with
$\mathcal{O}(\frac{\Lambda^{3}}{r^{3}})$.

& 1 [-1em] [-1em] [-1em] [-1em] 1
1 1
1 1
R_z(θ)

(a) $\hat{P}_{k}=\hat{Z}\hat{Z}\hat{Z}\hat{Z}$.

& H 2 [-1em] [-1em] 2 H

HS^† 1 1 SH
R_z(θ)

(b) $\hat{P}_{k}=\hat{X}\hat{I}\hat{Y}\hat{Z}$.

Figure 5: Circuits to implement $\hat{\Omega}_{k}=\exp(i\theta\hat{P}_{k}/2)$.
$R_{z}$ is the standard single-qubit rotation about the $z$-axis, $H$ is the
standard Hadamard gate, and $S$ is the standard phase gate. The solitary $H$
gate serves to rotate the qubits corresponding to an $\hat{X}$ operator to and
from the $X$ basis, and the $HS^{\dagger}$ gate serves the same purpose for
the $Y$ basis. Note that qubits corresponding to an $\hat{I}$ operator are
ignored.

It now suffices to specify how to implement the factor
$\hat{\Omega}_{k}=\exp(i\theta\hat{P}_{k}/2)$ as a quantum circuit. The
circuit for $\hat{P}_{k}=\hat{Z}\otimes\hat{Z}\otimes\hat{Z}\otimes\hat{Z}$ is
given in Fig. 5(a). To account for $\hat{I}$ operators, one simply omits the
CNOT gate on the corresponding qubit. To account for $\hat{X}$ and $\hat{Y}$
operators, one must apply basis-rotation gates before and after: Fig. 5(b)
illustrates the circuit for
$\hat{P}_{k}=\hat{X}\otimes\hat{I}\otimes\hat{Y}\otimes\hat{Z}$. See [28] for
a more thorough explanation. The full operator $\hat{\Omega}$ is constructed
by applying each $\hat{\Omega}_{k}$ one after the other.

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# Multipliers of the Hilbert spaces of Dirichlet series

Chaman Kumar Sahu  Department of Mathematics and Statistics
Indian Institute of Technology Kanpur, India<EMAIL_ADDRESS>
###### Abstract.

For a sequence $\mathbf{w}=\\{w_{j}\\}_{j=2}^{\infty}$ of positive real
numbers, consider the positive semi-definite kernel
$\kappa_{\mathbf{w}}(s,u)=\sum_{j=2}^{\infty}w_{j}j^{-s-\overline{u}}$ defined
on some right-half plane $\mathbb{H}_{\rho}$ for a real number $\rho.$ Let
$\mathscr{H}_{\mathbf{w}}$ denote the reproducing kernel Hilbert space
associated with $\kappa_{\mathbf{w}}.$ Let

$\displaystyle\delta_{\mathbf{w}}=\inf\Bigg{\\{}\Re(s):\sum\limits_{\begin{subarray}{c}j\geqslant
2\\\ \tiny{\mbox{\bf gpf}}(j)\leqslant
p_{n}\end{subarray}}w_{j}j^{-s}<\infty~{}\text{for
all}~{}n\in\mathbb{Z}_{+}\Bigg{\\}},$

where $\\{p_{j}\\}_{j\geqslant 1}$ is an increasing enumeration of prime
numbers and $\text{\bf gpf}(n)$ denotes the greatest prime factor of an
integer $n\geqslant 2.$ If $\mathbf{w}$ satisfies

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant
0,\quad n\geqslant 2,$

where $\mu$ is the M$\ddot{\mbox{o}}$bius function, then the multiplier
algebra $\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ of $\mathscr{H}_{\mathbf{w}}$
is isometrically isomorphic to the space of all bounded and holomorphic
functions on $\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}}$ that are
representable by a convergent Dirichlet series in some right half plane. As a
consequence, we describe the multiplier algebra
$\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ when $\mathbf{w}$ is an additive
function satisfying $\delta_{\mathbf{w}}\leqslant 0$ and

$\displaystyle\frac{w_{p^{j-1}}}{w_{p^{j}}}\leqslant
p^{-\delta_{\mathbf{w}}}~{}\text{for all integers}~{}~{}j\geqslant
2~{}\mbox{and all prime numbers}~{}p.$

Moreover, we recover a result of Stetler that classifies the multipliers of
$\mathscr{H}_{\mathbf{w}}$ when $\mathbf{w}$ is multiplicative. The proof of
the main result is a refinement of the techniques of Stetler.

###### Key words and phrases:

Dirichlet series, Reproducing kernel Hilbert space, Multiplier,
M$\ddot{\mbox{o}}$bius function, Additive function

###### 2020 Mathematics Subject Classification:

Primary 30B50, 46E22; Secondary 11Z05.

###### Contents

  1. 1 Preliminaries
    1. 1.1 Multipliers of a reproducing kernel Hilbert space
  2. 2 The multiplier algebra of $\mathscr{H}_{\mathbf{w}}$
  3. 3 Applications

##  1\. Preliminaries

Let $\mathbb{Z}_{+}$ and $\mathbb{N}$ denote the sets of positive and non-
negative integers, respectively. For $k\in\mathbb{N},$ let
$\mathbb{N}_{k}=\\{m\in\mathbb{N}:m\leqslant k\\}.$ Denote by $\mathbb{R}$ and
$\mathbb{C},$ the sets of real and complex numbers, respectively. For
$s\in\mathbb{C},$ let $\Re(s),$ $|s|,$ $\overline{s}$ and $\arg(s)$ denote the
real part, the modulus, the complex conjugate and the argument of $s,$
respectively. The open unit disc $\\{z\in\mathbb{C}:|z|<1\\}$ is denoted by
$\mathbb{D}.$ For $\rho\in\mathbb{R},$ let $\mathbb{H}_{\rho}$ denote the
right half-plane $\\{s\in\mathbb{C}:\Re(s)>\rho\\}.$

We invoke here some known arithmetic functions on $\mathbb{Z}_{+}$ used in the
sequel. The prime omega function $\omega$ that counts the total number of all
distinct prime factors of a positive integer. The divisor function
$\mathbf{d}$ counts all divisors of a positive integer. The M$\ddot{o}$bius
function $\mu:\mathbb{Z}_{+}\rightarrow\mathbb{C}$ is given by

$\displaystyle\mu(n)=\begin{cases}1&\text{if n = 1},\\\ (-1)^{j}&\text{if $n$
is a product of $j$ distinct primes},\\\ 0&\text{otherwise}.\end{cases}$ (1)

A Dirichlet series is a series of the form

$\displaystyle f(s)=\sum_{j=1}^{\infty}a_{j}j^{-s},$

where $a_{j}\in\mathbb{C}.$ If $a_{j}=1$ for all $j\geqslant 1,$ then we have
the Riemann zeta function, denoted by $\zeta.$ If $f$ is convergent at
$s=s_{0}$, then it converges uniformly throughout the angular region
$\\{s\in\mathbb{C}:|\arg(s-s_{0})|<\frac{\pi}{2}-\delta\\}$ for every positive
real number $\delta<\frac{\pi}{2}.$ Consequently, $f$ defines a holomorphic
function on $\mathbb{H}_{\Re(s_{0})}$ (refer to [13, Chapter IX] for the basic
theory of Dirichlet series). Let $\mathcal{D}$ denotes the set of all
functions which are representable by a convergent Dirichlet series in some
right half plane.

The following proposition describes the product of two Dirichlet series (see
[10, Theorem 4.3.1 and discussion prior to Theorem 4.3.4]).

######  Proposition 1.1.

Let $f(s)=\sum_{j=1}^{\infty}a_{j}j^{-s}$ and
$g(s)=\sum_{j=1}^{\infty}b_{j}j^{-s}$ be two convergent Dirichlet series on
$\mathbb{H}_{\rho}.$ If $g$ converges absolutely on $\mathbb{H}_{\rho},$ then

$\displaystyle
f(s)g(s)=\sum_{j=1}^{\infty}\Big{(}\sum_{m|j}a_{m}b_{\frac{j}{m}}\Big{)}j^{-s},$

which converges on $\mathbb{H}_{\rho}.$

For an integer $n\geqslant 2,$ let $\text{\bf gpf}(n)$ denote the greatest
prime factor of $n.$ For a sequence $\mathbf{x}=\\{x_{j}\\}_{j\geqslant 2}$ of
non-negative real numbers, let

$\displaystyle\sigma_{\mathbf{x}}:=\inf\Big{\\{}\Re(s):\sum_{j=2}^{\infty}x_{j}j^{-s}<\infty\Big{\\}},$
$\displaystyle\delta_{\mathbf{x}}:=\inf\Bigg{\\{}\Re(s):\sum\limits_{\begin{subarray}{c}j\geqslant
2\\\ \tiny{\mbox{\bf gpf}}(j)\leqslant
p_{n}\end{subarray}}x_{j}j^{-s}<\infty~{}\text{for
all}~{}n\in\mathbb{Z}_{+}\Bigg{\\}},$ (2)

where $\\{p_{j}\\}_{j=1}^{\infty}$ is the increasing enumeration of prime
numbers. Note that $\delta_{\mathbf{x}}\leqslant\sigma_{\mathbf{x}}.$

###  1.1. Multipliers of a reproducing kernel Hilbert space

Let $X$ be a non-empty open subset of $\mathbb{C}$ and let $H^{\infty}(X)$
denote the Banach space of bounded holomorphic functions on $X$ endowed with
supremum norm. Let $\mathcal{H}$ be a reproducing kernel Hilbert space of
complex-valued holomorphic functions on $X.$ A function
$\varphi:X\rightarrow\mathbb{C}$ is called multiplier of $\mathcal{H}$ if
$\varphi\cdot f\in\mathcal{H}$ for every $f\in\mathcal{H}.$ Clearly,
$\mathcal{M}(\mathcal{H})$ is an algebra. Denote by
$\mathcal{M}(\mathcal{H}),$ the set of all multipliers of $\mathcal{H}.$ Note
that if the constant function equal to $1$ belongs to $\mathcal{H},$ then by
[9, Corollary 5.22], $\mathcal{M}(\mathcal{H})$ is contractively contained in
$H^{\infty}(X).$

For a sequence $\mathbf{w}=\\{w_{j}\\}_{j=2}^{\infty}$ of positive real
numbers, consider the weighted Hilbert space of the formal Dirichlet series

$\displaystyle\mathscr{H}_{\mathbf{w}}:=\Bigg{\\{}f(s)=\sum_{j=2}^{\infty}\hat{f}(j)j^{-s}:\|f\|^{2}_{\mathbf{w}}:=\sum_{j=2}^{\infty}\frac{|\hat{f}(j)|^{2}}{w_{j}}\textless\infty\Bigg{\\}}$

endowed with the norm $\|\cdot\|_{\mathbf{w}}.$ If
$\sigma_{\mathbf{w}}<+\infty,$ then $\mathscr{H}_{\mathbf{w}}$ is a
reproducing kernel Hilbert space associated with $\kappa_{\mathbf{w}}$ given
by

$\displaystyle\kappa_{\mathbf{w}}(s,u)=\sum_{j=2}^{\infty}w_{j}j^{-s-\overline{u}},\quad
s,u\in\mathbb{H}_{\frac{\sigma_{\mathbf{w}}}{2}}.$ (3)

For every $s\in\mathbb{H}_{\frac{\sigma_{\mathbf{w}}}{2}},$
$\kappa_{\mathbf{w}}(\cdot,s)\in\mathscr{H}_{\mathbf{w}}$ and satisfies

$\displaystyle\langle{f},\,{\kappa_{\mathbf{w}}(\cdot,s)}\rangle=f(s),\quad
f\in\mathscr{H}_{\mathbf{w}}.$ (4)

Note that $\kappa_{\mathbf{w}}$ converges absolutely on
$\mathbb{H}_{\frac{\sigma_{\mathbf{w}}}{2}}\times\mathbb{H}_{\frac{\sigma_{\mathbf{w}}}{2}}.$

The following is immediate from [12, Theorem 3.1] and the discussion prior to
[12, Theorem 2.7].

######  Theorem 1.2.

$\mathcal{M}(\mathscr{H}_{\mathbf{w}})\subseteq
H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}$ and

$\displaystyle\|\varphi\|_{\infty,\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}}}\leqslant\|M_{\varphi,\mathbf{w}}\|_{\mathscr{H}_{\mathbf{w}}},\quad\varphi\in\mathcal{M}(\mathscr{H}_{\mathbf{w}}).$

This raises the following question.

######  Question 1.1.

For which sequences $\mathbf{w},$ the multiplier algebra
$\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ of $\mathscr{H}_{\mathbf{w}}$ is
isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}?$

In the following cases, Question 1.1 is answered affirmatively.

  * (i)

If $\mathbf{1}$ denotes the constant sequence with value $1,$ then
$\mathcal{M}(\mathscr{H}_{\mathbf{1}})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}$ (see [5, Theorem 3.1]).

  * (ii)

If, for some positive Radon measure $\eta$ on $[0,\infty)$ with
$0\in\text{supp}(\eta)$ and $n_{0}\in\mathbb{Z}_{+},$

$\displaystyle\frac{1}{w_{j}}=\int_{0}^{\infty}j^{-2\sigma}d\eta(\sigma),\quad
j\geqslant n_{0},$ (5)

then $\mathcal{M}(\mathscr{H}_{\mathbf{w}_{\eta}})$ is isometrically
isomorphic to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}$ (see [6, Theorem
1.11]).

  * (iii)

If $\mathbf{w}$ is either $\\{\frac{1}{d_{\beta}(n)}\\}_{n=1}^{\infty},$ where
$\beta>0$ and $d_{\beta}(n)$ denotes $n$-th coefficient of the Dirichlet
series $\zeta^{\beta},$ or $\\{(\mathbf{d}(n))^{\alpha}\\}_{n=1}^{\infty}$ for
$\alpha\in\mathbb{R},$ then $\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ is
isometrically isomorphic to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}$ (see
[8, Section 8]).

  * (iv)

If $\mathbf{w}=\\{w_{n}\\}_{n=1}^{\infty}$ is a multiplicative function (i.e.,
$w_{mn}=w_{m}w_{n}$ for all integers $m,n\geqslant 1$ such that $\gcd(m,n)=1$)
satisfying

$\displaystyle\frac{w_{p^{j-1}}}{w_{p^{j}}}\leqslant
p^{-\delta_{\mathbf{w}}}~{}\text{for all integers}~{}~{}j\geqslant
1~{}\mbox{and prime numbers}~{}p,$ (6)

then $\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}$ (see
[12, Corollary 4.2]).

In all the cases above, the multipliers of $\mathscr{H}_{\mathbf{w}}$ are
extended beyond the common domain of the functions in
$\mathscr{H}_{\mathbf{w}}$. However, this is not always true (see [6, Theorem
3.6]).

The following is the main theorem of this paper, which generalizes the
aforementioned results (i) and $\mbox{(iv)}.$

######  Theorem 1.3.

Suppose that

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant
0,\quad n\geqslant 2,$ (7)

where $\mu$ and $\delta_{\mathbf{w}}$ are given by (1) and (2), respectively.
Then $\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$

######  Remark 1.4.

Here we make some observations:

  * (a)

If $\delta_{\mathbf{w}}=-\infty,$ then there are no non-constant multipliers
of $\mathscr{H}_{\mathbf{w}}.$ Indeed, if
$\varphi\in\mathcal{M}(\mathscr{H}_{\mathbf{w}}),$ then by Theorem 1.2,
$\varphi$ is entire and bounded. Hence, by the Liouville’s theorem $($see [11,
Corollary 4.5]$),$ $\varphi$ is constant.

  * (b)

If $\mathbf{w}=\\{w_{j}\\}_{j=2}^{\infty}$ in the above theorem is replaced by
$\\{w_{j}\\}_{j=k}^{\infty}\subseteq(0,\infty)$ for an integer
$k\in\mathbb{Z}_{+}$ and

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant k\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant
0,\quad n\geqslant k,$ (8)

then the conclusion of Theorem 1.3 will be same.

The following consequence of Theorem 1.3 deals with additive functions.

######  Corollary 1.5.

Let $\mathbf{w}$ be an additive function $($i.e., $w_{mn}=w_{m}+w_{n}$ for all
integers $m,n\geqslant 2$ such that $\gcd(m,n)=1)$ satisfying

$\displaystyle\frac{w_{p^{j-1}}}{w_{p^{j}}}\leqslant
p^{-\delta_{\mathbf{w}}}~{}\text{for all integers}~{}~{}j\geqslant
2~{}\mbox{and all prime numbers}~{}p.$ (9)

If $\delta_{\mathbf{w}}\leqslant 0,$ then
$\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$

The proofs of Theorem 1.3 and Corollary 1.5 will be presented in Section 2. In
Section 3, we discuss some consequences of Theorem 1.3 (see Theorem 3.1 and
Proposition 3.2). In particular, we recover [12, Corollary 4.2]. We also
present some examples of the Hilbert spaces $\mathscr{H}_{\mathbf{w}}$
illustrating the results of this paper.

##  2\. The multiplier algebra of $\mathscr{H}_{\mathbf{w}}$

In view of Remark 1.4 (a), we assume that $\delta_{\mathbf{w}}\in\mathbb{R}.$
To prove Theorem 1.3, we require several lemmas.

The first one relates the condition (7) with the positive semi-definiteness of
a kernel.

######  Lemma 2.1.

Let $\tilde{\mathbf{w}}=\\{n^{-\delta_{\mathbf{w}}}w_{n}\\}_{n=2}^{\infty}$
and $\beta=\frac{1}{2}\max\\{\sigma_{\mathbf{w}}-\delta_{\mathbf{w}},1\\}.$
Then the kernel
$\eta:\mathbb{H}_{\beta}\times\mathbb{H}_{\beta}\rightarrow\mathbb{C}$ defined
by

$\displaystyle\eta(s,u)=\frac{\kappa_{\tilde{\mathbf{w}}}(s,u)}{\kappa_{\mathbf{1}}(s,u)},\quad
s,u\in\mathbb{H}_{\beta}$ (10)

is positive semi-definite if and only if (7) holds.

###### Proof.

By [1, Example $11.4.1$],

$\displaystyle\frac{1}{\zeta(s)}=\sum_{j=1}^{\infty}\mu(j)j^{-s},\quad
s\in\mathbb{H}_{1}.$

This, combined with Proposition 1.1 and (3), yields

$\displaystyle\eta(s,u)$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{j=2}^{\infty}j^{-\delta_{\mathbf{w}}}w_{j}j^{-s-\overline{u}}\Big{)}\Big{(}\sum_{j=1}^{\infty}\mu(j)j^{-s-\overline{u}}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{n=2}^{\infty}\Big{(}\sum_{\begin{subarray}{c}j\geqslant
2\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\Big{)}n^{-s-\overline{u}},\quad
s,u\in\mathbb{H}_{\beta}.$

The desired equivalence is now immediate from [7, Lemma 20]. ∎

The following lemma is needed in the proof of Theorem 1.3.

######  Lemma 2.2.

For $t\in\mathbb{R},$ let
$\tilde{\mathbf{w}}=\\{n^{-t}w_{n}\\}_{n=2}^{\infty}.$ If
$\varphi\in\mathcal{M}(\mathscr{H}_{\mathbf{w}}),$ then
$\tilde{\varphi}:\mathbb{H}_{\frac{\sigma_{\mathbf{w}}-t}{2}}\rightarrow\mathbb{C}$
defined by

$\displaystyle\tilde{\varphi}(s)=\varphi\Big{(}s+\frac{t}{2}\Big{)},\quad
s\in\mathbb{H}_{\frac{\sigma_{\mathbf{w}}-t}{2}}$

is a multiplier of $\mathscr{H}_{\tilde{\mathbf{w}}}$ $($see (3)$).$

###### Proof.

Let $j:\mathscr{H}_{\mathbf{w}}\rightarrow\mathscr{H}_{\tilde{\mathbf{w}}}$ be
a map defined by

$\displaystyle j(f)(s)=f\Big{(}s+\frac{t}{2}\Big{)},\quad
s\in\mathbb{H}_{\frac{\sigma_{\mathbf{w}}-t}{2}}.$

Since $\varphi\in\mathcal{M}(\mathscr{H}_{\mathbf{w}}),$ for any
$f\in\mathscr{H}_{\mathbf{w}},$

$\displaystyle\tilde{\varphi}(s)j(f)(s)$ $\displaystyle=$
$\displaystyle\varphi\Big{(}s+\frac{t}{2}\Big{)}f\Big{(}s+\frac{t}{2}\Big{)}$
$\displaystyle=$ $\displaystyle(\varphi\cdot
f)\Big{(}s+\frac{t}{2}\Big{)}=j(\varphi\cdot
f)(s),~{}~{}s\in\mathbb{H}_{\frac{\sigma_{\mathbf{w}}-t}{2}}.$

As $j$ is surjective, $\tilde{\varphi}\cdot
g\in\mathscr{H}_{\tilde{\mathbf{w}}}$ for every
$g\in\mathscr{H}_{\tilde{\mathbf{w}}}.$ This concludes the proof. ∎

The following fact shows that the inclusion map
$\mathcal{M}(\mathscr{H}_{\mathbf{w}})\xhookrightarrow{}H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}$
is surjective (cf. [12, Theorem 4.5]).

######  Lemma 2.3.

If (7) holds, then
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}=\mathcal{M}(\mathscr{H}_{\mathbf{w}})$
$($as a set$).$

###### Proof.

Assume that (7) holds. In view of Theorem 1.2, it suffices to check that
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}\subseteq\mathcal{M}(\mathscr{H}_{\mathbf{w}}).$
To see that, let $\varphi\in
H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$ Then
$\tilde{\varphi}:\mathbb{H}_{0}\rightarrow\mathbb{C}$ defined by
$\tilde{\varphi}(s)=\varphi(s+\frac{\delta_{\mathbf{w}}}{2}),~{}s\in\mathbb{H}_{0},$
belongs to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$ Hence, by [5, Theorem
3.1], $\tilde{\varphi}$ is a multiplier of $\mathscr{H}_{\mathbf{1}}.$
Therefore, by [9, Theorem 5.21], there exists $c\geqslant 0$ such that

$\displaystyle(c^{2}-\tilde{\varphi}(s)\overline{\tilde{\varphi}(u)})\kappa_{\mathbf{1}}(s,u)\geqslant
0.$

Let $\tilde{\mathbf{w}}=\\{n^{-\delta_{\mathbf{w}}}w_{n}\\}_{n=2}^{\infty}.$
Then, by Lemma 2.1 and [9, Theorem 4.8],

$\displaystyle(c^{2}-\tilde{\varphi}(s)\overline{\tilde{\varphi}(u)})\kappa_{\tilde{\mathbf{w}}}(s,u)\geqslant
0.$

Thus, by [9, Theorem 5.21],
$\tilde{\varphi}\in\mathcal{M}(\mathscr{H}_{\tilde{\mathbf{w}}}).$ An
application of Lemma 2.2 now completes the proof. ∎

###### Proof of Theorem 1.3.

By Lemma 2.3,

$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}=\mathcal{M}(\mathscr{H}_{\mathbf{w}})$
$($as a set$).$ (11)

Hence, in view of Theorem 1.2, it is sufficient to check that

$\displaystyle\|M_{\varphi,\mathbf{w}}\|_{\mathscr{H}_{\mathbf{w}}}\leqslant\|\varphi\|_{\infty,\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}}},\quad\varphi\in
H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$

To see the above estimate, let $\varphi\in
H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$ Note
that $\tilde{\varphi}:\mathbb{H}_{0}\rightarrow\mathbb{C},$ defined by
$\tilde{\varphi}(s)=\varphi(s+\frac{\delta_{\mathbf{w}}}{2}),~{}s\in\mathbb{H}_{0},$
belongs to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$ Therefore, by an
application of [5, Theorem 3.1],

the multiplication operator $M_{\tilde{\varphi},\mathbf{1}}$ is bounded on
$\mathscr{H}_{\mathbf{1}}.$

In particular, the Hilbert space adjoint $M^{*}_{\tilde{\varphi},\mathbf{1}}$
of $M_{\tilde{\varphi},\mathbf{1}}$ is well-defined as a bounded linear
operator.

Note that Lemma 2.1 combined with (7) yields that the kernel $\eta$ defined by
(10), is positive semi-definite on
$\mathbb{H}_{\beta}\times\mathbb{H}_{\beta},$ where
$\beta=\frac{1}{2}\max\\{\sigma_{\mathbf{w}}-\delta_{\mathbf{w}},1\\}.$ Hence,
by [9, Theorem 2.14], there exists a reproducing kernel Hilbert space
$\mathcal{H}(\eta)$ associated with $\eta.$ Thus, by (4), for any
$s,u\in\mathbb{H}_{\beta},$

$\displaystyle\langle{\kappa_{\tilde{\mathbf{w}}}(\cdot,u)},\,{\kappa_{\tilde{\mathbf{w}}}(\cdot,s)}\rangle_{\mathscr{H}_{\tilde{\mathbf{w}}}}=\langle{\kappa_{\mathbf{1}}(\cdot,u)\otimes\eta(\cdot,u)},\,{\kappa_{\mathbf{1}}(\cdot,s)\otimes\eta(\cdot,s)}\rangle_{\mathscr{H}_{\mathbf{1}}\otimes\mathcal{H}(\eta)}.$

An application of the Lurking isometry Lemma (see [2, Lemma 2.18]) yields a
linear isometry
$V:\mathscr{H}_{\tilde{\mathbf{w}}}\rightarrow\mathscr{H}_{\mathbf{1}}\otimes\mathcal{H}(\eta)$
such that

$\displaystyle
V(\kappa_{\tilde{\mathbf{w}}}(\cdot,u))=\kappa_{\mathbf{1}}(\cdot,u)\otimes\eta(\cdot,u),\quad
u\in\mathbb{H}_{\beta}.$ (12)

Further, since $\varphi\in\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ (see (11)),
by Lemma 2.2,
$\tilde{\varphi}\in\mathcal{M}(\mathscr{H}_{\tilde{\mathbf{w}}}).$ This
implies that the multiplication operator
$M_{\tilde{\varphi},{\tilde{\mathbf{w}}}}$ is bounded on
$\mathscr{H}_{\tilde{\mathbf{w}}},$ and hence, by [9, Corollary 5.22],

$\displaystyle V^{*}(M_{\tilde{\varphi},\mathbf{1}}^{*}\otimes
I)V(\kappa_{\tilde{\mathbf{w}}}(\cdot,u))$
$\displaystyle\overset{\eqref{isometry}}{=}$ $\displaystyle
V^{*}(M_{\tilde{\varphi},\mathbf{1}}^{*}\otimes
I)(\kappa_{\mathbf{1}}(\cdot,u)\otimes\eta(\cdot,u))$ $\displaystyle=$
$\displaystyle\overline{\tilde{\varphi}(u)}V^{*}(\kappa_{\mathbf{1}}(\cdot,u)\otimes\eta(\cdot,u))$
$\displaystyle=$ $\displaystyle
M_{\tilde{\varphi},{\tilde{\mathbf{w}}}}^{*}(\kappa_{\tilde{\mathbf{w}}}(\cdot,u)),\quad
u\in\mathbb{H}_{\beta}.$

Since $V^{*}(M_{\tilde{\varphi},\mathbf{1}}^{*}\otimes I)V$ and
$M_{\tilde{\varphi},{\tilde{\mathbf{w}}}}^{*}$ are bounded linear operators on
$\mathscr{H}_{\tilde{\mathbf{w}}},$

$\displaystyle V^{*}(M_{\tilde{\varphi},\bf 1}^{*}\otimes
I)V=M_{\tilde{\varphi},\tilde{\mathbf{w}}}^{*}.$

Because $V$ is an isometry, we obtain

$\displaystyle\|M_{\tilde{\varphi},\tilde{\mathbf{w}}}\|_{{}_{{}_{\mathscr{H}_{\tilde{\mathbf{w}}}}}}\leqslant\|V^{*}\|\|M_{\tilde{\varphi},\bf
1}^{*}\|\|V\|\leqslant\|\tilde{\varphi}\|_{\infty,\,\mathbb{H}_{0}}=\|\varphi\|_{\infty,\,\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}}},$

which completes the proof. ∎

######  Remark 2.4.

Let $\kappa$ be given by

$\displaystyle\kappa(z,w)=\sum_{j=0}^{\infty}a_{j}z^{j}\overline{w}^{j}.$

Suppose that $\kappa$ converges on $\mathbb{D}\times\mathbb{D}.$ If
$\mathbf{a}$ is an increasing sequence of non-negative real numbers, then the
multiplier algebra $\mathcal{M}(\mathscr{H}(\kappa))$ of $\mathscr{H}(\kappa)$
is isometrically isomorphic to $H^{\infty}(\mathbb{D}).$ This can be shown
using the arguments in the proof of Theorem 1.3 with the essential change that
$\kappa_{\mathbf{1}}$ is replaced by the Szeg$\ddot{\mbox{o}}$ kernel
$S:\mathbb{D}\times\mathbb{D}\rightarrow\mathbb{C}$ defined by

$\displaystyle S(z,w)=\frac{1}{1-z\overline{w}},\quad z,w\in\mathbb{D}.$

###### Proof of Corollary 1.5.

Fix an integer $n\geqslant 2,$ let
$\prod\limits_{m=1}^{\omega(n)}p_{i_{m}}^{r_{m}},~{}r_{m}\in\mathbb{Z}_{+},$
be the prime factorization of $n.$ If $j\in\mathbb{Z}_{+}$ such that $j|n,$
then

$\displaystyle
j=\prod_{m=1}^{\omega(n)}p_{i_{m}}^{s_{m,j}},\quad(s_{m,j})_{m=1}^{\omega(n)}\in\prod_{m=1}^{\omega(n)}\mathbb{N}_{r_{m}}.$
(13)

Hence, there exists a bijective map
$\psi:\\!\\{j\in\mathbb{Z}_{+}\\!:j|n,~{}j\geqslant
2\\}\rightarrow\prod\limits_{m=1}^{\omega(n)}\mathbb{N}_{r_{m}}-\\{\bf 0\\}$
defined by $\psi(j)=(s_{m,j})_{m=1}^{\omega(n)}.$ Therefore, by (13),

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}$
(14) $\displaystyle=$
$\displaystyle\sum_{(s_{m})_{m=1}^{\omega(n)}\in\,\text{Im}(\psi)}\\!\\!\\!\Big{(}\prod_{m=1}^{\omega(n)}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\Big{)}\big{(}w_{{}_{\prod_{m=1}^{\omega(n)}p_{i_{m}}^{s_{m}}}}\big{)}\mu\Big{(}{\prod_{m=1}^{\omega(n)}p_{i_{m}}^{r_{m}-s_{m}}}\Big{)}.$

Let $\tilde{\mathbf{w}}$ be the extension of $\mathbf{w}$ to $\mathbb{Z}_{+}$
by letting $w_{1}=0.$ Since $\mathbf{w}$ is additive, $\tilde{\mathbf{w}}$ is
additive. This together with (14) and the multiplicativity of $\mu$ gives

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\\!\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\sum_{(s_{m})_{m=1}^{\omega(n)}\in\,\text{Im}(\psi)}\sum_{t=1}^{\omega(n)}\Bigg{(}w_{p_{i_{t}}^{s_{t}}}\Big{(}\prod_{m=1}^{\omega(n)}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\Big{)}\mu\Big{(}{\prod_{m=1}^{\omega(n)}p_{i_{m}}^{r_{m}-s_{m}}}\Big{)}\Bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{t=1}^{\omega(n)}\\!\sum_{(s_{m})_{m=1}^{\omega(n)}\in\,\text{Im}(\psi)}\\!\\!\\!\\!\\!\\!\Big{(}w_{p_{i_{t}}^{s_{t}}}\Big{(}\\!\\!\prod_{m=1}^{\omega(n)}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\Big{)}\Big{(}\prod_{m=1}^{\omega(n)}\mu(p_{i_{m}}^{r_{m}-s_{m}})\Big{)}\\!\Big{)}$
$\displaystyle\overset{\eqref{mu-fun}}{=}$
$\displaystyle\sum_{t=1}^{\omega(n)}\sum_{(s_{m})_{m=1}^{\omega(n)}\in\,\mathcal{U}_{n}}\\!\\!\\!\\!\\!\\!\Big{(}w_{p_{i_{t}}^{s_{t}}}\prod_{m=1}^{\omega(n)}\Big{(}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\mu(p_{i_{m}}^{r_{m}-s_{m}})\Big{)}\Big{)},$

where $\mathcal{U}_{n}=\prod\limits_{m=1}^{\omega(n)}\\{r_{m}-1,r_{m}\\}.$ For
any positive integer $t\leqslant\omega(n),$ let

$\displaystyle
T_{t}=\sum_{(s_{m})_{m=1}^{\omega(n)}\in\,\mathcal{U}_{n}}\\!\\!\\!\\!\\!\\!\Big{(}w_{p_{i_{t}}^{s_{t}}}\prod_{m=1}^{\omega(n)}\Big{(}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\mu(p_{i_{m}}^{r_{m}-s_{m}})\Big{)}\Big{)}.$

Then

$\displaystyle T_{t}\\!\\!$ $\displaystyle\\!\\!=$
$\displaystyle\sum_{s_{1}=\,r_{1}-1}^{r_{1}}\sum_{s_{2}=\,r_{2}-1}^{r_{2}}\ldots\sum_{s_{\omega(n)}=\,r_{\omega(n)}-1}^{r_{\omega(n)}}\Big{(}w_{p_{i_{t}}^{s_{t}}}\prod_{m=1}^{\omega(n)}\Big{(}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\mu(p_{i_{m}}^{r_{m}-s_{m}})\Big{)}\Big{)}$
$\displaystyle\\!\\!=$
$\displaystyle\\!\\!\Big{(}\\!\\!\sum_{s_{t}=r_{t}-1}^{r_{t}}\\!\\!\\!p_{i_{t}}^{-\delta_{{}_{\mathbf{w}}}s_{t}}w_{p_{i_{t}}^{s_{t}}}\mu(p_{i_{t}}^{r_{t}-s_{t}})\Big{)}\prod_{\begin{subarray}{c}m=1\\\
m\neq
t\end{subarray}}^{\omega(n)}\Big{(}\sum_{s_{m}=r_{m}-1}^{r_{m}}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\mu(p_{i_{m}}^{r_{m}-s_{m}})\Big{)}$
$\displaystyle=$
$\displaystyle\big{(}p_{i_{t}}^{-\delta_{{}_{\mathbf{w}}}(r_{t}-1)}w_{p_{i_{t}}^{r_{t}-1}}\mu(p_{i_{t}})+p_{i_{t}}^{-\delta_{\mathbf{w}}r_{t}}w_{p_{i_{t}}^{r_{t}}}\mu(1)\big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\prod_{\begin{subarray}{c}m=1\\\
m\neq
t\end{subarray}}^{\omega(n)}\big{(}p_{i_{m}}^{-\delta_{{}_{\mathbf{w}}}(r_{m}-1)}\mu(p_{i_{m}})+p_{i_{m}}^{-\delta_{\mathbf{w}}r_{m}}\mu(1)\big{)}$
$\displaystyle\\!\\!\overset{\eqref{mu-fun}}{=}$ $\displaystyle
p_{i_{t}}^{-\delta_{\mathbf{w}}(r_{t}-1)}(p_{i_{t}}^{-\delta_{\mathbf{w}}}w_{p_{i_{t}}^{r_{t}}}-w_{p_{i_{t}}^{r_{t}-1}})\prod_{\begin{subarray}{c}m=1\\\
m\neq
t\end{subarray}}^{\omega(n)}\big{(}p_{i_{m}}^{-\delta_{\mathbf{w}}(r_{m}-1)}(p_{i_{m}}^{-\delta_{\mathbf{w}}}-1)\big{)}.$

This combined with the assumptions $\delta_{\mathbf{w}}\leqslant 0$ and (9)
(which imply that $p_{i_{m}}^{-\delta_{\mathbf{w}}}\geqslant 1$ and
$p_{i_{t}}^{-\delta_{\mathbf{w}}}w_{p_{i_{t}}^{r_{t}}}\geqslant
w_{p_{i_{t}}^{r_{t}-1}}$) shows that $T_{t}\geqslant 0$ for every positive
integer $t\leqslant\omega(n).$ Since $n$ is arbitrary, (7) is valid for all
integers $n\geqslant 2.$ An application of Theorem 1.3 now completes the
proof. ∎

We provide an example illustrating Corollary 1.5.

######  Example 2.5.

Let $\mathbf{w}=\omega.$ Note that $\omega$ is additive and satisfies

$\displaystyle\text{$\omega(p^{j})=1$ for every prime $p$ and integer
$j\geqslant 1$}.$ (15)

By [13, Chap. I, Equations (1.6.1) and (1.6.2)],

$\displaystyle\sum_{j=2}^{\infty}\omega(j)j^{-s}=\zeta(s)\sum_{j=1}^{\infty}p_{j}^{-s},\quad
s\in\mathbb{H}_{1}.$ (16)

It follows from (15), (16) and [1, Theorem 1.13] that $\sigma_{\omega}=1.$ To
compute $\delta_{\omega},$ consider any $\epsilon>0$ and an integer
$n\geqslant 1,$

$\displaystyle\sum_{\tiny{\mbox{\bf gpf}}(j)\leqslant
p_{n}}\omega(j)j^{-\epsilon}\\!\\!$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\sum_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{j}=1\\\
i_{s}\neq i_{t},~{}s\neq
t\end{subarray}}^{n}\Bigg{(}\sum_{m_{i_{1}},\ldots,m_{i_{j}}=1}^{\infty}j(p_{i_{1}}^{m_{i_{1}}}p_{i_{2}}^{m_{i_{2}}}\ldots
p_{i_{j}}^{m_{i_{j}}})^{-\epsilon}\Bigg{)}$ (17) $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\Bigg{(}\sum_{\begin{subarray}{c}i_{1},i_{2},\ldots,i_{j}=1\\\
i_{s}\neq i_{t},~{}s\neq
t\end{subarray}}^{n}\Big{(}j\prod_{r=1}^{j}\sum_{m_{i_{r}}=1}^{\infty}p_{i_{r}}^{-\epsilon
m_{i_{r}}}\Big{)}\Bigg{)}<\infty.$

In view of (15), $\sum_{\tiny{\mbox{\bf gpf}}(j)\leqslant p_{n}}\omega(j)$
diverges for all $n\geqslant 1.$ This combined with (17) yields
$\delta_{\omega}=0,$ and hence, (9) holds. Therefore, by Corollary 1.5,
$\mathcal{M}(\mathscr{H}_{\omega})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$ Moreover,

$\displaystyle\omega(pq)=2\overset{\eqref{imp-pro}}{\neq}\omega(p)\omega(q)\
\text{for distinct primes $p$ and $q.$}$

So, $\omega$ is not multiplicative. Also, because $\omega$ is not monotone,
(5) does not hold.

##  3\. Applications

In this section, we discuss some applications of Theorem 1.3. The first one
recovers [12, Corollary 4.2].

######  Theorem 3.1.

Let $\mathbf{w}=\\{w_{j}\\}_{j=1}^{\infty}$ be an multiplicative function
satisfying (6). Then $\mathcal{M}(\mathscr{H}_{\mathbf{w}})$ is isometrically
isomorphic to
$H^{\infty}(\mathbb{H}_{\frac{\delta_{\mathbf{w}}}{2}})\cap\mathcal{D}.$

###### Proof.

Fix an integer $n\geqslant 2,$ let
$\prod_{m=1}^{\omega(n)}p_{i_{m}}^{r_{m}},~{}r_{m}\in\mathbb{Z}_{+},$ be the
prime factorization of $n.$ Since any divisor of $n$ is given by
$\prod_{m=1}^{\omega(n)}p_{i_{m}}^{s_{m}},~{}s_{m}\in\mathbb{N}_{r_{m}},$

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 1\\\
j|n\end{subarray}}j^{-\delta_{\mathbf{w}}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}$
$\displaystyle=$
$\displaystyle\prod_{m=1}^{\omega(n)}\sum_{s_{m}=0}^{r_{m}}p_{i_{m}}^{-\delta_{\mathbf{w}}s_{m}}\,w_{p^{s_{m}}_{i_{m}}}\,\mu(p_{i_{m}}^{r_{m}-s_{m}})$
$\displaystyle\overset{\eqref{mu-fun}}{=}$
$\displaystyle\prod_{m=1}^{\omega(n)}(p_{i_{m}}^{-\delta_{\mathbf{w}}(r_{m}-1)})(p_{i_{m}}^{-\delta_{\mathbf{w}}}w_{p_{i_{m}}^{r_{m}}}-w_{p_{i_{m}}^{r_{m}-1}})\overset{\eqref{mult-
growth}}{\geqslant}0.$

Thus, (8) is valid for all integers $n\geqslant 2$ and is trivially true for
$n=1.$ Hence by Theorem 1.3 and Remark 1.4(b), the conclusion now follows. ∎

As a consequence of the previous theorem, we recover [3, Theorem 3]. Indeed,
for any $\alpha>0,$ consider
$\mathbf{w}_{\alpha}=\\{(\mathbf{d}(j))^{\alpha}\\}_{j=1}^{\infty}.$ Since
$\mathbf{d}(n)=o(n^{\epsilon})$ for every $\epsilon>0$ (see [1, Equation
(31)]), $\sigma_{\mathbf{w}_{\alpha}}=1$ and $\delta_{\mathbf{w}_{\alpha}}=0.$
For any prime $p$ and integer $j\in\mathbb{Z}_{+},\mathbf{d}(p^{j})=j+1.$
Thus, (9) holds, and hence by Theorem 3.1,
$\mathcal{M}(\mathscr{H}_{\mathbf{w}_{\alpha}})$ is isometrically isomorphic
to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$

The following proposition yields a family of Hilbert spaces
$\mathscr{H}_{\mathbf{w}}$ illustrating the main result.

######  Proposition 3.2.

Let $\mathbf{w}=\\{w_{j}\\}_{j=1}^{\infty}$ be a multiplicative function
satisfying

  * $(i)$

$w_{j}=\mathcal{O}(j^{\delta})$ for every $\delta>0,$

  * $(ii)$

for each prime $p,$ $\\{w_{p^{j}}\\}_{j=0}^{\infty}$ is increasing.

If $\tilde{\mathbf{w}}=\\{1+w_{j}\\}_{j=1}^{\infty},$ then
$\mathcal{M}(\mathscr{H}_{\tilde{\mathbf{w}}})$ is isometrically isomorphic to
$H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$

###### Proof.

For a real number $r>1,$ let $\delta$ be a positive number such that
$r>\delta+1.$ Then by $(i),$

$\displaystyle\sum_{j=2}^{\infty}(1+w_{j})j^{-r}\leqslant\sum_{j=2}^{\infty}j^{-r}+C_{\delta}\sum_{j=2}^{\infty}j^{-(r-\delta)}<\infty,$
(18)

which yields $\sigma_{\tilde{\mathbf{w}}}\leqslant 1.$ Moreover, since
$\sum_{j=2}^{\infty}j^{-1}$ diverges, $\sigma_{\tilde{\mathbf{w}}}=1.$ By
using $(i)$, we also get that $\delta_{\tilde{\mathbf{w}}}=0.$ For any
$\sigma>0,$ let $\delta$ be such that $0<\delta<\sigma.$ Then, [12, Lemma 3.2]
combined with $(i)$ yields

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\ \tiny{\mbox{\bf
gpf}}(j)\leqslant
p_{n}\end{subarray}}w_{j}j^{-\sigma}\leqslant\sum_{\begin{subarray}{c}j\geqslant
2\\\ \tiny{\mbox{\bf gpf}}(j)\leqslant
p_{n}\end{subarray}}j^{-(\sigma-\delta)}<\infty,\quad n\geqslant 1.$ (19)

In view of $(ii),$ the series $\sum_{j=0}^{\infty}w_{2^{j}}$ diverges. This
together with (19) yields $\delta_{\mathbf{w}}=0.$ Hence, by $(ii),$ (9)
holds. Thus, by using the steps in the proof of Theorem 3.1, we obtain

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 1\\\
j|n\end{subarray}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant 0,\quad
n\geqslant 1.$

Combining this with [1, Theorem 2.1] gives

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 1\\\
j|n\end{subarray}}(1+w_{j})\mu\Big{(}\frac{n}{j}\Big{)}\geqslant 0,\quad
n\geqslant 1.$

The desired conclusion is now immediate from Theorem 1.3 together with Remark
1.4(b). ∎

Note that $\tilde{\mathbf{w}}$ defined above is never multiplicative.
Moreover, Proposition 3.2 applies to
$\mathbf{w}=\\{\mathbf{d}(j)\\}_{j=1}^{\infty}.$ We conclude this paper by
revealing that Theorem 1.3 also recovers [6, Theorem 1.11] for some particular
sequences.

######  Example 3.3.

Let $\mathbf{w}$ be given by (5). Then by [6, Equation (1.4)],
$\sigma_{\mathbf{w}}\leqslant 1.$ Since $\mathbf{w}$ is an increasing
sequence, $\sigma_{\mathbf{w}}=1.$ So, $\kappa_{\mathbf{w}}$ converges
absolutely on $\mathbb{H}_{1/2}\times\mathbb{H}_{1/2}.$ For an integer
$\alpha>0,$ consider the sequence
$\mathbf{w}_{\alpha}=\\{(\log(j))^{\alpha}\\}_{j=2}^{\infty}.$ By [6, Section
1],

$\displaystyle(\log(j))^{-\alpha}=\int_{[0,\infty)}j^{-2\sigma}\frac{2^{\alpha}}{\Gamma(\alpha)}\sigma^{-1+\alpha}d\sigma,\quad
j\geq 2,$

where $\Gamma$ denotes the Gamma function, and hence
$\sigma_{\mathbf{w}_{\alpha}}=1.$ Also, combining [6, Equation (1.4)] with
[12, Lemma 2] yields $\delta_{\mathbf{w}_{\alpha}}=0.$ In addition, by [4,
Equation (3.16)], we obtain the non-negativity of the Generalized von Mangoldt
function:

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant 2\\\
j|n\end{subarray}}(\log(j))^{\alpha}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant 0.$
(20)

An application of Theorem 1.3 now shows that
$\mathcal{M}(\mathscr{H}_{\mathbf{w}_{\alpha}})$ is isometrically isomorphic
to $H^{\infty}(\mathbb{H}_{0})\cap\mathcal{D}.$

The previous example yields that for any integer $\alpha>0,$
$\mathbf{w}_{\alpha}$ satisfies (20). In general, we don’t know whether
$\mathbf{w}$ satisfying (5) guarantees that

$\displaystyle\sum_{\begin{subarray}{c}j\geqslant n_{0}\\\
j|n\end{subarray}}w_{j}\mu\Big{(}\frac{n}{j}\Big{)}\geqslant 0,\quad
n\geqslant n_{0}.$

Acknowledgment. I am grateful to Sameer Chavan for suggesting this topic.
Needless to say, this paper would not have seen the present form without his
continuous support. I am also thankful to the anonymous referee for several
valuable inputs improving the presentation of the paper.

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# Music Genre Classification: Training an AI model

1st Keoikantse Mogonediwa  Academy of computer science
University of Johannesburg
Auckland Park JHB 2092, South Africa

###### Abstract

Music genre classification is an area that utilizes machine learning models
and techniques for the processing of audio signals, in which applications
range from content recommendation systems to music recommendation systems. In
this research I explore various machine learning algorithms for the purpose of
music genre classification, using features extracted from audio signals.The
systems are namely, a Multilayer Perceptron (built from scratch), a k-Nearest
Neighbours (also built from scratch), a Convolutional Neural Network and
lastly a Random Forest wide model. In order to process the audio signals,
feature extraction methods such as Short-Time Fourier Transform, and the
extraction of Mel Cepstral Coefficients (MFCCs), is performed. Through this
extensive research, I aim to asses the robustness of machine learning models
for genre classification, and to compare their results.

###### Index Terms:

Multi-layer perceptron, Convolutional Neural Network, K-Nearest Neighbours,
Random Forest Classifier

##  I Introduction

Music is a form of expression, a universal language that is easy to translate
into cultural stories and different emotions. Communities and societies all
over the world unite through music. The emergence of digital music in the
industry has resulted in music being easily accessible from any technology
device [9]. Online streaming platforms consist of different types of musical
genre’s. This research paper studies a gap in this industry, where genre
classification algorithms are developed to categorize musical genres as a
added feature to enhance a user’s experience. Genre classification can be used
in different areas within the music industry, such as assisting music
producers in identifying which musical genres are more popular.

A significant amount of work has gone into music genre classification models.
Genre classification models that have been developed depend on the quality of
the provided labels as part of the training process. Even though these models
are advanced, there remains a gap in real world applications, where some data
may not contain the necessary text labels required for genre classification
[2].

###  I-A Is the problem solved?

Classification models mainly follow a supervised learning approach, thus
meaning they require text labels that are clear and easily definable. The
study of this paper, aims to cover an area where ambiguity could exist within
genre classification. The niche research area is to establish if different
machine learning algorithms can perform more accurately if they use input data
extracted as features from audio files, instead of using text labels.

###  I-B A possible solution

A possible solution is to use ”Short-Time Fourier Transform” (STFT). STFT is
the practice of using audio signals to process and analyse the frequencies of
an audio file over a certain time period [6]. Fourier transformation consists
of different concepts such as overlapping windows, which is a practice that
helps to capture smoother transitions between the time segments. This study
proposes the use of Short-Time Fourier Transformation to extract features from
an audio file. The features are then used as an input into different machine
learning algorithms that are trained to classify audio files into their
respective genre.

Figure 1: Typical flowchart demonstrating the steps involved in music genre
classification

##  II Intended Experiment methods

Music genre classification within the machine learning niche creates a
specialized research area that aims to develop algorithms that can
automatically detect and categorize music. The algorithms include the
Multilayer Perceptron, a Convolutional Neural Network (CNN), which is part of
the deep learning models that are applied for various tasks such as computer
vision or classification, the K-Nearest Neighbour (K-NN), and finally a Random
Forest wide model that incorporates feature engineering.

The four models are compared to identify which model produces the most
accurate results. This research paper provides a deep understanding of the
challenges with regards to genre classification. The strategy used in this
research integrates machine learning algorithms and windows forms applications
as a user interface to make an impact on the music industry.

To ensure the results of this study are efficient, this study first looks at a
Multilayer Perceptron (MLP) algorithm as a baseline model, which has an
standard architecture of just 3 layers. The 3 layers comprise of an input,
which receives pre-processed data, typically presented to the model as a
feature vector, a hidden and output layer, where the model is trained to
reduce errors between the target output and the computed values [5]. An
example of the features used for a multilayer perceptron are Mel-Frequency
Cepstral Coefficients which are mostly used for the purpose of music analysis.
The second algorithm is a CNN, which is a deep neural network that has
multiple layers, with reduced parameters which provides researchers with the
opportunity to work with larger models [1]. To name a few, the hyper-
parameters typically used for these models are, namely batch sizes, epochs and
learning rates. CNN models also make use of a split between the training,
validation and test data.

K-Nearest Neighbour is a classification model that classifies instances based
on their similarities. It assigns a label to data point by identifying the
class that is predominantly recognized within the nearest k-neighbours [3].
K-Nearest Neighbour models includes hyper-parameter tuning of the different
values of k, to find the optimal value that provide the best classification
result. The last model is a wider model, which is considered as a type of
neural network that consists of nodes and layers. Wide models focus on the
different patterns within the data.

Figure 2: Metadata information retrieved from the dataset.

The proposed dataset for this research is the GTZAN Dataset, which is applied
specifically for the purpose of training machine learning models for
classification. The dataset consists of 10 genres, with 100 audio files for
each genre, that are 30 seconds long [4]. The different genres are ”blues”,
”classical”, ”country”, ”disco”, ”hiphop”, ”jazz”, ”metal”, ”pop”, ”reggae”,
”rock”. During the course of this research, I discovered that the dataset
actually has 1 or 2 corrupted files that could not be processed, therefore
entailing that those files had to be excluded from the dataset. The files in
question were identified within the ”jazz” genre.

##  III Data Pre-Processing

Figure 3 is the Short-Time Fourier Transform (STFT) representation that is
extracted from a randomly selected audio file. It contains the meaningful
information that can be used to classify audio. It is basically the analysis
of frequency data of a signal over short time intervals. Padding is applied to
the extracted features, to ensure that the audio features that are extracted
are of the same length, which is represented in figure 4. This also ensures
that the model captures only the necessary information for the training
process.

The process of STFT includes the division of the input audio signal into short
frames, that overlap each other, capturing temporal information. Fourier
Transform is then applied to each window frame, whereby a conversion from a
time domain to a frequency domain occurs, which reveals the frequency of the
signal in each frame. It is also key to highlight that STFT operates with
fixed length segments (windows), therefore it may be necessary to apply
padding, to ensure that the length of the signal remains constant during the
transformation process.

The GTZAN dataset also consists csv metadata that is utilized for the purpose
of this research. The training data is split into 80% for the training data,
and 20% for the testing phase. Figure 2 presents a sample of the metadata
information, whereby a complete view of the metadata is a total of 60 columns
which all contain information extracted from audio signals through STFT.

Figure 3: STFT of a randomly selected Reggae audio file Figure 4: STFT of a
randomly selected Reggae audio file in which padding has been applied

The spectrogram representation which is obtained through STFT, contains the
necessary features required for classification. The most important feature to
highlight in this regard is the Mel-Frequency Cepstral Coefficients (MFCCs).
MFCCS is typically used for speech recognition, speaker identification and
musical genre classification. This feature contains the necessary frequency
information that used to process audio signals of different genres which is
important for classification. The numerical values determined from this
features are converted into a 1D feature vector and provided as input into the
models. Figure 5 represents the spectrogram.

Figure 5: Spectrogram of the selected reggae genre audio file Figure 6: Wave
form representation of a randomly selected Reggae audio file

##  IV Baseline Multilayer Perceptron

###  IV-A Model Specifications

The MLP used in this research comprises of an input layer, a hidden layer with
1024 neurons, and an output layer in which a softmax function is applied to
it. A ReLU activation function is applied, along with gradient descent with
the purpose of fine-tuning hyperparameters.

In the context of this model, the complexity limits the model’s capabilities
to learn meaningful representations. The model has not been tested with
techniques such as batch normalization, or drop out. The results presented by
the model showcase that the prescribed idea proposed in this research of using
an MLP with only 3 hidden layers for classification, is not an efficient
solution.

The similarly related work by [8], outlines the use of techniques such as
feature selection, or a combination of computing space-time decomposition. In
their research paper, the MLP’s performance is able to produce results up to
56.40%, which signifies the idea of using different techniques to improve
performance [8].

##  V Deep Neural Network: CNN

###  V-A Model Specifications and results

The model consists of 7 layers. The first convolutional layer uses 32 filters
with a kernel size of (3,3). The second convolutional layer consists of 64
filters, also with the kernel size of (3,3). A max pooling layer is included,
and the output is then flattened to ensure that it is contained within a 1D
array, which is the feature vector containing the necessary input values
required for computation. The selected activation function is a Rectifier
Linear Unit. When process the data, because the CNN uses the windows frames as
input for performing classification, which are the time segments, the feature
extraction process had to include padding, in which a specified max length is
determined, and if the length of the time segment exceeds the max length, the
values are truncated to fit the max length, or to add 0 values in a case where
the time segments are not equal to the max length. This is done to ensure that
the values remain constant for the CNN algorithm.

In terms of calculating the loss, a categorical cross entropy function is used
as it is most suitable for multiple output classifications. The selected
optimizer is ”adam” optimizer, with a softmax function being applied on the
last layer of the model. The results from this architecture return a 0.25%
accuracy when the model is trained for 10 epochs, with a dense layer
consisting of 128 neurons. The possible reason for this low accuracy rate
could be that the model might benefit from an increase in complexity, which is
what I further researched and noted, and indeed the accuracy rating increased.

When the model is provided with 2 dense layers of 256 and 128 neurons, along
with a dropout rate of 0.3 which randomly sets the input values to 0 to
prevent overfitting, its accuracy rate increases to around 0.275%. This could
mean that increasing the dense layers improves the performance of the model.

Figure 7: Training accuracy of CNN Model Figure 8: Training loss of CNN model.

Figure 7 essentially represents an accuracy rating of about 23%, in which the
validation data shows that it is possible that, at some point, the model was
not learning any information before a significant rise in accuracy. In
comparison to the loss outlined in figure 8, this could mean the model was
stuck in a local minima before continuing onwards. The change in accuracy is
affected by the addition of dropout and an inclusion of another layer with 64
neurons. The model is trained for 10 epochs, whereby the increase in accuracy
displayed on the chart, could signify that provided the training time is
increased, the model accuracy could increase.

##  VI K-Nearest Neighbours From Scratch

###  VI-A Model Specifications and results

The K-Nearest model presented a decent result of 55% for music genre
classification. The models specifications include a K = 1 value, in which,
increasing this value decreases the performance of the model significantly.

K-fold cross validation is technique that can be used for improving the KNN’s
performance. A similar study highlighted the use of K-fold cross validation
for performing classification [10], which improves the performance, and also
ensures that the accuracy is not biased.

The KNN model is essentially a model that requires less training time compared
to other existing models. Selecting a k value ranges from 1 to 100 in which
the different values can be tested to identify the most optimal accuracy. The
similarly related research work on genre classification presented a KNN with
an accuracy rating of 77.18%. The author opted for the Spotify dataset which
contains all the values for the metadata of each genre [7].

##  VII Random Forest Wide model

###  VII-A Model Specifications and results

The random forest is defined with 35 estimators and a depth of 25 for
complexity. In this regard, the more complex the model is, the more it is able
to interpret data and improve its accuracy, similarly for its number of
estimators. The model’s results are present a positive outcome of about 84%
for genre classification.

The confusion matrix entails that a lighter color essentially represent how
well the model has learned meaningful representations. Interestingly, with
regards to this particular model, the confusion matrix demonstrates the models
ability to classify classical music most accurately compared to other genres.
Figure 9 presents the confusion matrix.

Figure 9: Confusion Matrix for Random Forest.

###  VII-B Comparisons

Model | Results
---|---
MLP | 0%
CNN | 40%
KNN | 55%
RandomForest | 84%

Figure 10: Count of the audio files for each genre

##  VIII Conclusion

To summarise the findings, the Random Forest model presented a higher accuracy
rating compared to the MLP, CNN and KNN. The Random Forest’s ensemble of
decision trees allowed it to capture complex relationships in the data more
effectively than the other models. In contrast, the MLP’s architecture and
limited capacity prevent the model from learning intricate patterns and useful
information to perform classification. This is also similar to the
covolutional neural network, whereby the models complexity plays a role in its
ability to classify accurately. It is also possible that the CNN’s fixed
kernel sizes were not the most optimal solution for the purpose of this
research.

In terms of feature engineering, STFT is a feature extraction method that can
be applied to audio files to depict information that can be transposed into
feature vectors, to train machine learning models. The dataset contains a few
files that are corrupt, and it is possible that those files can impact the
ability of the models to generalise when shown new data. The identified files
were recognised in the ”jazz” class, which could also result in a class
imbalance compared to the rest of the classes.

Figure 10 present a count of the number of audio files for each class that are
processed and used for this research. The distribution of samples across
classes provides considerations when interpreting the model’s performance and
generalization. To conclude this research, machine learning models present an
avenue in music that can be further explored and utilized.

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