# Systematic input scheme for many-boson Hamiltonians via quantum walk

Weijie Du<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50010, USA  James P. Vary  Department of Physics and
Astronomy, Iowa State University, Ames, Iowa 50010, USA

###### Abstract

We develop a novel, systematic input scheme for many-boson Hamiltonians in
order to solve field theory problems within the light-front Hamiltonian
formalism via quantum computing. We present our discussion of this input
scheme based on the light-front Hamiltonian of the two-dimensional $\phi^{4}$
theory. In our input scheme, we employ a set of quantum registers, where each
register encodes the occupation of a distinct boson mode as binaries. We
squeeze the boson operators of each mode and present the Hamiltonian in terms
of unique combinations of the squeezed boson operators. We design the circuit
modules for these unique combinations. Based on these circuit modules, we
block encode the many-boson Hamiltonian utilizing the idea of quantum walk. We
demonstrate our input scheme by solving the low-lying spectra of the
Hamiltonian utilizing the IBM Qiskit quantum simulator. We can incorporate the
input scheme in this work with the input scheme for many-fermion Hamiltonians;
they jointly offer new pathways to solving the structure and dynamics of field
theory problems on future fault-tolerant quantum computers.

##  I Introduction

Quantum field theory (QFT) reconciles quantum mechanics and special relativity
and it has important applications in many fundamental areas of physics [1].
Nonperturbative, ab initio structure and dynamics solutions of QFT are
challenging on classical computers due to the nature of quantum many-body
problems [2]. Quantum computing hardwares and algorithms hold the promise to
address such numerical challenges [3].

Although, it is not currently possible to solve realistic problems on quantum
computers in a way that outcompetes the best classical computers, it is an
important and challenging task to develop new quantum algorithms that are
suitable for solving QFT problems via ab initio methods on future fault-
tolerant quantum computers. Following pioneering work by Jordan, Lee and
Preskill [4], many quantum algorithms have been proposed for treating the QFT
problems via the lattice-based Hamiltonian formalism [5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 8, 20]. In this cutting-edge area, major
research focuses on reducing the error of digitization and latticing the
continuous field onto qubit degrees of freedom as well as reducing the scaling
of the computational resources [6, 8].

Fewer works employ the alternative promising light-front (LF) Hamiltonian
formalism [21, 22] in formulating the QFT problems in quantum computing. The
LF Hamiltonian formalism has the attractive feature that the LF vacuum has a
simple structure as the Fock vacuum is an exact eigenstate of the full normal-
ordered LF Hamiltonian; this enables the systematic Fock-space expansion of
the physical states [23]. To date, the LF Hamiltonian formalism has witnessed
successful initial applications in describing relativistic bound-state
problems and dynamics in quantum electrodynamics, quantum chromodynamics, and
chiral field theories on classical computers [24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 28, 34, 35, 36, 37, 38, 39, 40]. Many quantum algorithms have also
been proposed to solve the structure and dynamics of hadronic systems [41, 42,
43, 44, 45, 46, 47, 48].

However, a systematic Hamiltonian input scheme is a missing corner stone for a
general-purpose quantum algorithm in computing the bound-state and scattering
properties of the QFT problems based on the LF Hamiltonian formalism. An
efficient input scheme is critical for large-scale calculations on future
quantum computers, where the construction of the many-body Hamiltonian matrix
is highly nontrivial. In Refs. [49, 50], we proposed a systematic input scheme
for second-quantized many-fermion Hamiltonians. This input scheme respects the
symmetries of the many-fermion Hamiltonians and is flexible for incorporating
the particle-number variations. Yet, it is still an important mission to
develop the counterpart Hamiltonian input scheme for many-boson systems. One
noticeable difficulty that hinders such a development comes from the Bose-
Einstein statistics, where multiple bosons can occupy the same mode and such
many-boson states require proper normalization [51]. This is in contrast to
the many-fermion case where the Pauli exclusion principle holds and the many-
fermion states do not encounter multiple-occupancy normalization factors.

In this work, we introduce a novel and systematic input scheme for the many-
boson Hamiltonians in quantum computing. We work in the second-quantized
representation. Compared to the prototype sparse-matrix-based Hamiltonian
input models [52, 53] that work with the row and column indices of the
Hamiltonian matrices in the framework of first quantization, our input model
operates directly on the Fock states and offers a convenient approach to
access the many-boson Hamiltonian matrix elements based on the Fock bases.
Compared to the Hamiltonian input scheme based on the approach of linear
combination of unitaries [54, 55], our input scheme avoids the nontrivial
“Prepare Oracle” for extracting the Hamiltonian matrix elements, which can be
demanding in gate cost.111We expect that the gate cost of the Prepare Oracle
for inputting the matrix elements of a general Hamiltonian can be as demanding
as that for preparing a high-dimensional state vector with arbitrary
coefficients. Our input scheme respects the symmetries of the many-boson
Hamiltonians by design. It also enables a straightforward approach to prepare
the many-boson states with desired symmetries on quantum computers. We provide
an explicit and systematic circuit design for our input model in this work.
This is to be compared with alternative input scheme of many-boson
Hamiltonians [56] that is based on abstract controlled arithmetic operations
that might be complicated to achieve on quantum computers.

We present the discussion of our input scheme utilizing the two-dimensional
$\phi^{4}$ [$(\phi^{4})_{2}$] theory as it is one of the simplest interacting
QFT problems, and illustrates the essential ideas and techniques. We address
the difficulty in normalizing the many-boson state by introducing the squeezed
(or scaled) boson operators. We also present explicit circuit designs of the
squeezed boson operators as well as their combinations, which properly treats
the normalization of many-boson states and encodes the normalization as the
phases of qubit states. Based on these techniques, we show the construction of
the quantum walk states [57, 52, 53] to block encode the many-boson
Hamiltonian. Utilizing this input scheme, we can quantum compute the structure
and dynamics of the many-boson system utilizing precise and efficient quantum
algorithms [58, 59, 60, 61]. We demonstrate our input model by solving the
spectrum of the LF Hamiltonian of the $(\phi^{4})_{2}$ theory, where such
spectral solutions shed lights on the phase transition of the $(\phi^{4})_{2}$
theory [62, 63].

The arrangement of this paper is as follows. We present the $(\phi^{4})_{2}$
theory within the LF Hamiltonian formalism and discuss the basis choice in
Sec. II. In Sec. III, we present the details of the Hamiltonian input scheme
of the LF Hamiltonian of the $(\phi^{4})_{2}$ theory. In Sec. IV, we
demonstrate the input scheme via a numerical example, where we compute the
eigenenergies based on our input scheme utilizing a quantum-classical
algorithm. We summarize in Sec. V, where we also present an outlook. We
provide additional details in the Appendix.

##  II Theory

###  II.1 LF Hamiltonian of the $(\phi^{4})_{2}$ theory

In brief, we present necessary details of the LF Hamiltonian formalism of the
$(\phi^{4})_{2}$ theory. Interested readers are referred to Refs. [62, 63] for
more details.

The Lagrangian density of the $(\phi^{4})_{2}$ theory is

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi-m^{2}\phi^{2})-\frac{\lambda}{4!}\phi^{4},$
(1)

where $m$ denotes the mass parameter while $\lambda$ is the coupling constant.
One chooses $\lambda>0$ and $m>0$ so that the corresponding LF Hamiltonian is
bounded and the vacuum state is the normal vacuum for small coupling in the
symmetric phase of the theory. Following the discrete light-cone quantization
(DLCQ) approach [64, 65, 66], one obtains the total longitudinal momentum $K$
and the LF Hamiltonian $H$. These are given by Eqs. (8-11) in Ref. [63] where
the zero mode is neglected as

$\displaystyle K=$ $\displaystyle\sum_{k=1}ka^{\dagger}_{k}a_{k},$ (2)
$\displaystyle H=$ $\displaystyle H_{1\rightarrow 1}+H_{2\rightarrow
2}+H_{3\rightarrow 1}+H_{1\rightarrow 3},$ (3)

with

$\displaystyle H_{1\rightarrow 1}=$
$\displaystyle\sum_{k}\frac{1}{k}\Big{[}m^{2}+\underbrace{\frac{\lambda}{4\pi}\frac{1}{2}\sum_{l}\frac{1}{l}}\Big{]}a_{k}^{\dagger}a_{k},$
(4) $\displaystyle H_{2\rightarrow 2}=$
$\displaystyle\frac{\lambda}{4\pi}\sum_{k\leq l}\sum_{m\leq
n}\frac{1}{N^{2}_{kl}}\frac{1}{N^{2}_{mn}}\frac{1}{\sqrt{klmn}}\Big{[}a_{k}^{\dagger}a_{l}^{\dagger}a_{m}a_{n}\Big{]}\delta_{(m+n),(k+l)},$
(5) $\displaystyle H_{3\rightarrow 1}=$
$\displaystyle\frac{\lambda}{4\pi}\sum_{k}\sum_{l\leq m\leq
n}\frac{1}{N^{2}_{lmn}}\frac{1}{\sqrt{klmn}}\Big{[}a^{\dagger}_{k}a_{l}a_{m}a_{n}\Big{]}\delta_{k,(m+n+l)},$
(6) $\displaystyle H_{1\rightarrow 3}=$
$\displaystyle\frac{\lambda}{4\pi}\sum_{k}\sum_{l\leq m\leq
n}\frac{1}{N^{2}_{lmn}}\frac{1}{\sqrt{klmn}}\Big{[}a^{\dagger}_{n}a^{\dagger}_{m}a^{\dagger}_{l}a_{k}\Big{]}\delta_{k,(m+n+l)},$
(7)

with $N_{kl}=1$ for $k\neq l$ and $N_{kl}=\sqrt{2!}$ for $k=l$, and

$\displaystyle N_{lmn}=\begin{cases}1,&\text{for}\ l\neq m\neq n,\\\
\sqrt{3!},&\text{for}\ l=m=n,\\\ \sqrt{2!},&\text{for any two of $l,m$, and
$n$ are equal}.\end{cases}$ (8)

The summations of $k$, $l$, $m$, and $n$ run over positive integers for
symmetric boundary conditions which is the choice adopted here. Here,
$H_{1\rightarrow 1}$ and $H_{2\rightarrow 2}$ are Hermitian and
$H^{\dagger}_{3\rightarrow 1}=H_{1\rightarrow 3}$. One notes that $H$ preserve
the evenness and oddness of the particle number of the many-boson system. The
total longitudinal momentum $K$ is a conserved quantity in the LF Hamiltonian
formalism as indicated by the Kronecker deltas in the above expressions and
$[H,K]=0$. According to $H$, one can compute the eigenvalues of the
$(\phi^{4})_{2}$ theory. These eigenvalues are related to the mass spectrum of
the invariant-mass operator

$\hat{M}^{2}=KH.$ (9)

One can typically scale $H$ with $m^{2}$ and rewrite the interaction theory as
a function of a single dimensionless variable $\lambda/m^{2}$ for convenience
[63]. With this scaling, $H$ has the dimension of [MeV]2 and $K$ is
dimensionless.

We define $a^{\dagger}_{k}$ and $a_{k}$ as the creation and annihilation
operators for the mode with the longitudinal momentum $k$, respectively. These
boson operators satisfy the commutation relations [51]

$[a_{m},a_{n}]=[a^{\dagger}_{m},a^{\dagger}_{n}]=0,\
[a_{m},a^{\dagger}_{n}]=\delta_{m,n}.$ (10)

For the LF Hamiltonian $H$ [Eq. (3)] retained in this work, we note that there
are at most four boson operators in all the monomials.

We denote $\ket{k^{r_{k}}}$ to be the normalized many-boson state with $r_{k}$
bosons being in the mode $k$. The actions of the boson operators on
$\ket{k^{r_{k}}}$ are [51]

$a_{k}^{\dagger}\ket{k^{r_{k}}}=\sqrt{r_{k}+1}\ket{k^{r_{k}+1}},\
a_{k}\ket{k^{r_{k}}}=\sqrt{r_{k}}\ket{k^{r_{k}-1}},$ (11)

where we have $a_{k}\ket{k^{0}}=0$ with $r_{k}=0$ denoting zero occupation of
the mode $k$. In this sense, we have the normalized state $\ket{k^{r_{k}}}$ as

$\ket{k^{r_{k}}}=\frac{1}{\sqrt{r_{k}!}}\big{(}a_{k}^{\dagger}\big{)}^{r_{k}}\ket{k^{0}}.$
(12)

We note that the underbraced term in Eq. (4) is logarithmically divergent [62,
67]. One can remove this logarithmic divergence by considering the normal-
ordered Hamiltonian [68], which means that one ignores a divergent constant
from the definition of the mass parameter in the Hamiltonian.

The $(\phi^{4})_{2}$ theory has also been used for studying spontaneous
symmetry breaking. However, spontaneous symmetry breaking arises at a critical
coupling ${\lambda}/{m^{2}}$ whose value depends on the treatment of the zero
mode [69, 70, 63, 71]. Evaluating the precise value for the critical coupling
involves taking the continuum limit which is very challenging on classical
computers due to the nature of the strong correlation in the quantum many-body
problems. Quantum computing might be promising for addressing such challenges.

###  II.2 Basis choice and eigenvalue problem

Following the notation of Ref. [62], we denote a general Fock state as
$\ket{k^{r_{k}},\ l^{r_{l}},\ m^{r_{m}},\ \cdots}$. The total longitudinal
momentum of the Fock state is $kr_{k}+lr_{l}+mr_{m}+\cdots$, which receives
contributions from every single mode in the Fock state.

As $K$ commutes with $H$, the LF Hamiltonian matrix elements between the Fock
states with different $K$ values vanish. Therefore, in practical DLCQ
calculations, one can enumerate and retain the Fock bases with the fixed total
longitudinal momentum [24, 40, 33, 34, 35, 36]. Within the Fock bases set
$\mathcal{B}_{K}\equiv\\{\ket{k^{r_{k}},\ l^{r_{l}},\ m^{r_{m}},\ \cdots}\\}$
of a specific $K$ value, one evaluates the matrix of $H$, or equivalently, the
matrix of $\hat{M}^{2}$ [Eq. (9)], and computes the mass spectrum of the
$(\phi^{4})_{2}$ theory. Based on the mass spectrum, one can interrogate the
critical coupling constant for the vanishing mass gap of the many-boson system
as a function of $K$, and extrapolate for the critical coupling in the limit
of $K\rightarrow\infty$, i.e., the continuum limit [62, 63].

##  III Hamiltonian input scheme

In this section, we present the elements of our Hamiltonian input scheme. We
begin with our strategy to squeeze the boson operators, and rewrite the many-
boson Hamiltonian with the squeezed boson operators as well as their
combinations. Then, we discuss the scheme to encode the many-boson states, and
present the design of the circuit modules for the squeezed boson operators as
well as their combinations. Based on these modules, we show our Hamiltonian
input scheme utilizing the idea of quantum walk. Finally, we preset the
analysis of the qubit and gate cost of our input scheme.

###  III.1 Modified Hamiltonian

For the $(\phi^{4})_{2}$ theory, we can rewrite Eq. (3) according to the
unique types of monomials as

$\displaystyle
H=\underbrace{B_{j_{0}}a^{\dagger}_{k}a_{k}+\cdots}_{H_{1\rightarrow
1}}+\underbrace{B_{j_{1}}a^{\dagger}_{k}a^{\dagger}_{l}a_{m}a_{n}+\cdots}_{H_{2\rightarrow
2}}+\underbrace{B_{j_{2}}a^{\dagger}_{k}a_{l}a_{m}a_{n}+\cdots}_{H_{3\rightarrow
1}}+\underbrace{B_{j_{3}}a^{\dagger}_{n}a^{\dagger}_{m}a^{\dagger}_{l}a_{k}+\cdots}_{H_{1\rightarrow
3}},$ (13)

where we employ the subscripts $k$, $l$, $m$, $n\in[1,\ K]$ to specify the
modes of the boson operators. $B_{j_{i}}$ denotes the coefficient, with

$B_{j_{0}}=\frac{1}{k},\
B_{j_{1}}=\frac{\lambda}{4\pi}\frac{1}{N^{2}_{kl}N^{2}_{mn}}\frac{\delta_{m+n,k+l}}{\sqrt{klmn}},\
B_{j_{2}}=B_{j_{3}}=\frac{\lambda}{4\pi}\frac{1}{N^{2}_{lmn}}\frac{\delta_{m+n+l,k}}{\sqrt{klmn}}.$
(14)

The dots denote the other monomials with the same types of operator structures
as those shown explicitly.

In order to develop our input scheme for the many-boson Hamiltonian, we
squeeze (or scale) the boson operators for the mode $k$ according to the
maximal occupation value $\Lambda_{k}=\lfloor K/k\rfloor$ that is consistent
with the total longitudinal momentum $K$ for the system as

$b_{k}\equiv{a_{k}}/{\sqrt{\Lambda_{k}}},\
b^{\dagger}_{k}={a^{\dagger}_{k}}/{\sqrt{\Lambda_{k}}}.$ (15)

This leads to the following identities

$\displaystyle
b_{k}\ket{k^{r_{k}}}=\sqrt{\frac{r_{k}}{\Lambda_{k}}}\ket{k^{r_{k}-1}},\ $
$\displaystyle\ \text{for}\ r\in[1,\Lambda_{k}],$ (16) $\displaystyle
b^{\dagger}_{k}\ket{k^{r_{k}}}=\sqrt{\frac{r_{k}+1}{\Lambda_{k}}}\ket{k^{r_{k}+1}},\
$ $\displaystyle\ \text{for}\ r\in[0,\Lambda_{k}-1],$ (17)

with $b_{k}\ket{k^{0}}=0$. According to this scaling, we have the scaled
normalization factor $\sqrt{r_{k}/\Lambda_{k}}\in(0,1]$ for
$r_{k}\in[1,\Lambda_{k}]$ in Eq. (16), and the factor
$\sqrt{(r_{k}+1)/\Lambda_{k}}\in(0,1]$ for $r_{k}\in[0,\Lambda_{k}-1]$ in Eq.
(17). These scaled normalization factors, denoted as $\xi_{k}$, can be encoded
in terms of the angles of the rotational gates [72] in the quantum circuit, as
shown in the following text.

With the squeezed boson operators, we can rewrite $H$ [Eq. (13)] as

$H=\underbrace{B^{\prime}_{j_{0}}b^{\dagger}_{k}b_{k}+\cdots}_{H_{1\rightarrow
1}}+\underbrace{B^{\prime}_{j_{1}}b^{\dagger}_{k}b^{\dagger}_{l}b_{m}b_{n}+\cdots}_{H_{2\rightarrow
2}}+\underbrace{B^{\prime}_{j_{2}}b^{\dagger}_{k}b_{l}b_{m}b_{n}+\cdots}_{H_{3\rightarrow
1}}+\underbrace{B^{\prime}_{j_{3}}b^{\dagger}_{n}b^{\dagger}_{m}b^{\dagger}_{l}b_{k}+\cdots}_{H_{1\rightarrow
3}},$ (18)

with

$\displaystyle B^{\prime}_{j_{0}}=$ $\displaystyle\frac{\Lambda_{k}}{k},$ (19)
$\displaystyle B^{\prime}_{j_{1}}=$
$\displaystyle\frac{\lambda}{4\pi}\frac{1}{N^{2}_{kl}N^{2}_{mn}}\sqrt{\frac{\Lambda_{k}\Lambda_{l}\Lambda_{m}\Lambda_{n}}{klmn}}\delta_{m+n,k+l},$
(20) $\displaystyle B^{\prime}_{j_{2}}=$ $\displaystyle
B^{\prime}_{j_{3}}=\frac{\lambda}{4\pi}\frac{1}{N^{2}_{lmn}}\sqrt{\frac{\Lambda_{k}\Lambda_{l}\Lambda_{m}\Lambda_{n}}{klmn}}\delta_{m+n+l,k}.$
(21)

The boson modes $k,\ l,\ m,\ n$ in $H$ are not necessarily different. In this
sense, we can group the squeezed boson operators according to the modes in
each monomial, while preserving the ordering of the creation and annihilation
operators in view of the commutation relations. For example, with $k\neq l\neq
m$, we can rewrite $b^{\dagger}_{k}b^{\dagger}_{l}b_{l}b_{k}$ and
$b^{\dagger}_{k}b^{\dagger}_{k}b_{l}b_{m}$ as
$(b^{\dagger}_{k}b_{k})(b^{\dagger}_{l}b_{l})$ and
$(b^{\dagger}_{k}b^{\dagger}_{k})(b_{l})(b_{m})$, respectively. With this
grouping for all the monomials in $H$, we find that there are eight unique
combinations $\\{W_{k}\\}$ of the squeezed boson operators that act on one
single mode. We list these unique combinations in Table 1, where we also
present other details of these combinations.

Table 1: Eight unique combinations of the squeezed boson operators. The first column lists all the unique combinations $W_{k}$ of the squeezed boson operators. The second column presents the notations $(\cdot)_{k}$ of the circuit modules for $W_{k}$. The third column presents the scaled normalization factors $\xi_{r_{k}}$ of the many-boson state obtained from the action of $W_{k}$ on $\ket{k^{r_{k}}}$ [Eq. (24)] for $r_{k}\in[r_{k,i},r_{k,f}]$ (fourth column). The fifth column presents the $r_{k}$ values that belong to the vanishing cases in the practical circuit design of $(\cdot)_{k}$. The rightmost column presents the increment and decrement $\Delta_{k}$ in the occupation $r_{k}$ when $W_{k}$ acts on $\ket{k^{r_{k}}}$. $W_{k}$ | $(\cdot)_{k}$ | scaled normalization factors $\xi_{r_{k}}$ | $[r_{k,i},r_{k,f}]\ \ \ \ \ $ | vanishing cases | $\Delta_{k}$
---|---|---|---|---|---
$b^{\dagger}_{k}$ | $(+)_{k}$ | $\sqrt{(r_{k}+1)/\Lambda_{k}}$ | $[0,\ \Lambda_{k}-1]$ | $\Lambda_{k}$ | ${+1}$
$b^{\dagger}_{k}b^{\dagger}_{k}$ | $(++)_{k}$ | $\sqrt{(r_{k}+1)(r_{k}+2)/\Lambda_{k}^{2}}$ | $[0,\ \Lambda_{k}-2]$ | $\Lambda_{k}-1,\ \Lambda_{k}$ | ${+2}$
$b^{\dagger}_{k}b^{\dagger}_{k}b^{\dagger}_{k}$ | $(+++)_{k}$ | $\sqrt{(r_{k}+1)(r_{k}+2)(r_{k}+3)/\Lambda_{k}^{3}}$ | $[0,\ \Lambda_{k}-3]$ | $\Lambda_{k}-2,\ \Lambda_{k}-1,\ \Lambda_{k}$ | ${+3}$
$b_{k}$ | $(-)_{k}$ | $\sqrt{r_{k}/\Lambda_{k}}$ | $[1,\ \Lambda_{k}]$ | $0$ | ${-1}$
$b_{k}b_{k}$ | $(--)_{k}$ | $\sqrt{r_{k}(r_{k}-1)/\Lambda^{2}_{k}}$ | $[2,\ \Lambda_{k}]$ | $0,\ 1$ | ${-2}$
$b_{k}b_{k}b_{k}$ | $(---)_{k}$ | $\sqrt{r_{k}(r_{k}-1)(r_{k}-2)/\Lambda^{3}_{k}}$ | $[3,\ \Lambda_{k}]$ | $0,\ 1,\ 2$ | ${-3}$
$b^{\dagger}_{k}b_{k}$ | $(+-)_{k}$ | $r_{k}/\Lambda_{k}$ | $[1,\ \Lambda_{k}]$ | $0$ | ${0}$
$b^{\dagger}_{k}b^{\dagger}_{k}b_{k}b_{k}$ | $(++--)_{k}$ | $r_{k}(r_{k}-1)/\Lambda_{k}^{2}$ | $[2,\ \Lambda_{k}]$ | $0,\ 1$ | ${0}$

###  III.2 Basis encoding scheme

Recall that we retain the Fock basis set $\mathcal{B}_{K}$ with fixed total
longitudinal momentum $K$ and omit the zero mode in this work. In order to
encode the Fock states in $\mathcal{B}_{K}$, we choose to implement the
quantum register $s$ that consists of a set of subregisters $\\{s_{k}\\}$: for
the mode $k\in[1,K]$, we apply the compact mapping to encode the occupation
$r_{k}$ as binaries in the subregister $s_{k}$. Following this scheme, as an
example, we can encode the Fock state
$\ket{\mathcal{F}}=\ket{k^{r_{k}},l^{r_{l}},m^{r_{m}}}$ with $1\leq k<l<m\leq
K$ as the register state

$\ket{0}_{s_{1}}\cdots\ket{{r_{k}}}_{s_{k}}\cdots\ket{{r_{l}}}_{s_{l}}\cdots\ket{{r_{m}}}_{s_{m}}\cdots\ket{0}_{s_{K}},$
(22)

where the occupations $r_{k}$, $r_{l}$, and $r_{m}$ are encoded in the binary
representation. We adopt the notation $\ket{\cdot}_{reg}$ to specify the state
of the quantum register “$reg$”. We occasionally omit the subscript when there
is no risk of confusion.

###  III.3 Circuit representation of $W_{k}$

We discuss the design of the circuit module $(\cdot)_{k}$ for the combination
$W_{k}$ of the squeezed boson operators in this section. Indeed, this module
can be understood as the “circuit representation” of $W_{k}$. A schematic
circuit design of $(\cdot)_{k}$ is shown in Fig. 1(a).

Figure 1: (color online) Schematic design of the circuit module $(\cdot)_{k}$
of $W_{k}$ in Table 1. Panel (a) presents the general scheme of the circuit
design of $(\cdot)_{k}$, where $v_{1}$, $v_{2}$, $\cdots$ denote the
occupation numbers for the vanishing cases. Panel (b) shows an illustration of
the circuit design of the module $(--)_{k}$ that corresponds to
$W_{k}=b_{k}b_{k}$. Panels (c) and (d) are the examples of the circuit designs
of the quantum adder “$+1$” and subtractor “$-1$” for one-unit increment and
decrement in the three-qubit case, respectively. The slashed line denotes a
register that may contain several qubits. The half-filled squares denote the
controlling conditions for the quantum operations; these conditions can be
constructed according to corresponding occupation values above the squares.
The dot in each $R_{y}$ gate denotes the corresponding rotational angle.

We recall that the occupation $r_{k}$ of the mode $k$ is encoded as binaries
in the subregister $s_{k}$. For each $s_{k}$, we employ a single qubit
$ph_{k}$, which is initialized as $\ket{0}$. Jointly, the set of qubits
$\\{ph_{k}\\}$ make up the register $ph$. Then, we design the circuit module
$(\cdot)_{k}$ to conduct the following operations on the register states

$\ket{{r_{k}}}_{s_{k}}\ket{0}_{ph_{k}}\xrightarrow{(\cdot)_{k}}\ket{{r_{k}+\Delta_{k}}}_{s_{k}}\ket{\gamma_{r_{k}}}_{ph_{k}},$
(23)

where
$\ket{\gamma_{r_{k}}}=\ket{\xi_{r_{k}}}=\xi_{r_{k}}\ket{0}+\sqrt{1-\xi^{2}_{r_{k}}}\ket{1}$
for $r_{k}\in[r_{k,i},\ r_{k,f}]$ (fourth column in Table 1). Here,
$\xi_{r_{k}}\in(0,1]$ is the scaled normalization factor and $\Delta_{k}$
denotes the change in $r_{k}$. The values of $\xi_{r_{k}}$ and $\Delta_{k}$
are shown in the third column and the rightmost column in Table 1,
respectively. In practical design of the circuit module $(\cdot)_{k}$, it is
convenient to specify the “vanishing cases” (fifth column in Table 1). These
correspond to the cases where $W_{k}\ket{k^{r_{k}}}=0$ or
$W_{k}\ket{k^{r_{k}}}\propto\ket{k^{r_{k}+\Delta}}$ with
$(r_{k}+\Delta_{k})>\Lambda_{k}=\lfloor K/k\rfloor$. We set
$\ket{\gamma_{r_{k}}}=\ket{1}$ for the $r_{k}$ values that belong to such
vanishing cases.

The circuit module $(\cdot)_{k}$ of the combination $W_{k}$ can be achieved by
the following analyses and constructions.

#### 1\. Vanishing cases.

For each $r_{k}$ value that belongs to the vanishing cases (fifth column in
Table 1), we flip the state of $ph_{k}$ from $\ket{0}$ to $\ket{1}$ controlled
by the subregister $s_{k}$ being in the state $\ket{r_{k}}_{s_{k}}$.

#### 2\. Nonvanishing cases.

For the nonvanishing cases with $r_{k}\in[r_{k,i},\ r_{k,f}]$ (fourth column
in Table 1), we have

$W_{k}\ket{k^{r_{k}}}=\xi_{r_{k}}\ket{k^{r_{k}+\Delta_{k}}}.$ (24)

As for the corresponding circuit construction, we apply a set of
$R_{y}(\theta_{r_{k}})$ gates [72] to the qubit $ph_{k}$, each of which is
controlled by the state $\ket{r_{k}}_{s_{k}}$ with $r_{k}\in[r_{k,i},\
r_{k_{f}}]$. We take the rotational angles as

$\theta_{r_{k}}=2\arccos\xi_{r_{k}},$ (25)

such that the state of $ph_{k}$ becomes $\ket{\xi_{r_{k}}}_{ph_{k}}\equiv
R_{y}(\theta_{r_{k}})\ket{0}_{ph_{k}}$. That is, $\xi_{r_{k}}$ is encoded as
the coefficient of the component $\ket{0}$ of the state
$\ket{\xi_{r_{k}}}_{ph_{k}}$.

#### 3\. Changes in the occupation.

To account for the change $\Delta_{k}$ in the occupation $r_{k}$ recorded as
the state $\ket{r_{k}}_{s_{k}}$, we can perform a sequence of quantum adders
or subtractors [73] to the subregister $s_{k}$. These quantum arithmetic
operators perform the following operation on the state of $s_{k}$:

$\ket{r_{k}}_{s_{k}}\rightarrow\ket{r_{k}+\Delta_{k}}_{s_{k}}.$ (26)

We illustrate the circuits of the quantum adder and subtractor in Figs. 1(c)
and (d) for the three-qubit case, respectively.

The above completes the construction of the circuit module $(\cdot)_{k}$
corresponding to Eq. (23). As a specific example, we present in Fig. 1(b) the
circuit module $(--)_{k}$ of $W_{k}=b_{k}b_{k}$.

###  III.4 Encoding many-boson Hamiltonian via quantum walk

We now present the input scheme of the LF Hamiltonian of the $(\phi^{4})_{2}$
theory utilizing the idea of the quantum walk. We first construct the forward
walk state. Provided the Fock state $\ket{\mathcal{F}}$, we start with the
state of multiple quantum registers

$\ket{\Phi_{0}}=\ket{0}_{id}\ket{\mathcal{F}}_{s}\ket{0}_{ph}\ket{0}_{me}\ket{1}_{ac},$
(27)

where the index register “$id$” is initialized as $\ket{0}$. The Fock state
$\ket{\mathcal{F}}$ is encoded by the system register $s$ that consists of a
set of subregisters $\\{s_{k}\\}$ (with $k\in[1,K]$). The phase register
“$ph$” contains $K$ qubits $\\{ph_{k}\\}$. Each single-qubit subregister
$ph_{k}$ is initialized as $\ket{0}$ and is coupled with the subregister
$s_{k}$. In addition, we adopt the single-qubit register “$me$” that is
initialized as $\ket{0}$ to encode the coefficients $\\{B^{\prime}_{j}\\}$ of
the monomials of $H$ [Eq. (18)]. Finally, we employ the single-qubit action
register “$ac$” and initialize it as $\ket{1}$.

Based on $\ket{\Phi_{0}}$, we introduce the diffusion operator (which consists
of a set of Hadamard gates) to the register $id$ and obtain

$\ket{\Phi_{1}}=\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\ket{j}_{id}\ket{\mathcal{F}}_{s}\ket{0}_{ph}\ket{0}_{me}\ket{1}_{ac}.$
(28)

We label each monomial in $H$ [Eq. (18)] by a specific index
$j\in[0,\mathcal{D}-1]$. For the clarity of explanation, we take the $j$th
monomial of $H$ [Eq. (18)] to be $B^{\prime}_{j}W_{k}W_{l}$ (with $1\leq
k<l\leq K$) as an example. Then, we implement the following operations
controlled by $\ket{j}_{id}$.

#### Step 1.

We apply the circuit module $(\cdot)_{k}$ of $W_{k}$ to the subregisters
$s_{k}$ and $ph_{k}$, and the circuit module $(\cdot)_{l}$ of $W_{l}$ to
$s_{l}$ and $ph_{l}$. According to Eq. (23), we have the operations on the
state of registers $s$ and $ph$ as

$\ket{\mathcal{F}}_{s}\ket{0}_{ph}\xrightarrow{(\cdot)_{l}(\cdot)_{k}}\ket{\mathcal{F}_{j}}_{s}\ket{\Upsilon_{j}}_{ph}.$
(29)

Here, the Fock state $\ket{\mathcal{F}}$ is encoded as
$\ket{r_{1}}_{s_{1}}\cdots\ket{r_{k}}_{s_{k}}\cdots\ket{r_{l}}_{s_{l}}\cdots\ket{r_{K}}_{s_{K}}$
in the register $s$. With the changes of the occupations in the modes $k$ and
$l$ determined by $W_{k}$ and $W_{l}$, we have the modified Fock state
$\ket{\mathcal{F}_{j}}$ encoded as
$\ket{r_{1}}_{s_{1}}\cdots\ket{r_{k}+\Delta_{k}}_{s_{k}}\cdots\ket{r_{l}+\Delta_{l}}_{s_{l}}\cdots\ket{r_{K}}_{s_{K}}$
in the register $s$. We also have the operation on the phase register $ph$ as

$\displaystyle\ket{0}_{ph}$
$\displaystyle=\ket{0}_{ph_{1}}\cdots\ket{0}_{ph_{k}}\cdots\ket{0}_{ph_{l}}\cdots\ket{0}_{ph_{K}}$
$\displaystyle\rightarrow\ket{\Upsilon_{j}}_{ph}$
$\displaystyle\equiv\ket{0}_{ph_{1}}\cdots\ket{\gamma_{r_{k}}}_{ph_{k}}\cdots\ket{\gamma_{r_{l}}}_{ph_{l}}\cdots\ket{0}_{ph_{K}}.$
(30)

#### Step 2.

We operate on the single-qubit register $me$ as

$\ket{0}_{me}\rightarrow e^{i\beta_{j}}\ket{\rho_{j}}_{me},$ (31)

where $\ket{\rho_{j}}\equiv\rho_{j}\ket{0}+\sqrt{1-\rho^{2}_{j}}\ket{1}$. The
parameters $\beta_{j}$ and $\rho_{j}$ are determined from the identity

$\rho_{j}e^{i\beta_{j}}=B_{j}^{\prime}/\Xi,$ (32)

with $\Xi\geq\max_{j}|B_{j}^{\prime}|$ and $\rho_{j}=|B_{j}^{\prime}/\Xi|\leq
1$. We take $\beta_{j}=0$ as $B_{j}^{\prime}>0$ in the $(\phi^{4})_{2}$ theory
discussed in this work.222In general applications, we take
$\beta_{j}=\arg[B_{j}^{\prime}/\Xi]\in(-\pi,\pi]$ as $B_{j}^{\prime}$ can be a
complex number. In other words, the coefficient $B_{j}^{\prime}$, scaled by
$\Xi$ that is at least the max norm of all the coefficients
$\\{B^{\prime}_{j}\\}$ of $H$ [Eq. (18)], is encoded by the register $me$ as
the coefficient of the component $\ket{0}$ of the state
$e^{i\beta_{j}}\ket{\rho_{j}}_{me}$. Practically, the operation in Eq. (31)
can be achieved by applying the phase gate $P_{X}(\beta_{j})$ followed by a
$R_{y}(\alpha_{j})$ gate to the register $me$, where we take

$P_{X}(\beta_{j})\ket{0}=e^{i\beta_{j}}\ket{0},\
P_{X}(\beta_{j})\ket{1}=\ket{1},\ \alpha_{j}=2\arccos\rho_{j}.$ (33)

#### Step 3.

We flip $\ket{1}_{ac}$ to $\ket{0}_{ac}$ by a Pauli $X$ gate.

The above completes the operations on $\ket{\Phi_{1}}$, where these operations
are based on the assumed form of the $j$th monomial
$B^{\prime}_{j}W_{k}W_{l}$.

Analogously, we apply the same steps as described above for all the monomials
in $H$ [Eq. (18)], where each of these monomials is labeled by a distinct $j$.
With these operations, we can construct the forward walk state

$\ket{\Phi_{2}}=\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}\ket{j}_{id}\ket{\mathcal{F}_{j}}_{s}\ket{\Upsilon_{j}}_{ph}e^{i\beta_{j}}\ket{\rho_{j}}_{me}\ket{b_{j}}_{ac},$
(34)

where $b_{j}=0$ if the index $j$ indeed corresponds to a monomial in $H$;
$b_{j}$ remains as $1$ otherwise.333Note that some of the $j$ values encoded
in the register $id$ may not relate to any term in $H$ in practical
applications. In Fig. 2, we illustrate the circuit to construct the forward
walk state via a simple example.

Figure 2: (color online) Illustration of the circuit design of the forward
walk state according to the Hamiltonian $H_{x}$ of a simple many-boson system.
The combinations of the squeezed boson operators and their circuit modules are
shown in Table 1 and Fig. 1(a), respectively. The coefficient in the $j$th
monomial (with $j=0,1$) is expressed as $\rho_{j}e^{i\beta_{j}}$ with
$\rho_{j}\in(0,1]$; it is encoded by a phase gate $P_{X}(\beta_{j})$ with
$P_{X}(\beta_{j})\ket{0}=e^{i\beta_{j}}\ket{0}$ and
$P_{X}(\beta_{j})\ket{1}=\ket{1}$. The angle of the $R_{y}$ gate [72] is
$\alpha_{j}=2\arccos\rho_{j}$. For clarity, the subregisters $s_{k}$ and
$ph_{k}$ with $k=1,2$ are coupled in the circuit.

Next, we construct the backward walk state based on the Fock state
$\ket{\mathcal{G}}$ as

$\ket{\Psi_{0}}=\ket{0}_{id}\ket{\mathcal{G}}_{s}\ket{0}_{ph}\ket{0}_{me}\ket{0}_{ac},$
(35)

where the register $ac$ is initialized as $\ket{0}$ in this case. With the
action of the diffusion operator on $\ket{0}_{id}$, we get

$\ket{\Psi_{1}}=\frac{1}{\sqrt{\mathcal{D}}}\sum_{k=0}^{\mathcal{D}-1}\ket{k}_{id}\ket{\mathcal{G}}_{s}\ket{0}_{ph}\ket{0}_{me}\ket{0}_{ac}.$
(36)

The walk states $\ket{\Phi_{2}}$ and $\ket{\Psi_{1}}$ block encode the many-
boson Hamiltonian. In particular, the inner product of the walk states is

$\displaystyle\langle\Psi_{1}|\Phi_{2}\rangle=$
$\displaystyle\frac{1}{\mathcal{D}}\sum_{j=0}^{\mathcal{D}-1}\sum_{k=0}^{\mathcal{D}-1}e^{i\beta_{j}}\langle
k|j\rangle_{id}\langle\mathcal{G}|\mathcal{F}_{j}\rangle_{s}\langle
0|\Upsilon_{j}\rangle_{ph}\langle 0|\rho_{j}\rangle_{me}\langle
0|b_{j}\rangle_{ac},$ (37)

which correctly produces the many-boson matrix element of the scaled
Hamiltonian $H^{\prime}\equiv H/(\mathcal{D}\Xi)$ based on the Fock states
$\ket{\mathcal{F}}$ and $\ket{\mathcal{G}}$. The kernel $\langle
0|b_{j}\rangle_{ac}=\delta_{b_{j},0}$ removes the null contributions indexed
by the $j$ values that do not correspond to any monomial in the LF
Hamiltonian.

We can rewrite the walk states based on the isometries $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ as

$\ket{\Phi_{2}}=\mathcal{T}_{1}\ket{\mathcal{F}}_{s}\otimes\ket{0}_{a},\
\ket{\Psi_{1}}=\mathcal{T}_{2}\ket{\mathcal{G}}_{s}\otimes\ket{0}_{a},$ (38)

where $a$ denotes all the registers except the system register $s$. These walk
states block encode the many-boson Hamiltonian as

$(\bra{\mathcal{G}}_{s}\otimes\bra{0}_{a})\mathcal{T}^{\dagger}_{2}\mathcal{T}_{1}(\ket{\mathcal{F}}_{s}\otimes\ket{0}_{a})=\frac{1}{\mathcal{D}\Xi}\langle\mathcal{G}|H|\mathcal{F}\rangle.$
(39)

We comment that this Hamiltonian input scheme is non-Hermitian, i.e.,
$\mathcal{T}^{\dagger}_{2}\mathcal{T}_{1}\neq\big{(}\mathcal{T}^{\dagger}_{2}\mathcal{T}_{1}\big{)}^{\dagger}$.
Compared to the typical sparse-matrix input scheme discussed in Refs. [57, 52,
53] that access the Hamiltonian matrix elements according to their row and
column indices, our input scheme dynamically computes the matrix element of
the scaled many-boson Hamiltonian $H^{\prime}$ based on the Fock states [Eq.
(37)]. Finally, we remark that our input scheme does not interact with any
precomputed database. There is no need to uncompute any ancilla qubits, where
such uncomputation procedures are necessary in the typical Hamiltonian input
scheme [57, 52, 53].

###  III.5 Analysis of the gate and qubit cost

We estimate the gate cost of our Hamiltonian input scheme with the following
analysis. We retain $K$ modes in our calculations by setting the total
longitudinal momentum to be $K$. In the most complicated case, there are
$O(K^{4})$ monomials in $H$ [Eq. (18)].444We adopt the conventional notations
in computer science for the complexity analysis in this work. For the
functions $w(x)$ and $v(x)$, $w(x)\in\Theta(v(x))$ denotes
$\lim_{x\rightarrow\infty}\frac{w(x)}{v(x)}=c$ with $c$ being some constant.
The notation $w(x)\in O(v(x))$ means
$\lim_{x\rightarrow\infty}\frac{v(x)}{w(x)}=0$.  Expressed in terms of
$W_{k}$, each monomial may contain four combinations of the squeezed boson
operators (i.e., in the form of $W_{k}W_{l}W_{m}W_{n}$). It takes at most
$\Theta(K)$ gates to construct the circuit module for each combination
according to our design (Sec. III.3). These are the dominant gate costs
required to construct the forward walk state. In comparison, the gate cost to
construct the backward walk state is much less than the gate cost required for
constructing the forward walk state. Hence, the total gate cost to input the
many-boson Hamiltonian scales as $\widetilde{O}(K^{5})$, where the tilde
notation denotes that we suppress the logarithmic factor from compiling the
multiple controlled gates [1, 74, 75]. We expect that this gate complexity can
be significantly reduced in realistic implementations as (1) the number of
monomials is much less than $O(K^{4})$ due to the symmetries of the system
(e.g., the momentum conservation); and (2) the maximal occupation
$\Lambda_{k}=\lfloor K/k\rfloor$ of the mode $k$ can be much less than $K$
when $k$ is large, which reduces the gate cost to construct the circuit module
$(\cdot)_{k}$.

The qubit cost to input the Hamiltonian is dominated by the number of qubits
necessary for the registers $s=\\{s_{k}\\}$ and $ph=\\{ph_{k}\\}$ for
$k\in[1,K]$. According to our encoding scheme, we need
$\log_{2}(\Lambda_{k}+1)$ qubits to encode all the possible occupation values
in the subregister $s_{k}$. As the maximal occupation of the mode $k$ is
$\Lambda_{k}=\lfloor K/k\rfloor$, the number of qubits required for the
register $s$ is

$Q_{s}=\sum_{k=1}^{K}\lceil\log_{2}(\Lambda_{k}+1)\rceil=\sum_{k=1}^{K}\lceil\log_{2}(\lfloor
K/k\rfloor+1)\rceil.$ (40)

We note that $Q_{s}$ grows approximately as $2K$ for large $K$. In addition,
it takes $K$ qubits for the register $ph$. In sum, the total number of qubits
to block encode the many-boson Hamiltonian is approximately $\Theta(K)$, where
we omit the minor qubit cost: (1) the two qubits required for the registers
$me$ and $ac$; and (2) the $O(4\log_{2}K)$ qubits required for the register
$id$ to encode all the possible indices $j$ of the monomials in $H$ [Eq.
(18)].

Kirby et al. [56] introduced a general-purpose Hamiltonian input scheme for
many-particle systems, where the authors employ the compact basis encoding and
the log-local operations for the controlled arithmetic calculations in order
to reduce the cost of the quantum computing resources. As for the LF
Hamiltonian of the $(\phi^{4})_{2}$ theory discussed in this work, their input
scheme [56] reports the gate complexity to be $O(K^{3})$ and the qubit cost to
be $O(K\log K)$. Our scaling of the quantum computing resources is close to
theirs. However, it can be challenging to compile the abstract log-local
operations employed in Ref. [56] to primitive gates in quantum computing. In
comparison, our input scheme presents an explicit circuit based on
straightforward applications of the elementary gates and their controlled
versions. In this sense, it provides a realistic alternative to the input
scheme in Ref. [56] for the many-boson Hamiltonian.

##  IV Numerical demonstration and discussions

We present the spectral calculations of the LF Hamiltonian of the
$(\phi^{4})_{2}$ theory based on our Hamiltonian input scheme in order to
demonstrate its validity and utility. Though we restrict our discussions to
the spectral calculations, we comment that our Hamiltonian input scheme can
also be implemented with other efficient algorithms, such as the quantum
signal processing [58], the quantum singular value transformation [59, 60],
and the quantum eigenvalue transformation [61], etc., for dynamics simulations
with optimal gate complexities with respect to the simulation error and time.

We adopt the algorithm of the Chebyshev-polynomial-based quantum Krylov
subspace diagonalization (CP-QKSD) [76] for the eigenvalue calculations in our
numerical example. The CP-QKSD algorithm makes use of the quantum computer to
deal with the quantum many-body problems that can be intractable on classical
computers, and cast the many-body problems onto a Hilbert subspace spanned by
a limited set of Krylov bases that can be handled on the classical computer.
Then, one solves the Hamiltonian eigenvalue problem in the Krylov subspace for
the eigenenergies of the low-lying states on the classical computer. We
present more details of the CP-QKSD method in Appendix A.

Table 2: Information of the many-boson Hamiltonian $H_{K=4}$ for the circuit design of the Hamiltonian input scheme. The first, second, and third columns list the index $j$, the circuit modules of the combinations of the squeezed boson operators, and the coefficient of each monomial, respectively.  index $j$ | circuit modules |  $B^{\prime}_{j}$ [MeV2] |  |  | index $j$ | circuit modules |  $B^{\prime}_{j}$ [MeV2]
---|---|---|---|---|---|---|---
0 | $(+-)_{1}$ | $4$ |  |  | 7 | $(-)_{1}(++)_{2}(-)_{3}$ | $4.24866$
1 | $(+-)_{2}$ | $1$ |  |  | 8 | $(+)_{1}(--)_{2}(+)_{3}$ | $4.24866$
2 | $(+-)_{3}$ | $1/3$ |  |  | 9 | $(++--)_{2}$ | $1.83972$
3 | $(+-)_{4}$ | $1/4$ |  |  | 10 | $(---)_{1}(+)_{3}$ | $5.66488$
4 | $(++--)_{1}$ | $29.4356$ |  |  | 11 | $(--)_{1}(-)_{2}(+)_{4}$ | $7.35889$
5 | $(+-)_{1}(+-)_{2}$ | $29.4356$ |  |  | 12 | $(+++)_{1}(-)_{3}$ | $5.66488$
6 | $(+-)_{1}(+-)_{3}$ | $9.81186$ |  |  | 13 | $(++)_{1}(+)_{2}(-)_{4}$ | $7.35889$

We retain a limited model problem with $K=4$ in our numerical demonstration.
We elect $m^{2}=1$ MeV2 and $\lambda/m^{2}=92.4746$ for the $(\phi^{4})_{2}$
theory, where the reasons for this parameter setup will be clear in the
following text. The LF Hamiltonian for the $(\phi^{4})_{2}$ theory in terms of
the squeezed boson operators [Eq. (15)] reads

$\displaystyle H_{K=4}=$ $\displaystyle
B^{\prime}_{0}b^{\dagger}_{1}b_{1}+B^{\prime}_{1}b^{\dagger}_{2}b_{2}+B^{\prime}_{2}b^{\dagger}_{3}b_{3}+B^{\prime}_{3}b^{\dagger}_{4}b_{4}+B^{\prime}_{4}b^{\dagger}_{1}b^{\dagger}_{1}b_{1}b_{1}+B^{\prime}_{5}b^{\dagger}_{1}b_{1}b^{\dagger}_{2}b_{2}$
$\displaystyle+B^{\prime}_{6}b^{\dagger}_{1}b_{1}b^{\dagger}_{3}b_{3}+B^{\prime}_{7}b_{1}b^{\dagger}_{2}b^{\dagger}_{2}b_{3}+B^{\prime}_{8}b^{\dagger}_{1}b_{2}b_{2}b^{\dagger}_{3}+B^{\prime}_{9}b^{\dagger}_{2}b^{\dagger}_{2}b_{2}b_{2}+B^{\prime}_{10}b_{1}b_{1}b_{1}b^{\dagger}_{3}$
$\displaystyle+B^{\prime}_{11}b_{1}b_{1}b_{2}b^{\dagger}_{4}+B^{\prime}_{12}b^{\dagger}_{1}b^{\dagger}_{1}b^{\dagger}_{1}b_{3}+B^{\prime}_{13}b^{\dagger}_{1}b^{\dagger}_{1}b^{\dagger}_{2}b_{4}.$
(41)

For clarity, we present the indices of the monomials of $H_{K=4}$ in Table 2,
where we also list the coefficient and the corresponding circuit modules
(Table 1) for each monomial. With this, we design the quantum walk states and
present the corresponding input scheme of the second-quantized many-boson
Hamiltonian according to Eq. (39), where we take $\Xi=B^{\prime}_{4}=29.4356$
MeV2.

We note that there are five many-boson bases for our model problem with $K=4$.
These bases can be sorted into two sectors according to the evenness and
oddness of the particle number as $\mathcal{B}_{\rm
even}=\\{\ket{3^{1},1^{1}},\ket{2^{2}},\ket{1^{4}}\\}$ and $\mathcal{B}_{\rm
odd}=\\{\ket{4^{1}},\ket{2^{1},1^{2}}\\}$. The many-boson Hamiltonian matrix
in the basis $\mathcal{B}_{K=4}=\mathcal{B}_{\rm even}\oplus\mathcal{B}_{\rm
odd}$ is

$\displaystyle H_{K=4}=\begin{pmatrix}3.78630&1.50213&3.46902&&\\\
1.50213&1.91986&0&&\\\ 3.46902&0&26.0767&&\\\ &&&0.25&1.83972\\\
&&&1.83972&13.5383\end{pmatrix},$ (42)

in units of MeV2. We find that the Hamiltonian matrix is block diagonal. This
is due to the fact that the LF Hamiltonian [Eq. (3)] preserves the symmetry of
oddness and evenness. By straightforward matrix diagonalization, we obtain the
eigenvalues of $H_{K=4}$ to be $E_{0}=1.61752\times 10^{-7}$ MeV2,
$E_{1}=0.958969$ MeV2, $E_{2}=4.21772$ MeV2, $E_{3}=13.7883$ MeV2,
$E_{4}=26.6062$ MeV2, where $E_{0}$ and $E_{3}$ are the eigenvalues for the
odd-particle-number sector (lower-right submatrix), while the others are the
eigenvalues for the even-particle-number sector (upper-left submatrix).

We adopt the CP-QKSD method [76] in the eigenvalue calculations of $H_{K=4}$
with the strategy shown in Ref. [50]. We note that our Hamiltonian input
scheme respects the symmetry (the evenness and oddness) of the many-boson
system. Therefore, we can generate the Krylov bases with specific symmetry by
preparing the many-boson pivot state $\ket{\psi_{0}}$ to be of the desired
symmetry (this preparation is straightforward in the framework of second
quantization) on the quantum computer. Based on the symmetry-adapted Krylov
basis set, we evaluate the Hamiltonian and the overlap matrix elements
implementing the the noiseless Statevector simulator of the IBM Qiskit package
[72]. These matrix elements are used to construct the generalized eigenvalue
problems on the classical computer to solve for the ground-state eigenvalues
in the symmetry-adapted Hilbert subspaces.

Within the Hilbert subspace of odd particle numbers, we obtain the ground-
state eigenenergy to be $1.61752\times 10^{-7}$ MeV2. On the other hand, the
ground state eigenvalue is $0.958969$ MeV2 when we work in the subspace of
even particle numbers. These eigenenergies agree with those from the classical
calculations. The small ground-state eigenvalue in the odd-particle-number
sector implies a vanishing mass gap [Eq. (9)]. This indicates that the
coupling constant $\lambda/m^{2}=92.4746$ is close to the critical coupling
for the $(\phi^{4})_{2}$ theory at $K=4$.

For the $(\phi^{4})_{2}$ theory in the DLCQ framework, we expect that our
input scheme can face numerical challenges in the setup when we do not
truncate the number of particles and when the strong coupling constant
$\lambda/m^{2}\gg 1$ is retained in the many-body interaction term. However,
one does not expect these challenging cases in many field theory problems of
higher spatial dimensions, such as the $(d+1)$-dimensional $\phi^{4}$ (with
$d\geq 2$) theory, and other realistic field theory problems such as quantum
electrodynamics, quantum chromodynamics, or chiral effective field theories.
In such field theory problems, the coupling strengths are small, and it is
reasonable to restrict the number of constituent particles in a physical
system utilizing the schemes of the Fock-sector expansion and truncation
within the LF Hamiltonian formalism. With such considerations, we expect that
our input scheme can be adopted to treat these theories [21, 23, 24] in a
straightforward manner.

##  V Summary and outlook

In this work, we propose a novel input scheme for second-quantized many-boson
Hamiltonians in quantum computing. For clarity, we present the input scheme
utilizing the simple two-dimensional $\phi^{4}$ theory within the Hamiltonian
formalism obtained by the discretized light-cone discretization approach.

In our input scheme, we employ a set of registers to encode the many-boson
states, where each register encodes the occupation of a specific boson mode in
binaries. We also squeeze the boson operators of each mode with the maximal
occupation number of the mode. For all the monomials in the light-front (LF)
many-boson Hamiltonian, we find a limited set of unique combinations of the
squeezed-boson operators, and we rewrite the Hamiltonian in terms of these
unique combinations. In this work, we present explicit design for the circuit
modules (or circuit representation) for these combinations. Based on these
modules, we prepare the quantum walk states according to the many-boson
Hamiltonian. With these walk states, we block encode the many-boson
Hamiltonian.

For demonstration purposes, we present the spectral calculations of the LF
Hamiltonian of the two-dimensional $\phi^{4}$ theory in a restricted model
space based on our Hamiltonian input scheme and implementing the hybrid
algorithm of the quantum Krylov subspace diagonalization [76]. We perform our
quantum computations utilizing the noiseless Statevector simulator of the IBM
Qiskit package [72], of which the results are post-processed on the classical
computer for the eigenenergies of the low-lying states. Our results of the
hybrid quantum-classical calculations agree with those from pure classical
calculations.

Our input scheme can be applied to treat other field theory problems, e.g.,
quantum electrodynamics, quantum chromodynamics, or chiral field theories
[e.g., Ref. [38]]. Combined with the input scheme for the second-quantized
many-fermion Hamiltonians [49, 50], our input scheme for the second-quantized
many-boson Hamiltonians opens up a novel and systematic path for solving the
structure and dynamics of quantum field theory on future fault-tolerant
quantum computers.

## Acknowledgements

We acknowledge fruitful discussions with Peter Love, Chao Yang, Pieter Maris,
Michael Kreshchuk, and Shreeram Jawadekar. This work was supported by US DOE
Grant DE-SC0023707 under the Office of Nuclear Physics Quantum Horizons
program for the “NuNuclei and Hadrons with Quantum computers (NuHaQ)” project.

##  Appendix A CP-QKSD method

We present the details of the CP-QKSD algorithm [76] applied in our numerical
demonstration. The CP-QKSD algorithm computes the eigenvalues of the low-lying
states of the Hamiltonian based on the block-encoded Hamiltonian in quantum
computing. Interested readers are also referred to Ref. [77] and references
therein for a general review of various other quantum-subspace methods for
structure calculations.

As for this work, our input scheme is designed to block encode the second-
quantized LF Hamiltonian as [Eq. (39)]

$(\bra{\mathcal{G}}_{s}\otimes\bra{0}_{a})\mathcal{T}^{\dagger}_{2}\mathcal{T}_{1}(\ket{\mathcal{F}}_{s}\otimes\ket{0}_{a})=\frac{1}{\mathcal{D}\Xi}\langle\mathcal{G}|H|\mathcal{F}\rangle.$
(43)

With this Hamiltonian input model, we can construct the input scheme of the
Chebyshev polynomials of the first kind [78] as functions of
$H^{\prime}=H/(\mathcal{D}\Xi)$:

$\displaystyle\langle\psi_{0}|T_{2k+1}(H^{\prime})|\psi_{0}\rangle=$
$\displaystyle(\langle{\psi_{0}}|_{s}\otimes\langle
0|_{a})\big{[}U_{H}{\Pi}(U^{\dagger}_{H}{\Pi}U_{H}{\Pi})^{k}\big{]}(|{\psi_{0}}\rangle_{s}\otimes\ket{0}_{a}),$
(44) $\displaystyle\langle\psi_{0}|T_{2k}(H^{\prime})|\psi_{0}\rangle=$
$\displaystyle(\langle\psi_{0}|_{s}\otimes\langle
0|_{a})(U^{\dagger}_{H}{\Pi}U_{H}{\Pi})^{k}(|\psi_{0}\rangle_{s}\otimes\ket{0}_{a}),$
(45)

with $k=0,\ 1,\ 2,\cdots$. Here we define
$U_{H}\equiv\mathcal{T}^{\dagger}_{2}\mathcal{T}_{1}$. The reflection operator
${\Pi}\equiv(2|0\rangle_{a}\langle
0|_{a}-\mathbb{I}_{a})\otimes\mathbb{I}_{s}$ produces the reflection around
$\ket{0}_{a}$ in the auxiliary space. The quantum circuit to input
$T_{j}(H^{\prime})$ is achieved via alternating applications of $U_{H}\Pi$ and
$U_{H}^{\dagger}\Pi$ [79, 50].

We can construct the Krylov basis as [76]

$\\{\ket{\psi_{i}}=T_{i}(H^{\prime})\ket{\psi_{0}}|i=0,\ 1,\ 2,\ \cdots,\
\mathcal{K}-1\\},$ (46)

where $\ket{\psi_{0}}$ denotes the pivot many-boson state.

In the Krylov basis, the matrix element of $H^{\prime}$ is

$H^{\prime}_{ij}=\bra{\psi_{i}}H^{\prime}\ket{\psi_{j}}=\bra{\psi_{0}}T_{i}(H^{\prime})H^{\prime}T_{j}(H^{\prime})\ket{\psi_{0}},$
(47)

while the element of the overlap matrix $S$ is

$S_{ij}=\langle\psi_{i}|\psi_{j}\rangle=\bra{\psi_{0}}T_{i}(H^{\prime})T_{j}(H^{\prime})\ket{\psi_{0}}.$
(48)

Utilizing the identities of the Chebyshev polynomials [78]

$T_{0}(x)=1,\ T_{1}(x)=x,\
T_{i}(x)T_{j}(x)=\big{[}T_{i+j}(x)+T_{|i-j|}(x)\big{]}/2,$ (49)

one can express $H^{\prime}_{ij}$ and $S_{ij}$ in terms of the expectation
$\langle
T_{k}(H^{\prime})\rangle_{0}\equiv\bra{\psi_{0}}T_{k}(H^{\prime})\ket{\psi_{0}}$.

The expectation values $\\{\langle T_{k}(H^{\prime})\rangle_{0}\\}$ are, in
general, challenging to calculate on classical computers for quantum many-body
systems, while they are expected to be evaluated with efficiency on quantum
computers [3]. We can then compute these expectation values on quantum
computers and input them to the classical computer to compute
$\\{H^{\prime}_{ij}\\}$ and $\\{S_{ij}\\}$, based on which we construct the
generalized eigenvalue problem

$H^{\prime}\ket{\varphi}=\mathcal{E}S\ket{\varphi},$ (50)

where $\mathcal{E}$ denotes the eigenvalue of the eigenstate $\ket{\varphi}$.

A general bound of $\mathcal{K}$ [Eq. (46)] to compute the ground state energy
with the error $\epsilon$ is [76]

$\mathcal{K}=\Theta[(\log|\kappa_{0}|^{-1}+\log\epsilon^{-1})\cdot\min(\epsilon^{-1},{\delta}^{-1})],$
(51)

where $\kappa_{0}$ denotes the overlap between the pivot $\ket{\psi_{0}}$ and
the ground state. $\delta$ is the spectral gap between the ground and first-
excited state. One notes also that the overlap matrix is, in general, ill
conditioned. In practical calculations, one can adopt the canonical
orthogonalization [80] for approximating the lowest eigenvalue of $H^{\prime}$
[81].

As noted in Ref. [50], our Hamiltonian input scheme, which operates in the
framework of second-quantization, offers a natural advantage in implementing
the CP-QKSD algorithm. Indeed, the occupation representation makes it
straightforward and convenient to prepare a pivot $\ket{\psi_{0}}$ to be of
specific symmetries of the Hamiltonian. Therefore, we can naturally generate a
Krylov basis set $\\{\ket{\psi_{i}}\\}$ [Eq. (46)] with elected symmetries. In
this way, we are able to work within a particular symmetry-adopted Hilbert
subspace and solve for the eigenenergy of corresponding ground state with the
elected symmetries, where the unwanted states that do not respect the elected
symmetries are naturally excluded.

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