# Hamiltonian input model and spectroscopy on quantum computers

Weijie Du  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 present a novel input model for general second-quantized Hamiltonians of
relativistic or non-relativistic many-fermion systems. This input model
incorporates the fermionic anticommutation relations, particle number
variations, and respects the symmetries of the Hamiltonian. Based on our input
model, we propose a hybrid framework for spectral calculations on future
quantum hardwares. We provide explicit circuit designs and the associated gate
cost and circuit depth. We demonstrate our framework by solving the low-lying
spectra of ${{}^{42}}Ca$ and ${{}^{46}}Ca$. Our input model provides new
pathways to solving the spectra and time evolutions of the relativistic and
nonrelativistic many-fermion systems.

Introduction. Quantum computing has the potential to solve the structure and
dynamics of many-body problems that are intractable with classical computers
[1, 2] in the fields such as quantum chemistry [3, 4, 5, 6], nuclear physics
[7, 8, 9, 10, 11, 12, 13, 14], elementary particles physics [15, 16, 17],
field theories [18, 19, 20, 21, 22, 23, 24, 25, 26, 27], and condensed matter
physics [28, 29, 30]. One of the key questions in solving the structure and
dynamics of relativistic and nonrelativistic many-body theories is: how to
design a systematic Hamiltonian input model that is able to treat particle
number variations and respects the symmetries of the underlying theory?

We present a novel Hamiltonian input model of the many-fermion system in this
work. Working in the second-quantized representation, we create the quantum
walk states [31, 32, 33] from the Hamiltonian and Fock states of the many-
fermion system. We then project the walk states to the sub-Hilbert space that
encodes the Hamiltonian by pruning the walk-state components according to the
symmetries of the system. Our input model is constructed based on oracles. It
incorporates the Pauli principle, anticommutation relations, and particle
number variations that may be present.

With the input model, we also propose a hybrid quantum-classical framework to
solve the spectra of relativistic and non-relativistic many-fermion systems.
Our input model enables constructing the Krylov subspace [34] set with elected
symmetries, with which we evaluate the many-fermion matrix elements on the
quantum computer. These elements are then input to the classical computer for
the ground- and excited-state energies in a greatly reduced matrix eigenvalue
problem. We demonstrate our framework by solving the low-lying spectra of the
Calcium isotopes. Compared to the conventional variational quantum eigensolver
[35, 36] where measurement overhead and optimization is in general
challenging, our framework does not require extensive parameter optimizations
and enables access to excited states as well.

Many-fermion Hamiltonian. The general Hamiltonian $H$ of a many-fermion system
with possible particle number variations is

$H=\sum_{j}\langle Q_{j}|H|P_{j}\rangle b^{\dagger}_{Q_{j}}b_{P_{j}},$ (1)

with the mappings $Q_{j}\mapsto\\{p_{j},q_{j},\cdots,r_{j}\\}$ and
$P_{j}\mapsto\\{u_{j},v_{j},\cdots,w_{j}\\}$ such that $\langle
Q_{j}|H|P_{j}\rangle=\langle p_{j}q_{j}\cdots r_{j}|H|u_{j}v_{j}\cdots
w_{j}\rangle$,
$b^{\dagger}_{Q_{j}}=a_{p_{j}}^{\dagger}a_{q_{j}}^{\dagger}\cdots
a_{r_{j}}^{\dagger}$, $b_{P_{j}}=a_{w_{j}}\cdots a_{v_{j}}a_{u_{j}}$, and
$j\in[0,\mathcal{D}-1]$ runs over all possible “P-Q” pairs for which matrix
elements of $H$ are non-vanishing. $H$ can contain terms which change the
particle number. The anti-commutation relations hold for the fermion
operators, i.e., $\\{a^{\dagger}_{p},a_{q}\\}=\delta_{pq}$, and
$\\{a^{\dagger}_{p},a^{\dagger}_{q}\\}=\\{a_{p},a_{q}\\}=0$, with $p$ and $q$
labeling the single-particle (SP) basis. For the convenience of discussion, we
write $H^{\dagger}$ as

$H^{\dagger}=\sum_{k}\langle Q_{k}|H|P_{k}\rangle
b^{\dagger}_{Q_{k}}b_{P_{k}},$ (2)

with the mappings $Q_{k}\mapsto\\{u_{k},v_{k},\cdots,w_{k}\\}$ and
$P_{k}\mapsto\\{p_{k},q_{k},\cdots,r_{k}\\}$ such that
$b^{\dagger}_{Q_{k}}=a^{\dagger}_{u_{k}}a^{\dagger}_{v_{k}}\cdots
a^{\dagger}_{w_{k}}$ and $b_{P_{k}}=a_{r_{k}}\cdots a_{q_{k}}a_{p_{k}}$. We
coordinate the indices of the monomials in Eqs. (1) and (2), and set $j=k$
when $P_{j}=Q_{k}$ and $Q_{j}=P_{k}$, which provide
$(b^{\dagger}_{Q_{j}}b_{P_{j}})^{\dagger}=b^{\dagger}_{Q_{k}}b_{P_{k}}$.

Direct encoding scheme. We employ the direct encoding scheme [37, 38] to map
the SP basis set to a set of qubits in the quantum register. Provided a total
set of $N_{\rm sp}$ SP states, each SP state corresponds to a specific qubit
in the quantum register, while the vacancy and occupation of individual SP
state correspond to the qubit states $\ket{0}$ and $\ket{1}$, respectively. In
this sense, the Fock states are mapped to binary strings in the quantum
register.

Enumerator oracle. We define the enumerator oracle $O_{F}$ as

$O_{F}\ket{\mathcal{F}}_{s}\ket{j}_{id}\ket{0}_{cp}\ket{1}_{e_{p}}\ket{1}_{e_{q}}\\\
=\ket{\mathcal{F}}_{s}\ket{j}_{id}\ket{\mathcal{F}_{j}}_{cp}\ket{y^{P}_{\mathcal{F},j}}_{e_{p}}\ket{y^{Q}_{\mathcal{F},j}}_{e_{q}},$
(3)

where each register is labeled by a subscript. With the input index $j$ and
many-fermion (Fock) state $\ket{\mathcal{F}}$, $O_{F}$ computes the output
state $\ket{\mathcal{F}_{j}}=b^{\dagger}_{Q_{j}}b_{P_{j}}\ket{\mathcal{F}}$.
Each of “$e_{p}$” and “$e_{q}$” denotes a single-qubit register that is
initialized as $\ket{1}$. If the operation $b_{P_{j}}\ket{\mathcal{F}}$ is
valid, we flip the state of $e_{p}$ from $\ket{1}$ to $\ket{0}$; $e_{p}$
remains in $\ket{1}$ otherwise. Similarly, we flip the state of $e_{q}$ from
$\ket{1}$ to $\ket{0}$ if the operation
$b^{\dagger}_{Q_{j}}\ket{\mathcal{F}_{P_{j}}}$ is valid; $e_{q}$ remains in
$\ket{1}$ otherwise. As a result, $y^{P}_{\mathcal{F},j}$ and
$y^{Q}_{\mathcal{F},j}$ in Eq. (3) can be either 0 or 1; it is only when the
operation $b^{\dagger}_{Q_{j}}b_{P_{j}}\ket{\mathcal{F}}$ is valid that
$y^{P}_{\mathcal{F},j}=y^{Q}_{\mathcal{F},j}=0$.

We construct $O_{F}$ as follows (see illustration in Fig. 1).

## Step 1:

Provided the Fock state $\ket{\mathcal{F}}$ in the system register “$s$”
represented in terms of $0$’s and $1$’s, we duplicate it to the register
“$cp$” that is initiated in the all-zero state via a sequence of qubit-wise
CNOT gates. While the duplication of a general quantum state is not possible,
it is possible to duplicate a simple Fock state represented in terms of $0$’s
and $1$’s.

## Step 2:

With the index $j$ in the “$id$” register, we identify the $j{\rm th}$
monomial $\langle Q_{j}|H|P_{j}\rangle b^{\dagger}_{Q_{j}}b_{P_{j}}$ in the
Hamiltonian, where $P_{j}\mapsto\\{u_{j},v_{j},\cdots,w_{j}\\}$ specifies the
SP states in $\ket{\mathcal{F}}$ of which the occupations are to be removed.
However, one needs to check if these orbitals in $\ket{\mathcal{F}}$ are
indeed occupied before the removal: if any of them is vacant, the operation
$b_{P_{j}}\ket{\mathcal{F}}$ is invalid. We record the error message with the
$e_{p}$ qubit that is prepared as $\ket{1}$: only if the qubits corresponds to
the orbitals $\\{u_{j},v_{j},\cdots,w_{j}\\}$ are all in the $\ket{1}$ states
(i.e., occupied), we flip $e_{p}$ from $\ket{1}$ to $\ket{0}$; otherwise,
$e_{p}$ remains in $\ket{1}$.

We have the error and validity information recorded as
$y^{P}_{\mathcal{F},j}=1$ and $0$ by $e_{p}$, respectively. Then, we proceed
to flip all the qubits corresponding to the orbitals
$\\{u_{j},v_{j},\cdots,w_{j}\\}$ in the duplicated $\ket{\mathcal{F}}$ state
in the $cp$ register by a sequence of NOT gates. The resultant state of $cp$
is $\ket{\mathcal{F}_{P_{j}}}=b_{P_{j}}\ket{\mathcal{F}}$.

## Step 3:

With the index $j$, we determine a sequence of creation operators denoted by
$b^{\dagger}_{Q_{j}}$, where $Q_{j}$ specifies the orbitals
$\\{p_{j},q_{j},\cdots,r_{j}\\}$ in $\ket{\mathcal{F}_{P_{j}}}$ on which these
creation operators act. We apply a set of multiple-open-controlled-NOT gates
(i.e., those with the NOT gate controlled by a set of zero states) to check if
these orbitals in $\ket{\mathcal{F}_{P_{j}}}$ are all vacant; that is, the
qubits corresponds to these orbitals in the $cp$ register are all in the
$\ket{0}$ states. The error/validity message is recorded by the $e_{q}$
register which is prepared in $\ket{1}$. If the orbitals
$\\{p_{j},q_{j},\cdots,r_{j}\\}$ in $\ket{\mathcal{F}_{P_{j}}}$ are all
vacant, we flip the state of $e_{q}$ from $\ket{1}$ to $\ket{0}$, denoting
$b^{\dagger}_{Q_{j}}\ket{\mathcal{F}_{P_{j}}}$ to be valid. Otherwise, $e_{q}$
remains in $\ket{1}$, which denotes that the operation
$b^{\dagger}_{Q_{j}}\ket{\mathcal{F}_{P_{j}}}$ is invalid. With the message
state $\ket{y^{Q}_{\mathcal{F},j}}$ kept in $e_{q}$, we apply a sequence of
NOT gates to flip the qubits that correspond to the orbitals
$\\{p_{j},q_{j},\cdots,r_{j}\\}$. The resultant state of $cp$ is
$\ket{\mathcal{F}_{j}}=b^{\dagger}_{Q_{j}}b_{P_{j}}\ket{\mathcal{F}}$.

This completes the construction of $O_{F}$.

Matrix element oracle. We define the matrix element oracle $O_{H}$ as

$O_{H}\ket{\mathcal{F}}_{s}\ket{j}_{id}\ket{\mathcal{F}_{j}}_{cp}\ket{0}_{\zeta}\ket{0}_{me}\\\
=\ket{\mathcal{F}}_{s}\ket{j}_{id}\ket{\mathcal{F}_{j}}_{cp}{\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}}\ket{0}_{\zeta}e^{i\theta_{j}}\ket{\rho_{j}}_{me},$
(4)

where each register is labeled by a subscript. We have
$\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}=\langle\mathcal{F}_{j}|b^{\dagger}_{Q_{j}}b_{P_{j}}|\mathcal{F}\rangle$
and $\ket{\rho_{j}}\equiv{\rho}_{j}\ket{0}+(1-\rho_{j})\ket{1}$. The
parameters $\theta_{j}$ and $\rho_{j}$ are determined from the matrix element
$\langle Q_{j}|H|P_{j}\rangle$ of the $j$th monomial of $H$:

$h(Q_{j},P_{j})=\langle Q_{j}|H|P_{j}\rangle/\Lambda=\rho_{j}e^{i\theta_{j}},$
(5)

with $\Lambda\geq\max_{j}|\langle Q_{j}|H|P_{j}\rangle|$,
$\rho_{j}=|h(Q_{j},P_{j})|\leq 1$, and
$\theta_{j}=\arg[h(Q_{j},P_{j})]\in(-\pi,\pi]$.

$O_{H}$ takes the monomial index $j$ in the Hamiltonian, and the Fock state
$\ket{\mathcal{F}}$ and $\ket{\mathcal{F}_{j}}$. It computes the factor
$\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}$, and encodes the matrix element as
$e^{i\theta_{j}}\ket{\rho_{j}}$. $O_{H}$ is constructed as follows (see
illustration in Fig. 1).

## Step 1:

The factor $\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}$ is determined by two
actions:
$b_{P_{j}}\ket{\mathcal{F}}=c(P_{j},\mathcal{F})\ket{\bar{\mathcal{F}}^{\prime}_{j}}$
and
$b_{Q_{j}}\ket{\mathcal{F}_{j}}=c(Q_{j},\mathcal{F}_{j})\ket{\bar{\mathcal{F}}^{\prime}_{j}}$,
where the phase is
$\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}=c(P_{i},\mathcal{F})\cdot
c(Q_{j},\mathcal{F}_{j})$.

Without loss of generality, we illustrate the idea of computing
$\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}=\langle{\mathcal{F}_{j}}|b^{\dagger}_{Q_{j}}b_{P_{j}}|\mathcal{F}\rangle$
by taking $b_{Q_{j}}=a_{q_{j}}a_{p_{j}}$ (with $q_{j}>p_{j}\geq 0$) and
$b_{P_{j}}=a_{w_{j}}a_{v_{j}}a_{u_{j}}$ (with $w_{j}>v_{j}>u_{j}\geq 0$). For
the case $a_{w_{j}}a_{v_{j}}a_{u_{j}}\ket{\mathcal{F}}$, we group the fermion
operators in pairs from left to right, and obtain
$(a_{w_{j}}a_{v_{j}})a_{u_{j}}$ with $a_{u_{j}}$ being unpaired. Then, in the
Fock state $\ket{\mathcal{F}}$, we count the occupancy $N^{\rm
occ}_{0\rightarrow u_{j}}$ in all the orbitals with the indices
$\kappa\in(0,u_{j})$, and the occupancy $N^{\rm occ}_{v_{j}\rightarrow w_{j}}$
in all the orbitals with the indices $\kappa\in(v_{j},w_{j})$. The same
technique applies to the other case $a_{q_{j}}a_{p_{j}}\ket{\mathcal{F}_{j}}$.
We first group the string of the fermion operators in pairs from the left to
right, where no operator is left unpaired in this case. We then count the
occupancy $N^{\rm occ}_{p_{j}\rightarrow q_{j}}$ in all the orbitals with the
indices $\eta\in(p_{j},q_{j})$ in $\ket{\mathcal{F}_{j}}$.

With that, we record $N^{\rm occ}_{{\rm tot},j}=N^{\rm occ}_{0\rightarrow
u_{j}}+N^{\rm occ}_{v_{j}\rightarrow w_{j}}+N^{\rm occ}_{p_{j}\rightarrow
q_{j}}$ as the state $\ket{{\rm mod}(N^{\rm occ}_{{\rm tot},j},2)}$ by the
single-qubit register $\zeta$ that is initialized as $\ket{0}$. This is
achieved by two sequences of CNOT gates: (1) in one set, each CNOT is
controlled by the $\kappa$th qubit in the $s$ register that encodes
$\ket{\mathcal{F}}$, and acts on $\zeta$, where
$\kappa\in(0,u_{j})\cup(v_{j},w_{j})$; and (2) in the other set, each CNOT is
controlled by the $\eta$th qubit in the $cp$ register that encodes
$\ket{\mathcal{F}_{j}}$, and acts on $\zeta$, where $\eta\in(p_{j},q_{j})$.

The factor $\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}=(-1)^{{\rm mod}(N^{\rm
occ}_{{\rm tot},j},2)}$ is obtained by applying a Pauli-$Z$ gate on the state
$\ket{{\rm mod}(N^{\rm occ}_{{\rm tot},j},2)}$ of $\zeta$. Finally, we
uncompute the state of $\zeta$ to $\ket{0}$.

The calculations of $\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}$ in general
cases follow the technique discussed above.

## Step 2.

The matrix element $\rho_{j}e^{i\theta_{j}}$ of the $j$th monomial is recorded
as the coefficient of the $\ket{0}$ component of the state
$e^{i\theta_{j}}\ket{\rho_{j}}$ by the single-qubit register “$me$”. This is
achieved by applying a phase gate $P_{X}(\theta_{j})$ and then a rotation-Y
gate $R^{\dagger}_{y}(\alpha_{j})$ to the $me$ qubit that is initialized in
$\ket{0}$. In particular, we define
$P_{X}(\theta_{j})\ket{0}=e^{i\theta_{j}}\ket{0}$ and
$P_{X}(\theta_{j})\ket{1}=\ket{1}$. The rotational gate is [39]

$R_{y}^{\dagger}(\alpha_{j})=\begin{pmatrix}\cos\frac{\alpha_{j}}{2}&-\sin\frac{\alpha_{j}}{2}\\\
\sin\frac{\alpha_{j}}{2}&\cos\frac{\alpha_{j}}{2}\end{pmatrix},$ (6)

with $\alpha_{j}=2\arccos{\rho_{j}}$.

This completes the construction of the $O_{H}$ oracle.

Walk-state constructions. With the oracles, we introduce the isometries to
construct the walk states. We define the isometry $\mathcal{T}_{f}$ to create
the “forward” walk state with the Fock state $|\mathcal{F}\rangle$ as

$\displaystyle\mathcal{T}_{f}|\mathcal{F}\rangle_{s}\otimes$
$\displaystyle\ket{0}_{a}=\frac{1}{\sqrt{\mathcal{D}}}\sum_{j=0}^{\mathcal{D}-1}{\zeta^{{\mathcal{F}}}_{{\mathcal{F}_{j}}}}e^{i\theta_{j}}\ket{\mathcal{F}}_{s}\ket{j}_{id}\ket{\mathcal{F}_{j}}_{cp}$
$\displaystyle\otimes\ket{y^{P}_{\mathcal{F},j}}_{e_{p}}\ket{y^{Q}_{\mathcal{F},j}}_{e_{q}}\ket{0}_{\zeta}\ket{\rho_{j}}_{me}\ket{0}_{b_{p}}\ket{0}_{b_{q}},$
(7)

with one query to both the $O_{F}$ and $O_{H}$ oracles constructed based on
$H$. Except for the register $s$, the ancilla register (denoted by “$a$”)
consists of the other eight registers. Here, the “$b_{p}$” and “$b_{q}$”
registers are both prepared in $\ket{0}$. $\mathcal{D}$ denotes the total
number of monomials in $H$.

Figure 1: (color online) Illustration: the circuit of $\mathcal{T}_{f}$ via
the $O_{\rm F}$ and $O_{H}$ oracles designed from the Hamiltonian $H_{x}$. The
qubit indices of the $s$ and $cp$ registers correspond to the SP bases.

Similarly, we define the isometry $\mathcal{T}_{b}$ to create the “backward”
walk state with the Fock state $|\mathcal{G}\rangle$ as

$\displaystyle\mathcal{T}_{b}|\mathcal{G}\rangle_{s}\otimes\ket{0}_{a}=\frac{1}{\sqrt{\mathcal{D}}}$
$\displaystyle\sum_{k=0}^{\mathcal{D}-1}\ket{\mathcal{G}}_{s}\ket{k}_{id}\ket{\mathcal{G}_{k}}_{cp}\ket{y^{P}_{\mathcal{G},k}}_{e_{p}}$
$\displaystyle\otimes\ket{y^{Q}_{\mathcal{G},k}}_{e_{q}}\ket{0}_{\zeta}\ket{0}_{me}\ket{0}_{b_{p}}\ket{0}_{b_{q}},$
(8)

with one query to the $O_{F}$ oracle that is constructed based on
$H^{\dagger}$. We recall that the monomial labeling schemes in $H$ and
$H^{\dagger}$ are coordinated.

Hamiltonian input model. These walk states block encode $H$ as

$(\langle\mathcal{G}|_{s}\otimes\langle
0|_{a})\mathcal{T}^{\dagger}_{b}{S}\mathcal{T}_{f}(|\mathcal{F}\rangle_{s}\otimes\ket{0}_{a})=\frac{1}{\mathcal{D}\Lambda}\langle\mathcal{G}|H|\mathcal{F}\rangle,$
(9)

with ${1}/{(\mathcal{D}\Lambda)}$ being the scaling coefficient. This defines
our Hamiltonian input model
$U_{H}\equiv\mathcal{T}^{\dagger}_{b}{S}\mathcal{T}_{f}$. $S$ denotes a set of
bit-wise swap operations: (1) between the $s$ and $cp$ registers; (2) between
the single-qubit registers $e_{p}$ and $b_{p}$; and (3) between single-qubit
registers $e_{q}$ and $b_{q}$. $S$ prunes the walk-state components according
to the symmetries of $H$ and error messages, such that only those physical
walk-state components are retained.

Input model of the Chebyshev polynomials. By the reflection operator
${\Pi}\equiv(2|0\rangle_{a}\langle
0|_{a}-\mathbb{I}_{a})\otimes\mathbb{I}_{s}$ that produces the reflection
around $\ket{0}_{a}$ in the auxiliary space, we can block encode the Chebyshev
polynomial of the first kind of $H^{\prime}\equiv H/(\mathcal{D}\Lambda)$ by
implementing the qubitization [40, 41, 42] based on $U_{H}$ and
$U_{H}^{\dagger}$. In particular, for a pivot Fock state $\ket{\psi_{0}}$ in
the $s$ register, we have

$\displaystyle\langle\psi_{0}|T_{2k+1}(H^{\prime})|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle(\langle{\psi_{0}}|_{s}\otimes\langle
0|_{a})\big{[}U_{H}{\Pi}(U^{\dagger}_{H}{\Pi}U_{H}{\Pi})^{k}\big{]}(|{\psi_{0}}\rangle_{s}\otimes\ket{0}_{a}),$
(10) $\displaystyle\langle\psi_{0}|T_{2k}(H^{\prime})|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle(\langle\psi_{0}|_{s}\otimes\langle
0|_{a})(U^{\dagger}_{H}{\Pi}U_{H}{\Pi})^{k}(|\psi_{0}\rangle_{s}\otimes\ket{0}_{a}),$
(11)

with $k=0,1,2,\cdots$. The circuit to block encode $T_{j}(H^{\prime})$ is
achieved by alternatively applying $U_{H}\Pi$ or $U_{H}^{\dagger}\Pi$ [43].

Spectral calculations. We propose a quantum-classical framework for the
spectral calculation of the relativistic and nonrelativistic many-fermion
systems by leveraging our input model with the idea of the quantum Krylov
subspace diagonalization (QKSD) [44, 45, 46, 47, 48, 49, 50].

Based on the pivot state $\ket{\psi_{0}}$ and the block-encoded
$T_{j}(H^{\prime})$, we construct the Krylov basis on the quantum computer as
[50]

$\\{\ket{\psi_{i}}=T_{i}(H^{\prime})\ket{\psi_{0}}|i=0,1,2,\cdots,\mathcal{K}-1\\}.$
(12)

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

$H^{\prime}_{ij}=\langle\psi_{i}|H^{\prime}|\psi_{j}\rangle=\langle\psi_{0}|T_{i}(H^{\prime})H^{\prime}T_{j}(H^{\prime})|\psi_{0}\rangle,$
(13)

and the overlap matrix element is

$\Upsilon_{ij}=\langle\psi_{i}|\psi_{j}\rangle=\langle\psi_{0}|T_{i}(H^{\prime})T_{j}(H^{\prime})|\psi_{0}\rangle.$
(14)

$H^{\prime}_{ij}$ and $\Upsilon_{ij}$ can be expressed in terms of $\langle
T_{k}(H^{\prime})\rangle_{0}\equiv\langle\psi_{0}|T_{k}(H^{\prime})|\psi_{0}\rangle$
utilizing the identities
$T_{i}(x)T_{j}(x)=\big{(}T_{i+j}(x)+T_{|i-j|}(x)\big{)}/2$, and $T_{1}(x)=x$
[51]. We employ the quantum computer to evaluate the expectation $\langle
T_{k}(H^{\prime})\rangle_{0}$ via the Hadamard test [52] or other methods such
as the multifidelity estimation protocol [46], which addresses the many-body
calculations that can be challenging for classical computers. With these
expectations $\langle T_{k}(H^{\prime})\rangle_{0}$, we obtain
$H^{\prime}_{ij}$ and $\Omega_{ij}$, which are input to the classical computer
to solve the generalized eigenvalue equation

$H^{\prime}\ket{\Psi}=\lambda\Upsilon\ket{\Psi},$ (15)

for the eigenvalues $\lambda$. In general, the overlap matrix can be ill
conditioned; one can adopt the canonical orthogonalization [53] in practical
calculations, which is capable of approximating the lowest eigenvalue of
$H^{\prime}$ [48].

We highlight the following feature of our input model from which the QKSD
benefits. Working with the second-quantized Hamiltonian and the occupation
representation of the Fock states, we have direct control on the symmetries of
the Hilbert space (e.g., the particle number and the projection $M_{J}$ of the
total angular momentum $J$ of the system) spanned by the Krylov basis through
manipulating the pivot state $\ket{\psi_{0}}$. Specifically, in the cases
where $H$ preserves $M_{J}$, we can generate the Krylov basis with specific
$M_{J}$ by electing $\ket{\psi_{0}}$ to be of fixed $M_{J}$. With this $M_{J}$
regulation, those eigenstates of $H$ with $J<M_{J}$ are excluded in the
resultant spectrum. This symmetry-adapted screening technique is useful to
probe the excited states.

Gate complexity and circuit depth. We analyze the gate complexity of our input
model $U_{H}$ and $U^{\dagger}_{H}$. We consider the extreme case where all
the $\mathcal{D}$ monomials in $H$ are terms with $\mathcal{X}$ creation and
$\mathcal{Y}$ annihilation operators, together with their conjugate terms. The
gate cost to construct $U_{H}$ and $U^{\dagger}_{H}$ is dominated by $O_{H}$,
which is $\widetilde{O}\big{(}\mathcal{D}(N_{\rm
sp}^{\lceil\frac{\mathcal{X}}{2}\rceil}+N_{\rm
sp}^{\lceil\frac{\mathcal{Y}}{2}\rceil})\big{)}$ with $\mathcal{D}\in
O(N^{\mathcal{X}+\mathcal{Y}}_{\rm sp})$. Here the $\widetilde{O}$ notation
means that the logarithmic factor from compiling the multiple controlled gates
[54, 55] is suppressed. The circuit depth of the input model, which is
determined by the operations necessary for the phase evaluation on the
register $\zeta$, also scales as $\widetilde{O}\big{(}\mathcal{D}(N_{\rm
sp}^{\lceil\frac{\mathcal{X}}{2}\rceil}+N_{\rm
sp}^{\lceil\frac{\mathcal{Y}}{2}\rceil})\big{)}$.

The gate cost and the circuit depth to block encode $T_{k}(H^{\prime})$ are
dominated by $k$ applications of $U_{H}$ and $U_{H}^{\dagger}$. They both
scale as $\widetilde{O}\big{(}k\cdot N^{\mathcal{X}+\mathcal{Y}}_{\rm
sp}(N_{\rm sp}^{\lceil\frac{\mathcal{X}}{2}\rceil}+N_{\rm
sp}^{\lceil\frac{\mathcal{Y}}{2}\rceil})\big{)}$.

As for our framework of spectral calculations, we evaluate $\langle
T_{k}(H^{\prime})\rangle_{0}$ with $k=0,1,2,\cdots,2\mathcal{K}-1$ on the
quantum computer. A general bound on $\mathcal{K}$ to compute the ground state
energy within the error $\mathcal{E}$ is [50]

$\mathcal{K}=\Theta[(\log|\gamma_{0}|^{-1}+\log{\mathcal{E}}^{-1})\cdot\min({\mathcal{E}}^{-1},{\Delta}^{-1})],$
(16)

with $\Delta$ being the spectral gap. $\gamma_{0}$ denotes the overlap between
the pivot $\ket{\psi_{0}}$ and the ground state of $H^{\prime}$ in the
symmetry-adapted Hilbert space.

The total number of the quantum evaluations of $\langle
T_{k}(H^{\prime})\rangle_{0}$ scales as $\Theta(\mathcal{K}^{2})$. The gate
complexity and circuit depth of the most complex evaluation both scale as
$\widetilde{O}\big{(}\mathcal{K}\cdot N^{\mathcal{X}+\mathcal{Y}}_{\rm
sp}(N^{\lceil\frac{\mathcal{X}}{2}\rceil}+N^{\lceil\frac{\mathcal{Y}}{2}\rceil})\big{)}$.

Application. We demonstrate our framework by solving the low-lying spectra of
42Ca and 46Ca. These isotopes are considered as the two- and six-neutron
systems outside of a closed 40Ca core [56]. We model the Hamiltonian as the
pairing plus quadrupole-quadrupole interaction [57]

$\displaystyle
H_{A}=g\Big{[}-\sum_{p,q}\xi_{p}\xi_{q}a_{p}^{\dagger}a_{\bar{p}}^{\dagger}a_{\bar{q}}a_{q}$
$\displaystyle+\chi\sum_{p<q}\sum_{u<v}\sum_{\mu=-2}^{2}\langle
p|\mathcal{Q}_{2\mu}|u\rangle\langle q|\mathcal{Q}^{\ast}_{2\mu}|v\rangle
a^{\dagger}_{p}a^{\dagger}_{q}a_{v}a_{u}\Big{]},$ (17)

with the quadrupole operator $\mathcal{Q}_{2\mu}\equiv
r^{2}Y_{2\mu}(\Omega_{\hat{r}})$. We take the $0f_{7/2}$ valence space for the
calculation and adopt the harmonic oscillator basis [56, 58] as the SP basis.
$a_{p}$ annihilates a neutron on the $p$th SP basis labeled by
$p\mapsto(n_{p},l_{p},j_{p},m_{p})$ with the radial quantum number $n_{p}$,
the orbital angular momentum $l_{p}$, the total angular momentum $j_{p}$ and
its projection $m_{p}$ (with $m_{p}>0$), where the quantum numbers of the spin
and isospin are omitted. We also define the label
$\bar{p}\mapsto(n_{p},l_{p},j_{p},-m_{p})$, and the coefficient
$\xi_{p}\equiv(-1)^{j_{p}-m_{p}}$. We take $g=0.147439$ MeV and
$\chi=-3.934\times 10^{8}$ by fitting the excitation spectrum of 42Ca. The
same $g$ and $\chi$ values are adopted to compute the spectra of 46Ca.

We elect the oscillator strength and nucleon mass to be 12 MeV and 938.919
MeV, respectively. We evaluate the two-body matrix elements of $H_{A}$ with
the retained SP basis set in the $0f_{7/2}$ valence shell [56]. We construct
the circuits to block encode the Chebyshev polynomials of the target
Hamiltonian based on our input model. By selecting the symmetries of the pivot
state $\ket{\psi_{0}}$, we construct the symmetry-adapted Krylov basis, with
which we evaluate the Hamiltonian and overlap matrix elements utilizing the
noiseless Statevector simulator in IBM Qiskit [39]. These matrix elements are
input to the classical computer to solve the eigenenergies.

Table 1: The low-lying eigenenergies and excitation spectra (in MeV’s) of 42Ca and 46Ca compared to the experiment [59]. $J^{\pi}$ | $E_{{}^{42}Ca}$ | $E^{\rm ex}_{{}^{42}Ca}$ | Expt. | $E_{{}^{46}Ca}$ | $E^{\rm ex}_{{}^{46}Ca}$ | Expt.
---|---|---|---|---|---|---
$0^{+}$ | -2.34280 | 0 | 0 | 0.868409 | 0 | 0
$2^{+}$ | -0.818086 | 1.52471 | 1.52471 | 2.39312 | 1.52471 | 1.346
$4^{+}$ | 0.584347 | 2.92714 | 2.7524 | 3.79555 | 2.92714 | 2.5747
$6^{+}$ | 0.584347 | 2.92714 | 3.18926 | 3.79555 | 2.92714 | 2.9739

We obtain the eigenenergies of the 42Ca and 46Ca with our hybrid framework,
where we prepare the pivot state $\ket{\psi_{0}}$ to be of specific $M_{J}$ to
eliminate those states with $J<M_{J}$ from the spectrum. We find the results
via our framework (bold font in Table 1) agree with the classical calculations
via the full configuration interaction method; they also compare well with the
experiment [59].

The excitation spectra of the two isotopes are identical. This is expected
from the fact that, in the $0f_{7/2}$ valence space, the two- and six-neutron
systems are conjugate two-particle and two-hole systems, respectively.

Summary and outlook. In sum, we propose a novel input model for general
second-quantized Hamiltonians of many-fermion systems. Working in the
occupation representation, our input model respects the symmetries of
Hamiltonians, and incorporates the Pauli principle, anticommutation relations,
and particle number variations. Based on our input model, we propose a hybrid
framework for the spectral solutions of the many-fermion systems in the
relativistic and nonrelativistic quantum theories on future quantum hardwares.
We present explicit circuit designs of our input model and framework, where
the gate cost and circuit depth are analyzed. We showcase our framework by
computing the low-lying spectra of 42Ca and 46Ca. Our input model provides new
pathways to solving the time-evolution unitary [60, 41, 13] of the
relativistic and nonrelativistic many-fermion systems whose Hamiltonians and
basis states are best described in second quantization for the structure and
dynamics via efficient quantum algorithms.

Acknowledgments. We acknowledge fruitful discussions with Peter Love, Chao
Yang, Pieter Maris, Michael Kreshchuk, and William Kirby. 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.

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