Article Title]Supplementary information for "Quantum Zeno Monte Carlo for observable measurement"

1]Mancheon<EMAIL_ADDRESS>
[2,3]Hyowon<EMAIL_ADDRESS>
[1]Sangkook<EMAIL_ADDRESS>
*[1]Department of Computational Sciences, Korea Institute For Advanced Study (KIAS), 85 Hoegiro Dongdaemun-gu, Seoul, 02455, Republic of Korea

*[2]Materials Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

*[3]Department of Physics, University of Illinois at Chicago, Chicago, IL, 60607, USA

In this supplementary information, we discuss details of the methods
and calculations in the main text. Then, we discuss
the initial state preparation for the QZMC and compare our approach with others.

§ QUANTUM ZENO MONTE CARLO

This section discuss details of the quantum Zeno Monte Carlo.

§.§ Static observables

First, we show the derivation of Eq. (<ref>).
By substituting $\mathcal{P}_\alpha$ by $\mathcal{P}^\beta_\alpha$,
$\braket{\Psi_\alpha|A|\Psi_\alpha} =
\braket{\Phi_0|\mathcal{P}^\dagger_{\alpha} A \mathcal{P}_\alpha|\Phi_0}$ becomes
\vspace{-10pt}
\begin{align}
\braket{\Psi_{\alpha}|A|\Psi_{\alpha}} &= \frac{1}{(2\pi\beta^2)^\alpha} \int_{-\infty}^{\infty}\dots \int_{-\infty}^{\infty} e^{-\sum\limits_{a=1}^{2\alpha}\frac{t^2_{a}}{2\beta^2}+i\phi_\alpha(\mathbf{t})} z_\alpha(A,\mathbf{t}) dt_1 \dots dt_{2\alpha}, \nonumber \\
z_\alpha(A,\mathbf{t})&= \braket{\Phi_0| e^{-i H_1 t_{2\alpha}} \dots
e^{-i H_\alpha t_{\alpha+1}} A e^{-i H_\alpha t_{\alpha}}\dots
e^{-i H_1 t_{1}}|{\Phi_0}}, \nonumber
\\
\phi_\alpha(\mathbf{t}) & =\sum_{\alpha'=1}^{\alpha} E_{\alpha'} (t_{\alpha'} + t_{2\alpha+1-\alpha'}).
\label{eq:obsmc_supp}
\end{align}
This multi-dimensional integration can be efficiently calculated
by using the Monte Carlo method~\cite{Kroese2011} with
This yields Eq.~(\ref{eq:obsmc}).
The measurement of $z_α(O,𝐭)$ is achieved
through the Pauli string~\cite{Tilly2022} expansion of operator $O$.
$⟨Ψ_α|Ψ_α|=⟩⟨Ψ_α|𝕀|Ψ_α|$⟩ becomes

\begin{align}
\braket{\Psi_\alpha|\Psi_\alpha} &= \frac{1}{(2\pi\beta^2)^\alpha} \int_{-\infty}^{\infty}\dots \int_{-\infty}^{\infty} e^{-\sum\limits_{a=1}^{2\alpha}\frac{t^2_{a}}{2\beta^2}+i\phi_\alpha(\mathbf{t})} z_\alpha(\mathbb{I},\mathbf{t}) dt_1 \dots dt_{2\alpha}.
\label{eq:psimc}
\end{align}

So it is computed as in Eq. (<ref>).
Then, the expectation value of the observable $O$
is obtained by

\begin{align}
\braket{O}_\alpha = \frac{\braket{{\Psi_\alpha}|O|{\Psi_\alpha}}}{\braket{\Psi_\alpha|\Psi_\alpha}}. \label{eq:obs_exp}
\end{align}

§.§ Green's function

Next, we derive the Quantum Zeno Monte Carlo method
for computing the electronic Green's function.
The computation of other Green's function,
such as the spin-spin correlation function,
follows a similar approach.
The retarded electronic Green's function is defined as
= - i \theta(t) \braket{\Phi|\{c_j(t),c^\dagger_k(0)\}|\Phi}$,
where ${  ,  }$ represents the anticommutator
and $c_j$ ($c^†_j$) is the annihilation (creation)
operator for the $j$-th orbital.
This Green's function is expressed compactly
using the spectral function $A_jk(ω)$,
\begin{align}
G^{\mathrm{R}}_{jk}(\omega+i0^+) = \int_{-\infty}^{\infty}\, \frac{A_{jk}(\omega')}{\omega+i0^{+}-\omega'} d \omega'.
\end{align}
The spectral function $A_jk(ω)$ is expanded in~\cite{Negele}
\begin{align}
A_{jk}(\omega)   &= A_{jk,1}(\omega) + A_{jk,2}(\omega), \nonumber \\
A_{jk,1}(\omega) &= \sum_{m}\braket{\Phi|c_j|m}\braket{m|c^\dagger_k|\Phi} \delta(\mathcal{E}_m-\mathcal{E}_\Phi-\omega), \nonumber \\
A_{jk,2}(\omega) &= \sum_{m}\braket{\Phi|c_k^{\dagger}|m}\braket{m|c_j|\Phi} \delta(\mathcal{E}_m-\mathcal{E}_\Phi+\omega).
\end{align}
While $|m⟩$ and $ℰ_m$ are the energy eigenstate
and corresponding energy eigenvalue.
We focus on computing $A_jk,1(ω)$,
as $A_jk,2(ω)$
can be computed similarly.
The dirac delta function $δ(x)$ is approximated
the gaussian function $1/√(2πσ^2) e^-x^2/2σ^2$.
So, with the gaussian broadening,
$A_jk,1(ω)$ is computed as
\begin{align}
A_{jk,1}(\omega) &= \frac{1}{\sqrt{2\pi\sigma^2}}\sum_{m}\braket{\Phi|c_j|m}\braket{m|c^\dagger_k|\Phi} e^{-(\mathcal{E}_m-\mathcal{E}_\Phi-\omega)^2/2\sigma^2} \nonumber \\
&= \frac{1}{\sqrt{2\pi\sigma^2}} \braket{\Phi|c_j e^{-(H-\mathcal{E}_\Phi-\omega)^2/2\sigma^2} c^\dagger_k|\Phi}  .
\end{align}
For QZMC, we consider the Green's function at $α$.
Then, like Eq.~(\ref{eq:obs_exp}), we can represent $A_jk,1(ω)$ as
\begin{align}
A_{jk,1}(\omega) = \frac{1}{\sqrt{2\pi\sigma^2}}\frac{\tilde{A}_{jk,1}(\omega)}{\braket{\Psi_\alpha|\Psi_\alpha}}, \quad \tilde{A}_{jk,1}(\omega) =
\braket{\Psi_\alpha|c_je^{-(H_\alpha-E_\alpha-\omega)^2/2\sigma^2} c^\dagger_k |\Psi_\alpha}.
\end{align}
By fourier expansion of $e^-(H_α-E_α-ω)^2$,
we can compute $Ã_jk,1$ like Eq.~(\ref{eq:obsmc}),
\begin{align}
\tilde{A}_{jk,1}(\omega) = \frac{1}{N_\nu}\sum_\nu g_{\alpha}(c_j,c^\dagger_k,\mathbf{t}_\nu) e^{i\phi_\alpha(\mathbf{t_\nu},\omega)},\label{eq:greenmc}
\end{align}
while $g_α(O_1,O_2,𝐭)
=⟨Φ_0| e^-i H_1 t_2α+1
…e^-i H_αt_α+2
O_1 e^-i H_αt_α+1 O_2 e^-i H_αt_α…e^-i H_1 t_1|Φ_0|$⟩
=\sum_{\alpha'=1}^{\alpha} E_{\alpha'} (t_{\alpha'} + t_{2\alpha+2-\alpha'}) + (E_\alpha + \omega) t_{\alpha+1}$.
Here, $t_{\nu,\alpha+1}\sim\mathcal{N}(0,1/\sigma)$ and
$t_{\nu,l}\sim\mathcal{N}(0,\beta)$ for all $l\neq\alpha+1$.
Since $g_{\alpha}(O_1,O_2,\mathbf{t})$
is not dependent on $\omega$,
we can compute $A_{jk}(\omega)$ for any $\omega$
by storing $g_{\alpha}(c_j,c^\dagger_k,\mathbf{t}_\nu)$.

§.§ Corrector for the energy difference

The energy difference formula,
Eq. (<ref>), can be derived as follows.

\begin{align}
E_{\alpha}&= \frac{\braket{\Phi_0|\mathcal{P}^\dagger_{\alpha} H_{\alpha}
\mathcal{P}_{\alpha}|\Phi_0
\mathcal{P}_{\alpha}|\Phi_0}}\nonumber\\
&= \frac{\braket{\Phi_0|\mathcal{P}^\dagger_{\alpha} \ket{\Phi_\alpha}\bra{\Phi_\alpha} H_{\alpha}
\mathcal{P}_{\alpha-1}|\Phi_0
\mathcal{P}_{\alpha}|\Phi_0}} \nonumber \\
\frac{\braket{\Phi_0|\mathcal{P}^\dagger_{\alpha} \ket{\Phi_\alpha}\bra{\Phi_\alpha} (H_{\alpha-1} + h_{\alpha-1})
\mathcal{P}_{\alpha-1}|\Phi_0
\mathcal{P}_{\alpha}|\Phi_0}} \nonumber \\
&= E_{\alpha-1} + \frac{\braket{\Psi_\alpha\ket{\Phi_\alpha}\bra{\Phi_\alpha} h_{\alpha-1}
\end{align}

By following same steps as obtaining Eq. (<ref>),
we obtain the Monte Carlo formula of the numerator,

\begin{align}
&\braket{\Psi_\alpha\ket{\Phi_\alpha}\bra{\Phi_\alpha} h_{\alpha-1}|\Psi_{\alpha-1}} = \frac{1}{N_\nu}\sum_\nu \hat{z}_{\alpha}(h_{\alpha-1},\mathbf{t}_\nu) e^{i\phi_\alpha(\mathbf{t_\nu})} \nonumber \\
&\hat{z}_{\alpha}(O,\mathbf{t}) = \braket{\Phi_0| e^{-i H_1 t_{2\alpha}} \dots
e^{-i H_{\alpha} t_{\alpha+1}} e^{-i H_{\alpha} t_{\alpha}} O \dots
e^{-i H_1 t_{1}}|{\Phi_0}}, \label{eq:correctormc}
\end{align}

=\sum_{\alpha'=1}^{\alpha} E_{\alpha'} (t_{\alpha'} + t_{2\alpha+1-\alpha'})$.

§.§ Computational complexity of QZMC

This section provides details on the complexity
analysis of our method, which is briefly outline in the main text.
We consider the hamiltonian $H(\lambda) = H_0 + \lambda H'$.
And we begin with $\ket{\Phi_0}$, an eigenstate of the $H_0=H(\lambda=0)$
with eigenvalue $E_0$.
For simplicity, we assume that $\ket{\Phi(\lambda)}$, adiabatically
connected to $\ket{\Phi_0}$, is non-degenerate and
has an eigenvalue $E(\lambda)$ satisfying $|E(\lambda)-\mathcal{E}_m|\geq \Delta_g$
for all eigenenergies of $\mathcal{E}_m$ of $H(\lambda)$ other than $E(\lambda)$.
We then consider $H_\alpha = H(\lambda_\alpha), \, \alpha=1,\dots,N_\alpha$
with $\lambda_\alpha=\alpha/N_\alpha$,
as in the main text.
We denote $E_\alpha=E(\lambda_\alpha),\, \ket{\Phi_\alpha}=\ket{\Phi(\lambda_\alpha)}$.

§.§.§ $N_\alpha$

First, we compute $N_\alpha$ using exact projection.
Utilizing perturbation theory <cit.>, we have

\begin{align}
\braket{\Phi_{\alpha}|\Phi_{\alpha+1}} &= \left(1-\frac{(\Delta\lambda)^2}{2}\sum_{m_\alpha\neq\Phi_\alpha}
\frac{|\braket{m_\alpha|H'|\Phi_\alpha}|^2}{(E_\alpha-\mathcal{E}_{m,\alpha})^2}\right),\nonumber \\
\braket{m_{\alpha}|\Phi_{\alpha+1}} &= \Delta\lambda \frac{\braket{m_\alpha|H'|\Phi_\alpha}}{E_\alpha-\mathcal{E}_{m,\alpha}}\,\,\,\text{for }m_\alpha\neq\Phi_\alpha, \label{eq:perturb}
\end{align}

up to the leading order of $\Delta\lambda=\lambda_{\alpha+1}-\lambda_{\alpha}$.
Here, $\ket{m_\alpha}$ represent an eigenstate of
$H_\alpha$ with an eigenvalue of

\begin{align}
\sum_{m_\alpha\neq\Phi_\alpha}\frac{|\braket{m_\alpha|H'|\Phi_\alpha}|^2}{(E_\alpha-\mathcal{E}_{m,\alpha})^2}\leq\sum_{m_\alpha\neq\Phi_\alpha}\frac{|\braket{m_\alpha|H'|\Phi_\alpha}|^2}{\Delta_g^2}\leq\frac{1}{\Delta_g^2}\|H'\|^2. \label{eq:coeffbound}
\end{align}

Here, we employ the matrix norm induced by vector $2$-norm <cit.>,
$\|A\| = \operatorname{sup}_{\|x\|=1}\|Ax\|$.
Combining Eq. (<ref>) and Eq. (<ref>), we obtain

\begin{align}
\braket{\Psi_\alpha|\Psi_\alpha} = |\braket{\Phi_0|\Phi_1}|^2 \dots|\braket{\Phi_{\alpha-1}|\Phi_{\alpha}}|^2\geq 1 - \frac{\|H'\|^2}{\Delta_g^2}\frac{1}{N_\alpha}.
\end{align}

Thus, we have $\|\Psi_\alpha\|^2\geq 1-\eta_0$ for all $\alpha$ by setting

\begin{align}
N_\alpha \geq \frac{\|H'\|^2}{\Delta_g^2}\frac{1}{\eta_0}. \label{eq:N_alpha}
\end{align}

§.§.§ $\beta$

The impact of finite $\beta$ on the projected state is
described by

\begin{align}
\ket{\delta\Psi^\beta_\alpha} = (\mathcal{P}^\beta_\alpha-\mathcal{P}_\alpha)\ket{\Phi_0}.
\end{align}

Here, we show

\begin{align}
\|\delta\Psi^\beta_\alpha\|\leq \frac{\alpha}{N_\alpha}e^{-\beta^2\Delta_g^2/2}\frac{\|H'\|}{\Delta_g} \label{eq:beta_norm}
\end{align}

inductively, where $\|\delta\Psi^\beta_\alpha\|^2=\braket{\delta\Psi^\beta_\alpha|\delta\Psi^\beta_\alpha}$
First, we address the case of $α=1$.
Employing the perturbation approach as in Eq.(\ref{eq:perturb}),
we obtain
\begin{align}
\|\delta\Psi^\beta_1\|^2 &= \braket{\Phi_0|(P^\beta_{H_1}(E_1)-\ket{\Phi_1}\bra{\Phi_1})^2|\Phi_0}\nonumber \\
&= (\Delta\lambda)^2 \sum_{m_1\neq\Phi_1} e^{-\beta^2(\mathcal{E}_{m,1}-E_1)^2}\frac{|\braket{\Phi_1|H'|m_1}|^2}{(\mathcal{E}_{m,1}-E_{1})^2}\nonumber \\
&\leq (\Delta\lambda)^2 e^{-\beta^2\Delta_g^2}\frac{\|H'\|^2}{\Delta^2_g} \label{eq:beta1}
\end{align}
So, $δΨ^β_1≤1/N_α e^-β^2Δ_g^2/2H'/Δ_g$.
Suppose for $α$,
\begin{align}
\ket{\delta\Psi^\beta_{\alpha+1}} = (P^\beta_{\alpha+1}-\ket{\Phi_{\alpha+1}}\bra{\Phi_{\alpha+1}})\ket{\Psi_\alpha} +
\end{align}
For simplicity, we expressed $P^β_α+1=P^β_H_α+1(E_α+1)$.
\begin{align}
\|\delta\Psi^\beta_{\alpha+1}\|^2  &= \braket{\Psi_\alpha|
|\Psi_\alpha} \nonumber \\
&+ \braket{\Psi_\alpha|(P^\beta_{\alpha+1}-\ket{\Phi_{\alpha+1}}\bra{\Phi_{\alpha+1}}) P^\beta_{\alpha+1}
|\delta\Psi^\beta_\alpha}\nonumber \\
&+\braket{\delta\Psi_\alpha| P^\beta_{\alpha+1} (P^\beta_{\alpha+1}-\ket{\Phi_{\alpha+1}}\bra{\Phi_{\alpha+1}})
|\Psi_\alpha} + \braket{\delta\Psi^\beta_\alpha|(P^\beta_{\alpha+1})^2|\delta\Psi^\beta_\alpha}.
\end{align}
The first term is smaller than $(Δλ)^2 e^-β^2Δ_g^2H'^2/Δ^2_g$
by following the same procedure as with Eq.~(\ref{eq:beta1}).
And the last term is smaller than $δΨ^β_α^2≤α^2/N^2_αe^-β^2Δ_g^2H'^2/Δ^2_g$
since $P^β_α$ has eigenvalues smaller than $1$.
Consider the second term.
\begin{align}
&|\braket{\Psi_\alpha|(P^\beta_{\alpha+1}-\ket{\Phi_{\alpha+1}}\bra{\Phi_{\alpha+1}}) P^\beta_{\alpha+1}
|\delta\Psi^\beta_\alpha}|\nonumber \\
&\leq\Delta\lambda\sum_{\substack{m_{\alpha+1}\\\neq\Phi_{\alpha+1}}}\frac{|\braket{\Phi_{\alpha+1}|H'|m_{\alpha+1}}|}{|\mathcal{E}_{m,\alpha+1}-E_{\alpha+1}|} e^{-\beta^2(\mathcal{E}_{m,\alpha+1}^2-E_{\alpha+1}^2)}
|\braket{m_{\alpha+1}|\delta\Psi^\beta_\alpha}| \nonumber \\
& \leq \Delta\lambda \frac{e^{-\beta^2\Delta_g^2}}{\Delta_g} \|H'\ket{\Phi_{\alpha+1}}\|\|\delta\Phi^\beta_{\alpha}\| \nonumber \\
& \leq \Delta\lambda \frac{e^{-\beta^2\Delta_g^2}}{\Delta_g} \|H'\|\|\delta\Phi^\beta_{\alpha}\|
\leq \Delta\lambda\frac{\alpha}{N_\alpha}  e^{-\beta^2\Delta_g^2} \frac{\|H'\|^2}{\Delta_g^2}.
\label{eq:beta_corrector_second}
\end{align}
Similar argument provides the same bound for the third term.
By using $Δα=1/N_α$, we have
\begin{align}
\|\delta\Psi^\beta_{\alpha+1}\|^2 \leq \frac{(\alpha+1)^2}{N^2_\alpha}e^{-\beta^2\Delta_g^2}\frac{\|H'\|^2}{\Delta^2_g},
\end{align}
which proves Eq.~(\ref{eq:beta_norm}).
So, $|⟨Ψ_α|Ψ_α|-⟩⟨Ψ^β_α|Ψ^β_α||⟩$
becomes smaller
than $δη$ by using
\begin{align}
\beta \geq \frac{1}{\Delta_g} \sqrt{2\operatorname{log}\left(\frac{\|H'\|}{\Delta_g}\frac{2}{\delta\eta}\right)}. \label{eq:beta_lowerbound_1}
\end{align}
If we are interested in the expectation value of the observable $O$,
has an error of

\begin{align}
\delta{\braket{\Psi^\beta_\alpha|O|\Psi^\beta_\alpha}}=\braket{\delta\Psi_\alpha^\beta|O|\Psi_\alpha} +\braket{\Psi_\alpha|O|\delta\Psi_\alpha^\beta} +\braket{\delta\Psi^\beta_\alpha|O|\delta\Psi_\alpha^\beta}.
\end{align}

which is smaller than

\begin{align}
2\|\delta\Psi^\beta_\alpha\|\|O\| + \|\delta\Psi^\beta_\alpha\|^2\|O\|.
\end{align}

By considering error up to the first order of $\|\delta\Psi^\beta_\alpha\|$, we have

\begin{align}
\delta{\braket{\Psi^\beta_\alpha|O|\Psi^\beta_\alpha}}\leq2\|\delta\Psi^\beta_\alpha\|\|O\|.
\end{align}

Then, the error of the expectation value $\braket{\Psi^\beta_\alpha|O|\Psi^\beta_\alpha}/\braket{\Psi^\beta_\alpha|\Psi^\beta_\alpha}$ is estimated by
\begin{align}
\delta{\left(\frac{\braket{\Psi^\beta_\alpha|O|\Psi^\beta_\alpha}}{\braket{\Psi^\beta_\alpha|\Psi^\beta_\alpha}}\right)}&= \frac{\delta\braket{\Psi^\beta_\alpha|O|\Psi^\beta_\alpha}}{\braket{\Psi_\alpha|\Psi_\alpha}} +\frac{\braket{\Psi_\alpha|O|\Psi_\alpha}}{\braket{\Psi_\alpha|\Psi_\alpha}^2} \delta{\braket{\Psi^\beta_\alpha|\Psi^\beta_\alpha}}\nonumber \\
&\leq 4\frac{\|\delta\Psi^\beta_\alpha\|}{\braket{\Psi_\alpha|\Psi_\alpha}} \|O\|,
\end{align}
because $⟨Ψ_α|O|Ψ_α|/⟩⟨Ψ_α|Ψ_α|≤⟩O$.
So, if $⟨Ψ_α|Ψ_α|≥⟩1-η$,
$⟨O|$⟩ can be estimated within an
error of $\epsilon$ by using

\begin{align}
\beta\geq\frac{1}{\Delta_g}\sqrt{2\operatorname{log}\left( \frac{\|H'\|}{\Delta_g}\frac{4\|O\|}{1-\eta}\frac{1}{\epsilon}\right)}
\end{align}

The energy estimator in Eq. (<ref>) is a slightly different case
since we are computing the energy difference
at each $\alpha$.
With finite $\beta$, the numerator in Eq. (<ref>) have an error of

\begin{align}
\braket{\Psi^\beta_\alpha|P^\beta_{\alpha+1}P^\beta_{\alpha+1}(H_{\alpha+1}-H_{\alpha}-(E_{\alpha+1}-E_{\alpha}))|\Psi^\beta_\alpha}.
\end{align}

Here, we considered error at $\alpha+1$ in Eq. (<ref>) instead of $\alpha$
for the notational simplicity.
Up to first order of $\Delta\lambda$ and $\|\delta\Psi^\beta_\alpha\|$, above equation is equal to

\begin{align}
\braket{\delta\Psi^\beta_\alpha|P^\beta_{\alpha+1}P^\beta_{\alpha+1}(H_{\alpha+1}-H_{\alpha}-(E_{\alpha+1}-E_{\alpha}))|\Psi_\alpha} \nonumber \\
+ \braket{\Psi_\alpha|P^\beta_{\alpha+1}P^\beta_{\alpha+1}(H_{\alpha+1}-H_{\alpha}-(E_{\alpha+1}-E_{\alpha}))|\delta\Psi^\beta_\alpha}. \label{eq:beta_energy_1}
\end{align}

By using $H_{\alpha+1}-H_{\alpha}=\Delta\lambda H'$ and $E_{\alpha+1}-E_{\alpha} = \Delta\lambda E'_\alpha + \mathcal{O}(\Delta\lambda^2)$,
Eq. (<ref>) is equal to

\begin{align}
\Delta\lambda (\braket{\delta\Psi^\beta_\alpha|P^\beta_{\alpha+1}P^\beta_{\alpha+1}(H'-E'_\alpha)|\Psi_\alpha}
+ \braket{\Psi_\alpha|P^\beta_{\alpha+1}P^\beta_{\alpha+1}(H'-E'_\alpha)|\delta\Psi^\beta_\alpha}),
\end{align}

which is smaller than

\begin{align}
4\Delta\lambda \|\delta\Psi^\beta_\alpha\| \| H'\|,
\end{align}

as $E'_\alpha \leq \|H'\|$.
If $\|\Psi^\beta_\alpha\|^2 \geq 1-\eta$,
each energy difference estimator has the error no greater than

\begin{align}
4 \Delta\lambda \frac{\|\delta\Psi^\beta_\alpha\| \| H'\|}{1-\eta} \leq 4 \frac{\alpha}{N^2_\alpha}e^{-\beta^2\Delta^2_g/2}\frac{ \| H'\|^2}{\Delta_g}\frac{1}{1-\eta}.
\end{align}

Thus, the energy estimation $E_\alpha$ has the error bound of

\begin{align}
2 \frac{\alpha(\alpha-1)}{N^2_\alpha}e^{-\beta^2\Delta^2_g/2}\frac{ \| H'\|^2}{\Delta_g}\frac{1}{1-\eta},
\end{align}

which increases as $\alpha$ increase and is smaller than
$2e^{-\beta^2\Delta^2_g/2}\frac{ \| H'\|^2}{\Delta_g}\frac{1}{1-\eta}$.
we can compute the energy within an error of $\epsilon$
by using

\begin{align}
\beta \geq \frac{1}{\Delta_g}\sqrt{2\operatorname{log}\left(\frac{2\|H'\|^2}{\Delta_g}\frac{1}{1-\eta}\frac{1}{\epsilon}\right)}. \label{eq:beta_energy}
\end{align}

§.§.§ $N_\nu$

Consider a random variable $\mathbf{t}$
with a continuous
probability distribution $g(\mathbf{t})$.
The expectation value of $x(\mathbf{t})$ is given by

\begin{align}
\operatorname{E}[x] = \int x(\mathbf{t}) g(\mathbf{t}) d\mathbf{t}.
\end{align}

The variance of the $x$ is calculated as

\begin{align}
\operatorname{Var}[x] = \operatorname{E}[x^2] - (\operatorname{E}[x])^2.
\end{align}

If $|x(\mathbf{t})|\leq L$ for all $\mathbf{t}$,
we have $\operatorname{E}[x^2]\leq L^2$,
which gives

\begin{align}
\sigma^2_x = \operatorname{Var}[x] \leq L^2 - (\operatorname{E}[x])^2 \leq L^2.
\end{align}

Consequently, if we generate $N_\nu$ samples of $\mathbf{t}$ and use the
average of $x(\mathbf{t})$ as our estimator $\bar{x}$ of $\operatorname{E}[x]$,
standard error $\sigma_{\bar{x}}$ of $\bar{x}$ is bounded by
$\frac{L}{\sqrt{N_\nu}}$ <cit.>.
This discussion extends to for Eq. (<ref>)
with $L=\|A\|$.
The observable expectation Eq. (<ref>)
can be seen as
$\bar{x}$ divided by $\bar{y}$,
where $|x(\mathbf{t})|\leq L$ and $|y(\mathbf{t})|\leq 1$.
Then, by using the the delta method <cit.>
and independent sampling of $\mathbf{t}$
for $\bar{x}$ and $\bar{y}$,
the variance $\sigma^2_{\bar{x}/\bar{y}}$
of $\bar{x}/\bar{y}$
is estimated as

\begin{align}
\sigma^2_{\bar{x}/\bar{y}}
= \frac{1}{\bar{y}^2}
\left(\sigma^2_{\bar{x}}+
\frac{\bar{x}^2}{\bar{y}^2}\sigma^2_{\bar{y}}\right)
\leq \frac{1}{\bar{y}^2}
\left(\frac{L^2}{N_\nu}+\frac{\bar{x}^2}
\leq \frac{L^2}{N_\nu\bar{y}^2}\left(1+\frac{1}{\bar{y}^2}\right).
\end{align}

Therefore, the standard error $\sigma_{\braket{O}_\alpha}$
of $\braket{O}_\alpha$ becomes

\begin{align}
\sigma_{\braket{O}_\alpha} \leq \frac{\|O\|}{\sqrt{N_\nu}} \frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}}\sqrt{1+\frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}^2}}.
\label{eq:montecarlo_error_obs}
\end{align}

the energy difference
calculated by Eq. (<ref>) has a standard error of

\begin{align}
\sigma_{E_\alpha-E_{\alpha-1}} \leq \Delta\lambda\frac{\|H'\|}{\sqrt{N_\nu}} \frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}}\sqrt{1+\frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}^2}}.
\label{eq:montecarlo_error_corrector}
\end{align}

§.§.§ The number of shots, $N_s$

In real quantum computers,
the estimation of unitary expectation value involves
with a finite number $N_s$ of repetitions
of the same circuits,
commonly referred as shots.
This introduces statistical errors
with a standard deviation of $1/\sqrt{N_s}$
to each measurement.
when measuring observable $\braket{O}$ using the pauli string expansion,
the finite number of shots modifies Eq.~(\ref{eq:montecarlo_error_obs}) to
\begin{align}
\sigma_{\braket{O}_\alpha} \leq \frac{1}{\sqrt{N_\nu}}\left(\|O\|+\frac{\sqrt{\sum_\gamma o^2_\gamma}}{\sqrt{N_s}}\right) \frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}}\sqrt{1+\frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}^2}},
\label{eq:montecarlo_error_obs_with_ns}
\end{align}
where $O = ∑_γo_γP_γ$ represents a pauli string expansion of $O$.
Similarly, Eq.~(\ref{eq:montecarlo_error_corrector}) transforms to
\begin{align}
\sigma_{E_\alpha-E_{\alpha-1}} \leq \Delta\lambda\frac{1}{\sqrt{N_\nu}} \left(\|H'\|+\frac{\sqrt{\sum_\gamma w^2_\gamma}}{\sqrt{N_s}}\right)\frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}}\sqrt{1+\frac{1}{\braket{\Psi_\alpha^\beta|\Psi_\alpha^\beta}^2}}.
\label{eq:montecarlo_error_corrector_with_ns}
\end{align}
Here, $H' = ∑_γw_γP_γ$ stands as a pauli string expansion of $H'$.

\subsubsection{Complexity of the energy estimation}

In this section, we explore the complexity of obtaining the energy eigenvalue with an additive error of $ϵ$
by determining suitable computational parameters $N_α$, $β$, and $N_ν$. Similarly,
complexities for observable calculations can be derived. The computational complexity of
the quantum algorithm is mainly influenced by the circuit depth and the number of repetitions.
To quantify the circuit depth, we examine the distribution of the consecutive time evolution
length $L_ν= |t_ν,1|+…+|t_ν,2N_α|$.
Since each $|t_ν,1|,…, |t_ν,2N_α|$ follows a half-normal
distribution with a mean of $β√(2/π)$ and
a standard deviation of $β√(1-2/π)$, $L_ν$ has
a mean $2 N_αβ√(2/π)$ and a standard deviation
of $√(2 N_α) β√(1-2/π)$. As a measure of
the maximum time evolution length, we employ $C_90N_αβ$,
where the probability of $L_ν$ being smaller than $C_90N_αβ$
is $90%$.
$C_90$ is numerically determined by testing the cumulative distribution of $L_ν$,
and it monotonically decreases as $N_α$ increases.
For $N_α=1$, $C_90=2.7545$, which reduces to $2.0978$ at $N_α=5$,
and eventually approaches $2√(2/π)$ as $N_α→∞$.

Then, $β$ and $N_α$ determines the circuit depth.
We find appropriate values for $β$ and $N_α$
by ensuring that $⟨Ψ_α^β|Ψ_α^β|≥⟩1-η$,
where $η=0.5$, in order to limit error amplification.
To achieve this, we set $η_0=0.4$ in Eq.~(\ref{eq:N_alpha})
and $δη$ in Eq.~(\ref{eq:beta_lowerbound_1}) to $0.1$,
resulting in $η=η_0+δη=0.5$.
Considering an energy error of
$ϵ$ in Eq.~(\ref{eq:beta_energy}),
this gives
\begin{align}
N_\alpha \beta \geq 2.5\frac{\|H'\|^2}{\Delta_g^3}\sqrt{2\operatorname{log}\left(\max\left(\frac{4\|H'\|^2}{\Delta_g}\frac{1}{\epsilon},\frac{20\|H'\|}{\Delta_g}\right)\right)}.
\end{align}
So, in terms of $n$, the number of qubits,
we have the maximum evolution time of
\begin{align}
C_{90} N_\alpha \beta = \mathcal{O}(\Delta_g^{-3}\operatorname{poly}(n))\sqrt{\operatorname{log}(\mathcal{O}(\Delta_g^{-1}\epsilon^{-1}\operatorname{poly}(n)))}.
\end{align}

Next, we consider the number of repetitions.
Given that we compute $N_ν$ circuits with $N_s$
shots each, the total number of repetitions
is $N_νN_s$. Utilizing
with $⟨Ψ^β_α|Ψ^β_α|≥⟩1-η=0.5$,
we can compute Eq.(\ref{eq:corrector}) within an
error of $ϵ$ with the Monte Carlo samples
\begin{align}
N_\nu \geq \left(\|H'\|+\frac{\sqrt{\sum_\gamma w^2_\gamma}}{\sqrt{N_s}}\right)^2 \frac{20}{\epsilon^2}
\end{align}
for each $α$.
By fixing $N_s$, we find
$N_ν= 𝒪(ϵ^-2poly(n))$,
as $H'^2$ and $∑_γw_γ^2$ are $𝒪(poly(n))$.
Consequently, the total number of repetitions required to obtain the energy
eigenvalue within an error of $ϵ$ is
\begin{align}
N_\nu N_s = \mathcal{O}(\epsilon^{-2}\operatorname{poly}(n)).
\end{align}
Table.~\ref{tab:S1} summarize and compare the computational
complexity of our method with other methods.

\begin{table} % Complexity table
\caption{Computational complexity of QZMC and other quantum algorithms}
\label{tab:S1}%
\begin{tabular}{@{}lll@{}}
\toprule
& Maximum time evolution length & Repetitions\\
\midrule
QZMC         &$\mathcal{O}(\Delta_g^{-3}\operatorname{poly}(n))\sqrt{\operatorname{log}(\mathcal{O}(\Delta_g^{-1}\epsilon^{-1}\operatorname{poly}(n)))}$ & $\mathcal{O}(\epsilon^{-2}\operatorname{poly}(n))$  \\
QPE~\cite{Berry2009,Higgins2007}  &$\tilde{\mathcal{O}}(\epsilon^{-1}p_0^{-1})$ & $\tilde{\mathcal{O}}(p_0^{-1}\operatorname{polylog}(\epsilon^{-1}))$ \\
QEEA~\cite{Somma2019} &$\tilde{\mathcal{O}}(\epsilon^{-1}\operatorname{polylog}(p_0^{-1}))$  & $\tilde{\mathcal{O}}(\epsilon^{-3}p_0^{-2})$  \\
Ref.~\cite{LinLin2022}&$\tilde{\mathcal{O}}(\epsilon^{-1}\operatorname{polylog}(p_0^{-1}))$  & $\tilde{\mathcal{O}}(p_0^{-2}\operatorname{polylog}(\epsilon^{-1}))$ \\
Ref.~\cite{Wang2023}  &$\mathcal{O}(\Delta_g^{-1}\operatorname{polylog}(\epsilon^{-1}p_0^{-1}\Delta_g))$  & $\mathcal{O}(p_0^{-2}\epsilon^{-2}\Delta_g^2)$ \\
\botrule
\end{tabular}
\footnotetext{
This table summarize the complexity of QZMC to compute
energy eigenvalues and compares it with
several other quantum algirithms that computes
energy eigenvalues
within a single ancilla qubit. Compelxity analysis of
QPE and QEEA imported from Ref.~\cite{LinLin2022}. Here, $p_0$ the probability of getting exact eigenstate
from the initial states, $\epsilon$ is a desired precision in the energy,
and $n$ is the number of qubits. Optimized algorithms for highly overlapped
initial states~\cite{Ding2023,Ni2023} shows similar dependence with algorithm
of Ref.~\cite{LinLin2022}, only constant factor is different.
\end{table}

\subsection{The effect of the inaccurate energy eigenvalues}
In practical calculations,
we encounter a finite error $δ$ in energy estimation.
The effect of this energy error is $e^-β^2δ^2$
for each projection.
However, this effect cancels out for the calculation
of the expectation values and the energy
difference formula. Thus, one should only consider that
\begin{align}
e^{-2N_\alpha\beta^2\delta^2} \label{eq:error_from_precision}
\end{align}
is not too small to completely destroy $⟨Ψ^β_α|Ψ^β_α|$⟩.
To ensure that this value is greater than $1/2$, we require
$\beta$ to satisfy

\begin{align}
\beta \leq \frac{1}{\delta}\sqrt{\frac{\operatorname{log}{2}}{2N_\alpha}}.
\end{align}

§.§ Quantum Circuits for QZMC

We illustrate several quantum circuits utilized
in the quantum Zeno Monte Carlo (QZMC) method.
Figure <ref> depicts the QZMC circuits
employing controlled time-evolution.
If noise is too large so that the noise that
are significanlty different to each other,
we can use  for the calculation of $\braket{\Psi|\Psi}$
instead of \textbf{\textsf{a}} to mitigate the influence of the noise.
QZMC with the circuits shown in Fig.~\ref{fig:S1} is general
and applicable for any Hamiltonian.
However, it may suffer from substantial device noise
due to numerous controlled-time evolutions.
To circumvent this, we can adopt the quantum
circuits depicted in Figure~\ref{fig:S2}.
This approach eliminates the need for controlled
time-evolution when a common eigenstate
$|Φ_ref⟩$ exists
for $H_1, …, H_N_α$.
While this appears to be a stringent restriction,
many practical physical and chemical systems possess such a common eigenstate, such as the vacuum.
For the Green's function calculation, we have to compute
$g_α$ in Eq.~(\ref{eq:greenmc}). Through
the pauli string decomposition,
this translates into a weighted sum of
$⟨Φ_0| e^-i H_1 t_2α+1
…e^-i H_αt_α+2
P_γe^-i H_αt_α+1 P_γ' e^-i H_αt_α…e^-i H_1 t_1|Φ_0|$⟩.
Therefore, the Green's function calculation
can be executed using the quantum circuits
depicted in Figure <ref>.

…    1
…    1

…    e^-iH_αt_α
…    e^-iH_1t_2α

[anchor=north west] at (current bounding box.north west) ;

…    1
…    1

…   e^-iH_αt_α
P_γ   e^-iH_αt_α+1
…    e^-iH_1t_2α

[anchor=north west] at (current bounding box.north west) ;

…    1
…    1

…    P_γ
…    e^-iH_1t_2α

[anchor=north west] at (current bounding box.north west) ;

…    [2]_
…    1

…    e^-iH_1t_2α

[anchor=north west] at (current bounding box.north west) ;

[-0.5ex]0.5pt2.5ex Quantum circuits for QZMC without a reference state.
is the circuit for $\braket{\Psi|\Psi}$,
\textsf{\textbf{b}} for $⟨Ψ|O|Ψ|$⟩,
for $\braket{\Psi_\alpha\ket{\Phi_\alpha}{\bra{\Phi_\alpha}}h_{\alpha-1}|\Psi_{\alpha-1}}$.
The noise effect difference in $⟨Ψ|Ψ|$⟩
and $\braket{\Psi_\alpha\ket{\Phi_\alpha}{\bra{\Phi_\alpha}}h_{\alpha-1}|\Psi_{\alpha-1}}$ can be
reduced by using the circuit \textsf{\textbf{d}},
which contains noisy identity that mimics
the noise effect of controlled-$P_γ$,
instead of \textsf{\textbf{a}}.
The circuit \fbox{\textbf{\textsf{init}}}
refers the circuit that transforms
$|0^n⟩$ to $|Φ_0⟩$.
\label{fig:S1}
\end{figure}

\begin{figure}
\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad  \ket{0}
& \gate{\textsf{H}}
& \gate[2]{\textsf{Init}}
& \qw
& \qw \ldots
& \qw
& \qw
& \qw \ldots
& \qw
& \gate[2]{\textsf{Init}^\dagger}
& \gate{\textsf{P}(\phi_\alpha)}
& \gate{\textsf{H}} &\meter{}
\\
\quad\quad  \ket{0^{n}}
& \qw
& \gate{e^{-iH_{1}t_1}}
& \qw \ldots
& \gate{e^{-iH_{\alpha}t_\alpha}}
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \qw \ldots
& \gate{e^{-iH_{1}t_{2\alpha}}}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{a}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad \ket{0}
& \gate{\textsf{H}}
& \gate[2]{\textsf{Init}}
& \qw
& \qw \ldots
& \qw
& \ctrl{1}
& \qw
& \qw \ldots
& \qw
& \gate[2]{\textsf{Init}^\dagger}
& \gate{\textsf{P}(\phi_\alpha)}
& \gate{\textsf{H}} &\meter{}
\\
\quad\quad \ket{0^{n}}
& \qw
& \gate{e^{-iH_{1}t_1}}
& \qw \ldots
& \gate{e^{-iH_{\alpha}t_\alpha}}
& \gate{P_\gamma}
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \qw \ldots
& \gate{e^{-iH_{1}t_{2\alpha}}}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{b}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad \ket{0}
& \gate{\textsf{H}}
& \gate[2]{\textsf{Init}}
& \qw
& \qw \ldots
& \ctrl{1}
& \qw
& \qw
& \qw \ldots
& \qw
& \gate[2]{\textsf{Init}^\dagger}
& \gate{\textsf{P}(\phi_\alpha)}
& \gate{\textsf{H}}
& \meter{}
\\
\quad\quad \ket{0^{n}}
& \qw
& \gate{e^{-iH_{1}t_1}}
& \qw \ldots
& \gate{P_\gamma}
& \gate{e^{-iH_{\alpha}t_\alpha}}
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \qw \ldots
& \gate{e^{-iH_{1}t_{2\alpha}}}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{c}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad  \ket{0}
& \gate{\textsf{H}}
& \gate[2]{\textsf{Init}}
& \qw
& \qw \ldots
& \gate[2]{\textsf{Id}_{\textsf{n}}}
& \qw
& \qw
& \qw \ldots
& \qw
& \gate[2]{\textsf{Init}^\dagger}
& \gate{\textsf{P}(\phi_\alpha)}
& \gate{\textsf{H}} &\meter{}
\\
\quad\quad  \ket{0^{n}}
& \qw
& \gate{e^{-iH_{1}t_1}}
& \qw \ldots
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \qw \ldots
& \gate{e^{-iH_{1}t_{2\alpha}}}
& \qw
& \qw
& \qw \\
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{d}}};
\end{tikzpicture}

\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Quantum circuits for QZMC with a reference state}.
\textsf{\textbf{a}} is the circuit for $\braket{\Psi|\Psi}$,
\textsf{\textbf{b}} for $\braket{\Psi|O|\Psi}$,
\textsf{\textbf{c}} for $\braket{\Psi_\alpha\ket{\Phi_\alpha}{\bra{\Phi_\alpha}}h_{\alpha-1}|\Psi_{\alpha-1}}$.
\textsf{\textbf{d}} is the circuit for the $\braket{\Psi|\Psi}$
that includes
noisy identity \fbox{$\textbf{\textsf{Id}}_{\textsf{n}}$},
making noise effect on $\braket{\Psi|\Psi}$
and $\braket{\Psi_\alpha\ket{\Phi_\alpha}{\bra{\Phi_\alpha}}h_{\alpha-1}|\Psi_{\alpha-1}}$
similar to each other.
The circuit \fbox{\textbf{\textsf{init}}}
transforms $\ket{0^{n}}$ to a reference state $\ket{\Phi_{\textrm{ref}}}$
if the control qubit is at $\ket{0}$,
and $\ket{0^{n}}$ to $\ket{\Phi_0}$
if the control qubit is at $\ket{1}$.
\label{fig:S2}
\end{figure}

\begin{figure}
\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
% qubit # 0
\quad\quad  \ket{0}
& \gate{\textsf{H}}
& \ctrl{1}
& \ctrl{1}
& \ctrl{1}
& \ctrl{1}
& \ctrl{1}
& \gate{\textsf{P}(\tilde{\phi}_\alpha)}
& \gate{\textsf{H}}
& \meter{}
\\
% other qubits
\quad\quad  \ket{0^{n}}
& \gate{\textsf{init}}
& \gate{e^{-iH_{1}t_1}\cdots e^{-iH_{\alpha}t_\alpha}}
& \gate{P_{\gamma}}
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \gate{P_{\gamma'}}
& \gate{e^{-iH_{\alpha}t_{\alpha+2}} \cdots e^{-iH_{1}t_{2\alpha+1}}}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{a}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad \ket{0}
& \gate{\textsf{H}}
& \gate[2]{\textsf{Init}}
& \qw
& \ctrl{1}
& \qw
& \ctrl{1}
& \qw
& \gate[2]{\textsf{Init}^\dagger}
& \gate{\textsf{P}(\tilde{\phi}_\alpha)}
& \gate{\textsf{H}} &\meter{}
\\
\quad\quad \ket{0^{n}}
& \qw
& \gate{e^{-iH_{1}t_1}\cdots e^{-iH_{\alpha}t_\alpha}}
& \gate{P_{\gamma}}
& \gate{e^{-iH_{\alpha}t_{\alpha+1}}}
& \gate{P_{\gamma'}}
& \gate{e^{-iH_{\alpha}t_{\alpha+2}} \cdots e^{-iH_{1}t_{2\alpha+1}}}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{b}}};
\end{tikzpicture}

\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Quantum circuits for Green's function}.
\textsf{\textbf{a}} is the circuit without a reference state,
\textsf{\textbf{b}} is the circuit with a reference state.
$\tilde{\phi}_\alpha=\phi_\alpha$ for a real part,
$\tilde{\phi}_\alpha=\phi_\alpha-\pi/2$ for an imaginary part.
\label{fig:S3}
\end{figure}

\section{Details of NISQ calculations}
In this section, we describe details of NISQ simulations considered
in the main text. Throughout the NISQ simulations,
we used $N_s=4000$.
\subsection{Compressed quantum circuit}
First, we present the compressed circuits for the $1$ and $2$
qubit systems, used for computations in
Because any $1$ and $2$ qubit unitary
can be represented with a few operations~\cite{Shende2004},
we can use compressed circuits Figure~\ref{fig:S4}
instead of Fig.~\ref{fig:S1} or Fig.~\ref{fig:S2}
for those cases.
For the $1$-qubit system, parameters
$θ$ for the unitary matrix $U$
is found from~\cite{Shende2004}
\begin{align}
U = e^{i\theta_4} \begin{bmatrix}
&\cos(\theta_1/2) & -\sin(\theta_1/2)e^{i\theta_3} \\
&\sin(\theta_1/2)e^{i\theta_2}  &\cos(\theta_1/2)e^{i(\theta_2+\theta_3)}
\end{bmatrix}.
\end{align}
For the $2$-qubit system,
we used the two-qubit Weyl decomposition~\cite{Cross2019}
as implemented in the qiskit.

\begin{figure}
\begin{tikzpicture}
\node[scale=0.7] {
\begin{quantikz}
\quad\quad  \ket{0}
& \gate{\textsf{H}}
& \gate{\textsf{P}(\theta_4)}
& \gate{\textsf{P}\left(\frac{\theta_3+\theta_2}{2}\right)}
& \ctrl{1}
& \qw
& \ctrl{1}
& \gate{\textsf{P}(\phi_\alpha)}
& \gate{\textsf{H}}
\\
\quad\quad  \ket{0}
& \gate{\textsf{init}}
& \qw
& \gate{\textsf{P}\left(\frac{\theta_3-\theta_2}{2}\right)}
& \gate{\textsf{X}}
& \gate{U\left(-\frac{\theta_1}{2},0,-\frac{\theta_2+\theta_3}{2}\right)}
& \gate{\textsf{X}}
& \gate{U\left(\frac{\theta_1}{2},\theta_2,0\right)}
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{a}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=0.6] {
\begin{quantikz}
\quad\quad  \ket{0} & \gate{\textsf{H}} & \gate{\textsf{P}(\theta_1)}
& \ctrl{1}
& \ctrl{2}
& \ctrl{1}
& \ctrl{1}
& \ctrl{1}
& \ctrl{1}
& \ctrl{2}
& \gate{\textsf{P}(\phi_\alpha)} & \gate{\textsf{H}} &\meter{}
\\
\quad\quad  \ket{0} & \gate[2]{\textsf{init}}  & \qw
& \gate{U(\theta_2,\theta_3,\theta_4)}
& \qw
& \gate[2]{\textsf{R}_{\textsf{XX}}(\theta_8)}
& \gate[2]{\textsf{R}_{\textsf{YY}}(\theta_9)}
& \gate[2]{\textsf{R}_{\textsf{YY}}(\theta_{10})}
& \gate{U(\theta_{11},\theta_{12},\theta_{13})}
& \qw
& \qw
& \qw
& \qw
\\
\quad\quad  \ket{0} &  & \qw
& \qw
& \gate{U(\theta_5,\theta_6,\theta_7)}
& \qw
& \gate{U(\theta_{14},\theta_{15},\theta_{16})}
& \qw
& \qw
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{b}}};
\end{tikzpicture}

\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Quantum circuits for QZMC that uses 1 and 2 qubit circuit compression}
\textsf{\textbf{a}} is the compressed circuit for $1$-qubit systems,
and \textsf{\textbf{b}} is the compressed circuit for $2$-qubit systems.
\label{fig:S4}
\end{figure}

\subsection{Two-level system\label{sec:S1Q}}
In the main text, we investigated the two level system
described by the Hamiltonian
$H(λ) = X/2 + (2λ-1) Z$,
traversing from $λ=0$ to $λ=1$
via a discrete path with
Here, with noiseless simulator,
we assess the parameter dependencies of
QZMC by varying $N_α$ and $β$, and
also evaluating the impact of finite
trotterization steps.
Figure~\ref{fig:S5} illustrates these dependencies
for $⟨Ψ|Ψ|$⟩ and the ground state energy
$E$ of $H(\lambda=1)$.  and
of Fig. <ref>
demonstrate the $\beta$ dependence.
This reveals an increasing accuracy with $\beta$
until $\beta=10$.
Large error for $\beta>10$ can be
understood from Eq. (<ref>).
Next,  and
depict the variation of $\braket{\Psi|\Psi}$ and $E$
with $N_α$, showcasing convergence to exact
values as $N_α$ increases, with $E$
attaining accuracy as soon as $⟨Ψ|Ψ|$⟩
takes a non-trivial value, at $N_\alpha=5$.
display the results with trotterization,
where the trotterized time evolution
$U(\lambda,\tau,N_T)$ for $H(\lambda)$ is defined as

\begin{align}
U(\lambda,\tau,N_T) = \prod_{l}^{N_T} e^{-i\Delta \tau X/2} e^{-i\Delta \tau (2 \lambda -1) Z}
\end{align}

with $\Delta \tau = \tau/N_T$.
With increasing $N_T$, $\braket{\Psi|\Psi}$
converges to its exact value,
while $E$ converges to the exact values more rapidly,
indicating the noise resilience of $E$ in the
presence of inaccurate time-evolution.

\begin{figure} % Figure S1
\centering
\centerline{\includegraphics[width=18cm]{./Figures/figS5.pdf}}
\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{$\beta$, $N_\alpha$, and $N_{T}$ dependence of QZMC applied on the two-level system}.
$\beta$ dependence of (\textbf{\textsf{a}}) $\braket{\Psi|\Psi}$ and (\textbf{\textsf{b}}) $E$,
$N_\alpha$ dependence of (\textbf{\textsf{c}}) $\braket{\Psi|\Psi}$ and (\textbf{\textsf{d}}) $E$,
$N_T$ dependence of (\textbf{\textsf{e}}) $\braket{\Psi|\Psi}^2$ and (\textbf{\textsf{f}}) $E$
for a ground state are plotted. In (\textbf{\textsf{a}}) and (\textbf{\textsf{b}}), $N_\alpha=10$ and we used the exact time evolution.
For (\textbf{\textsf{c}}) and (\textbf{\textsf{d}}), we used $\beta=5$ and the exact time evolution.
And (\textbf{\textsf{e}}) and (\textbf{\textsf{f}}) are computed with $\beta=5$ and $N_\alpha=10$.
In this figure,
squares represents data points and shaded region is the
numerically estimated error-bar.
Here, We fixed $N_\nu =400$.
\label{fig:S5}
\end{figure}

\subsection{Hubbard dimer}
Here, we demonstrate the application of QZMC to
the Hubbard dimer using trotterized time evolution
on an \textbf{\textit{ibm}}\underline{ }\textbf{\textit{perth}}
quantum computer. We focus solely on the ground state
for simplicity, employing second-order trotterization
for the time evolution.
For $H=H_1+H_2$, the second order trotterized
time evolution $U(τ,N_T)$ is
\begin{align}
U(\tau,N_T) = \prod_{l}^{N_T}
e^{-i\Delta \tau H_1/2} e^{-i\Delta \tau H_2}e^{-i\Delta \tau H_1/2}.\label{eq:trotter2}
\end{align}
For Figure~\ref{fig:S6} \textbf{\textsf{a}} and
\textbf{\textsf{b}}, we employed the compressed
circuit detailed in Fig.\ref{fig:S4}
\textbf{\textsf{b}}.
The trotterization using Eq.~(\ref{eq:trotter2})
is used with $N_T=4$
and $H_1=-U/2 (I + Z_1Z_2)$ and $ H_2=-t (X_1+X_2)$.
These figures illustrate the robustness of our algorithm
even in the presence of both of trotterization errors
and device noise.
However, circuit compression may not always be feasible.
To address this, we tested our algorithm with an
uncompressed implementation of the trotterization.
This approach notably reduces $⟨Ψ|Ψ|$⟩
due to the increased circuit depth compared to the
compressed version.

To manage this depth, we employed QZMC with
a reference state (Fig.<ref>).
The reference state used was $\ket{\Psi_\text{ref}}=[1/\sqrt{2}\quad 0\quad0\quad -1/\sqrt{2}]^T$,
representing the first excited state of the Hamiltonian.
So, in Fig. <ref>
and , we used $N_\alpha=4$,
with $N_T=1$.
We averaged two different trotterization choices
(one with $H_2=-\frac{U}{2} (I + Z_1Z_2)$ and the
other with $H_2=-t (X_1+X_2)$) as this choice
significantly influences computational results when
trotterization steps are small.
We can see that, with the circuit Fig. <ref> ,
QZMC gives accurate energy values, though it reduces $\braket{\Psi|\Psi}$.
The circuits that we used for the QZMC with
uncompressed trotterization are depicted in
Fig.~\ref{fig:S7} \textbf{\textsf{a}}
represents the initialization circuit, while
Fig.~\ref{fig:S7} \textbf{\textsf{c}} mimics
the noise of the controlled-\textsf{ZZ}
operation depicted in
Figure~\ref{fig:S7} \textbf{\textsf{b}}.

\begin{figure} % Figure S6
\centering
\centerline{\includegraphics[width=18cm]{./Figures/figS6.pdf}}
\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{QZMC for the Hubbard dimer with the trotterized time evolution}.
In \textbf{\textsf{a}} and \textbf{\textsf{b}},
we used compressed circuit (Fig.~\ref{fig:S4} \textsf{\textbf{b}}),
while the time evolution is computed
with the trotter steps of $4$.
In \textbf{\textsf{c}} and \textbf{\textsf{d}},
we used uncompressed trotterized time evolutions.
In \textbf{\textsf{c}} and \textbf{\textsf{d}},
red square calculated from the circuit
Fig.~\ref{fig:S2} \textbf{\textsf{d}},
while blue ones are
calculated with Fig.~\ref{fig:S2} \textbf{\textsf{a}}.
In this figure,
squares are computed using \textbf{\textit{ibm}}\underline{ }\textbf{\textit{perth}},
while gray crosses are for the noiseless simulation
and the dotted line represent exact results.
In every calculations, we used $\beta=0.5,\, N_\nu=100$.
\label{fig:S6}
\end{figure}

\begin{figure}

\begin{tikzpicture}
\node[scale=1.0] {

\begin{quantikz}[row sep = 0.3cm]
\quad\quad
& \gate[3]{\textsf{Init}}
& \qw
\\
\quad\quad
& \qw
\\
\quad\quad
\qw
\end{quantikz}=
\begin{quantikz}[row sep = 0.3cm]
& \ctrl{1}
& \ghost{X}\qw
& \qw
& \qw
& \qw
& \qw
\\
& \gate{\textsf{X}}
& \ctrl{1}
& \gate{\textsf{X}}
& \gate{\textsf{H}}
& \ctrl{1}
& \qw
\\
& \qw
& \gate{\textsf{H}}
& \qw
& \qw
& \gate{\textsf{X}}
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{a}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=1.0] {
\begin{quantikz}[row sep = 0.3cm]
\quad\quad
& \ctrl{1}
& \ghost{X} \qw
\\
\quad\quad
& \gate[2]{\textsf{ZZ}}
& \qw
\\
\quad\quad
& \qw
\end{quantikz} =
\begin{quantikz}[row sep = 0.3cm]
& \qw
& \ghost{X}\qw
& [0.19cm]\qw
& \ctrl{1}
& [0.19cm]\qw
& \qw
& \qw
& \qw
\\
& \gate{\textsf{H}}
& \ctrl{1}
& \qw
& \gate{\textsf{X}}
& \qw
& \ctrl{1}
& \gate{\textsf{H}}
& \qw
\\
& \gate{\textsf{H}}
& \gate{\textsf{X}}
& \qw
& \qw
& \qw
& \gate{\textsf{X}}
& \gate{\textsf{H}}
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{b}}};
\end{tikzpicture}

\begin{tikzpicture}
\node[scale=1.0] {
\begin{quantikz}[row sep = 0.3cm]
\quad\quad
& \gate[3]{\textsf{Id}_{\textsf{n}}}
& \qw
\\
\quad\quad
& \ghost{X}
& \qw
\\
\quad\quad
& \qw
\end{quantikz} =
\begin{quantikz}[row sep = 0.25cm]
& \qw
& \ghost{X}\qw
& \gate[3]{\textsf{Delay}(t_{\textsf{CX}})}
& \qw
& \qw
& \qw
\\
& \gate{\textsf{H}}
& \ctrl{1}
& \ctrl{1}
& \gate{\textsf{H}}
& \qw
\\
& \gate{\textsf{H}}
& \gate{\textsf{X}}
& \gate{\textsf{X}}
& \gate{\textsf{H}}
& \qw
\end{quantikz}
\node[anchor=north west] at (current bounding box.north west) {\textsf{\textbf{c}}};
\end{tikzpicture}

\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Quantum circuits for the Hubbard dimer}
\textsf{\textbf{a}} is the initialization circuit for the ground state
of the Hubbard dimer used in the QZMC with a reference state.
The implementation of controlled-\textsf{ZZ} is drawn in \textsf{\textbf{b}}.
To mimic error induced by controlled-\textsf{ZZ},
we used the noisy identity circuit \textsf{\textbf{c}}.
In \textbf{\textsf{c}},
$t_{\textsf{CX}}$ represent the duration that takes
to CNOT operation the Delay($t$) means the machine wait the duration of $t$.
\label{fig:S7}
\end{figure}

\section{Details of noiseless simulator calculations}
Here, we discuss more detailed information
of the large size Hubbard models consdiered
in the main text.
The Hubbard model is described by the Hamiltonian
\begin{equation}
H = -\sum_{<ij>\sigma} t_{ij} c^{\dagger}_{i\sigma} c^{ }_{j\sigma}
- \sum_{i} \mu (n_{i\uparrow} + n_{i\downarrow})
+  \sum_{i} U n_{i\uparrow} n_{i\downarrow}.
\end{equation}
The first two terms represent the kinetic energy and are denoted as $H_t$,
while the last term represents electron-electron interaction and is referred to as $H_U$.
First, we present the quantum circuit for
the ground state of the Hubbard dimer
in Figure.~\ref{fig:S8}.
Here, the angle $θ_d$ is given by
\begin{align}
\theta_d = -2\arctan\left(\frac{1}{2t}\left(\frac{U}{2}
+ \sqrt{\frac{U^2}{4} + 4 t^2} \right)\right). \label{eq:thetad}
\end{align}
In the simulation, we used $N_ν$
and trotter steps $N_T$ that varies with
system size, and we fixed the number of shots
Based on Eq.~(\ref{eq:montecarlo_error_corrector}),
we determined $N_ν$ to proportional to $H'^2$,
which can be calculated  as
$H'=t ×(# of sites)$
for a chain and
$H'=4t/π×(# of sites)$
for a ladder.
And the proportionality constant is determined by testing $6×1$ system.
The first order trotterized time evolution $U_1(τ)$
of the Hubbard model
with $N_T$ trotter steps gives a Trotter error of~\cite{Layden2022}
\begin{align}
\|e^{-iH\tau}-U_1(\tau)\|\leq \frac{\tau^2}{2N_T} \|[H_t,H_U]\|.
\end{align}
\begin{align}
\|[H_t,H_U]\|\leq \sum_{<ij>\sigma} t_{ij} U \|
[c^{\dagger}_{i\sigma} c^{ }_{j\sigma},
\sum_{i} n_{i\uparrow} n_{i\downarrow}]\|.
\end{align}
Since all orbital indices are equivalent, $[c^†_iσ c^ _jσ,
∑_i n_i↑ n_i↓]$
remains constant fo any $i$ and $j$.
Consequently, $[H_t,H_U] ≤C U
(t_intraN_intra+ t_interN_inter)$
for some constant $C$.
$N_intra$ denotes the number of intra-dimer hoppings
and $N_inter$ represents the number of inter-dimer hoppings.
Based on this, we determined $N_T$ as
\begin{align}
N_T = \operatorname{int}\left[75\times
\frac{\beta^2}{9} \frac{U}{5}
\frac{\left( t_{\textrm{intra}}N_{\textrm{intra}}+
\right)}{8}
\right]
\end{align}
with a minimum $N_T=20$.

We depict $⟨Ψ|Ψ|$⟩ for Hubbard models in
Figure <ref> -.
These figures demonstrate that increasing $N_\alpha$
proportionally to the number of sites yields similar
values of $\braket{\Psi|\Psi}$ across various Hubbard models we studied.
Fig.~\ref{fig:S9} \textbf{\textsf{g}} and \textbf{\textsf{h}}
display spectral functions calculated with QZMC for $6$-site
Hubbard models.

\begin{figure} % Figure S7
\begin{tikzpicture}
\node[scale=1.0] {
\begin{quantikz}[row sep = 0.3cm]
\fbox{$1_\uparrow$}:\, \ket{0}
& \gate{\textsf{X}}
& \qw
& \qw
& \gate{\textsf{X}}
& \qw
& \qw
\\
\fbox{$1_\downarrow$}:\, \ket{0}
& \gate{\textsf{X}}
& \qw
& \qw
& \qw
& \gate{\textsf{X}}
& \qw
\\
\fbox{$2_\uparrow$}:\, \ket{0}
& \gate{\textsf{R}_\textsf{Y}(\theta_d)}
& \ctrl{0}
& \gate{\textsf{X}}
& \ctrl{-2}
& \qw
& \qw
\\
\fbox{$2_\downarrow$}:\, \ket{0}
& \gate{\textsf{H}}
& \ctrl{-1}
& \ctrl{-1}
& \qw
& \ctrl{-2}
& \qw
\end{quantikz}
\end{tikzpicture}
\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Quantum circuit for the ground state of the Hubbard dimer}.
The Quantum circuit that prepares the ground state of the
Hubbard dimer is dipicted. Here, $\theta_d$ is defined in Eq.~(\ref{eq:thetad}).
\label{fig:S8}
\end{figure}

\begin{figure} % Figure S8
\centering
\centerline{\includegraphics[width=18cm]{./Figures/figS9.pdf}}
\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{The half-filled Hubbard model in various sizes with QZMC}.
We plotted the $\braket{\Psi|\Psi}$ for various sized Hubbard model
in \textbf{\textsf{a}}-\textbf{\textsf{f}}, and the spectral function
for $6$-site Hubbard models are plotted in \textbf{\textsf{g}}
and \textbf{\textsf{h}}.
\label{fig:S9}
\end{figure}

\section{State preparation for QZMC}
Our method relies on the assumption that we have
$H_0$, which
can be easily solved with a classical computer, and
that the eigenstate is readily preparable as a
quantum circuit. However, arbitrary unitary
synthesis can incur exponential quantum time
costs~\cite{Shende2006}, making the preparation of
the eigenstate $\ket{\Phi_0}$ challenging
even when $H_0$ can be exactly solved
with a classical computer.
In such cases, we prepare an easily accessible state $\ket{\tilde{\Phi}_0}$
which has a non-trivial overlap with $\ket{\Phi_0}$ (e.g. $|\braket{\Phi_0|\tilde{\Phi}_0}|^2>0.5$).
Then, we project $\ket{\tilde{\Phi}_0}$ onto $\ket{\Phi_0}$
at the first time by using Eq.~(\ref{eq:gaussproj})
and perform QZMC in a equivalent way.
\begin{align}
\ket{\Psi_\alpha} = \mathcal{P}_\alpha \ket{\Phi_0}\braket{\Phi_0|\tilde{\Phi}_0}, \label{eq:state_prep}
\end{align}
instead of Eq.~(\ref{eq:conproj}).
This costs $N_\alpha\rightarrow N_{\alpha}+1$
for the QZMC.
As $\ket{\Phi_0}$ is known and
can be processed with a classical computer,
finding $\ket{\tilde{\Phi}_0}$ can
be efficiently accomplished using
classical computing resources.

\section{Comparison with other works}
Here, we compare the maximum time evolution length of QZMC with other methods
under similar initial conditions.
We prepared the initial state $\ket{\tilde{\Phi}_0}$
with $|\braket{\Phi|\tilde{\Phi}_0}|^2 = 0.4$
and applied QZMC with Eq.~(\ref{eq:state_prep}).
We fixed $N_\alpha=4$ and increased $\beta$.
And we determined the
maximum time evolution length of QZMC as
$C_{90} 5 \beta = 10.489 \beta$.
compares our approach with Ref.~\cite{LinLin2022}.
As shown in the figure,
our method requires significantly smaller
maximum time evolution length than the reference.
Comparision with Ref.~\cite{Ding2023} and QPE is shown in Fig.~\ref{fig:S10}\textbf{b}.
As depicted in the figure, our method requires a
much smaller maximum time evolution length than QPE,
and it is similar to Ref.\cite{Ding2023}.

\begin{figure} % Figure S9
\centering
\centerline{\includegraphics[width=18cm]{./Figures/figS10.pdf}}
\caption {\protect\rule[-0.5ex]{0.5pt}{2.5ex} \textbf{Comparison with other methods}.
\textbf{a-c} plots ground state energy error $\epsilon$
as a function of the maixmum time evolution length $T$.
\textbf{a} considers $4\times1$ Hubbard model with $U=4t$,
\textbf{b} computes $4\times1$ Hubbard model with $U=10t$,
and $8\times1$ Hubbard model with $U=10t$ is considered in
\textbf{c}. In \textbf{a}, blue points represents the data
from Ref.~\cite{LinLin2022}. And QCELS and QPE data
in \textbf{b-c} are imported from Ref.~\cite{Ding2023}
with unit conversion.
\label{fig:S10}
\end{figure}

<EMAIL_ADDRESS>\end{document}


(Heidelberg, Germany), Magdalena Kryczek (Bochum, Germany), Martin Welham
(Yatton, UK), Marvin W. Huddleston (Mesquite, USA), Megan Powell (Cobham, UK),
Melany Van Every (Lisbon, USA), Melody (Largo, USA), Melina Thévenot
(Belgium), Michael Jason Pearson (Hattiesburg, USA), Michele T. Mazzucato
(Florence, Italy), Milton K. D. Bosch MD (Napa, USA), Monisha Uriti (Puyallup,
USA), Nazir Ahmad (Birmingham, UK), Panagiotis J. Ntais (Philothei, Greece),
Patricia MacMillan (Fredericksburg, USA), Petyerák Jánosné (Fót, Hungary),
Phil Todd (Harpenden, UK), R. Banfield (Bad Tölz, Germany), Robert Bankowski
(Sanok, Poland), Robert Pickard (Grove Hill, USA), Robert Zach Moseley
(Worcester, USA), Rosemary Billington (Wilmslow, UK), Sarah Barratt (New
Mills, UK), Sarah Grissett (Tallahassee, USA), Scott Virtes (Escondido, USA),
Sergey Y. Tumanov (Glazov, Russia), Shalabh Shukla (Seattle, USA), Shelley-
Anne Lake (Johannesburg, South Africa), Simon Lund Sig Bentzen (Kolding,
Denmark), Somsikova Liudmila Leonidovna (Chirchik, Uzbekistan), Steven Green
(Witham, UK), Stikhina Olga Sergeevna (Tyumen, Russia), Tami Lyon (Gypsum,
USA), Thomas Fercho (Heidelberg, Germany), Thorsten Eschweiler (Übach-
Palenberg, Germany), Tiffany Shaw-Diaz (Dayton, USA), Timothy Scott (Baddeck,
Canada), Tomasz Konecki (Warsaw, Poland), Tommy Mattecheck (Tualatin, USA),
Vincent Decker (Saverne, France), Vinutha Karanth (Bengaluru, India), Virgilio
Gonano (Udine, Italy), Washington Kryzanowski (Montevideo, Uruguay), and Zac
Pujic (Brisbane, Australia).

A special thanks to our Superclassifiers: Angelina A. Reese (Sequim, USA),
Antonio Pasqua (Catanzaro, Italy), Carl L. King (Ithaca, USA), Dan Crowson
(Dardenne Prairie, USA), @EEZuidema (Driezum, Netherlands), Eric Fabrigat
(Velaux, France), @graham_d (Hemel Hempstead, UK), Henryk Krawczyk (Czeladż,
Poland), Marvin W. Huddleston (Mesquite, USA), Robert Zach Moseley (Worcester,
USA), Thorsten Eschweiler (Übach-Palenberg, Germany), and Washington
Kryzanowski (Montevideo, Uruguay).

We are very grateful for our Active Asteroids Champion Classifiers: @WRSunset
(Shaftesbury, UK), Amit Raka (Chhatrapati Sambhaji Nagar, India), Angela Hoffa
(Greenfield, USA), Arttu Sainio (Järvenpää, Finland), Ashok Ghosh (Howrah,
India), Brian K Bernal (Greeley, USA), C. D’silva (Mumbai, India), C. M.
Kaiser (Parker, USA), Clara Garza (West Covina, USA), Cledison Marcos da Silva
(Luminárias, Brazil), Dawn Boles (Bakersfield , USA), Dr. Elisabeth Chaghafi
(Tübingen, Germany), Elisabeth Baeten (Leuven, Belgium), Ernest Jude P. Tiu
(Pototan, Philippines), Gordon Ward (Castleford, UK), Graeme Aitken (Towen
Mountain, Australia), Graham Mitchell (Chilliwack, Canada), I. Carley (Gold
Coast, Australia), Jan Jungmann (Chyňava, Czech Republic), Jayanta Ghosh
(Purulia, India), Konstantinos Dimitrios Danalis (Athens, Greece), Leah
Mulholland (Peoria (IL), USA), Martin Welham (Yatton, UK), Melody (Largo,
USA), Melany Van Every (Lisbon, USA), Melina Thévenot (Belgium), Monisha Uriti
(Puyallup, USA), Panagiotis J. Ntais (Philothei, Greece), Petyerák Jánosné
(Fót, Hungary), R. Banfield (Bad Tölz, Germany), Robert Bankowski (Sanok,
Poland), Sergey Y. Tumanov (Glazov, Russia), Shalabh Shukla (Seattle, USA),
Shelley-Anne Lake (Johannesburg, South Africa), Somsikova Liudmila Leonidovna
(Chirchik, Uzbekistan), Stikhina Olga Sergeevna (Tyumen, Russia), Timothy
Scott (Baddeck, Canada), and Zac Pujic (Brisbane, Australia).

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship Program under grant No. 2018258765 and grant No.
2020303693. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation. C.O.C., H.H.H., and
C.A.T. also acknowledge support from the NASA Solar System Observations
program (grant 80NSSC19K0869). W.J.O. acknowledges support from NASA grant
80NSSC21K0114.

This research received support through the generosity of Eric and Wendy
Schmidt by recommendation of the Schmidt Futures program. Chandler and
Sedaghat acknowledge support from the DiRAC Institute in the Department of
Astronomy at the University of Washington. The DiRAC Institute is supported
through generous gifts from the Charles and Lisa Simonyi Fund for Arts and
Sciences, and the Washington Research Foundation.

Computational analyses were run on Northern Arizona University’s Monsoon
computing cluster, funded by Arizona’s Technology and Research Initiative
Fund. This work was made possible in part through the State of Arizona
Technology and Research Initiative Program.

“GNU’s Not Unix!” (GNU) Astro astfits (Akhlaghi & Ichikawa, 2015) provided
command-line FITS file header access. C Flexible Image Transport System Input
Output (CFITSIO) enabled FITS compression and more Pence (1999). World
Coordinate System (WCS) corrections facilitated by the Astrometry.net software
suite (Lang et al., 2010).

This work was supported in part by NSF award 1950901 (NAU REU program in
astronomy and planetary science). This research has made use of data and/or
services provided by the International Astronomical Union’s Minor Planet
Center. This research has made use of NASA’s Astrophysics Data System. This
research has made use of the Institut de Mécanique Céleste et de Calcul des
Éphémérides (IMCCE) SkyBoT Virtual Observatory tool (Berthier et al., 2006).
This work made use of the FTOOLS software package hosted by the NASA Goddard
Flight Center High Energy Astrophysics Science Archive Research Center. This
research has made use of SAOImageDS9, developed by Smithsonian Astrophysical
Observatory (Joye, 2006). This work made use of the Lowell Observatory
Asteroid Orbit Database astorbDB (Bowell et al., 1994; Moskovitz et al.,
2021). This work made use of the astropy software package (Robitaille et al.,
2013).

These results made use of the Lowell Discovery Telescope (LDT) at Lowell
Observatory. Lowell is a private, non-profit institution dedicated to
astrophysical research and public appreciation of astronomy and operates the
LDT in partnership with Boston University, the University of Maryland, the
University of Toledo, Northern Arizona University and Yale University. The
Large Monolithic Imager was built by Lowell Observatory using funds provided
by the National Science Foundation (AST-1005313). NIHTS was funded by NASA
award #NNX09AB54G through its Planetary Astronomy and Planetary Major
Equipment programs.

We thank Gemini Observatory Director Jennifer Lotz for granting our DDT
request for observations, German Gimeno for providing science support, and
Pablo Prado for observing. Proposal ID GS-2022A-DD-103, PI Chandler.

The VATT referenced herein refers to the Vatican Observatory’s Alice P. Lennon
Telescope and Thomas J. Bannan Astrophysics Facility. We are grateful to the
Vatican Observatory for the generous time allocations (Proposal ID S165, PI
Chandler). A special thanks to Vatican Observatory Director Br. Guy
Consolmagno, S.J. for his guidance, Vice Director for Tucson Vatican
Observatory Research Group Rev. Pavel Gabor, S.J. for his constant support,
Telescope Scientist Rev. Richard P. Boyle, S.J. for his patient VATT training,
and for including us in small body discovery observations, Chris Johnson for
innumerable consultations, Michael Franz and Summer Franks for on-site
troubleshooting, and Gary Gray for everything from telescope balance to
drinking water, without whom we would have been completely lost.

Based on observations obtained with the Apache Point Observatory 3.5-meter
telescope, which is owned and operated by the Astrophysical Research
Consortium. Observations made use of Astrophysical Research Consortium
Telescope Imaging Camera (ARCTIC) imager (Huehnerhoff et al., 2016b). ARCTIC
data reduction made use of the acronym software package (L. Weisenburger et
al., 2017).

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and CEA/DAPNIA, at the Canada France Hawaii Telescope (CFHT) which is operated
by the National Research Council (NRC) of Canada, the Institut National des
Sciences de l’Univers of the Centre National de la Recherche Scientifique
(CNRS) of France, and the University of Hawaii. The observations at the CFHT
were performed with care and respect from the summit of Maunakea which is a
significant cultural and historic site. Magellan observations made use of the
IMACS instrument (Dressler et al., 2011).

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the US Department of Energy, the US National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute for Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho
de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Científico e Tecnológico and the Ministério da Ciência,
Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Enérgeticas, Medioambientales y Tecnológicas–Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenössische Technische Hochschule (ETH), Fermi National
Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the
Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de Física d’Altes
Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians
Universität München and the associated Excellence Cluster Universe, the
University of Michigan, NSF’s NOIRLab, the University of Nottingham, the Ohio
State University, the OzDES Membership Consortium, the University of
Pennsylvania, the University of Portsmouth, SLAC National Accelerator
Laboratory, Stanford University, the University of Sussex, and Texas A&M
University.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David
Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID
#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy
Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS
together include data obtained, respectively, at the Blanco telescope, Cerro
Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward
Observatory, University of Arizona; and the Mayall telescope, Kitt Peak
National Observatory, NOIRLab. The Legacy Surveys project is honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham Nation. BASS is a
key project of the Telescope Access Program (TAP), (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
11433005). The Legacy Survey team makes use of data products from the Near-
Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project
of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE
is funded by the National Aeronautics and Space Administration. The Legacy
Surveys imaging of the DESI footprint is supported by the Director, Office of
Science, Office of High Energy Physics of the U.S. Department of Energy under
Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility under the same
contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

Based in part on data collected at Subaru Telescope and obtained from the
SMOKA, which is operated by the Astronomy Data Center, National Astronomical
Observatory of Japan (Baba et al., 2002). Based in part on observations made
with the Large Binocular Camera (Speziali et al., 2008) on the Large Binocular
Telescope (LBT) at the Mount Graham International Observatory (MGIO), in
association with Steward Observatory and the University of Arizona (Hill,
2010). This research used the facilities of the Italian Center for
Astronomical Archive (IA2) operated by INAF at the Astronomical Observatory of
Trieste.

This research uses services or data provided by the Astro Data Archive at
NSF’s NOIRLab. NOIRLab is operated by the Association of Universities for
Research in Astronomy (AURA), under a cooperative agreement with the National
Science Foundation.

Based on observations obtained at the international Gemini Observatory, a
program of NSF’s NOIRLab, which is managed by the AURA under a cooperative
agreement with the National Science Foundation on behalf of the Gemini
Observatory partnership: the National Science Foundation (United States),
National Research Council (Canada), Agencia Nacional de Investigación y
Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación
(Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações
(Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).
We thank Gemini Observatory Director Jennifer Lotz for granting our Director’s
Discretionary Time (DDT) request for observations, German Gimeno for providing
science support, and Pablo Prado for observing. Proposal ID GS-2022A-DD-103,
PI Chandler.

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT
and Commissariat a l’Energes Atomique (CEA)/ Département d’Astrophysique, de
physique des Particules, de physique Nucléaire et de l’Instrumentation
Associée (DAPNIA), at the Canada France Hawaii Telescope (CFHT) which is
operated by the National Research Council (NRC) of Canada, the Institut
National des Science de l’Univers of the Centre National de la Recherche
Scientifique (CNRS) of France, and the University of Hawaii. The observations
at the Canada-France-Hawaii Telescope were performed with care and respect
from the summit of Maunakea which is a significant cultural and historic site.

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## Acronyms

AA

    active asteroid
ACO

    asteroid on a cometary orbit
AI

    artificial intelligence
API

    Application Programming Interface
APT

    Aperture Photometry Tool
ARC

    Astrophysical Research Consortium
ARCTIC

    Astrophysical Research Consortium Telescope Imaging Camera
APO

    Apache Point Observatory
ARO

    Atmospheric Research Observatory
AstOrb

    Asteroid Orbital Elements Database
ASU

    Arizona Statue University
AURA

    Association of Universities for Research in Astronomy
BASS

    Beijing-Arizona Sky Survey
BLT

    Barry Lutz Telescope
CADC

    Canadian Astronomy Data Centre
CASU

    Cambridge Astronomy Survey Unit
CATCH

    Comet Asteroid Telescopic Catalog Hub
CBAT

    Central Bureau for Astronomical Telegrams
CBET

    Central Bureau for Electronic Telegrams
CCD

    charge-coupled device
CEA

    Commissariat a l’Energes Atomique
CFHT

    Canada France Hawaii Telescope
CFITSIO

    C Flexible Image Transport System Input Output
CNEOS

    Center for Near Earth Object Studies
CNRS

    Centre National de la Recherche Scientifique
CPU

    Central Processing Unit
CTIO

    Cerro Tololo Inter-American Observatory
DAPNIA

    Département d’Astrophysique, de physique des Particules, de physique Nucléaire et de l’Instrumentation Associée
DART

    Double Asteroid Redirection Test
DDT

    Director’s Discretionary Time
DECaLS

    Dark Energy Camera Legacy Survey
DECam

    Dark Energy Camera
DES

    Dark Energy Survey
DESI

    Dark Energy Spectroscopic Instrument
DCT

    Discovery Channel Telescope
DiRAC

    Data Intensive Research and Computing
DOE

    Department of Energy
DR

    Data Release
DS9

    Deep Space Nine
ESO

    European Space Organization
ETC

    exposure time calculator
ETH

    Eidgenössische Technische Hochschule
FAQ

    frequently asked questions
FITS

    Flexible Image Transport System
FOV

    field of view
GEODSS

    Ground-Based Electro-Optical Deep Space Surveillance
GIF

    Graphic Interchange Format
GMOS

    Gemini Multi-Object Spectrograph
GPU

    Graphics Processing Unit
GRFP

    Graduate Research Fellowship Program
HARVEST

    Hunting for Activity in Repositories with Vetting-Enhanced Search Techniques
HSC

    Hyper Suprime-Cam
IAU

    International Astronomical Union
IMACS

    Inamori-Magellan Areal Camera and Spectrograph
IMB

    inner Main-belt
IMCCE

    Institut de Mécanique Céleste et de Calcul des Éphémérides
INAF

    Istituto Nazionale di Astrofisica
INT

    Isaac Newton Telescopes
IP

    Internet Protocol
IRSA

    Infrared Science Archive
ITC

    integration time calculator
JAXA

    Japan Aerospace Exploration Agency
JD

    Julian Date
JFC

    Jupiter Family Comet
JPL

    Jet Propulsion Laboratory
KBO

    Kuiper Belt object
KOA

    Keck Observatory Archive
KPNO

    Kitt Peak National Observatory
LBC

    Large Binocular Camera
LCO

    Las Campanas Observatory
LBCB

    Large Binocular Camera Blue
LBCR

    Large Binocular Camera Red
LBT

    Large Binocular Telescope
LDT

    Lowell Discovery Telescope
LINEAR

    Lincoln Near-Earth Asteroid Research
LINCC

    LSST Interdisciplinary Network for Collaboration and Computing
LMI

    Large Monolithic Imager
LONEOS

    Lowell Observatory Near-Earth-Object Search
LSST

    Legacy Survey of Space and Time
MBC

    Main-belt Comet
MGIO

    Mount Graham International Observatory
ML

    machine learning
MMB

    middle Main-belt
MOST

    Moving Object Search Tool
MzLS

    Mayall z-band Legacy Survey
MPC

    Minor Planet Center
NAU

    Northern Arizona University
NEA

    near-Earth asteroid
NEAT

    Near-Earth Asteroid Tracking
NEO

    near-Earth object
NEOWISE

    Near-Earth Object Wide-field Infrared Survey Explorer
NIHTS

    Near-Infrared High-Throughput Spectrograph
NOAO

    National Optical Astronomy Observatory
NOIRLab

    National Optical and Infrared Laboratory
NRC

    National Research Council
OMB

    outer Main-belt
OSIRIS-REx

    Origins, Spectral Interpretation, Resource Identification, Security, Regolith Explorer
NSF

    National Science Foundation
PA

    position angle
PANSTARRS

    Panoramic Survey Telescope and Rapid Response System.
PI

    Principal Investigator
PNG

    Portable Network Graphics
PSI

    Planetary Science Institute
PSF

    point spread function
PTF

    Palomar Transient Factory
QH

    Quasi-Hilda
QHA

    Quasi-Hilda Asteroid
QHC

    Quasi-Hilda Comet
QHO

    Quasi-Hilda Object
RA

    Right Ascension
REU

    Research Experiences for Undergraduates
RNAAS

    Research Notes of the American Astronomical Society
SAFARI

    Searching Asteroids For Activity Revealing Indicators
SDSS

    Sloan Digital Sky Survey
SMOKA

    Subaru Mitaka Okayama Kiso Archive
SAO

    Smithsonian Astrophysical Observatory
SBDB

    Small Body Database
SDSS DR-9

    Sloan Digital Sky Survey Data Release Nine
SLAC

    Stanford Linear Accelerator Center
SOAR

    Southern Astrophysical Research Telescope
SNR

    signal to noise ratio
SSOIS

    Solar System Object Information Search
SQL

    Structured Query Language
SUP

    Suprime Cam
SWRI

    Southwestern Research Institute
TAP

    Telescope Access Program
TNO

    Trans-Neptunian object
UA

    University of Arizona
UCSC

    University of California Santa Cruz
UCSF

    University of California San Francisco
VATT

    Vatican Advanced Technology Telescope
VIA

    Virtual Institute of Astrophysics
VIRCam

    VISTA InfraRed Camera
VISTA

    Visible and Infrared Survey Telescope for Astronomy
VLT

    Very Large Telescope
VST

    Very Large Telescope (VLT) Survey Telescope
WFC

    Wide Field Camera
WIRCam

    Wide-field Infrared Camera
WISE

    Wide-field Infrared Survey Explorer
WCS

    World Coordinate System
YORP

    Yarkovsky–O’Keefe–Radzievskii–Paddack
ZTF

    Zwicky Transient Facility
  *[JFCs]: Jupiter Family Comet
  *[DECam]: Dark Energy Camera
  *[MBC]: Main-belt Comet
  *[YORP]: Yarkovsky–O’Keefe–Radzievskii–Paddack
  *[DART]: Double Asteroid Redirection Test
  *[MBCs]: Main-belt Comet
  *[HARVEST]: Hunting for Activity in Repositories with Vetting-Enhanced Search Techniques
  *[SAFARI]: Searching Asteroids For Activity Revealing Indicators
  *[PI]: Principal Investigator
  *[IMCCE]: Institut de Mécanique Céleste et de Calcul des Éphémérides
  *[WCS]: World Coordinate System
  *[NOIRLab]: National Optical and Infrared Laboratory
  *[CADC]: Canadian Astronomy Data Centre
  *[MPC]: Minor Planet Center
  *[ETC]: exposure time calculator
  *[SNR]: signal to noise ratio
  *[IMB]: inner Main-belt
  *[MMB]: middle Main-belt
  *[OMB]: outer Main-belt
  *[JFC]: Jupiter Family Comet
  *[TNO]: Trans-Neptunian object
  *[KBO]: Kuiper Belt object
  *[FITS]: Flexible Image Transport System
  *[FOV]: field of view
  *[PNG]: Portable Network Graphics
  *[CTIO]: Cerro Tololo Inter-American Observatory
  *[PA]: position angle
  *[FAQ]: frequently asked questions
  *[RNAAS]: Research Notes of the American Astronomical Society
  *[SSOIS]: Solar System Object Information Search
  *[KPNO]: Kitt Peak National Observatory
  *[SOAR]: Southern Astrophysical Research Telescope
  *[LCO]: Las Campanas Observatory
  *[ESO]: European Space Organization
  *[VST]: Very Large Telescope (VLT) Survey Telescope
  *[NEAT]: Near-Earth Asteroid Tracking
  *[GEODSS]: Ground-Based Electro-Optical Deep Space Surveillance
  *[SDSS]: Sloan Digital Sky Survey
  *[HSC]: Hyper Suprime-Cam
  *[WISE]: Wide-field Infrared Survey Explorer
  *[CASU]: Cambridge Astronomy Survey Unit
  *[INT]: Isaac Newton Telescopes
  *[WFC]: Wide Field Camera
  *[MOST]: Moving Object Search Tool
  *[IRSA]: Infrared Science Archive
  *[ZTF]: Zwicky Transient Facility
  *[PTF]: Palomar Transient Factory
  *[KOA]: Keck Observatory Archive
  *[CATCH]: Comet Asteroid Telescopic Catalog Hub
  *[ARCTIC]: Astrophysical Research Consortium Telescope Imaging Camera
  *[APO]: Apache Point Observatory
  *[GMOS]: Gemini Multi-Object Spectrograph
  *[IMACS]: Inamori-Magellan Areal Camera and Spectrograph
  *[LBCB]: Large Binocular Camera Blue
  *[LBCR]: Large Binocular Camera Red
  *[LBT]: Large Binocular Telescope
  *[MGIO]: Mount Graham International Observatory
  *[LMI]: Large Monolithic Imager
  *[NIHTS]: Near-Infrared High-Throughput Spectrograph
  *[LDT]: Lowell Discovery Telescope
  *[VATT]: Vatican Advanced Technology Telescope
  *[ACO]: asteroid on a cometary orbit
  *[QHO]: Quasi-Hilda Object
  *[LSST]: Legacy Survey of Space and Time
  *[AA]: active asteroid
  *[QHC]: Quasi-Hilda Comet
  *[PIs]: Principal Investigator
  *[QHA]: Quasi-Hilda Asteroid
  *[ARC]: Astrophysical Research Consortium
  *[CFHT]: Canada France Hawaii Telescope
  *[NEO]: near-Earth object
  *[CNEOS]: Center for Near Earth Object Studies
  *[NEOs]: near-Earth object
  *[ACOs]: asteroid on a cometary orbit
  *[AI]: artificial intelligence
  *[GRFP]: Graduate Research Fellowship Program
  *[CPU]: Central Processing Unit
  *[GPU]: Graphics Processing Unit
  *[LINCC]: LSST Interdisciplinary Network for Collaboration and Computing
  *[DiRAC]: Data Intensive Research and Computing
  *[NAU]: Northern Arizona University
  *[CFITSIO]: C Flexible Image Transport System Input Output
  *[NSF]: National Science Foundation
  *[REU]: Research Experiences for Undergraduates
  *[DDT]: Director’s Discretionary Time
  *[CEA]: Commissariat a l’Energes Atomique
  *[DAPNIA]: Département d’Astrophysique, de physique des Particules, de physique Nucléaire et de l’Instrumentation Associée
  *[CNRS]: Centre National de la Recherche Scientifique
  *[DES]: Dark Energy Survey
  *[ETH]: Eidgenössische Technische Hochschule
  *[SLAC]: Stanford Linear Accelerator Center
  *[DECaLS]: Dark Energy Camera Legacy Survey
  *[BASS]: Beijing-Arizona Sky Survey
  *[NOAO]: National Optical Astronomy Observatory
  *[MzLS]: Mayall z-band Legacy Survey
  *[TAP]: Telescope Access Program
  *[NEOWISE]: Near-Earth Object Wide-field Infrared Survey Explorer
  *[DESI]: Dark Energy Spectroscopic Instrument
  *[DOE]: Department of Energy
  *[SMOKA]: Subaru Mitaka Okayama Kiso Archive
  *[INAF]: Istituto Nazionale di Astrofisica
  *[AURA]: Association of Universities for Research in Astronomy
  *[NRC]: National Research Council