#  Designing exceptional-point-based graphs yielding topologically guaranteed
quantum search

Quancheng Liu  Department of Physics, Institute of Nanotechnology and Advanced
Materials, Bar-Ilan University, Ramat-Gan 52900, Israel  David A. Kessler
Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel  Eli
Barkai  Department of Physics, Institute of Nanotechnology and Advanced
Materials, Bar-Ilan University, Ramat-Gan 52900, Israel

(27th August 2024)

###### Abstract

Quantum walks underlie an important class of quantum computing algorithms, and
represent promising approaches in various simulations and practical
applications. Here we design stroboscopically monitored quantum walks and
their subsequent graphs that can naturally boost target searches. We show how
to construct walks with the property that all the eigenvalues of the non-
Hermitian survival operator, describing the mixed effects of unitary dynamics
and the back-action of measurement, coalesce to zero, corresponding to an
exceptional point whose degree is the size of the system. Generally, the
resulting search is guaranteed to succeed in a bounded time for any initial
condition, which is faster than classical random walks or quantum walks on
typical graphs. We then show how this efficient quantum search is related to a
quantized topological winding number and further discuss the connection of the
problem to an effective massless Dirac particle.

Quantum walks Aharonov  _et al._ (1993); Mülken and Blumen (2011), the quantum
analog of the well-known classical random walks, have attracted increasing
attention due to its importance both in fundamental physics and applications
for quantum information processing Kempe (2003). Taking advantage of coherent
superposition and interference, the quantum walk in many respects is superior
to its classical counterpart and finds applications in quantum algorithms
Ambainis (2003); Qiang  _et al._ (2016), universal quantum computation Childs
(2009); Childs  _et al._ (2013), quantum simulation Mülken and Blumen (2011);
Xu  _et al._ (2021), and bio-chemical processes et al. (2020); Dudhe  _et al._
(2021). One main challenge of the quantum walk is to maximize the detection
probability on a predetermined target state $\lvert\psi_{\rm d}\rangle$ given
some initial state $\lvert\psi_{0}\rangle$ Childs and Goldstone (2004); Tang
_et al._ (2018). With unitary evolution, nearly perfect quantum search with
detection probability approaching unity was found in several graphs for some
special initial states at some particular time $t$, including a glued binary
tree Tang  _et al._ (2018), a hypercube, and high-dimensional lattices Childs
and Goldstone (2004), while typical systems fall far from this limit. In a
broad sense, the transmission of a known initial state to another state is
called quantum state transfer Cirac  _et al._ (1997); Reiserer and Rempe
(2015); Chakraborty  _et al._ (2016). For instance, Kostak et al. designed
permutation operations that propagate the system from one specific node of the
graph to another at a predetermined time $t$ Kostak  _et al._ (2007). However,
if one does not know what the initial condition is, as is typical in many
search problems, we cannot rely on the quantum state transfer or one-shot
measurement quantum walks. Therefore, we herein design special graphs and
measurement protocols, with the aim to achieve what we call a guaranteed
search. Namely, the quantum walker should be successfully detected in a
bounded time for any initial condition. We describe how to construct such
quantum graphs and corresponding measurement strategies. We further
investigate whether these measurements, that destroy the unitary evolution,
are either harmful or useful for the search, and in what sense.

Our work is motivated by the state-of-the-art technology advances in
experiments Daley  _et al._ (2022) that allow clever engineering of
Hamiltonians with superconducting circuits et al. (2021a), waveguide arrays
Mittal  _et al._ (2014); Keil  _et al._ (2016); Caruso  _et al._ (2016); Boada
_et al._ (2017); Mittal  _et al._ (2019); Chen  _et al._ (2021), trapped ions
Manovitz  _et al._ (2020); et al. (2021b), and arrays of neutral atom
generated either in an optical cavity Periwal  _et al._ (2021) or via optical
tweezers et al. (2022). For instance, using photons carrying information
between atomic spins, programmable non-local interactions in an array of
atomic ensembles are realized in an optical cavity Periwal  _et al._ (2021).
These advances allow us to consider the option of constructing a device with
non-trivial matrix elements of the Hamiltonian and thus design special types
of graphs to speed up the quantum search.

We find that the designed quantum graphs, together with the stroboscopic
search protocol, have remarkable search capabilities, either with or without
control of the initial state. The ability to search an unknown initial state,
i.e., a black box initial state, is a significant step forward, in contrast to
previous works that considered quantum walks that start from a uniform or
specific localized initial state. Physically, one of the features of efficient
quantum search we find here is that it is intimately related to the study of
exceptional points. The latter are degenerate eigenvalues of the non-Hermitian
operators that are studied, for example, in optics and laser physics Miri and
Alù (2019); Liertzer  _et al._ (2012); et al. (2016); Xu  _et al._ (2016), and
topological phases Lee (2016); Leykam  _et al._ (2017); Shen  _et al._ (2018);
Bergholtz  _et al._ (2021), and are fundamentally related to parity-time
symmetry breaking et al. (2009); El-Ganainy  _et al._ (2018); Xiao  _et al._
(2021). Here, we design graphs and a search protocol with an exceptional point
of unusually high degeneracy, namely the size of the entire Hilbert space,
which can be made as large as we wish. We highlight the idea that exceptional
search is found when all the eigenvalues of the survival operator, defined
below, coalesce to zero, creating a large degeneracy. We then explore the
topology of the model at the exceptional point and show that efficient quantum
search is related to the quantitation of certain topological winding numbers.
Towards the end of the paper, we show how the search problem and the
corresponding degeneracy of the exceptional points and the topology property
are related to an effective massless Dirac particle, though all along we use
Schrödinger dynamics. We also show how our search strategies are related to
quantum state transfer.

## Stroboscopic search protocol and $N$-th order exceptional point

To perform efficient quantum walks, we use the strategy of stroboscopic
measurements, which as we show later can be made into an efficient tool. In
the stroboscopic protocol, the quantum walker starts from an unknown initial
state $|\psi_{0}\rangle$ and evolves unitarily according to the graph
Hamiltonian $H$. We projectively measure the system at times
$\tau,2\tau,\cdots$, at each measurement asking if the system is found at its
target, namely at $|\psi_{\rm d}\rangle$. The search target can be a localized
node on the graph, however in general this is not a requirement. This yields a
string of $n-1$ successive No’s followed by a Yes from the $n$-th measurement.
Once we record a Yes, the system is at the target state $|\psi_{\rm d}\rangle$
and in that sense, we have a successful quantum search. The time $n\tau$ is
the search time of the target state $\lvert\psi_{{\rm d}}\rangle$, which is
clearly a random variable whose statistical properties ultimately depend on
the initial state of the system $\lvert\psi_{{0}}\rangle$, the unitary
evolution between measurements, and the choice of $\tau$.

Figure 1: Designed quantum graphs. Schematic presentation of the crawl graph
(a) and funnel model (b). Here $N=20$. The thickness of the connecting line
represents the strength of the matrix element connecting two nodes [(a) and
(b)]. The colors represent the phases of the hopping rates (a). In (b) we
utilize colors to represent the magnitude of the on-site energies, whose
matrix elements are real (see details in SI Appendix). The search on both
graphs is guaranteed to succeed within a bounded time, for any initial
condition.  Figure 2: Topological winding. Plot of the generating function
$\hat{\Phi}(\theta)$ versus $\theta$ for $N=3$. Here we choose the crawl
Hamiltonian in Eq. (16) and the generating function is given by Eq. (13). The
initial states are the three $|Q_{k}\rangle$s which span the full Hilbert
space. Due to the topology of the model, generating function forms a closed
circle in the Laplace domain, hence the winding number is quantized as
predicted in Eq. (14). As shown in the Figure, for the winding number of
initial state $|Q_{0}\rangle$, we have $\Omega=3$ (a). The winding number of
$|Q_{1}\rangle$ is 2 (b) and the winding number for $|Q_{2}\rangle$ is 1 (c).
These windings give the number of the measurements for the successful search.

Let $F_{n}$ be the probability of detecting the system in state
$\lvert\psi_{{\rm d}}\rangle$ for the first time at $n\tau$. Then the total
search probability of finding the quantum walker on the target state is
$P_{{\rm det}}=\sum_{n=1}^{\infty}F_{n}$. If $P_{{\rm det}}=1$ the mean search
time is $\langle t\rangle=\tau\sum_{n=1}^{\infty}nF_{n}$. The search
probability $F_{n}$ is given in terms of the amplitudes $\phi_{n}$ of first
detection, namely $F_{n}=|\phi_{n}|^{2}$ with Grünbaum  _et al._ (2013); Dhar
_et al._ (2015); Friedman  _et al._ (2017); Thiel  _et al._ (2018); Liu  _et
al._ (2022); Dubey  _et al._ (2021a)

$\phi_{n}=\langle\psi_{{\rm d}}\rvert U(\tau){\cal
S}^{n-1}(\tau)\lvert\psi_{{\rm in}}\rangle,$ (1)

where the survival operator is ${\cal S}(\tau)=(1-\lvert\psi_{{\rm
d}}\rangle\langle\psi_{{\rm d}}\rvert)U(\tau)$, with $U(\tau)=\exp(-iH\tau)$
and $\hbar=1$. Here the back-action of the first $n-1$ repeated measurements
is to repeatedly project out the amplitude of the target state $|\psi_{\rm
d}\rangle$. In Eq. (1) we have used the basic postulates of quantum theory
with the projection $1-\lvert\psi_{{\rm d}}\rangle\langle\psi_{{\rm
d}}\rvert$.

As usual with these types of problems, the eigenvalues of the non-Hermitian
operator ${\cal S}(\tau)$ are essential for the characterization of the
process. The eigenvalues of ${\cal S}(\tau)$, denoted $\xi$, are all on or
inside the unit circle $|\xi|\leq 1$, and the eigenvalues with $|\xi|=1$
correspond to dark states Krovi and Brun (2006); Thiel  _et al._ (2020); Liu
_et al._ (2022). Our goal is to find $U(\tau)$ and the corresponding $H$ so
that all the eigenvalues of ${\cal S}(\tau)$ are equal to zero. Intuitively,
if all the eigenvalues are very small, the decay of $F_{n}$ is expected to be
fast and the quantum search time will be minimized. It is also clear that if
we find such an $H$, all the eigenvalues $\xi$ coalesce to the value $\xi=0$,
meaning that we are engineering a method that yields a survival operator with
a $N$-th order exceptional point, $N$ being the size of the system.

The eigenvalues $\xi$ are given implicitly by

$\mbox{det}|\xi-{\cal S}(\tau)|=\xi\mbox{det}|\xi-U(\tau)|\langle\psi_{{\rm
d}}\rvert\frac{1}{\xi-U(\tau)}\lvert\psi_{{\rm d}}\rangle=0$ (2)

where we have used the matrix determinant lemma, see Materials and Methods.
Clearly, the system always has at least one solution $\xi=0$. Let $H\lvert
E_{k}\rangle=E_{k}\lvert E_{k}\rangle$ where $k=0,\cdots,N-1$ and as usual we
may expand $\lvert\psi_{{\rm d}}\rangle=\sum_{k=0}^{N-1}\langle
E_{k}\rvert\psi_{{\rm d}}\rangle\lvert E_{k}\rangle$, and then

$\langle\psi_{{\rm d}}\rvert\frac{1}{\xi-U(\tau)}\lvert\psi_{{\rm
d}}\rangle=\sum_{k=0}^{N-1}{p_{k}\over\xi-\exp(-iE_{k}\tau)}$ (3)

with $\mbox{det}|\xi-U(\tau)|=\prod_{k=0}^{N-1}[\xi-\exp(-iE_{k}\tau)]$. Here
$p_{k}=|\langle E_{k}\rvert\psi_{{\rm d}}\rangle|^{2}$ is the square of the
overlap between the energy state $\lvert E_{k}\rangle$ and the detected state.
Our first requirement is that the system is such that $p_{k}\neq 0$ for all
the energy states $\lvert E_{k}\rangle$, and that there is no degeneracy i.e.,
$\exp(-iE_{k}\tau)\neq\exp(-iE_{m}\tau)$ for any choice of $m\neq k$.
Physically this demand means that we exclude dark states so that $|\xi|<1$ and
hence the eigenvalues satisfy $\mbox{det}|\xi-U(\tau)|\neq 0$. Using Eqs.
(2,3) it is not difficult to show that the eigenvalue problem reduces to
finding the solution of

$\xi\sum_{k=0}^{N-1}{p_{k}\over\xi-\exp(-iE_{k}\tau)}=0.$ (4)

We now engineer the system in such a way that the only solution is the
degenerate solution with $\xi=0$. As we will shortly show, the following
requirement is sufficient

$p_{k}={1\over N}\ \ \mbox{and}\ \ E_{k}\tau={2\pi k\over N}.$ (5)

We see that the energy levels are equally spaced, which intuitively is
expected as this causes the periodicities in the dynamics to resonate at
specific times, enhancing constructive interference. More specifically, we
will soon choose $E_{k}=\gamma k$ where $\gamma$ has units of energy, and then
$\tau=2\pi/\Delta E$ where $\Delta E=E_{max}-E_{min}$ is the energy gap
between the ground and largest energy in the spectrum. We note here that a
relativistic mass-less free particle, with energy
$E=\sqrt{m^{2}C^{4}+c^{2}p^{2}}$, and $m=0$, has a dispersion $E_{k}\propto
p\propto k$, instead of the well-known Schrödinger dispersion of a free
particle $E_{k}\sim k^{2}$. Hence the energy spectrum we find in Eq. (5) is
essentially relativistic; the consequence of this for search will be discussed
later. We also see that the overlaps $p_{k}$ are $k$-independent. To verify
these requirements, insert Eq. (5) in Eq. (4) and then with summation formulas
(see SI Appendix) we have

${\xi\over N}\sum_{k=0}^{N-1}{1\over\xi-\exp(-i2\pi k/N)}=-{\xi^{N}\over
1-\xi^{N}}=0$ (6)

and the only possible solution is $\xi=0$. We see that for a quantum system
satisfying Eq. (5), the survival operator has a $N$-fold degenerate
eigenvalues at $\xi=0$, as we aimed for. The order of the exceptional point is
equal to the size of the Hilbert space $N$, namely

$\xi=0,\quad\quad N\text{-th order exceptional point}.$ (7)

It can also be shown that Eq. (5) is in fact a necessary condition for a
degree $N$ exceptional point (see SI Appendix). Further, all the right and
left eigenvectors also coalesce with
$|\xi^{R}\rangle=U(\tau)^{-1}|\psi_{d}\rangle=U(-\tau)|\psi_{d}\rangle$ and
$\langle\xi^{L}|=\langle\psi_{\rm d}|$. Before constructing the graph that
yields this result we study its general consequences for search.

## Efficient quantum search and quantized topological winding number

We denote $H_{s}$, $U_{s}$ and ${\cal S}_{s}$ the Hamiltonian, unitary and
survival operator for a system that satisfies the efficient search conditions
Eq. (5) and in this notation we omit the dependence on $\tau$. We define the
states $\lvert Q_{k}\rangle=(U_{s})^{k}\lvert\psi_{\rm d}\rangle$ with
$k=0,\cdots,N-1$. The operators $U_{s}$ and ${\cal S}_{s}$ acting on these
states give

$\begin{array}[]{l l}U_{s}\lvert Q_{N-1}\rangle=\lvert\psi_{{\rm
d}}\rangle,&U_{s}\lvert Q_{k}\rangle=\lvert Q_{k+1}\rangle\ \mbox{if}\ k\neq
N-1,\\\ {\cal S}_{s}\lvert Q_{N-1}\rangle=0,&{\cal S}_{s}\lvert
Q_{k}\rangle=\lvert Q_{k+1}\rangle\ \mbox{if}\ k\neq N-1.\end{array}$ (8)

These formulas are mathematically straightforward, for example $U_{s}\lvert
Q_{N-1}\rangle=(U_{s})^{N}\lvert\psi_{\rm
d}\rangle=\sum_{k=0}^{N-1}(U_{s})^{N}\langle E_{k}\rvert\psi_{{\rm
d}}\rangle\lvert E_{k}\rangle=\sum_{k=0}^{N-1}\exp(-iE_{k}\tau N)\langle
E_{k}\rvert\psi_{{\rm d}}\rangle\lvert E_{k}\rangle=\lvert\psi_{{\rm
d}}\rangle=\lvert Q_{0}\rangle$ where we used Eq. (5) and hence
$\exp(-iE_{k}\tau N)=1$. We see that both ${\cal S}_{s}$ and $U_{s}$ are shift
operators, their difference being the action on the boundary term $\lvert
Q_{N-1}\rangle$. We can also show that the states $\lvert Q_{k}\rangle$ are
orthonormal $\langle Q_{l}\rvert Q_{m}\rangle=\delta_{lm}$ (see SI Appendix)
and they form a complete set spanning any initial condition in the Hilbert
space. From here we reach the following conclusions. First, consider an
initial condition which is a $\lvert Q_{k}\rangle$ state, then following Eq.
(1) we consider the operation $({\cal S}_{s})^{n}\lvert Q_{k}\rangle$ and
using Eq. (8) we obtain

$\phi_{n}=\left\\{\begin{array}[]{l l}1&\ \mbox{if}\ \ n=N-k,\\\ 0&\
\mbox{otherwise}.\end{array}\right.$ (9)

This means that we detect the target with probability one at time $(N-k)\tau$,
hence the detection process is deterministic as the fluctuations of the
detection time vanish. Then when $|\psi_{0}\rangle=|Q_{k}\rangle$, we have

$P_{{\rm det}}=1,\quad\langle t\rangle=t=\tau(N-k),\quad\text{Var}(t)=0.$ (10)

For a more general initial condition, exploiting the fact that the states
$\lvert Q_{k}\rangle$ form a complete set and the linearity of Eq. (1) with
respect to the initial condition, the probability of first detection
$F_{n}=|\phi_{n}|^{2}$ is

$F_{n}=\left\\{\begin{array}[]{l l}|\langle Q_{N-n}\rvert\psi_{{\rm
in}}\rangle|^{2}&\ \mbox{for}\ \ n=1,\cdots,N,\\\ 0&\
\mbox{otherwise}.\end{array}\right.$ (11)

This implies a guaranteed search, since even in the absence of knowledge about
the initial condition, the search will find the target with at most $N$
operations. From here it also follows that we have an upper bound on the
search time for any initial state

$t\leq\tau N=\frac{2\pi k}{E_{k}}=\frac{2\pi}{\gamma}.$ (12)

This upper bound is $N$ independent, so the maximum search time does not
increase with the system size. The upper limit is found when the initial
condition is the target state $\lvert\psi_{{\rm in}}\rangle=\lvert\psi_{{\rm
d}}\rangle$. To conclude, for a quantum walker starting from an unknown
initial state, i.e., a black-box problem, our strategy will find this walker
at the target state within a fixed time with probability one.

We next explore the topological properties of the efficient quantum search. In
spatially periodic systems, such as the topological materials, their
topologies are revealed by the Chern number or winding number of the Bloch
Hamiltonian in the band-theory framework Chiu  _et al._ (2016). For our model,
the periodicity originates from the stroboscopic measurements. Hence instead
of a Brillouin zone in $k$ space, here we investigate the topology of the
system in the Laplace domain Friedman  _et al._ (2017). Using the $Z$
transform, the generating function of the search amplitude $\phi_{n}$ reads:

$\hat{\Phi}(\theta)=\sum_{n=1}^{\infty}e^{in\theta}\phi_{n}=\frac{\langle\psi_{\rm
d}|\hat{{\cal U}}(\theta)|\psi_{\rm 0}\rangle}{1+\langle\psi_{\rm
d}|\hat{{\cal U}}(\theta)|\psi_{\rm d}\rangle},$ (13)

where $\hat{{\cal
U}}(\theta)=\sum_{n=1}^{\infty}e^{in\theta}U(n\tau)=e^{i\theta}U(\tau)/(1-e^{i\theta}U(\tau))$
is the generating function of $U(n\tau)$ Thiel  _et al._ (2018). The
statistics of the search process can be calculated in terms of the generating
function. For example, the total search probability
$P_{det}=1/(2\pi)\int_{0}^{2\pi}d\theta|\hat{\Phi}(\theta)|^{2}$ and the mean
search time $\langle t\rangle=\tau/(2\pi
i)\int_{0}^{2\pi}d\theta[\hat{\Phi}\left(\theta\right)]^{*}[\partial_{\theta}\hat{\Phi}\left(\theta\right)],$
where $*$ is the complex conjugate of the generating function Friedman  _et
al._ (2017).

With Eq. (13), we calculate the winding number when the quantum system meets
the conditions in Eq. (5), corresponding to an $N$-th order exceptional point.
The winding number is quantized and characterized by the choice of the initial
state. When $|\psi_{0}\rangle=|Q_{k}\rangle$, the winding number $\Omega$
reads

$\Omega=\frac{1}{2\pi i}\int_{0}^{2\pi}d\theta\
\partial_{\theta}\ln[\hat{\Phi}(\theta)]=N-k.$ (14)

Using Eq. (11), this quantized winding number equals the number of measurement
attempts needed to detect the walker with probability unity. It is in this
sense that the search process is related to the topology of the model in
Laplace space and the search times for states $|Q_{k}\rangle$ are given in
terms of $\tau$ multiplied by the number of windings, i.e., $t=\Omega\tau$. We
plot $\hat{\Phi}(\theta)$ in Fig. 2 for $N=3$. Note here we use the crawl
Hamiltonian later derived in Eq. (16) for illustration. As shown in the
Figure, the $\hat{\Phi}(\theta)$ forms closed circles and the number of times
it rotates around the center is equal to $\Omega$.

Figure 3: Guaranteed and fast quantum search. Detection probability $F_{n}$
versus $n$ for the crawl (a) and funnel (b) model. Here the graph has $N=50$
nodes and we present results with initial states localized on one of the nodes
$\lvert\psi_{{\rm in}}\rangle=\lvert x\rangle$, $x=0,1,\cdots$ and
$|\psi_{{\rm d}}\rangle=|0\rangle$. For the crawl search we find a
deterministic outcome of the process, where $F_{n}=1$ when $n=x$ (for $x=0$,
$F_{50}=1$). For the funnel model (b) notice the sharp cutoff of $F_{n}$ for
$n>N=50$. For any initial condition, the detection of the state is guaranteed
with probability one, within at most $N$ measurements, which we call
guaranteed search. In (b) notice the peak of height one for $n=50$, when the
initial condition is the same as the detected state. The upper bound for the
search time is $t_{max}=\tau N=2\pi$.

## Examples of designed quantum graphs: crawl and funnel models

What are the tight-binding Hamiltonians of size $N\times N$, that yield a
guaranteed search? The condition Eq. (5) admits many types of solutions, and
here we present two that have certain advantages.

### Crawl Model

First, we present an approach where the nodes of the graph are the states
$\lvert Q_{k}\rangle$. This is clearly useful since this means that we can
start the process with the wave packet on one node of the graph and find the
walker with probability one after a fixed time at any other node, which we
call deterministic search, as the fluctuations vanish. We use
$H=\sum_{k}E_{k}\lvert E_{k}\rangle\langle E_{k}\rvert$ and for the equal
distance energies we set $E_{0}=0,E_{1}=\gamma,\cdots,E_{N-1}=(N-1)\gamma$ and
Eq. (5) gives $\tau=2\pi/N\gamma$. More generally
$\tau=2\pi/N\gamma+2j\pi/\gamma$ and $j$ is a non-negative integer. In this
system the states $\lvert x\rangle$ with $x=0,1,\cdots,N-1$ are the nodes of
the graph, see Fig. 1. To perform this trick let

$\lvert
E_{k}\rangle=\left\\{1,e^{i\theta_{k}},e^{i2\theta_{k}},\cdots,e^{i(N-1)\theta_{k}}\right\\}^{T}/\sqrt{N}$
(15)

where $\theta_{k}=2\pi k/N$. This eigenstate is a discrete Fourier wave, which
is related to the “relativistic” linear dispersion in Eq. (5), and Dirac
physics as discussed below. Clearly Eq. (15) gives $p_{k}=1/N$ and hence

$\begin{array}[]{c}H_{{\rm Crawl}}=\\\ \gamma\left[\begin{array}[]{c c c c
c}0&{1\over 1-e^{i\theta_{1}}}&{1\over 1-e^{i\theta_{2}}}&\cdots&{1\over
1-e^{i\theta_{N-1}}}\\\ {1\over 1-e^{-i\theta_{1}}}&0&{1\over
1-e^{i\theta_{1}}}&\cdots&{1\over 1-e^{i\theta_{N-2}}}\\\ {1\over
1-e^{-i\theta_{2}}}&{1\over 1-e^{-i\theta_{1}}}&0&\cdots&{1\over
1-e^{i\theta_{N-3}}}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ {1\over
1-e^{-i\theta_{N-1}}}&{1\over 1-e^{-i\theta_{N-2}}}&{1\over
1-e^{-i\theta_{N-3}}}&\cdots&0\end{array}\right]\end{array}$ (16)

which we call the Crawl Hamiltonian, see a schematic diagram in Fig. 1(a).
This system, as discussed below, breaks time-reversal symmetry. Namely, the
uni-directional movement of the packet can be reversed by changing $H_{{\rm
Crawl}}$ to its complex $H_{{\rm Crawl}}^{*}$, which is a feature of time-
reversal. In Fig. 3(a) we plot $F_{n}$ for a system with $N=50$ and where the
target state is $\lvert\psi_{{\rm d}}\rangle=\lvert 0\rangle$. We choose local
initial conditions such that $\lvert\psi_{{\rm in}}\rangle=\lvert x\rangle$,
hence we are considering a transition from $x$ to $0$ and in the plot we
choose $x=0,1,\cdots,49$. We see that $F_{n}$ is sharply peaked and is equal
to unity when $n=x$ [Fig. 3(a)]. This type of deterministic search is not
found for classical random walks, and relies on the fact that the quantum wave
packet can, at specific times of the evolution, be localized on a single node,
while at prior measurement times the wave packet vanishes on the node. This
$H_{{\rm Crawl}}$ is very much related to unitary state transfer in the
absence of measurements, as discussed below.

The efficient quantum walks we found here are reminiscent of the physics of a
massless Dirac particle in dimension one. First, the energy is linear in $k$,
Eq. (5). Second, in the crawl model the energy states are discrete free waves,
and finally, due to time reversal breaking, the wave packet can travel either
clockwise or anti-clockwise, somewhat similar to a particle and anti-particle.
But why do we find this relation between our problem and these relativistic
effects? We started this work with the demand that the eigenvalues of ${\cal
S}$, are real and all coalesce to zero, to speed up the search process. We
then added rotational invariance of the searching process, such that all nodes
of the graph are identical, namely no matter what the detected state,
$p_{k}=1/N$ on every node of the specially designed graph. We then naturally
find the ideal search for a quasi-particle with no dispersion, at least at the
measurement times. Namely, a wave packet that is widening will create a less
efficient search, in the sense that it renders impossible the absolute
detection of the particles in a single measurement made at a node, a feature
that is also revealed by the quantized winding number in the Laplace domain.
Similarly inspired by a massless Dirac particle, consider the trivial wave
equation in continuous space and time in dimension one,
$\partial_{x}\psi(x,t)=\partial_{t}\psi(x,t)$, the solution is $\psi(x,t)=\int
g(k)\exp(i(kx-w_{k}t))$ and $g(k)$ is the initial packet in momentum space.
For a localized initial condition and using $w=k$, we get a delta traveling
wave, in close analogy with what we find in discrete space. Of course, the
underlying dynamics in our case are controlled by the Schrödinger equation,
but the Hamiltonian under study yields effective motion in which space and
time are treated on the same footing. Finally, Dirac’s wave function in
dimension one has two components. Similarly, we have a particle traveling
clockwise and anti-clockwise, in fact, at least in principle, we can switch
between these modes, if in the middle of the experiment we replace the
Hermitian crawl $H$ with its complex conjugate.

### Funnel Model

An alternative approach that uses on-site energies to direct the search to a
specific node denoted $\lvert\psi_{\rm d}\rangle=\lvert 0\rangle$ is now
considered. Here the process does not break time-reversal symmetry. As before,
the spatial nodes of the graph are denoted $|x_{i}\rangle$, and
$i=0,1,\cdots,N-1$. We still have condition to fulfill Eq. (5) and we start
with the normalised state $\lvert
E_{0}\rangle=\left\\{1/\sqrt{N},-\sqrt{N-1}/\sqrt{N},0,\cdots\right\\}^{T}$ in
agreement with the second condition in Eq. (5). The next energy state is
constructed such that it is normalized and orthogonal to the first one and has
an overlap $1/N$ with the detected state, $\lvert
E_{1}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},-\sqrt{N-2}/\sqrt{N-1}.0,0,\cdots\right\\}^{T}$.
The process of constructing these states is then continued (SI Appendix), and
exploiting the demand that energy levels be equidistant, Eq. (5), we find

$\begin{array}[]{c}H_{{\rm Funnel}}=\frac{\gamma}{2}\left[\begin{array}[]{c c
c c c c}N-1&\sqrt{N-1}&\cdots&\sqrt{\frac{2}{N}}\\\
\sqrt{N-1}&1&\cdots&\sqrt{\frac{2}{N^{2}-N}}\\\ \vdots&\vdots&\ddots&\vdots\\\
\sqrt{\frac{2}{N}}&\sqrt{\frac{2}{N^{2}-N}}&\cdots&2N-3\end{array}\right]\end{array}$
(17)

with matrix elements $H_{0,m}=\sqrt{(N-m)(N-m+1)/N}$ ($m\neq 0$) and
$H_{j,m}=\sqrt{(N-m)(N-m+1)/[(N-j+1)(N-j)]}$ ($j\neq 0,m$). We call this
approach the funnel model. This type of system is schematically shown in Fig.
1(b) while in Fig. 3(b) we present the detection probabilities for localized
initial conditions. Fig. 3(b) illustrates a sharp cutoff, namely $F_{n}=0$ for
any $n>N$ and thus the search is guaranteed to succeed in a finite time, a
feature completely absent in classical random walks or quantum walks on non-
specialized graphs. Interestingly, if the initial state is the same as the
detected one, corresponding to what is known as the return problem Grünbaum
_et al._ (2013), $F_{50}=1$, otherwise it is zero. This means the system is
detected exactly after $N$ attempts, and this feature is universal, namely for
any search $H_{s}$, satisfying Eq. (5), and for $|\psi_{\rm
d}\rangle=|\psi_{0}\rangle$, $F_{N}=1$ and $F_{n\neq N}=0$. We will discuss
this surprising effect in more detail in the discussion.

Figure 4: Comparison with typical graph. We plot the mean measurement numbers
for a full randomly connected SK graph for different choices of $\tau$ with
stroboscopic search protocol. Here we choose $N=50$ and for the designed graph
the upper bound of the measurement number is $50$ with time $t\leq
2\pi/\gamma$ for $\tau=2\pi/\Delta E$. The search on the typical graph is much
slower than that. As shown in the Figure, the lower bound of the SK graph is
much bigger that the upper bound of our models. It is clear the designed
graphs boost the quantum search process.  Figure 5:  Noise robustness. We plot
the search probability $F_{n}$ of the $50$-site funnel model with the initial
state being a node of the graph. We introduce random noise to $\tau$ with
different magnitudes from $0.1\%$ (green dots) to $10\%$ (blue hexagrams). As
shown in the Figure, the $F_{n}$ is robust to the noise and the search
probability is nearly zero for $n>50$. We calculate the total search
probability $P_{\rm det}$ within the first 50 measurements for 1000
realizations. The guaranteed search reminds even for comparable large noise,
where the $P_{\rm det}\sim 98\%$ with noise $10\%$.

## Comparison to the SK graph, influence of noise, and connections to quantum
state transfer

As a comparison, we calculate the search time for finding the quantum walker
on a full randomly connected Sherrington-Kirkpatrick (SK) model Harrigan and
et al. (2021) with $50$ nodes, in the sense that the designed coupling in our
model is replaced by the random connections. As we have shown, for our
designed graphs with $50$ nodes, the walker will be detected within $50$
measurement attempts and the upper bound of the search time is $2\pi/\gamma$.
As shown in Fig. 4, on the SK model, the mean number of the measurement to
find the walker is much larger than our designed graphs ($\langle n\rangle\sim
10^{3}\gg 50$) and fluctuates strongly for different choices of $\tau$. When
$\tau$ is small, the detection times diverge due to the quantum Zeno effect
Misra and Sudarshan (1977); Itano  _et al._ (1990); Facchi  _et al._ (2001);
Dubey  _et al._ (2021b).

We also plot the cases when there is noise on $\tau$ in Fig. 5. Here we choose
the funnel model with $N=50$ and the initial state is $|\psi_{\rm
0}\rangle=|49\rangle$. For each sampling interval $\tau$, we use the designed
$\tau$ [Eq. (5)] with the noise generated from a uniform distribution, see
Materials and Methods. We plot the detection probability versus measurements
for different noise strengths. The model shows robustness to the effects of
the noise, where the search is still guaranteed to succeed, namely
$\sum_{n=1}^{50}F_{n}\sim 1$. The detection probability is close to zero for
$n>50$ as shown in Fig. 5. As a more realistic test of the guaranteed search,
we calculate the search probability $P_{\rm det}$ within 50 measurements for
$1000$ realizations. We find $P_{\rm det}\sim 0.98$ even under $10\%$ noise,
hence the approach is robust.

We investigate the effects of repeated measurements on the search for an
unknown initial state by designing special graphs. In some special cases, the
measurement does not interact with the unitary dynamics. To see this special
case, consider a walker that starts from a node of the graph, in other words,
we now assume the complete knowledge of the initial state, which is a special
localized state. As mentioned, using the crawl model Eq. (16) the unitary
$U(\tau)$ is the shift operator, which shifts the particle from one node to
the other (see Fig. 6). Throughout the evolution, the wave function
$|\psi(t)\rangle$ is zero on $|\psi_{\rm d}\rangle$. So, the measurements do
not interfere with the walker. Only after a number of shifts (depending on the
distance between nodes), the shift operations transport the system to the
target state, and the measurement records the system with probability 1.
Namely, if we start on a node $\lvert\psi_{0}\rangle=\lvert x_{0}\rangle$ and
focus on the final node $\lvert\psi_{\rm d}\rangle=\lvert x_{f}\rangle$ the
(non-monitored) success probability is

$|\langle\psi(t)|\psi_{{\rm d}}\rangle|^{2}=1$ (18)

at times $t=(x_{f}-x_{0})\tau$. Therefore, since measurements do not destroy
the unitarity till the final measurement, our result is equivalent to quantum
state transfer with unitary dynamics for this special case, which has been
considered by Kostak et al. Kostak  _et al._ (2007). For comparison for the
funnel model, even if the system starts from a special localized node, we see
that the wave function spreads to all the nodes of the graph. So the
measurements will interact and disturb the evolution of the funnel walker
during the whole search process, until it is caught by the detector.

Figure 6: Non-monitored quantum walk and perfect quantum state transfer. The
time evolution of the wave packet on the crawl (a) and funnel (b) graphs for
non-monitored quantum walks. The color code gives the probability of finding
the walker on a node. For the crawl graph, the wave function is fully
localized one node after the other at times $\tau,2\tau,\cdots$, a feature
which is vital for state transfer to any node in the system. Both quantum
walks exhibit revival, namely the wave function returns to its initial state,
at time $N\tau$.

## Discussion

We have designed a survival operator ${\cal S}(\tau)$, with an exceptional
point whose degeneracy is the size of the Hilbert space. Such an exceptional
point reaches the highest order of degeneracy possible in the model and can be
designed as large as possible. This is certainly an advance in exceptional
physics compared to previous results considering second or third order
exceptional points. In general, for an $N$-dimensional system, to find the
highest order exceptional point, one needs to solve an $N$-th order
characteristic polynomial Xiao  _et al._ (2019); Hodaei  _et al._ (2017),
which, in principle, is difficult when the system is large. Here we show that
the high order exceptional point can be designed by exploiting the symmetry of
the model, which leads to the two conditions we discussed in Eq. (5). At the
exceptional point, the vector space is severely skewed, as all the eigenstates
of ${\cal S}(\tau)$ coincide. Obviously, this means that the single eigenstate
of ${\cal S}(\tau)$ cannot be used to construct a full basis. So can we find a
new basis that spans the Hilbert space, which should also be connected to the
exceptional properties of the model? This challenge is solved by the states
$|Q_{k}\rangle$ we proposed, which form a full basis that can be used to
expand any initial state of the walker, and is determined by the system
parameters at the exceptional point. In this new basis, the survival operator
${\cal S}(\tau)$ becomes a shift operator Eq. (8). Roughly speaking, this new
basis can be considered to play the role of the energy eigenbasis found for
unitary dynamics. The new basis $|Q_{k}\rangle$ is an efficient tool for
studying the quantum search process. We find a full basis using the
exceptional point, which is related to the fact that the degree of the
exceptional point here is the size of the Hilbert space, while if the degree
of the exceptional point is less than that, the effect would not have been
found.

Remarkably, the search probability is sharply peaked, i.e., $F_{n=N}=1$, for
the case $|\psi_{\rm d}\rangle=|\psi_{\rm in}\rangle$, as shown in Fig. 3(b).
Since the initial state here is the same as the target one, this is called in
the literature the return problem Grünbaum  _et al._ (2013). This is a generic
property for all the designed graphs and describes the special recurrence
property in repeatedly monitored quantum walks. To see this note that choosing
$k=0$, $|Q_{0}\rangle=|\psi_{\rm d}\rangle$, and then use Eq. (15).
Physically, this shows the wave function always has destructive interference
on the search target at times $n\tau$ with $n=1,\cdots,N-1$ and fully
constructive interference, which collects all the amplitude of wave function,
at time $N\tau$ at $|\psi_{\rm d}\rangle$. In connection with previous
results, Grünbaum et al. have shown that the mean search time for the return
problem is quantized and equals the effective dimension of the system Grünbaum
_et al._ (2013); Bourgain  _et al._ (2014), which is related to the topology
of the Schur functions. This result in our case means that the average
$\langle n\rangle=N$ for the return problem. However, as shown in our work,
for purposely designed graphs, we have $F_{n}=0$ for $n>N$ for the guaranteed
search. These two results together mean that in our case we must have
$F_{N}=1$ for the return problem. The result is therefore the sharp peak seen
in Fig. 3. This mean that in general for the return problem $\text{Var}(n)\neq
0$, while we have $\text{ Var}(n)=0$, namely no fluctuations at all of the
return time. This indicates a specialized recurrence on our designed graphs,
which is absent in previous results.

The resulting special purpose family of Hamiltonians allows for guaranteed
search. The main condition Eq. (5) still allows for further freedom in the
design of the search process. For specialized states $\lvert Q_{i}\rangle$,
which are used to expand the Hilbert space as we discussed, the monitored
search process is deterministic as the fluctuations in the detection attempt
vanish. Our work shows a connection between guaranteed search and the
topology, see Fig. 2. Very generally, starting with state $|Q_{i}\rangle$ the
generating function winds, the number of windings gives the number of
measurements till detection. Hence the search time can also be expressed in
terms of the winding number times $\tau$, i.e., $t=\Omega\tau$. For a random
unknown initial state, the quantum walks we designed here are guaranteed to
succeed in a bounded time. Usually, topology is related to some protected
physical reality that is insensitive to sources of noise. Also in our case,
the topology is related to a physically robust result, i.e. protecting the
search in the sense that it is secured to be detected, up until a fixed time,
no matter what is the initial state.

For the crawl model, the search is effectively uni-directional in a system
that conserves energy. The Hamiltonian is independent of the choice of the
target state, and in that sense, the system exhibits universal search. The
$H_{{\rm Crawl}}$ breaks time-reversal symmetry, and is related to the Dirac
equation. In contrast, the funnel model does not break time-reversal symmetry,
but the target state is unique.

To conclude, our search is globally optimal with respect to any other search
algorithm, in the sense that the success probability is unity within a finite
time even for large systems and for all the initial conditions.

##  I Methods

### Matrix determinant lemma

We provide details on the derivation of Eq. (2) using the matrix determinant
lemma. Suppose $A$ is an invertible square matrix and $u$, $v$ are column
vectors, then the matrix determinant lemma states:

$\text{det}|A+uv^{T}|=(1+v^{T}A^{-1}u)\text{det}|A|,$ (19)

where $uv^{T}$ is the outer product of the vectors $u$ and $v$. We are
interested in the eigenvalues of the survival operator $\text{det}|\xi-{\cal
S}(\tau)|=0$ with ${\cal S}(\tau)=(1-|\psi_{\rm d}\rangle\langle\psi_{\rm
d}|)U(\tau)$ as defined in the main text. Substituting ${\cal S}$ into the
matrix determinant, we have

$\text{det}|\xi-{\cal S}(\tau)|=\text{det}|\xi-U(\tau)+|\psi_{\rm
d}\rangle\langle\psi_{\rm d}|U(\tau)|.$ (20)

As discussed in the main text, to prevent the appearance of dark states, which
are not optimal for the quantum search, we have conditioned
$\text{det}|\xi-U(\tau)|\neq 0$. Hence $\xi-U(\tau)$ is an invertible square
matrix. If we denote $\xi-U(\tau)$ as the matrix $A$, it fits the condition
for the matrix determinant lemma. We then let $u=|\psi_{\rm d}\rangle$, and
$v^{T}=\langle\psi_{\rm d}|U(\tau)$. Using Eq. (19), we have get Eq. (2) used
in the main text.

### Numerical Simulation Approach

To prepare the plots, we simulate the search process directly based on Eq.
(1). We first construct the search Hamiltonians for the crawl graph and funnel
graph using Eqs. (16) and (17). In the simulation, we set $N=50$ (for
preparation of Fig. 3). The initial state of the system is usually a node of
the graph namely $|\psi_{\rm in}\rangle=|x\rangle$. We represent it by a
vector of dimension $N$. For example, if the system is initially localized on
node 0, we set the first entry of the vector to be one and all the rest
remains zero. With the initial state and funnel/crawl Hamiltonians, we
numerically calculate $\phi_{1}$, which is the overlap between the wave
function at time $\tau$ and the search target $|\psi_{\rm d}\rangle$, namely
$\phi_{1}=\langle\psi_{\rm d}|U(\tau)|\psi_{\rm in}\rangle$. The square of
$|\phi_{1}|$ is the probability that we detect the particle in the first
measurement at time $t=\tau$, which is recorded for plotting Fig. 3. We then
turn to the calculation of $F_{2}$. In the first step, the failed measurement
(at time $\tau$) projects out the state on $|\psi_{\rm d}\rangle$. This is
done by setting the state that overlap with $|\psi_{\rm d}\rangle$ to zero, in
other words, we mimic the back-action of projection $(1-|\psi_{\rm
d}\rangle\langle\psi_{\rm d}|)$. For example, let $|\psi_{\rm
d}\rangle=|0\rangle$, then after the measurement, the state of the system on
node 0 is zero. The measured state is the new initial state for the
calculation of $F_{2}$. Similar to the calculation of $F_{1}$, we let the
system evolve for time $\tau$ by $U(\tau)$, then calculate the overlap between
the state of the system and search target, which is $\phi_{2}$. The search
probability $F_{2}=|\phi_{2}|^{2}$. Such procedure is repeated, we numerically
calculate $F_{3},F_{4},\cdots,F_{n}$. The results are plotted in Fig. 3 for
the crawl (a) and funnel (b) models. In Fig. 4, we utilize the same process
for the calculation of $F_{n}$ and use the random SK Hamiltonian. The mean
measurement times are given by $\langle n\rangle=\sum_{n=1}^{M}nF_{n}$. In the
numerical simulation we choose $M=100000$. In Fig. 5, the time interval
between two measurements is random, depending on the magnitude of the noises.
For each $\tau$, we choose $\tau=(2\pi/N)\\{1+a*uniform[-0.5,0.5]\\}$ with $a$
being the magnitude of the noise. For the calculation of the search
probability, we use the simulated $F_{n}$ and sum the first 50 measurements,
i.e., $P_{\rm det}=\sum_{n=1}^{n=N=50}F_{n}$. For each magnitude of the noise,
$P_{\rm det}$ is averaged over 1000 realizations.

E.B. thanks Shimon Yankelevich, Moshe Goldstein, and Lev Khaikovich for
comments and suggestions. The support of Israel Science Foundation’s grant
1614/21 is acknowledged.

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Supplemental Information for
“Designing exceptional-point-based graphs yielding topologically guaranteed
quantum search”

Quancheng Liu, David A. Kessler, and Eli Barkai

##  S2 Figure preparation

We now provide further details of the figures.

  * •

Fig. 1(a). Schematic plot of the crawl graph Hamiltonian Eq. (16) of size
$20\times 20$. Here we set $\gamma=1$ and subsequently. In Eq. (16), we have a
typical matrix element $1/[1-\exp(i\theta)]$, which can be formally written as
$1/[1-\exp(i\theta)]=R\exp(i\Phi)$. The $R$ represents the coupling strength
between the nodes, in the figure we utilize the thickness of the connecting
line to represent its magnitude. The $R$ decreases as the distance between the
nodes becomes longer. For example, $R_{0,1}=R_{0,19}\textgreater
R_{0,2}=R_{0,18}\textgreater R_{0,3}=R_{0,17}\textgreater\cdots\textgreater
R_{0,10}$. The colors represent the phases $\Phi$, where the phases
$\pi\textgreater\Phi\textgreater\pi/2$. The on site energy of the nodes are
equal to zero, hence all of them are plotted gray. Geometrically, the system
is rotationally invariant.

  * •

Fig. 1(b). Schematic plot of the funnel Hamiltonian Eq. (17) of size $20\times
20$. The matrix elements are real, and again we unitize the thickness of the
line connecting the nodes to represent the hopping rate magnitude. Now the on-
site energies are not identical, and we present this variation by the colors.
The detection node is specialized for this model, and we mark this node in the
graph. The on-site energies increase linearly for the nodes from 1 to $N-1$,
and for the search node, the on-site energy is approximately $N/2$.

  * •

Fig. 3(a). Search probability $F_{n}$ versus measurement steps $n$ for the
crawl graph. Here the graph has 50 nodes. We choose the initial states to be
the nodes of the graph, namely $|\psi_{\rm
in}\rangle=|0\rangle,|1\rangle,\cdots,|49\rangle$. As we discussed in the main
text, the search state can be any node of the graph, but here for
demonstration, we choose $|\psi_{\rm d}\rangle=|0\rangle$. We numerically
simulate the search process with the method we discussed above. As shown in
the figure, the search is deterministic, namely, we detect the walker with
probability one at some specified times, as described in the text.

  * •

Fig. 3(b). Search probability $F_{n}$ versus measurement steps $n$ for the
funnel model. Here we set $N=50$ and the search target is $|\psi_{\rm
d}\rangle=|0\rangle$. Again, we choose the initial state to be a node of the
graph and $x$ goes from 0 to 49. We then apply the simulation approach
discussed above, which gives the statistics of $F_{n}$ as shown in the figure.
For any initial state, the detection of the state is guaranteed with
probability one within $N$ measurements. There is a clear cutoff for $F_{n}$
when $n\textgreater N$, which drops to zero, namely $F_{n\textgreater N}=0$.
The upper bound of the search time is found when $|\psi_{\rm
in}\rangle=|0\rangle=|\psi_{\rm d}\rangle$, where $F_{50}=1$ and $F_{n\neq
50}=0$, and in this case the detection time is $2\pi$.

  * •

Fig. 6 describes the unitary evolution without measurements (non-monitored
quantum walks) for the crawl Hamiltonian (a) and funnel model (b). We plot the
probability of finding the walker on node $x$, $|\langle\psi(t)|x\rangle|^{2}$
versus $x$, and for continuous-time $t$ (in the unit $\tau/10$). Here, for
both graphs, we choose $N=20$ and the initial state is $|0\rangle$. We record
$|\langle\psi(t)|x\rangle|^{2}$ for all the nodes with sampling time interval
$\tau/10$. As shown in the figure, the wave function of the crawl graph is
localized at specific nodes of the graph at times $\tau,2\tau,3\tau,\cdots$.
In the funnel mode (b), starting for a localized state $|0\rangle$, the wave
function first spreads to the whole graph. Then it returns to the localized
state $|0\rangle$. The system is recurrent, which is rooted in the periodicity
of the energy spectrum we design.

##  S3 DETAILS ON THE DERIVATION OF EQ. 6

We present the derivation of Eq. (6). As discussed in the main text, the eigen
function of the survival operator ${\cal S}(\tau)$ can be written as
$\xi\sum_{k=0}^{N-1}p_{k}/[\xi-\exp(-iE_{k}\tau)]=0$. Here, we denote the
summation as $I$. With Eq. (5), we have

$I=\frac{\xi}{N}\sum_{k=0}^{N-1}\frac{1}{\xi-\exp(-i2\pi
k/N)}=-\frac{\xi}{N}\sum_{k=0}^{N-1}\frac{\exp(i2\pi k/N)}{1-\xi\exp(i2\pi
k/N)},$ (S1)

where we multiply both the numerator and denominator by $\exp(-i2\pi k/N)$ for
each term in the summation. We first Taylor expand $1/[1-\xi\exp(i2\pi k/N)]$
and get

$I=-\frac{\xi}{N}\sum_{k=0}^{N-1}\exp(i2\pi
k/N)\sum_{j=0}^{\infty}[\xi\exp(i2\pi
k/N)]^{j}=-\frac{\xi}{N}\sum_{k=0}^{N-1}\sum_{j=0}^{\infty}\xi^{j}\exp[i2\pi
k(j+1)/N].$ (S2)

We then calculate $I$ by changing the order of the summations. Namely we first
perform the summation over $k$, which is a geometric progression with common
ratio $\exp[i2\pi(j+1)/N]$. By calculating the geometric progression, we have:

$I=-\frac{\xi}{N}\sum_{j=0}^{\infty}\xi^{j}\sum_{k=0}^{N-1}\exp[i2\pi
k(j+1)/N]=\frac{\xi}{N}\sum_{j=0}^{\infty}\xi^{j}\frac{1-\exp(i2\pi
j)}{1-\exp[i2\pi(j+1)/N]}.$ (S3)

Since $j$ is an integer, the numerator $1-\exp(i2\pi j)$ always equals to
zero. The whole fraction is non-zero only when the denominator
$1-\exp[i2\pi(j+1)/N]$ also equals to zero. That is possible and happens when
$\exp[i2\pi(j+1)/N]=1$, namely $j=nN-1$, where $n$ is an integer and goes from
1 to infinity (if $n$ starts from 0, $j=-1$, which goes beyond the regime of
$j$). We replace the summation index $j$ with $n$, where $n$ goes from 1 to
infinity. Then for the summation $I$, we have:

$I=\frac{\xi}{N}\sum_{j=nN-1}\xi^{j}N=\frac{\xi}{N}\sum_{n=1}^{\infty}\xi^{nN-1}N=\sum_{n=1}^{\infty}\xi^{nN}=-\frac{\xi^{N}}{1-\xi^{N}}.$
(S4)

These are the details of the derivation of Eq. (6) in the main text.

##  S4 Necessary condition for the $N$-th order exceptional point

In this section, we will show Eq. (5) derived in the main text is a necessary
condition for the $N$-th order exceptional point. For following Eq. (3) in the
main text, the eigenvalue function for $\xi$ reads:

${\cal F}(\xi)=\langle\psi_{\rm d}|\frac{1}{\xi-U(\tau)}|\psi_{\rm
d}\rangle=\sum_{k=0}^{N-1}\frac{p_{k}}{\xi-e^{-iE_{k}\tau}}=0.$ (S5)

We now prove Eq. (5) in the main text is the only solution to have a
degeneracy of $N-1$ for $\xi_{0}=0$. Namely, this equation is a necessary
condition for the high order exceptional point we derived. Mathematically,
when $\xi_{0}=0$ is $N-1$ degenerate, we have

${\cal F}(\xi_{0})=0,\ {\cal F}^{\prime}(\xi_{0})=0,\ {\cal
F}^{\prime\prime}(\xi_{0})=0,\ {\cal F}^{(3)}(\xi_{0})=0,\ \cdots,\ {\cal
F}^{(N-2)}(\xi_{0})=0,$ (S6)

where $\prime$, $\prime\prime$, and $(3)$ denote the first, second, and third
order derivative. These conditions leads to $N-1$ equations for the $p_{k}$,
$E_{k}$, and $\tau$. Since $p_{k}=|\langle E_{k}|\psi_{\rm d}\rangle|^{2}$, we
also have the conditions for the form of the $p_{k}$s, where

$\sum p_{k}=1,\quad\forall k,\ p_{k}\ \text{is real and positive}.$ (S7)

Eqs. (S6) and (S7) ensure Eq. (5) derived in the main text is a necessary
condition for the exceptional point. To see that, let us start with the simple
case when $N=2$. Using Eqs. (S6) and (S7), the function for the $p_{1}$ and
$p_{2}$ are

$p_{1}e^{iE_{1}\tau}+(1-p_{1})e^{iE_{2}\tau}=0\ \longrightarrow\
p_{1}=\frac{e^{iE_{2}\tau}}{e^{iE_{2}\tau}-e^{iE_{1}\tau}}=\frac{e^{i\Delta_{21}}}{e^{i\Delta_{21}}-1},\quad
p_{2}=1-p_{1}.$ (S8)

Here we define the energy difference times $\tau$ as $\Delta_{21}$, i.e.,
$(E_{2}-E_{1})\tau=\Delta_{21}$. Since $p_{1}$ is real and finite [Eq. (S7)],
we have $e^{i\Delta_{21}}=-1$ in the complex plane. This gives the conditions
for the energy spectrum, i.e., $\Delta_{21}=(E_{2}-E_{1})\tau=\pi+2k\pi,k\in
Z$. Namely the phase between $E_{2}\tau$ and $E_{1}\tau$ is $\pi$, as given in
Eq. (5). Put the energy spectrum back into Eq. (S8), we have $p_{1}=1/2$ and
$p_{2}=1/2$, the equal magnitude as we presented in the main text. So for the
$N=2$ case, the only solution for the degenerate exceptional point is when
$p_{1}=p_{2}=1/2$ and $(E_{2}-E_{1})\tau=\pi+2k\pi$.

Similarly, for $N=3$, we have

$\\{\begin{aligned}
&p_{1}e^{iE_{1}\tau}+p_{2}e^{iE_{2}\tau}+(1-p_{1}-p_{2})e^{iE_{3}\tau}=0\\\
&p_{1}e^{2iE_{1}\tau}+p_{2}e^{2iE_{2}\tau}+(1-p_{1}-p_{2})e^{2iE_{3}\tau}=0\end{aligned},\rightarrow
p_{1}=\frac{e^{i(\Delta_{21}+\Delta_{31})}}{(-1+e^{i\Delta_{21}})(-1+e^{i\Delta_{31}})},p_{2}=\frac{e^{i\Delta_{31}}}{(e^{i\Delta_{21}}-1)(e^{i\Delta_{31}}-e^{i\Delta_{21}})}.$
(S9)

Here $\Delta_{21}=(E_{2}-E_{1})\tau$ and $\Delta_{31}=(E_{3}-E_{1})\tau$.
Using the conditions in Eq. (S7), we have

$\Delta_{21}=\frac{2\pi}{3}+2k_{1}\pi,\quad\Delta_{31}=\frac{4\pi}{3}+2k_{2}\pi,\quad
k_{1},k_{2}\in Z.$ (S10)

This is the energy spectrum condition we have in the main text. For the
magnitude of the $p$s, substituting Eq. (S10) back into Eq. (S9), we have
$p_{1}=p_{2}=p_{3}=1/3$.

So in general, for the $N$ dimensional system, using Eqs. (S6) and (S7), we
have

$p_{1}=\frac{e^{i\sum_{i=2}^{N}\Delta_{i1}}}{\Pi_{i=2}^{N}(e^{i\Delta_{i1}-1})},p_{2}=\frac{e^{i\sum_{i=3}^{N}\Delta_{i1}}}{(e^{i\Delta_{21}}-1)\Pi_{i=3}^{N}(e^{i\Delta_{i1}-e^{i\Delta_{31}}})},\cdots,p_{k}=\frac{e^{i\sum_{i=2,i\neq
k}^{N}\Delta_{i1}}}{(e^{i\Delta_{k1}}-1)\Pi_{i=2,i\neq
k}^{N}(e^{i\Delta_{i1}-e^{i\Delta_{k1}}})}.$ (S11)

Since these $p_{k}$ should be real and possible, we have the conditions for
the $\Delta_{i1}$s. This process leads to the energy spectrum conditions as
presented in Eq. (5). Substituting the energy level conditions back into Eq.
(S11). The corresponding magnitude of the $p_{k}$ are all equal, i.e.,
$p_{k}=1/N$. To conclude, Eq. (5) is a necessary condition for achieving an
$N$-th order exceptional point.

##  S5 Proof of the orthogonal of states $|Q_{k}\rangle$

We present the proof that the states
$|Q_{0}\rangle,|Q_{1}\rangle,|Q_{2}\rangle,\cdots,|Q_{N-1}\rangle$ are
orthogonal with each other, namely $\langle Q_{l}|Q_{m}\rangle=\delta_{lm}$.
The state $|Q_{k}\rangle$ is defined by the unitary evolution operator $U_{s}$
to the power $k$ times the search target $|\psi_{\rm d}\rangle$, where
$U_{s}=\exp(-iH_{s}\tau)$ and $H_{s}$ are search Hamiltonian. To show the
orthogonality, we first expanded $|\psi_{\rm d}\rangle$ in the energy basis,
which leads to:

$|Q_{m}\rangle=(U_{s})^{m}|\psi_{\rm
d}\rangle=\sum_{k=0}^{N-1}(U_{s})^{m}\langle E_{k}|\psi_{\rm
d}\rangle|E_{k}\rangle=\sum_{k=0}^{N-1}\exp(-imE_{k}\tau)\langle
E_{k}|\psi_{\rm d}\rangle|E_{k}\rangle=\sum_{k=0}^{N-1}\exp(-i2\pi
km/N)\langle E_{k}|\psi_{\rm d}\rangle|E_{k}\rangle.$ (S12)

Here we have used the fact $E_{k}\tau=2\pi k/N$. Similarly, we can find the
representation of the state $|Q_{l}\rangle$ in the energy basis. We then
calculate the overlap between the states $|Q_{m}\rangle$ and $|Q_{l}\rangle$.
We have

$\langle Q_{l}|Q_{m}\rangle=\sum_{k=0}^{N-1}\sum_{k^{\prime}=0}^{N-1}\langle
E_{k}|\langle\psi_{\rm d}|E_{k}\rangle\exp[i2\pi(kl-k^{\prime}m)/N]\langle
E_{k^{\prime}}|\psi_{\rm
d}\rangle|E_{k^{\prime}}\rangle=\sum_{k=0}^{N-1}|\langle\psi_{\rm
d}|E_{k}\rangle|^{2}\exp[i2\pi k(l-m)/N].$ (S13)

The square of the overlap between the detected state $|\psi_{\rm d}\rangle$
and the energy state $|E_{k}\rangle$ is denoted as $p_{k}$ in the main text,
namely $p_{k}=|\langle\psi_{\rm d}|E_{k}\rangle|^{2}$. Eq. (5) states this
value is a dependent of $k$ for the search Hamiltonian $H_{s}$, where
$p_{k}=1/N$. So for Eq. (S13), we only need to calculate the summation of
$\exp[i2\pi k(l-m)/N]$ from $k=0$ to $N-1$. This has been done also in Eq.
(S3), where $(j+1)$ in Eq. (S3) is replaced by $(l-m)$ here. We then have:

$\langle Q_{l}|Q_{m}\rangle=\frac{1}{N}\sum_{k=0}^{N-1}\exp[i2\pi
k(l-m)/N]=\frac{1}{N}\frac{\exp[i2\pi(l-m)]-1}{\exp[i2\pi(l-m)/N]-1}.$ (S14)

$\langle Q_{l}|Q_{m}\rangle$ is non-zero only when $l-m=0,N,2N,\cdots$. Here
$N-1\geq l\geq 0$ and $N-1\geq m\geq 0$. Hence only when $l=m$ we have
$\langle Q_{l}|Q_{m}\rangle=1$, otherwise $\langle Q_{l}|Q_{m}\rangle=0$.
Namely $\langle Q_{l}|Q_{m}\rangle=\delta_{lm}$. This is the conclusion we
used in the main text. Another thing to notice is that, since the
$|Q_{k}\rangle$ are generated by the unitary operators, it is naturally
normalized. Hence the states
$\\{|Q_{0}\rangle,|Q_{1}\rangle,|Q_{2}\rangle,\cdots,|Q_{N-1}\rangle\\}$ forms
a complete and normalized space.

##  S6 Funnel Model Hamiltonian

We provide details on the funnel Hamiltonian and its explicate presentation.
In this model, the search target $|\psi_{\rm d}\rangle=|0\rangle$. As before,
the spatial nodes of the graph are denoted $|x_{i}\rangle$, and
$i=0,1,\cdots,N-1$. We start with the energy state
$|E_{0}\rangle=\left\\{1/\sqrt{N},-\sqrt{(N-1)/N},0,0,\cdots\right\\}$, where
$1/\sqrt{N}$ fulfills the first condition in Eq. (5) and $-\sqrt{(N-1)/N}$
stands for normalization. We then construct the energy state $|E_{1}\rangle$.
It should be orthogonal with $|E_{0}\rangle$ and in agreement with the
condition in Eq. (5). We find
$|E_{1}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},-\sqrt{(N-2)/(N-1)},0,0,\cdots\right\\}$.
Again, the first term in $|E_{1}\rangle$ leads to $|\langle
E_{1}|0\rangle|^{2}=1/N$. The second entry guarantees $\langle
E_{0}|E_{1}\rangle=0$, and the third entry is for normalization. Following the
construction procedure, we have
$|E_{2}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},1/\sqrt{(N-2)(N-1)},-\sqrt{(N-3)/(N-2)},0,0,\cdots\right\\}$,
and in general $|E_{i\neq
N-1}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},\cdots,1/\sqrt{(N+2-k)(N+1-k)},\cdots,-\sqrt{(N-1-i)/(N-i)},0,0,\cdots\right\\}$,
where $k$ is the index for the $k$-th entry and $2\leq k\leq i$. This general
representation $|E_{i}\rangle$ can help us construct the states
$|E_{0}\rangle,|E_{1}\rangle,\cdots,|E_{N-2}\rangle$, but not
$|E_{N-1}\rangle$. Let us explain this and then show how to construct
$|E_{N-1}\rangle$. When $i=N-2$, we have
$|E_{N-2}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},\cdots,1/\sqrt{6},-1/\sqrt{2}\right\\}$.
Now we cannot utilize the procedure we did before to construct
$|E_{N-1}\rangle$, since roughly speaking there is no additional space for the
normalization. So how to construct the last energy state? We notice the last
term in $|E_{N-2}\rangle$ is $-1/\sqrt{2}$, which is special. Let us consider
a state where the first $N-1$ terms are the same as those in
$|E_{N-2}\rangle$, and the only different one is the last term, where we
change the $-1/\sqrt{2}$ to $1/\sqrt{2}$. We can see such a state is
orthogonal with $|E_{N-2}\rangle$ and also normalized. This is the last energy
state, i.e.,
$|E_{N-1}\rangle=\left\\{1/\sqrt{N},1/\sqrt{N(N-1)},\cdots,1/\sqrt{6},1/\sqrt{2}\right\\}$.
It is also easy to show that this state is orthogonal with respect to the
other state, $|E_{N-3}\rangle,|E_{N-4}\rangle,\cdots,|E_{0}\rangle$. We then
have $N$ orthogonal and normalized states fulfill the conditions in Eq. (5).

For the equal distance energies, we set
$E_{0}=0,E_{1}=\gamma,E_{2}=2\gamma,\cdots,E_{N-1}=(N-1)\gamma$, the resulting
Hamiltonian is

$H={\gamma\over 2}\left[\begin{matrix}N-1&H_{0,1}&H_{0,2}&\cdots&H_{0,N-1}\\\
H_{0,1}&1&H_{1,2}&\cdots&H_{1,N-1}\\\ H_{0,2}&H_{1,2}&3&\cdots&H_{2,N-1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
H_{0,N-1}&H_{1,N-1}&H_{2,N-1}&\cdots&2N-3\end{matrix}\right]$ (S15)

where $H_{0,m}=\sqrt{(N-m)(N-m+1)/N}$ ($m\neq 0$) and
$H_{j,m}=\sqrt{(N-m)(N-m+1)/[(N-j+1)(N-j)]}$ ($j\neq 0,m$). We call this
approach the funnel model.