#  First Hitting Times on a Quantum Computer: Topological Effects and Dark
States

Qingyuan Wang<EMAIL_ADDRESS>Department of Physics, Institute of
Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900,
Israel  Silin Ren  Department of Physics, Institute of Nanotechnology and
Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel  Ruoyu Yin
<EMAIL_ADDRESS>Department of Physics, Institute of Nanotechnology and
Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel  Klaus
Ziegler  Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
Eli Barkai  Department of Physics, Institute of Nanotechnology and Advanced
Materials, Bar Ilan University, Ramat-Gan 52900, Israel  Sabine Tornow
<EMAIL_ADDRESS>Research Institute CODE, University of the Bundeswehr
Munich, D-81739 Munich, Germany

###### Abstract

We investigate a quantum walk on a ring represented by a directed triangle
graph with complex edge weights and monitored at a constant rate until the
quantum walker is detected. To this end, the first hitting time statistics is
recorded using unitary dynamics interspersed stroboscopically by measurements,
which is implemented on IBM quantum computers with a midcircuit readout
option. Unlike classical hitting times, the statistical aspect of the problem
depends on the way we construct the measured path, an effect that we quantify
experimentally. First, we experimentally verify the theoretical prediction
that the mean return time to a target state is quantized, with abrupt
discontinuities found for specific sampling times and other control
parameters, which has a well-known topological interpretation. Second,
depending on the initial state, system parameters, and measurement protocol,
the detection probability can be less than one or even zero, which is related
to dark-state physics. Both, return-time quantization and the appearance of
the dark states are related to degeneracies in the eigenvalues of the unitary
time evolution operator. We conclude that, for the IBM quantum computer under
study, the first hitting times of monitored quantum walks are resilient to
noise. Yet, finite resolution effects lead to new topological, chirality, and
broadening effects, which disappear in the asymptotic theory with an infinite
number of measurements. Our results point the way for the development of novel
quantum walk algorithms that exploit measurement-induced effects on quantum
computers.

##  I Introduction

The option of midcircuit readout of qubits on state-of-the-art quantum
computers [1] opens opportunities to test the dynamics and statistics of
monitored quantum dynamics. The repeated monitoring at predetermined times
yields a string of measurement outputs. These can be viewed as a stochastic
trajectory varying in time which depends on many factors, including the
initial state, the dynamics of the measurement-free process, or the chosen
time intervals between measurements. Given a stochastic trajectory, the first
quantum hitting time, defined as the time taken for a monitored process or a
signal to reach a specific level or target for the first time, has attracted
considerable attention [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19]. The classical counterpart, known as the first passage time
problem, was the subject of a very large body of work; see [20, 21, 22] and
references therein.

Consider, as an example, a classical random walker on a lattice of dimension
$d$ [20, 23]. The classical walker starts at the origin, and the first hitting
time is the time it takes the classical walker to reach some other vertex. The
basic questions are: Will the classical walker reach the target after in
principle an infinite number of steps? What is the distribution of the first
passage times to the target state? The mean first hitting time is a widely
used quantifier of the process. In finite systems, excluding ergodicity
breaking, simple random walks are recurrent, and hence the classical walker is
detected with probability one, i.e., $P_{{\rm det}}=1$, in any dimension. For
infinite systems, the process is non-recurrent in dimension $3$ and above and
hence $P_{\rm det}<1$. For the purpose of search, the processes are diffusive
and hence non-efficient in the sense that paths resample previously visited
locations many times.

For quantum walks, with monitoring, the situation is vastly different, and
here we mention four such aspects.

  * 1

Constructive and destructive interference. Quantum walks, for example, tight-
binding models (see below), serve as a benchmark for quantum search. These
walks may exhibit a quantum speedup for the hitting time, which can be
exponential for specific initial states and highly symmetric graphs [2].
However, in some other cases, the quantum search can perform poorly due to
destructive interference, and the latter problem can be avoided in principle
using specially designed graphs [24] or with a restart strategy [25].

  * 2

Path definition matters. When we obtain quantum trajectories through repeated
strong measurements, the statistics of the first detection time will depend on
the observation scheme with which the path is constructed. Here, we address
this issue using IBM quantum computers. For that purpose, the first hitting
time for tracking and for localized (on-site) measurements at the target state
is defined and studied.

  * 3

Topological effects. A case study is the return or recurrence problem in
finite systems. The return problem addresses the time it takes for a process
to return to its initial state [26, 5, 27, 10, 14]. In the classical domain,
the mean recurrence time is described by Kac’s lemma [28]. This gives the mean
return time in terms of the steady-state measure (see below). In a pioneering
work, Grünbaum et al. [5] extended this result to the quantum domain. They
showed that the mean number of measurements until the detection of the quantum
walker in its initial state is quantized. This is connected to a topological
effect. Mathematically, the mean return time is related to a winding number of
the generating function of the first detection statistics. The mean return
time is topologically protected; however, sharp transitions are found for
special choices of the sampling time. These results are valid for local
measurements at the target state. The authors of Ref. [29] studied a different
protocol, when the trajectory is defined with a tracking protocol. In this
case, the position of the quantum walker is recorded along its path until its
first detection in the target state. Again, the mean return time is quantized,
being larger than or equal to the one found in Ref. [5] using local
monitoring.

  * 4

Zeno physics. In quantum mechanics, sampling cannot be done too fast, to avoid
freezing of the wave packet in its initial state [30]. Thus, the data
acquisition rate is an important parameter controlling the statistics of the
hitting time process.

The primary objective of this work is to enhance our understanding of the
aforementioned concepts by investigating the return time problem and dark
states on a quantum processor (IBM Sherbrooke). Specifically, our objective is
to explore the differences between monitoring one local site and monitoring
all sites. The fundamental questions are how the two monitoring methods affect
return times and detection probabilities on a quantum computer, whether they
are affected by noise, and how the known properties derived by the asymptotic
theory are changed due to a finite number of measurements. First, we discover
that IBM Sherbrooke is a well suited testbed to confirm the stochastic
trajectory of monitored quantum walks without considering a noise model.
Second, due to finite resolution effects (finite number of measurements) new
metastable quantization effects are found. These effects, while very visible
in our experiments, are expected to vanish in the limit of infinite number of
measurements, giving new insights into monitored quantum systems.

Figure 1:  Scheme of the tight-binding model for a ring with three sites
($\ket{\bf 0}$, $\ket{\bf 1}$ and $\ket{\bf 2}$) pierced by a magnetic flux
$\alpha$, corresponding to a directed or chiral triangle graph with complex
edge weights, for two different measurement protocols. $\gamma$ denotes the
strength of the hopping matrix element. Left panel: The on-site protocol
measures periodically only the target state $\ket{\bf{0}}$. Right panel: The
tracking protocol measures periodically all sites. The measurement is
indicated with a measuring device. In both cases, the hitting time is the
first time when the system is detected in state $\ket{\bf{0}}$.

The paper is organized as follows. Sec. II is the theoretical part that
describes the model, the measurement protocol, and how to implement it on a
quantum computer. In Sec. III and Sec. IV we present the asymptotic theory and
experiments, respectively, for the hitting time when the initial and target
states are identical for different control parameters. In Sec. V we vary the
initial state, which leads to interference effects and dark states. Finite
resolution effects are discussed in Sec.VI. We summarize our results in Sec.
VII and propose some ideas for future studies.

##  II Model and Measurement Protocols

###  II.1 Model

We consider a tight-binding chiral quantum walk [31] on a triangle graph with
complex edge weights in the presence of a magnetic flux (Fig. 1) for two
different monitoring protocols. The localized states are
$\ket{\bf{0}},\ket{\bf{1}}$ and $\ket{\bf{2}}$ and the initial state is
$\ket{\psi_{{\rm in}}}$. The measurement-free evolution is defined via unitary
dynamics

$\displaystyle U=\exp(-iH\tau)$ (1)

with the time intervall $\tau$ between detection attempts, and $\hbar=1$. The
Hamiltonian $H$ is

$H=-\gamma\
e^{i\alpha}\left(\ket{\bf{0}}\bra{\bf{1}}+\ket{\bf{1}}\bra{\bf{2}}+\ket{\bf{2}}\bra{\bf{0}}\right)+{\rm
h.c.}\ .$ (2)

Here h.c. stands for hermitian conjugate. $\alpha$ and $\gamma$ are control
parameters that model the effect of magnetic flux and hopping amplitude,
respectively. The time-reversal symmetry is broken when $\exp(i\alpha)\neq 1$.
The eigenvalues of the Hamiltonian are

$E_{k}=-2\gamma\cos\left({2\pi k\over 3}+\alpha\right),\ k=0,1,2\ .$ (3)

When $\alpha=0$, we have a pair of distinct energy levels, and more generally
when the flux $\alpha\neq 0$, we typically remove the two-fold degeneracy and
the number of distinct energy levels is three.

Figure 2:  Parameters where two or three eigenvalues of the unitary $U$ are
degenerate (phase factor matching diagram). In the plain
$(\gamma\tau,\alpha)$, the phase factors $\exp(-iE_{k}\tau)$ with $k=0,1,2$
match in pairs or triplets. Colors describe the matching of two phase factors,
for example, $\exp(-iE_{0}\tau)=\exp(-iE_{1}\tau)$ in red, and similarly for
the pairs $(E_{0},E_{2})$ (green) and $(E_{1},E_{2})$ (orange). The matching
of all three phase factors is shown as blue circles or a blue line at
$\gamma\tau=0$ indicating the Zeno regime. Examples where three phase factors
match are
$(\alpha,\gamma\tau)=(0,0),(0,2k\pi/3),(\pi/6,2\pi/\sqrt{3}),(\pi/3,2k\pi/3),(\pi/2,2\pi/\sqrt{3}),....$.
The two dashed vertical lines ($\alpha=0$ and $\alpha=0.5$) as well as the
horizontal bands ($\gamma\tau=2\pi/\sqrt{3}$ and $\gamma\tau=3\pi/\sqrt{3}$)
indicate the parameter regime considered for the computation of the first
hitting return time on IBM Sherbrooke. The phase factor matching diagram is
obtained experimentally by studying the dark states (on-site protocol) in Fig.
5 (c).

The removal of energy degeneracy will have a profound effect on the mean
return time and the dark states in the system as we will soon show.

To understand the physical properties of the monitored quantum walks, it is
useful to visualize at which parameters $\alpha$ and $\gamma\tau$ (our control
parameters), the eigenvalues of $U$ are degenerate

$\displaystyle\exp(-iE_{k}\tau)=\exp(-iE_{l}\tau)\ ,(k\neq l),\ k,l=0,1,2$ (4)

in a phase factor matching diagram; see Fig. 2. The analytical results are
given in Appendix A. For $\alpha=0$ two phase factors match due to the
degeneracies of the eigenenergies, while three phase factors match for
$\gamma\tau=2\pi j/3$ with $j\in\mathbb{N}$. Other points where three phase
factors match are, e.g.,
$(\alpha,\gamma\tau)=(\pi/6,2\pi/\sqrt{3}),(\pi/3,2\pi
j/3),(\pi/2,2\pi/\sqrt{3})$, etc. We will later see how this matching phase
factor diagram is related to the hitting time problem as well as to the dark
states. We define a target state $\ket{\bf{0}}$ which we monitor each $\tau$
units of time using two measurement protocols. These are described in the next
two subsections.

###  II.2 On-site protocol

First, we consider the stroboscopic measurement scheme [8], which we call the
local measurement protocol or the on-site protocol. Here, measurements are
performed every $\tau$ units of time. Between measurements the dynamics is
controlled by the unitary time evolution $U=\exp(-iH\tau)$. The measurement is
made locally at $\ket{\bf{0}}$ to detect this state; see Fig. 1 (left panel).
Here, the measurements yield either a no or a yes, that is, the quantum walker
is either not found or is found on $\ket{\bf{0}}$. Mathematically, the
measurement is described by the projector $\ket{\bf 0}\bra{\bf 0}$. A typical
string of measurements is, for instance,

$\\{\mbox{no},\mbox{no},\mbox{no},\mbox{yes},\cdots\\}.$

In this case, the first hitting time is $4\tau$. The first click yes gives the
random number of measurements, denoted $n$, needed to detect the quantum
walker in the target state $\ket{\bf{0}}$. In this case, unless we deal with a
two-state system, the information obtained in this way does not specify the
state function of the walker after each null measurement. Hence, after
obtaining a no-result, we do not know what the amplitudes are for states
$\ket{\bf{1}}$ and $\ket{\bf{2}}$.

###  II.3 Tracking protocol

Second, we consider the tracking protocol [29], with the initial and target
state $\ket{\bf{0}}$ where we record every $\tau$ units of time the position
of the walker on the graph; see Fig. 1 (right panel). In this stroboscopic
protocol, measurements are made at times $\tau,2\tau,\dots,20\tau$. In a
typical realization of the process, we find a result, which is given by the
eigenvalues of the position operator, for example

$\\{\bf{1},\bf{2},\bf{0},\cdots\\}$

or using the same initial condition and the same unitary

$\\{\bf{1},\bf{2},\bf{1},\bf{2},\bf{1},\bf{0},\cdots\\}.$

The first hitting time is clearly random and in these two examples it is
$3\tau$ and $6\tau$, respectively.

From the basic postulates of quantum measurement theory [32], we know the
state of the system immediately after each measurement. This will be the
eigenstate corresponding to the eigenvalue just recorded, assuming strong
measurements. For the first example, we will therefore find

$\\{\ket{\bf{1}},\ket{\bf{2}},\ket{\bf{0}},\cdots\\}.$

This implies that with this type of measurement we know precisely what the
state of the system along the measured path is.

Thus, we start with state $\ket{\psi_{\rm in}}=\ket{\bf{0}}$. Just before the
first measurement at time $\tau$, the state function is
$\ket{\psi(\tau)}=U\ket{\psi_{\rm in}}$. We then measure the location of the
walker on the triangle graph. Suppose that the first measurement yields
$\bf{1}$, i.e., the system is projected to state $\ket{\bf{1}}$. We then
continue and evolve the system until the second measurement according to
$U\ket{\bf{1}}$, and measure again.

###  II.4 Implementation on the Quantum Computer

To record the trajectory on a quantum computer we have to encode the tight-
binding Hamiltonian into a qubit Hamiltonian:

$\displaystyle
H=-\frac{1}{2}\gamma[\cos(\alpha)(\sigma_{1}^{x}+\sigma_{2}^{x}+\sigma_{1}^{z}\sigma_{2}^{x}+\sigma_{1}^{x}\sigma_{2}^{z}+\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y})$
$\displaystyle+\sin(\alpha)(\sigma_{1}^{y}-\sigma_{2}^{y}+\sigma_{1}^{y}\sigma_{2}^{z}-\sigma_{1}^{z}\sigma_{2}^{y}+\sigma_{1}^{x}\sigma_{2}^{y}-\sigma_{1}^{y}\sigma_{2}^{x})]$

where $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the Pauli matrices. The
Hamiltonian $H$ defines two disconnected subspaces, the first composed of
states $\ket{00},\ket{01},\ket{10}$ and the second of $\ket{11}$. The unitary
evolution operator $U(\tau)=exp(-iH\tau)$ is decomposed into unitary gates
using Cartan’s decomposition [33]. This allows us to vary $\tau$ without
increasing the depth of the circuits.

For the on-site protocol, we need to detect the state $\ket{\bf{0}}$ while we
do not want to receive the information that allows us to distinguish the
states $\ket{\bf{1}}$ and $\ket{\bf{2}}$. This can be achieved by measuring
only one qubit, which we define as the right qubit in the Dirac notation. We
use the following mapping between the qubits and the representation of the
spatial states:
$\ket{01}\rightarrow\ket{\bf{0}},\ket{10}\rightarrow\ket{\bf{2}}\ \mbox{and}\
\ket{00}\to\ket{\bf{1}}$. Measuring the right qubit in state $\ket{0}$ does
not give any information to distinguish the states $\ket{\bf{1}}=\ket{10}$ and
$\ket{\bf{2}}=\ket{00}$ but measuring the right qubit in state $\ket{1}$ the
system is in $\ket{\bf{0}}=\ket{01}$ with certainty.

For the tracking protocol, we measure the states of both qubits, and thus
after each measurement we find the outcome $00$, $01$ or $10$. For error
prevention reasons, we use a different mapping between the qubit and the
spatial state representation:
$\ket{00}\rightarrow\ket{\bf{0}},\ket{01}\rightarrow\ket{\bf{1}}\ \mbox{and}\
\ket{10}\to\ket{\bf{2}}$. The reason is that experimentally there is an
asymmetry in the measured spectrum. For example, it is much more likely to
migrate from a state $\ket{01}$ to $\ket{00}$ than from $\ket{00}$ to
$\ket{01}$ due to decay processes. Therefore, we define $\ket{00}$ as our
initial state.

As an error suppression strategy, we use dynamical decoupling by inserting two
$X$-gates at specific intervals on the qubit, which is not measured, to keep
it coherent [34]. We compute the time for the first detection or the first
hitting time $n\tau$ and evaluate its statistical properties as a function of
$\alpha$ and $\gamma$. Effectively, the measurement process has ended when we
find the target state for the first time. The result obtained from the
measurements is a string of size $N$ which yields what we call a trajectory (a
string of $N$ bits for the on-site protocol and $N$ times two bits for the
tracking protocol for 32,000 runs). In practice, we cannot stop the
experiments in the middle of the process, i.e., currently there is no
conditional abort option on IBM quantum computers. We therefore shorten the
trajectories using classical post-processing at the bit position where the
target site was measured.

Now we define statistical observables of interest. They can be estimated from
the sample paths and will depend on the parameters of the model, such as
$\gamma\tau$, $\alpha$, the initial state and the measurement protocol. Let
$F_{n}$ be the probability that the quantum walker is detected for the first
time in the target state at the $n$-th measurement [8]. Thus $F_{1}$ is the
probability that the target state is recorded at the first measurement event,
while $F_{2}$ describes the case where the state was recorded for the first
time in the second attempt. $\langle n\rangle$ is the conditional mean number
of events until detection

$\displaystyle\langle
n\rangle(N)=\frac{\sum_{n=1}^{N}nF_{n}}{\sum_{n=1}^{N}F_{n}}\ ,$ (5)

for $\sum_{n=1}^{N}F_{n}\neq 0$. Hence, $\tau\langle n\rangle$ is the mean
time until detection. From now on, we will call this physical quantity mean
return time for brevity. We also define the total detection probability
$P_{{\rm det}}$

$\displaystyle P_{{\rm det}}(N)=\sum_{n=1}^{N}F_{n}\ ,$ (6)

which is the probability that a detection event was recorded. The estimation
of this probability comes from the number of strings without detection. In an
ideal situation with infinite resolution, these are obtained through
$\lim_{N\to\infty}P_{{\rm det}}(N)=\sum_{n=1}^{\infty}F_{n}$ and if the latter
is unity, that is, if the system is detected with probability one,
$\lim_{N\to\infty}\langle n\rangle(N)=\sum_{n=1}^{\infty}nF_{n}$. As mentioned
in the Introduction, even for the triangle model under study we can find cases
where $P_{{\rm det}}<1$ due to destructive interference, as explained below.

In the experiment, we obtain the statistical properties of the first detection
time for a finite number of measurements $N$ giving rise to finite resolution
effects. Per trajectory, the first detection time is $n\tau$, where the
integer $n$ is the random number of measurements until the first detection of
the target state. Clearly in our experiments $n\leq N$. Note that the obtained
strings are vectors of size $N$. It is possible that the string did not
contain the target state $\ket{\bf{0}}$, a fact that will become important
later, close to topological transitions.

##  III Theory recap

We consider the return problem when the initial state $\ket{\psi_{{\rm
in}}}=\ket{\bf{0}}$ is detected for the first time. Furthermore, for now we
assume $N\to\infty$.

###  III.1 On-site protocol (theory)

The theorem of recurrence for on-site measurements reads [5]

$\langle n\rangle=\mbox{number of distinct phase factors}\ \
\exp(-iE_{k}\tau)$ (7)

where $E_{k}$ are the eigenstates of $H$ [Eq. (3)] for the triangle model.
From Eq. (3) it becomes clear why the phase factor matching diagram is a very
useful tool. For example, for the choice of parameters corresponding to blue
circles in Fig. 2 we expect $\langle n\rangle=1$, while for the colored lines
(besides blue) $\langle n\rangle=2$, and anywhere else $\langle n\rangle=3$.
This holds in general under the condition that the energy states, i.e.,
eigenstates of $H$, denoted $\ket{E_{k}}$ have finite overlap with the
detected state. For example, in the Zeno limit $\tau\to 0$, the three phase
factors merge, and hence $\langle n\rangle$ is equal to one. This is expected
for the return problem, since we start at the measured state, and hence the
first measurement detects it when $\tau=0$. We provide more details in
Appendix B.

###  III.2 Tracking protocol (theory)

In general, the mean return time $\langle n\rangle$ is the number of states in
the system, which is three for the triangle model under study. The result
$\langle n\rangle=3$ can be viewed as a classical result. More specifically,
consider a classical random walk coupled to a heat bath in the infinite
temperature limit. In this case, in a steady state, all states are equally
likely, namely the occupation probability is $1/3$. Then from the classical
Kac theorem for the return problem $\langle n\rangle=3$ [28]. At least in the
classical domain, this is easy to understand, since the probability of
measuring the walker in the target state in the first measurement is $1/3$, in
the second $(2/3)(1/3)$ etc. and hence $\langle
n\rangle=\sum_{n=1}^{\infty}n(2/3)^{n-1}(1/3)=3$. Thus, tracking, for most of
the choices of $\tau$, drives the system to a classical limit. The quantum
aspect of the hitting-time process is found for special values of $\tau$ that
capture the revival of the wave packet and hence the underlying periodicity of
the quantum dynamics. Namely, there exist revival times where the initial wave
function returns to its original state in addition to an unimportant phase.
Then if we measure at that time, the quantum walker is detected with
probability one in the target state in the first measurement and hence
$\langle n\rangle=1$. In particular, when $\tau=0$ we have $\langle
n\rangle=1$. In the diagram in Fig. 2 these special sampling times correspond
to three phase factors matching, shown as blue dots and lines. Clearly, the
idealized theory predicts that as we vary $\gamma\tau$ and $\alpha$, when
three phase factors match, we will see a dip or a transition in $\langle
n\rangle$. Thus, $\langle n\rangle$ will jump from $\langle n\rangle=3$ to
$\langle n\rangle=1$ and back. In Appendix C, following Ref. [29] we provide
more details of the theory by introducing the corresponding Markov or
stochastic matrix.

##  IV First hitting return times on IBM quantum computers

We now turn to the experiments for the first hitting return time where for the
initial condition we start at the target state

$\ket{\psi}_{0}=\ket{\bf{0}}.$

Theoretically, $\langle n\rangle$ exhibits pointwise discontinuous behavior as
mentioned. Can we see this on a quantum computer? At first glance, this might
seem hard since the width of these transitions is theoretically zero but in
reality for finite $N$ the transitions are broadened.

###  IV.1 On-site protocol (experiment)

Fig. 3 (a) shows the classical simulation of the mean return time as a
function of $\gamma\tau$ and $\alpha$ for $N=20$. In comparison to Fig. 2 the
areas for $\langle n\rangle=1$ (dark blue) and $\langle n\rangle=2$ (medium
blue) are broadened. For the experimental computation of $\langle n\rangle$ we
utilize the quantum circuit in Fig. 3 (b). The on-site protocol can be
emulated by measuring the upper qubit after each unitary $U$.

Let us first discuss the results of the asymptotic theory using Eq. (7). In
Fig. 3 (c) - (f), we consider $\langle n\rangle$ versus the dimensionless
sampling time $\gamma\tau$, as well as the magnetic flux $\alpha$ (black solid
line). For almost any $\tau$, on average, we return after two measurements,
i.e., $\langle n\rangle=2$. This holds when the magnetic flux is turned off
($\alpha=0$). In contrast, when $\alpha\neq 0$ we have besides very special
choices of $\tau$, $\langle n\rangle=3$ which is related to the fact that the
magnetic flux $\alpha>0$ removes the degeneracy by breaking the time-reversal
symmetry and the rotational invariance. In addition, one observes typical
sudden plunges in $\langle n\rangle$. These are found for special values of
$\gamma\tau$ and $\alpha$. As mentioned in the Introduction, these
transitions, e.g., dips in $\langle n\rangle$ when plotted versus $\gamma\tau$
or $\alpha$ are related to a discontinuous change of the topology of the
underlying generating function [5, 10].

Quantization of the mean return time of the experimental result (light blue
circles) is clearly evident for most values of $\gamma\tau$ and $\alpha$. Far
from transitions in $\langle n\rangle$, theory predicts, and the experiments
confirm, that for on-site measurements, with and without magnetic flux,
$\langle n\rangle\approx 2$ [Fig. 3 (c)] and $\langle n\rangle\approx 3$ [Fig.
3 (d)-(f)], respectively. Notwithstanding these successes, theory
($N\rightarrow\infty$) and experiment depart close to the parameters in which
the broadening of the dark blue and medium blue areas is obvious in Fig. 3
(a). When we choose path (III), in the ($\alpha$, $\gamma\tau$) plain, we are
in the vicinity of a phase factor matching curve, though strictly not exactly
on it. Since the number of measurements $N$ is finite in the experimental
study, we are effectively witnessing a topological state with $\langle
n\rangle=2$, instead of $\langle n\rangle=3$ predicted by theory, see Fig. 3
(e). We observe not only a broadening of the resonances, but also additional
resonances for finite $N$ that are not present in the theoretical graph
($N\rightarrow\infty$) [Fig. 3 (d)]. Finite resolution comes with a price.
This is relevant when triplets of dips of $\langle n\rangle$ are in the
vicinity of one another on path (II). Indeed, when $\alpha=0.5$, we see from
the theoretical prediction in Fig. 3 (d) three nearby dips (black solid line),
while in reality, we see one wide resonance (light blue circles). Thus, the
finite resolution wipes out the fine details of the model because all three
phase factors are almost matching.

The cause of the deviation between theory (black solid lines) and experiment
(light blue circles) close to these topological transitions is not entirely
clear. It might be due to noise, readout errors or finite resolution effects.
Without a doubt, all of these effects may play some role. However, for the
mean return time under study, we concluded that the main factor is the finite
resolution of the experiment. The unitary dynamics and measurements
implemented on the quantum computer work well on the time and size scale of
our experiment, indicated by the fact that the experimental points lie close
to the simulation on a classical computer (blue dashed lines). Finite
resolution effects are discussed in detail in Sec. VI.

Figure 3:  (a) Simulation of the mean return time $\langle n\rangle$ for
$N=20$ for the on-site protocol as a function of $\gamma\tau$ and $\alpha$.
For $\langle n\rangle=1$ three phase factors match (dark blue), for $\langle
n\rangle=2$ two phase factors match (medium blue) and for $\langle n\rangle=3$
all phase factors are different (white). The paths through the parameter space
labeled as (I), (II), (III) and (IV) are indicated with red dashed lines and
correspond to (c), (d), (e) and (f), respectively. Compared to the phase
factor matching diagram in Fig. 2 the lines are broadened due to finite $N$.
(b) Quantum circuit for two qubits representing the three localized states for
the on-site protocol with the initial state $\ket{\bf{0}}=\ket{01}$, where the
unitary $U=\exp{(-iH\tau)}$ and measurements are applied alternately. Only the
upper qubit is measured, since we need to detect the state $\ket{\bf{0}}$
while we do not want to receive the information that allows us to distinguish
the states $\ket{\bf{1}}$ and $\ket{\bf{2}}$. (c) - (f) Mean return time
$\langle n\rangle$ recorded on IBM Sherbrooke (light blue circles), obtained
with the asymptotic theory for $N\rightarrow\infty$ (black solid lines) and
simulated for $N=20$ (blue dashed lines): (c) [path (I)] $\langle n\rangle$
versus $\gamma\tau$ for $\alpha=0$. (d) [path (II)] When $\alpha=0.5$ we
almost always find $\langle n\rangle=3$. As explained in the text, this is
related to the removal of energy-level degeneracy when the magnetic flux is
turned on. Note that some fine structure details predicted by the asymptotic
theory, for $\alpha=0.5$, are washed out in the experiment, see the three
nearby dips that merge into one resonance here. (e) [path (III)] Mean return
time $\langle n\rangle$ as a function of $\alpha$ for
$\gamma\tau=3\pi/\sqrt{3}$. $\langle n\rangle$ develops a plateau, as well as
additional transitions that are absent for $N\rightarrow\infty$. (f) [path
(IV)] Mean return time $\langle n\rangle$ for $\gamma\tau=2\pi/\sqrt{3}$ as a
function of $\alpha$.  Figure 4:  (a) Simulation of the mean return time
$\langle n\rangle$ for $N=20$ for the tracking protocol as a function of
$\gamma\tau$ and $\alpha$. For $\langle n\rangle=1$ three phase factors match
(dark blue), and for $\langle n\rangle=3$ all phase factors are different
(white). Paths through the parameter space labeled (I), (II), (III) and (IV)
are indicated with red dashed lines and correspond to (c), (d), (e), and (f),
respectively. Compared to the phase factor matching diagram in Fig. 2 the dark
blue areas are broadened due to finite $N$. (b) Quantum circuit for two qubits
representing the three localized states for the tracking protocol with the
initial state $\ket{\bf{0}}=\ket{00}$, where the unitary $U=\exp{(-iH\tau)}$
and the measurements are applied alternately. (c) - (f) Mean return time
$\langle n\rangle$ recorded on IBM Sherbrooke (light blue circles), obtained
with the asymptotic theory for $N\rightarrow\infty$ (black solid line) and
simulated for $N=20$ (blue dashed line): (c) [path (I)] $\langle n\rangle$
versus $\gamma\tau$ for $\alpha=0$. $\langle n\rangle$ is quantized ($\langle
n\rangle=3$) and its value, for almost any choice of $\gamma\tau$, differs by
unity from the on-site measurement protocol, the case presented in Fig. 3 (c)
when $\langle n\rangle=2$. (d) [path (II)] For finite magnetic flux
$\alpha=0.5$ the transitions to $\langle n\rangle=1$ are absent except in the
Zeno limit at $\gamma\tau=0$ for $N\rightarrow\infty$. This is different for
$N=20$. The result for $\alpha=0.5$ shows a transition to approximately
$\langle n\rangle=1$ at $\gamma\tau=3.63$. (e) [path (III)] Mean return time
$\langle n\rangle$ for $\gamma\tau=3\pi/\sqrt{3}$. For $N=20$ $\langle
n\rangle$ develops additional transitions that are absent for
$N\rightarrow\infty$ (f) [path (IV)] Mean return time $\langle n\rangle$ for
$\gamma\tau=2\pi/\sqrt{3}$ as a function of $\alpha$.

###  IV.2 Tracking protocol (experiment)

Fig. 4 (a) shows the classical simulation of the mean return time as a
function of $\gamma\tau$ and $\alpha$ for $N=20$. In comparison to Fig. 2 the
areas for $\langle n\rangle=1$ (dark blue) are broadened and the lines where
two phase factors match are irrelevant. For the experimental computation of
$\langle n\rangle$ we implement the quantum circuit in Fig. 4 (b), where both
qubits are measured after applying the unitary $U$.

We start by discussing the theory for $N\rightarrow\infty$ (black solid line).
The tracking protocol gives with and without flux $\langle n\rangle=3$,
according to theoretical predictions. Note that any small deviation of $\tau$
from the special revival times or $\tau=0$, that is, the Zeno limit, yields
$\langle n\rangle=3$ as shown in Fig. 4 (c) - (f) as a black solid line. For
the finite magnetic flux $\alpha=0.5$ in Fig. 4 (d), there are no special
sampling times $\tau$ except $\tau=0$. (Similar for $\gamma\tau=3\pi/\sqrt{3}$
and finite $\alpha$ in Fig. 4 (e)). In the vicinity of the Zeno limit, when
revivals play a role, or more generally, when three phase factors merge, the
deviations between the theory valid for $N\to\infty$ and experiments (light
blue circles) for $N=20$ are large. We obtain a large deviation for the
parameters in which the broadening of the dark blue areas is observed in Fig.
4 (a). The broadening is present in direction of $\gamma\tau$ and $\alpha$
which leads not only to a broadening of the resonances, but also to additional
resonances for finite $N$ that are not present in the theoretical graph
($N\rightarrow\infty$) [Fig. 4 (d) and (e)] but in the finite $N$ simulation
(blue dashed line). Similarly to the on-site protocol, the deviation of the
experimental values from the asymptotic theory is due to finite-resolution
effects. These are discussed in detail in Sec. VI.

##  V Dark States on IBM quantum computers

So far we have considered the return problem, where the initial state was also
the target state. As mentioned in the Introduction, considering more general
initial conditions, we encounter dark states, and the eventual detection of
the walker, even after an infinite number of repeated measurements, is not
generally guaranteed [3, 11].

The detection probability is defined in Eq. 6. As mentioned in the
Introduction for classical random walks on finite graphs like the triangle
model $\lim_{N\to\infty}P_{{\rm det}}(N)=1$, namely, the walker is detected
with probability one. In our study, considering a finite graph, whenever
$\lim_{N\to\infty}P_{{\rm det}}(N)<1$ we say that the system exhibits dark-
state physics. Some initial conditions give $P_{{\rm det}}(N)=0$ and these
initial states are called dark states.

Figure 5:  Detection probability $0\leq P_{{\rm det}}\leq 1$ as a function of
the sampling time $\gamma\tau$ and the phase $\phi$ as well as $\gamma\tau$
and magnetic flux $\alpha$ obtained from IBM Sherbrooke with $N=10$ for the
on-site and tracking protocol. The dark blue color corresponds to $P_{\rm
det}=0$ (dark state), while the white color corresponds to $P_{\rm det}=1$
(bright state). (a) $P_{{\rm det}}$ as a function of $\gamma\tau$ and $\phi$
for the on-site protocol. Dark states are found when $\phi=\pi$ is the result
of destructive interference and for $\gamma\tau\to 0$ due to the Zeno effect.
Additional horizontal bands show dark states, which are found when the time of
revival in the initial state is the same as the sampling time $\gamma\tau$.
(b) $P_{\rm det}$ as a function of $\gamma\tau$ and $\phi$ for the tracking
protocol. The dark states found for $\phi=\pi$ for the on-site protocol turn
bright when the tracking method is used, since the latter breaks spatial
symmetry, while the former does not. The horizontal dark bands are due to
revivals that forbid detection of the system in the target state. (c) and (d)
$P_{\rm det}$ as a function of $\gamma\tau$ and magnetic flux $\alpha$. The
on-site protocol (c) and the tracking protocol (d) exhibit vastly different
trends, i.e., dark states are present in the tracking protocol where three
phase factors match and in the on-site protocol where two and three phase
factors match. As explained in the text, (c) yields the phase factor matching
diagram shown in Fig. 2, while (d) corresponds to three phase factors
matching, which are indicated by blue circles in Fig. 2.

###  V.1 Dark states for zero magnetic flux

Consider the case $\alpha=0$, the initial condition

$\ket{\psi}_{0}={\ket{\bf{1}}+e^{i\phi}\ket{\bf{2}}\over\sqrt{2}}$ (8)

and as before the hitting process ends when we detect the system in state
$\ket{\bf{0}}$. We study the influence of the phase $\phi$ on the detection
process.

For on-site measurement and when $\phi=0$ the system is detected with
probability one, except for special values of sampling times. In contrast, if
$\phi=\pi$, the detection probability is zero. The latter case is caused by
destructive interference and the symmetry of the problem. When $\phi=\pi$ the
current from $\ket{\bf{1}}$ to the detected state is minus the current from
$\ket{\bf{2}}$, which means that for all times, including between
measurements, the amplitude of the detected state is zero. This can break down
when $\alpha\neq 0$, i.e., when the symmetry is broken. A different behavior
is found for the tracking protocol. Then, after the first measurement, the
wave packet is spatially localized in the system; hence, the symmetry is
broken by the measurements, and the walker is eventually detected. Thus, a
dark state in the on-site protocol can be bright for the tracking method.

A comparison between the two protocols is presented in the three-dimensional
graphs in Fig. 5 (a) and (b). The dark color in the figure corresponds to dark
states. For the on-site protocol we find using methods in [11]

$\lim_{N\to\infty}P_{{\rm det}}(N)=\left\\{\begin{array}[]{l
l}0,&\gamma\tau=2\pi k/3\\\ &\\\ {1+\cos(\phi)\over
2},&\mbox{otherwise}\end{array}\right.$ (9)

The first line for $k=0$ corresponds to the well-studied Zeno limit. The
Schrödinger equation being first order in time implies that the amplitude in
the detected state $\ket{\bf{0}}$, at time $\tau$, is proportional to $\tau$
but according to the rules of quantum measurement theory, the probability of
detecting the quantum walker is proportional to $\tau^{2}$, and hence the
walker is not detected at all. It should be noted, however, that for any small
value of $\tau$ and when $N\to\infty$ the walker is eventually detected, which
means that the limits of large $N$ and small $\tau$ do not commute. Other
values of $k=1,2,3,...$ are choices of $\tau$ corresponding to revival times,
when the wave function returns to its original state, and hence the
stroboscopic measurements cannot click yes even after many measurements and
the quantum walker is never detected. The second line in Eq. (9) corresponds
to the destructive interference that is strongest when $\phi=\pi$. It is a
robust effect in the sense that, for on-site measurements, it will hold for
any choice of measurement times.

We recorded $P_{{\rm det}}$ on a quantum computer using on-site measurements;
see Fig. 5 (a). We use $N=10$ measurements for each value of $\gamma\tau$ and
$\phi$. The three-dimensional plot, Fig. 5 (a), clearly demonstrates the above
mentioned features as horizontal and vertical dark stripes. In contrast, in
the tracking protocol [Fig. 5 (b)], we see only the horizontal lines
corresponding to the Zeno effect and the revivals, since, as mentioned,
tracking breaks the symmetry in the system needed to maintain destructive
interference effects.

Clearly, the results in Fig. 5 (a) and (b) using the quantum computer and in
Fig. 12 in Appendix D using simulations indicate a very good agreement between
theory and experiment.

###  V.2 Dark states for a finite magnetic flux

When $\alpha\neq 0$ the detection probability plots presented so far, could in
principle be replaced with a study of $P_{{\rm det}}$ versus of $\alpha,\phi$
and $\gamma\tau$. To simplify the matter, in the experiments and simulations
below, the initial condition is

$\ket{\psi}_{0}=\ket{\bf{1}}$

and therefore the control parameters are $\gamma\tau$ and $\alpha$. In Fig. 5
(c) and (d), we plot the detection probability for the on-site and tracking
protocols, respectively, and find a striking difference. For comparison, the
corresponding simulation is shown in Fig. 12 in Appendix D.

The main features of these figures can be explained as follows. We start with
the on-site protocol. Consider the choice of parameters for which two phase
factors merge. For example, $\exp(-iE_{1}\tau)=\exp(-iE_{2}\tau)$. We also
denote the corresponding eigenstates with $\ket{E_{1}}$ and $\ket{E_{2}}$,
respectively. Then an initial state

$\ket{\psi}_{{\rm in}}\sim\
\bra{\bf{0}}E_{1}\rangle\ket{E_{2}}-\bra{\bf{0}}E_{2}\rangle\ket{E_{1}}$ (10)

is clearly orthogonal with respect to the detected state $\ket{\bf{0}}$. In
this case, every unitary step yields the same global phase shift
$\exp(-iE_{k}\tau)$ with $k=1,2$. Therefore, the amplitude of the target state
will be zero for any subsequent measurement. In other words, this initial
condition is completely dark and cannot be detected at all. Any initial state
that is not orthogonal to this state cannot be detected with probability one;
hence in our example, the initial state under study is not bright when two
phase factors merge.

When the three phase factors merge, a stronger effect will occur on the
triangle graph. Then the wave function, every $\tau$ units of time, returns to
its initial state and hence revives. Since in the case studied in this
section, the initial state is orthogonal to the detected state, clearly one
cannot detect the walker. The same holds for the tracking measurement. The
walker cannot be detected in the first measurement, nor in any of the repeated
measurements. Hence, for on-site measurements, nonbright states are found when
two or three phase factors merge. For the tracking protocol, dark states are
found only when three phase factors merge; otherwise, the state is detected
with probability one. Therefore, to better understand the detection, we return
to the phase factor matching diagram in Fig. 2, where we plot in the
($\gamma\tau$, $\alpha$) plane curves when two phase factors coincide. That
is, $\exp(-iE_{1}\tau)=\exp(-iE_{2}\tau)$ gives one branch of this relation
between $\gamma\tau$ and $\alpha$ and similarly for pairs $(E_{1},E_{3})$ and
$(E_{2},E_{3})$. The plot of this information requires the energy levels that
depend, of course, on $\alpha$. Different pairs of matching phase factors are
presented in different colors. When three phase factors merge (circles), i.e.,
when curves cross each other in Fig. 2, we see that for the tracking protocol,
the detection probability in Fig. 5 (d) is small or zero. These crossing
events appear as dark blue circles in Fig. 5 (d). In the small $\gamma\tau$
limit, we see a clear dark stripe, representing Zeno dark states. When
$\tau=0$ the three phase factors coincide.

When two phase factors match, in Fig. 2, we see for the on-site measurements,
i.e., in Fig. 5 (c), the corresponding darker color. An issue for future study
is whether one can distinguish between the colors in Fig. 2, using repeated
measurements. If we want to remove curves in Fig. 5 (c), at least partially,
then we have to choose an initial state orthogonal to the dark-energy state in
Eq. (10) which will be bright corresponding to the merging of the two phase
factors $\exp(-iE_{1}\tau)=\exp(-iE_{2}\tau)$. Adding a phase to the initial
state, like in Eq. (8) can also add features to these plots. Can any of this
be observed in the experiment? The results presented in Fig. 5 are very
convincing in this regard.

If we consider the return problem, the detection probability is at its
maximum, and hence a figure like Fig. 5 (c) would be colored white and,
therefore, not very informative. In the dark area where $P_{\rm det}=0$ the
mean return time is $\langle n\rangle=1$ for both protocols. On brighter lines
where the two phase factors match, we get $\langle n\rangle=2$ (return) and
$0<P_{\rm det}<1$ (transfer) for the on-site protocol. Also, here finite
resolution effects are important, as for $N\rightarrow\infty$ the dark lines
will become infinitely thin.

Finally, we pose the question of whether the total detection probability
changes if we choose $\ket{\psi}_{0}=\ket{\bf{2}}$. The total detection
probability for the transition from $\ket{\bf{1}}$ to $\ket{\bf{0}}$ and
$\ket{\bf{2}}$ to $\ket{\bf{0}}$, respectively, is different and depends on
the direction of the magnetic field for $N=10$. The difference of the total
detection probabilities is shown in Fig. 6, where an asymmetry is observed
around the broadened phase-matching areas and lines. The total detection time
of the chiral quantum walker for the transfer depends on the direction
(clockwise or counter clockwise) and also on the sign of the magnetic flux.
Breaking of the time-reversal symmetry can therefore enhance or suppress the
total detection and therefore provides a directional control for certain
parameters.

Figure 6: Simulation of the difference of the total detection probability
$\Delta P_{\rm det}=P_{\rm det}(\ket{\psi_{\rm in}}=\ket{\bf{1}})-P_{\rm
det}(\ket{\psi_{\rm in}}=\ket{\bf{2}})$ with the different initial states
$\ket{\bf{1}}$ and $\ket{\bf{2}}$, which are not the target states as a
function of the sampling time $\gamma\tau$ and magnetic flux $\alpha$ ($N=10$)
for the on-site (upper panel) and tracking protocol (lower panel).

##  VI Finite resolution

By finite resolution, we mean that measurements are performed $N$ times per
run. Typically, we record the target state for the first time in some random
measurement attempt numbered $n_{i}\leq N$. We repeat this process by
constructing an ensemble of trajectories to sample the mean return time which
is maximally the number of runs on the quantum computer that we choose to be
$32,000$. We have already explained that in some cases we might not find the
target within these $N$ measurements. Hence, in the figures presented, we show
sample averages that, in principle, depend on our choice of $N$. More
specifically, we define $K\leq 32,000$ as the number of paths for which we
record the target state. Then, the sample mean $n$ conditioned on the target
measurement state is $\langle n\rangle=\sum_{i=1}^{K}n_{i}\big{/}K$ where we
exclude the cases that we did not detect. Far from the topological
transitions, the option of zero detection per run (null measurement) of size
$N$, is rare, and then the conditional measurement and the theoretical
predictions yield similar results. However, as mentioned, close to topological
transitions, we see deviations, which are due to the finite value of $N$. It
should be noted that these deviations are a blessing in the sense that a
precise measurement of a jump of $\langle n\rangle$, for a special value of
$\tau$, as presented as black solid lines in Figs. 3 and 4, is non-physical,
as the width of the transition cannot be zero. In the asymptotic theory
($N\rightarrow\infty$) for a special value of model parameters (here
$\gamma\tau$ and $\alpha$), $\langle n\rangle$ exhibits a pointwise
discontinuity related to a jump in the winding number and the dark lines in
Fig. 5 become infinitely thin.

Remarkably, the simulations for the first hitting time (Figs. 3 and 4) and the
detection probability (Fig. 5) agree with the results provided by the IBM
quantum processor without fitting. This led us to the conclusion that noise is
not an essential factor here. In particular, the return time $\langle
n\rangle$ seems to be protected from noise for values until $N=20$, whereas
interference effects that lead to dark states are only unaffected by noise up
to $N=10$. The influence of noise for larger $N$ will be discussed in a
forthcoming publication.

Figure 7:  Simulation of the mean return time $\langle n\rangle$ versus
$\gamma\tau$ and $N$ for $\alpha=0.5$. (a) For $N=20$ (red dashed line) the
fine structure at $\gamma\tau\approx 3.49,\ 3.63$ and $3.78$ predicted by
asymptotic theory $N\to\infty$ is washed out, unlike the case where $N=1000$
(blue solid line). (b) At $\gamma\tau=3.63$ a transition from $\langle
n\rangle=3$ to $\langle n\rangle=1$ and back is present for $N=20$ (red dashed
line) which is absent for $N=1000$ (blue solid line) . (c) Mean $\langle
n\rangle$ versus $N$ for $\alpha=0.5$ and $\gamma\tau\approx 3.63$ (blue solid
line, on-site protocol), $\gamma\tau=2\pi/(\sin(0.5+\pi/6)+\cos(0.5))$ (green
dotted-dashed line, tracking protocol) and
$\gamma\tau=2\pi/(\sin(0.5+\pi/6)+\cos(0.5))$ (red dashed line, on-site
protocol). For an increasing number of midcircuit measurements, there is a
transition from $\langle n\rangle=1$ to $\langle n\rangle=3$ (tracking
protocol) that drives the system to the high-temperature limit. For the on-
site protocol exactly at $\gamma\tau=2\pi/(\sin(0.5+\pi/6)+\cos(0.5))$ a
crossover is observed from $\langle n\rangle=1$ to $\langle n\rangle=2$ and
for $\gamma\tau\approx 3.63$ a crossover between three topological phases from
$\langle n\rangle=1$ to $\langle n\rangle=2$ and $\langle n\rangle=3$.

For the finite $N$ case, e.g., in experiments, we gain three important
insights which are due to finite resolution: (i) the broadening of the
transition (Figs 3, 4 and 5), (ii) additional dips and plateaus [Figs. 3 (e)
and 4 (d) and (e)] and a chirality effect of the total detection probabilities
(Fig. 6).

We have already mentioned that for the on-site measurement and $\alpha=0.5$,
three dips found by theory are not found in the experiment; see Fig. 3 (d).
Furthermore, for the tracking protocol, a wide resonance is found in the
experiment which is absent in theory [Fig. 4 (d)]. We claim that this is due
to finte $N$ effects. To study this, we consider the mean return time,
obtained from simulations, for various $N$. As shown in Figs. 7 (a) and (b),
for $N=1000$ the resonances reveal themselves. However, that resolution is not
yet experimentally available. The effects of finite resolution lead to the
possibility of a crossover between different topological winding numbers [Fig.
7 (c)] from $\langle n\rangle=1$ to $\langle n\rangle=2$ and $\langle
n\rangle=3$ with an increasing number of measurements $N$, i.e., near the
special point $(\gamma\tau,\alpha)=(2\pi/\sqrt{3},\pi/6)$ for the on-site
protocol, which can be interpreted as a metastable topological effect. For the
tracking protocol, the crossover from $\langle n\rangle=1$ to $\langle
n\rangle=3$ can be understood as a partial thermalization for a small number
of measurements.

In Appendices B and C the eigenvalue spectrum of the survival operator and the
stochastic matrix are analyzed for the on-site and tracking protocol,
respectively. $\langle n\rangle$ is discontinuous at special points when the
absolute value of the eigenvalue is one. For $N=20$ resonances and plateaus
are also present when the absolute values of the eigenvalues are almost one,
leading to metastable topological effects and partial thermalization. This, as
mentioned, is because the observation is for a limited duration, i.e., $N\tau$
is finite, and because of the slow decay of the null measurement probability.
Both are discussed in the following two subsections.

###  VI.1 Broadening effect

We present an exact solution to the problem related to our experimental setup
for both on-site and tracking measurement protocols. Previously, the
broadening effect was studied in more generality for the on-site measurements,
although the asymptotic behavior of large $N$ was studied there [35].

###  VI.2 On-site protocol (broadening)

We first consider the broadening effect using the on-site measurement protocol
and $\alpha=0$. Let $F_{n}$ be the probability of finding the target for the
first time at the $n-$th measurement. The case under study is exactly solvable
[10, 36]. Specifically, with the eigenvalues of Eq. (3) we obtain

$F_{n}=\left\\{\begin{array}[]{c c}|z|^{2}&n=1\\\ &\\\
\left(1-|z|^{2}\right)^{2}|z|^{2(n-2)}&n\geq 2,\end{array}\right.$ (11)

where $|z|^{2}=5/9+4/9\cos(3\gamma\tau)$. Here, we stick to the notation in
[10], where in general, $z$ stands for the zeros of the generating function.
When $|z|^{2}=1$, we have $F_{1}=1$ and then $\langle n\rangle=1$. Otherwise,
$\langle n\rangle=\sum_{n=1}^{\infty}nF_{n}=2$, which is a behavior that was
presented in Fig. 3 (c). Clearly, when $|z|^{2}$ is slightly less than one, so
is $F_{1}$, yet from the recurrence theorem we have
$\lim_{N\rightarrow\infty}\langle n\rangle(N)=2$ implying a very slow decay of
$F_{n}\propto|z|^{2(n-2)}$ for $n>1$. Physically, close to $|z|^{2}=1$ we
usually detect the system with the first click, but in rare events, we click
no. Then the wave function does not overlap with the detected state, and
recording a yes becomes unlikely. On average we find that the return click yes
after $\langle n\rangle=2$ attempts.

In a finite-time experiment, we find $\langle n\rangle$ [Eq. (5)] using Eq.
(11)

$\langle n\rangle={2-|z|^{2(N-1)}\left[1+N(1-|z|^{2})\right]\over
1-|z|^{2(N-1)}\left(1-|z|^{2}\right)}.$ (12)

We will study this expression in the vicinity of the revival times, namely
when $|z|^{2}\to 1$, where we also include the Zeno limit in this category.
Note that if $|z|^{2}=1$ we have $\langle n\rangle(N)=1$ for any $N$.

We note that the expressions in Eqs. (11) and (12) have the same form for a
two-level system, except for a different function of $|z|^{2}$ [36]. This
indicates a universal behavior of $F_{n}$ for finite $N$ and their related
expectation values.

Figure 8:  (a) The transition from $\langle n\rangle=2$ to $\langle
n\rangle=1$ and back is widened due to the finite time of the experiment. We
compare IBM quantum processor results (light blue circles) and theory $\langle
n\rangle=2-\exp(-b)(1+b)$ with $b=N(2/9)(3\gamma\tau-2\pi)^{2}$ for $N=20$
(red dashed line) and simulation (blue solid line). (b) The transition from
$\langle n\rangle=3$ to $\langle n\rangle=1$ and back is widened due to the
finite time of the experiment. We compare IBM quantum processor results (light
blue circles) and theory $\langle n\rangle=3-2\exp(-b/2)(1+b/2)$ with
$b=N(2/9)(3\gamma\tau-2\pi)^{2}$ for $N=20$ (red dashed line) and simulation
(blue solid line).

Close to transitions we use the small parameter $\epsilon^{2}=1-|z|^{2}$. For
the model under study, $\epsilon^{2}=(2/9)(3\gamma\tau-2\pi k)^{2}$ where
$k=0,1,...$ is the index for the $k$-th transition of $\langle n\rangle$. For
example, $k=0$ is the Zeno limit which corresponds to the first dip from the
left in the upper panel in Fig. 3 (a). Taking the limit of large $N$ and
$\epsilon^{2}$ to be small, we find

$\langle n\rangle=2-\exp(-N\epsilon^{2})\left(1+N\epsilon^{2}\right).$ (13)

Recall that close to the transition point $\langle n\rangle=1$ for any large
though finite $N$. On the other hand, if we choose a control parameter like
$\gamma\tau$ to be far from the transition, then $\langle n\rangle\approx 2$,
since $N$ is large.

###  VI.3 Tracking protocol (broadening)

Now let us consider the broadening effect using the tracking protocol and
$\alpha=0$. The probability of finding the quantum walker the first time is
for the tracking protocol

$F_{n}=\left\\{\begin{array}[]{c c}|z|^{2}&n=1\\\ &\\\
2|\eta|^{4}\left(|\eta|^{2}+|z|^{2}\right)^{n-2}&n\geq 2,\end{array}\right.$
(14)

where $|z|^{2}=5/9+4/9\cos(3\gamma\tau)$ and
$|\eta|^{2}=2/9-2/9\cos(3\gamma\tau)$. With $|\xi|^{2}=|z|^{2}+|\eta|^{2}$ the
mean number is:

$\displaystyle\langle n\rangle$ $\displaystyle=$
$\displaystyle\frac{2|\eta|^{4}\left(|\xi|^{4}-2|\xi|^{2}+|\xi|^{2N}\left(-N|\xi|^{2}+N+1\right)\right)}{\left(|\xi|^{2}-1\right)\left(2|\eta|^{4}\left(|\xi|^{2}-|\xi|^{2N}\right)-|\xi|^{2}|z|^{2}\left(|\xi|^{2}-1\right)\right)}$
(15) $\displaystyle-$
$\displaystyle\frac{|\xi|^{2}|z|^{2}\left(|\xi|^{2}-1\right)}{\left(2|\eta|^{4}\left(|\xi|^{2}-|\xi|^{2N}\right)-|\xi|^{2}|z|^{2}\left(|\xi|^{2}-1\right)\right)}$

For small $\epsilon^{2}=(2/9)(3\gamma\tau-2\pi k)^{2}$ close to the transition
we get an expression for the tracking case:

$\displaystyle\langle n\rangle\approx
3-2\exp(-N\epsilon^{2}/2)(1+N\epsilon^{2}/2)$ (16)

Close to the transition $\langle n\rangle=1$ for any large though finite $N$.
If we choose a control parameter like $\gamma\tau$ to be far from the
transition, then $\langle n\rangle=3$. Fig. 8 compares the approximations for
the on-site and tracking protocol in Eq. 13 and Eq. 16 with the simulation and
experimental data.

Figure 9:  The null measurement probability versus $N$ for $\alpha=0.5$ for
the tracking and on-site protocol. The parameters for the tracking protocol
are $\gamma\tau=3.63$ and $\alpha=0.5$ (magenta, dashed), and $\gamma\tau=2$
and $\alpha=0.5$ (grey, dashed dotted), and for the on-site protocol
$\gamma\tau=3.63$ and $\alpha=0.5$ (black, dotted), and $\gamma\tau=2$ and
$\alpha=0.5$ (blue, solid). The null measurement rate slows down considerably
near special sampling rates at $(\gamma\tau,\alpha)=(3.63,0.5)$ for both
protocols.

###  VI.4 Slow decay of the null measurement probability

To explain the experimental findings of an additional dip (transition) in the
mean first return time $\langle n\rangle$ at $\alpha=0.5$, $\gamma\tau=3.63$
for both protocols [see Figs. 3 (d) and (e) and 4 (d) and (e)] as well as the
plateau near $\alpha=0.5$ for the on-site protocol in Fig. 3 (e), which are
absent in the theory for $N\rightarrow\infty$, we investigate the probability
of null measurement, i.e., the probability that the quantum walker is not
detected. The null measurement probability is defined as

$\displaystyle S_{N}=1-\sum_{n=1}^{N}F_{n}.$ (17)

It is zero in the limit $N\rightarrow\infty$ for the return problem [5, 8].
Near special points where the three phase factors merge, the null measurement
probabilities decay only very slowly to zero for increasing $N$, leading to
the transition of $\langle n\rangle$ to 3 (tracking protocol) and 2 (on-site
protocol) only for large $N>1000$ for
$\gamma\tau=2\pi/(\sin(0.5+\pi/6)+\cos(0.5))$ [Fig. 7 (c)]. If
$\gamma\tau\approx 3.63$, $\langle n\rangle$ shows a staircase behavior. The
system is only partially thermalized until, for a larger number of
measurements, the high-temperature limit is reached.

The probability of null measurement decreases exponentially to zero at non-
special sampling rates, whereas the decay rate slows down considerably at
special sampling rates. At the special point
$(\gamma\tau,\alpha)=(2\pi/\sqrt{3},\pi/6)$, the mean first return time
$\langle n\rangle=1$, since the three phase factors match. This is also
approximately the case near the special point $(\gamma\tau,\alpha)=(3.63,0.5)$
due to finite resolution. The null measurement probability is displayed in
Fig. 9 for the tracking and on-site protocol. This behavior is very similar to
the Zeno effect at $\tau\approx 0$ and is discussed in detail in the
Appendices B and C by analyzing the eigenvalues of the survival operator (on-
site protocol) and the stochastic matrix (tracking protocol), respectively.

##  VII Conclusion

We experimentally and theoretically investigated the monitored evolution of a
quantum walker on a ring represented by a directed triangle graph with complex
edge weights – a monitored chiral quantum walk. For this purpose, we used the
capabilities of midcircuit measurements on IBM quantum devices. We were able
to accurately confirm the predictions for the hitting times and detection
probabilities of the general theory for a finite number of measurements.
Remarkably, we utilize only dynamical decoupling as an error suppression
scheme and no other error mitigation methods.

Hitting times and detection probabilities are closely related to the number of
distinct eigenvalues (phase factors) of the unitary time-evolution operator
$U=\exp(-iH\tau)$ of the model Hamiltonian $H$ during the measurement-free
evolution time $\tau$ shown in Fig. 2. To investigate them, we swept through
the parameters of the phase-factor matching diagram and performed two
different stroboscopic measurement protocols. In the first protocol, readout
was performed only at the target site. The second protocol, known as the
tracking protocol, involves measurements at each site along the ring. The main
experimental result is that there are clear differences between both
protocols, a feature not known for classical random walks. The results show
that the postulates of measurement theory could be confirmed on the IBM
quantum computer, a non-trivial fact since $N$ repeated noisy measurements are
utilized.

We showed that finite resolution effects, which are clearly part of any
experimental study, are a key feature of the hitting time statistics. This
leads to a broadening of the mean return time and the detection probability
because the detection is imperfect near special points, where two or three
phase factors merge, due to finite $N\tau$. The finite resolution and the
broadening effect are accompanied by a slow relaxation of the null measurement
probability. The results show not only a broadening of the topological
transitions but also metastable topological [see Fig. 3 (e) and see Fig. 4
(e)] and chirality (see Fig. 6) effects, which disappear for the theory in the
asymptotic limit.

We derived an analytical expression for the broadening of the $\langle
n\rangle=2$ to $\langle n\rangle=1$ and back transition (on-site) and the
$\langle n\rangle=3$ to $\langle n\rangle=1$ and back transition (tracking)
for $\alpha=0$ as a function of $N$. For both protocols, the resonances narrow
as $N$ increases. Furthermore, the additional resonances near the special
points (but not exactly on them) disappear for growing $N$. The mean return
time $\langle n\rangle$ shows a crossover between different topological
phases, which is revealed, e.g., as the staircase behavior for the on-site
protocol [Fig. 7 (c)]. A similar effect was found for a perturbed ring [14].
The tracking protocol of the monitored quantum walk has more of the character
of a classical random walk. Because of the finite resolution, quantum effects
are present for a small number of midcircuit measurements close to the revival
time. This can be seen as partial thermalization [37] during a quantum-to-
classical crossover. Only for a large number of measurements the classical
limit and maximum $\langle n\rangle$ is reached.

There are several future directions in which our work can be extended. Here,
we focus on small systems. More generally, one could scale up the process by
increasing the number of measurements and/or increasing the size of the
system. One could then ask whether the quantum computer exhibits a transition
to a more classical behavior, for example, the elimination of the topological
effect found for the mean return time. Furthermore, one could explore the
possibility of measuring the readout rate-controlled oscillatory behavior of
the energy expectation values corresponding to a certain trajectory of the
monitored dynamics. In a future publication, we will show that noise in the
system enables transitions to theoretically forbidden states, an effect that
becomes particularly important as we increase $N$.

##  VIII Acknowledgments

We acknowledge the use of IBM Quantum services for this work. The views
expressed are those of the authors and do not reflect the official policy or
position of IBM or the IBM Quantum team. In this paper, we used the IBM
Sherbrooke processor which is an IBM Quantum Eagle Processor. Q.W., S.R. and
R.Y. acknowledge the use of the IBM Quantum Experience and the IBMQ-research
program. We acknowledge helpful discussions with David Kessler. The support of
the Israel Science Foundation grant 1614/21 is acknowledged.

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##  IX Author Contribution

E.B. and S.T. conceived and designed the project. S.T., S.R., and Q.W. ran
hardware, numerical experiments, and classical simulations. All authors
discussed the theory and the results. E.B. and S.T. drafted the manuscript
with input from all authors.

Figure 10:  (a) Phase factor matching diagram and paths through the parameter
space labeled (I), (II), (III), and (IV) which are indicated with red dashed
lines and correspond to (c), (d), (e), and (f), respectively. The eigenvalues
close to one lead to additional dips and plateaus in the mean return time,
whereas the eigenvalues exactly equal to one indicate transitions for the
theory $N\rightarrow\infty$. (b) Illustration of the on-site measurement
protocol. (c) - (f) Eigenvalue spectrum of the survival operator for the same
parameters used for the mean return time in Fig. 3. (c) For $\alpha=0$ the
eigenvalues are one for $\gamma\tau=0,2\pi/3,4\pi/3$ and $6\pi/3$, leading to
transitions in the mean return time [path (I)]. (d) For $\alpha=0.5$ the
eigenvalues are one for $\gamma\tau\approx 0,1.82,3.50,3.62,3.78,5.44$. The
three eigenvalues $|\lambda|=1$ for $\gamma\tau\approx 3.50,3.62,3.78$ are
very close, so almost three phase factors match leading to a transition to one
of the mean return times [path (II)]. In (e) [path (III)] and (f) [path (IV)]
$|\lambda|$ is close to one between $\alpha\approx 0.3$ and $\alpha\approx
0.7$ leading to a plateau for $N=20$.  Figure 11:  (a) Phase factor matching
diagram and paths through the parameter space labeled (I), (II), (III), and
(IV) which are indicated with red dashed lines and correspond to (c), (d),
(e), and (f), respectively. (b) Illustration of the tracking protocol. (c) -
(f) The eigenvalues of $G_{\rm tar}$ for the same parameters as the mean
return time in Fig. 4. Two nearly degenerate eigenvalues close to one lead to
additional dips in the mean return time, while two degenerate eigenvalues
exactly at one indicate transitions for the theory $N\rightarrow\infty$. (c)
For $\alpha=0$ the eigenvalues are degenerate, and one for
$\gamma\tau=0,2\pi/3,4\pi/3$ and $6\pi/3$ which leads to transitions of the
mean return time [path (I)]. (d) For $\alpha=0$ and $\gamma\tau=3.63$ the
eigenvalues are almost degenerate and one which leads to a transition for
$N=20$ due to finite resolution effects [path (II)]. (e) For
$\gamma\tau=3\pi/\sqrt{3}$ the eigenvalues are almost degenerate and one for
$\alpha\approx 0.3$ and $\alpha\approx 0.7$ [path (III)]. (f) For
$\gamma\tau=2\pi/\sqrt{3}$ the eigenvalues are degenerate and one for
$\gamma\tau=\pi/6$ indicating a transition for all $N$ [path (IV)].

##  Appendix A phase factor matching diagram and eigenstates

This appendix provides the analytical solutions of the phase factor matching
equations and the eigenstates. As mentioned in the text the eigenvalues of

$H=-\gamma\
e^{i\alpha}\left(\ket{\bf{0}}\bra{\bf{1}}+\ket{\bf{1}}\bra{\bf{2}}+\ket{\bf{2}}\bra{\bf{0}}\right)+{\rm
h.c.}$

are

$E_{k}=-2\gamma\cos\left({2\pi k\over 3}+\alpha\right),\ k=0,1,2\ .$

Its eigenvectors are for finite $\alpha$:

$\ket{E_{0}}=\frac{1}{\sqrt{3}}\left(\ket{\bf{0}}+\ket{\bf{1}}+\ket{\bf{2}}\right),$
$\ket{E_{1}}=\frac{1}{2}\left(\left(\frac{-1}{\sqrt{3}}-i\right)\ket{\bf{0}}+\left(\frac{-1}{\sqrt{3}}+i\right)\ket{\bf{1}}+\frac{2}{\sqrt{3}}\ket{\bf{2}}\right)$

and

$\ket{E_{2}}=\frac{1}{2}\left(\left(\frac{-1}{\sqrt{3}}+i\right)\ket{\bf{0}}+\left(\frac{-1}{\sqrt{3}}-i\right)\ket{\bf{1}}+\frac{2}{\sqrt{3}}\ket{\bf{2}}\right)$

The eigenstates for $\alpha=0$ are:

$\ket{E_{0}}=\frac{1}{\sqrt{3}}\left(\ket{\bf{0}}+\ket{\bf{1}}+\ket{\bf{2}}\right)$

and for the degenerate eigenvalue $E_{1}=E_{2}$:

$\ket{E_{1}}=\frac{1}{\sqrt{2}}\left(-\ket{\bf{0}}+\ket{\bf{1}}\right)$

and

$\ket{E_{2}}=\frac{1}{\sqrt{2}}\left(-\ket{\bf{0}}+\ket{\bf{2}}\right)$

The solutions of the following equations (matching of phase factors):

$\displaystyle\exp(-iE_{k}\tau)=\exp(-iE_{l}\tau)\ ,(k\neq l),\ k,l=0,1,2$
(18)

are for $k=1$ and $l=2$:

$\displaystyle\gamma\tau=\frac{2n\pi}{\sqrt{3}\sin(\alpha)}$ (19)

and

$\displaystyle\gamma\tau=-\frac{(2n+1)\pi}{\sqrt{3}\sin(\alpha)}$ (20)

for $k=1$ and $l=0$:

$\displaystyle\gamma\tau=\frac{n\pi}{\sin(\alpha+\pi/6)+\cos(\alpha)}$ (21)

for $k=2$ and $l=0$:

$\displaystyle\gamma\tau=\frac{n\pi}{\sin(-\alpha+\pi/6)+\cos(\alpha)}$ (22)

for $n\in\mathbb{Z}$.

##  Appendix B  Details of the on-site protocol

The probability $F_{n}$ to find the quantum walker for the first time is for
the on-site protocol [8]:

$\displaystyle
F_{n}=|\langle\psi_{tar}|\left(U(\bf{I}-|\psi_{tar}\rangle\langle\psi_{tar}|\right))^{n-1}U|\psi_{in}\rangle|^{2}$
(23)

where $\bf{I}$ is the identity matrix, $|\psi_{tar}\rangle$ is the target site
and $|\psi_{in}\rangle$ is the initial site. To explain the experimental
findings of a transition from $\langle n\rangle=3$ to $\langle n\rangle=1$ for
$\alpha=0.5$, $\gamma\tau=3.63$ and $N=20$ in Fig. 3 (c) and in the simulation
Fig. 7 (a) which is changes to three close transitions from $\langle
n\rangle=3$ to $\langle n\rangle=2$ (on-site protocol) for $N>1000$ we analyse
the survival operator for the on-site protocol.

The parameter regime is indicated by a dashed vertical line in Fig. 10 (a) at
$\alpha=0.5$. The line crosses five phase lines where two phase factors merge.
Three of them are very close. For $N$ very large in the on-site protocol, five
transitions from $\langle n\rangle=3$ to $\langle n\rangle=2$ are seen.

What is the effect of the nearby special point ($\alpha=\pi/6$,
$\gamma\tau=2\pi/\sqrt{3}$) where the three phase factors merge? Does it lead
to a slow relaxation of the null measurement probability and therefore to the
aforementioned transition which is present for $N=20$ measurements?

The survival operator for the on-site protocol is defined as

$\displaystyle S(\tau)=\left({\bf{I}}-\ket{\psi_{\rm tar}}\bra{\psi_{\rm
tar}}\right)U$ (24)

The eigenvalues of the survival operator are key to understanding the
processes and to answering the questions at the beginning of the section. The
absolute values of two eigenvalues $|\lambda_{1}|\approx|\lambda_{2}|$ are
close to one for $\gamma\tau=3.63$ [see Fig. 10 (d)]. The plateau in Fig. 3
(e) coincides with the eigenvalue $|\lambda|\approx 1$ between $\alpha=0.3$
and $\alpha=0.7$.

##  Appendix C  Details of the tracking protocol

For the time evolution that takes into account unitary evolution and global
measurements, we define the Markov or stochastic matrix $G$ [29]:

$\displaystyle G=\sum_{x,x^{\prime}\in
X}\left|\bra{x}U\ket{x^{\prime}}\right|^{2}\ket{x}\bra{x^{\prime}},$ (25)

where $X=\\{\ket{\bf{0}},\ket{\bf{1}},\ket{\bf{2}}\\}$ and
$\left|\bra{x}U\ket{x^{\prime}}\right|^{2}$ are the transition probabilities.
The probability $F_{n}$ of finding the quantum walker for the first time is
then:

$\displaystyle F_{n}=\langle\psi_{\rm tar}|\left(G({\bf{I}}-|\psi_{\rm
tar}\rangle\langle\psi_{\rm tar}|)\right)^{n-1}G|\psi_{\rm in}\rangle,$ (26)

For $\alpha=0$ the eigenvalues of $G$ are real and the absolute value is less
than or equal to one. $\langle n\rangle$ is discontinuous at special points
when the eigenvalue $\lambda=1$ becomes degenerate. For a finite magnetic
flux, the eigenvalues except one ($\lambda=1$) become complex and appear as
conjugated complex pairs. There are special points for finite
$\alpha=\pi/6,\pi/3,\pi/2,...,$ where for certain values of $\gamma\tau$ the
eigenvalues of $G$ are degenerate and one (see Fig. 11). Therefore, no special
points appear at $\alpha=0.5$ and $\langle n\rangle=3$ for all $\gamma\tau$
except in the Zeno regime for $\gamma\tau=0$. In Fig. 7 (b) $\langle n\rangle$
with a finite magnetic flux is shown. For $N=1000$ no transitions from 3 to 1
and back are present (except in the Zeno limit) while for $N=20$ the
transitions are clearly present due to finite resolution. Following Ref. [29]
we define:

$\displaystyle G_{\rm tar}=G\left({\bf{I}}-\ket{\psi_{\rm tar}}\bra{\psi_{\rm
tar}}\right)$ (27)

$G\ket{\psi_{\rm in}}$ is written in the eigenbasis of $G_{\rm tar}$. The left
and right eigenstates are defined in the usual way: $G_{\rm
tar}\ket{\mu_{R}}=\mu\ket{\mu_{R}}$ and $\bra{\mu_{L}}G_{\rm
tar}=\mu\bra{\mu_{L}}$ where $\mu$ is the eigenvalue. The null measurement
probability (probability that the quantum walker is not detected) is then
given by:

$\displaystyle
S_{N}=\sum_{\mu}\frac{e^{N\ln(\mu)}}{1-\mu}\frac{\bra{\mu_{L}}G\ket{\psi_{\rm
in}}}{\braket{\mu_{L}}{\mu_{R}}}\braket{\psi_{\rm tar}}{\mu_{R}}$ (28)

The null measurement probability is characterized by $\mu$ as well as the
overlap between $\ket{\mu_{R}}$ and the target state $\ket{\psi_{\rm tar}}$.
For the special point $(\pi/6,2\pi/\sqrt{3})$ the overlap is zero, and the
eigenvalue is $\mu=1$. Near the special point $(0.5,3.63)$ there is a finite
overlap and the eigenvalue is close to one, leading to a very slow relaxation
of the null measurement probability. Therefore, $\langle n\rangle=1$ for
$N=20$. For an increasing number of $N$, the system experiences a transition
from $\langle n\rangle=1$ to $\langle n\rangle=3$ driving the system into the
high-temperature limit. Thus, for $N=20$ we consider that the system is still
in the quantum regime.

##  Appendix D Simulated detection probability

Figure 12:  The same as Fig. 5 but as a simulation. We observe that the
experiment and the simulations agree very well for $N=10$.

In Fig. 12 we present the simulation of the detection probability which is in
a very good agreement with the experimental data shown in Fig. 5.