Measurement induced quantum walks on an IBM Quantum Computer
Sabine Tornow
Research Institute CODE, Universit¨at der Bundeswehr M¨unchen, Carl-Wery-Str. 22, D-81739 Munich, Germany
Klaus Ziegler
Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg, Germany
(Dated: October 19, 2022)
We study a quantum walk of a single particle that is subject to stroboscopic projective measure-
ments on a graph with two sites. This two-level system is the minimal model of a measurement
induced quantum walk. The mean first detected transition and return time are computed on an
IBM quantum computer as a function of the hopping matrix element between the sites and the
on-site potential. The experimentally monitored quantum walk reveals the theoretically predicted
behavior, such as the quantization of the first detected return time and the strong increase of the
mean first detected transition time near degenerate points, with high accuracy.
I. INTRODUCTION
Quantum walks are a central concept for quantum
information processing [1,2] as they are indispensable
for quantum algorithm development and for modeling
of physical processes. Furthermore, they provide a uni-
versal model of quantum computation [3] and can be
considered as a quantum version of the classical ran-
dom walk [4]. Measurement induced quantum walks [5]
present a special class of quantum walks for which the
unitary time evolution is supplemented by a (projective)
measurement, resulting in a non-unitary evolution. To
study this effect on a quantum computer, we consider a
closed quantum system that is subject to repeated iden-
tical projective (stroboscopic) measurements and that
evolves unitarily between two successive measurements.
The combined evolution of the system is non-unitary and
can be understood as a monitored evolution (ME) which
has some surprising properties. Assuming stroboscopic
measurements, where a projection is applied repeatedly
after a fixed time interval τ, we count the number of mea-
surements to observe a certain quantum state for the first
time. This number depends on the size of the underlying
Hilbert space, the time interval τ, the detected state as
well as the initial state, in which the quantum system was
prepared. We must distinguish two different cases: the
first detected return (FDR), where the initial state and
the measured state are identical and the first detected
transition (FDT), where the initial state and the mea-
sured state are different. The FDR has been intensively
studied and revealed some remarkable properties [6–14]:
The mean FDR time τhniis quantized, where hniis equal
to the number of energy levels [6,7]. Degenerate energy
levels count only once. This implies that hnijumps if we
tune the system through a degeneracy. The quantization
is related to the integer winding number of the Laplace
transform of the return amplitude [6,15] and exists also
for random time steps {τj}when we average with respect
to their distribution [16]. In the latter case, the mean
FDR time is formally a Berry phase integral due to the
time averaged measurements. The mean FDT time, on
the other hand, is not quantized but has characteristic
divergences near degenerate energy levels [17–19].
To the best of our knowledge, neither the quantization
of the mean FDR time nor the divergences of the mean
FDT time have been observed experimentally. However,
due to the fast improvement of current quantum com-
puters, including the possibility to implement mid-circuit
measurements, which are, e.g., crucial for the realization
of quantum error correction protocols [20], these comput-
ers provide an excellent platform for testing the theory
of the ME with stroboscopic measurements directly. For
this purpose, a tight-binding model on a finite graph is
realized on an IBM quantum computer to study the mean
FDR time and its fluctuations as well as the mean FDT
time experimentally. In this work, we focus on the sim-
plest case of a two-site graph with one particle which is
already sufficient to observe the characteristic features
of the ME, as described above. Such a system is im-
plemented on the IBM quantum computer with one and
with two qubits. For a small number of mid-circuit mea-
surements, the error-mitigated results are found to be
in very good agreement with the theoretically predicted
exact results.
The paper is organized as follows: Sect. II is the the-
oretical part that describes the model and the ME. A
detailed explanation of how the model is implemented
on the quantum computer and a discussion of an appro-
priate error mitigation scheme is provided in Sect. III. In
Sect. IV we present the experiments for the FDR/FDT
time as well as their variance. We summarize our results
in Sect. Vand propose some ideas for future studies.
II. MODEL
The tight-binding model for a quantum particle on a
finite chain of length lis described by the particle-number
conserving Hamiltonian
H=
l
X
j=1
(−γj,j+1(|ji hj+ 1|+|j+ 1i hj|) + Uj|ji hj|)
arXiv:2210.09941v1 [quant-ph] 18 Oct 2022