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

2025-05-02 0 0 995.33KB 12 页 10玖币
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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 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 [614]:
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 [1719].
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
2
FIG. 1. Scheme of the tight-binding model with two sites.
The quantum particle is prepared on the initial site 2 and pe-
riodically measured (indicated with the eye). Uand γdenote
the strength of potential and the hopping matrix element,
respectively.
with proper boundary conditions. This tight-binding
Hamiltonian is encoded by the qubit Hamiltonian
Hl=
l
X
j=1
[Ujσz,j γj,j+1(σx,j σx,j+1 +σy,j σy,j+1)],(1)
where σx,σyand σzare Pauli matrices. The states
|0...01i,|0...10i, ..., and |10...0iencode the position of
the particle at site 1, 2, ..., and lalong the chain. The
first term of the Hamiltonian represents the on-site en-
ergy Uion each site i, and the second term represents
the kinetic energy, parameterized by the hopping matrix
element γbetween neighboring sites.
Now we consider a particle moving on two sites and
prepared initially on site 1 or 2 at time t= 0, which
is measured stroboscopically on site 2 after the time τ,
2τetc. (see Fig. 1). The two-site Hamiltonian H2acts
on the computational basis states |10iand |01ias site
1 and 2 in our model, respectively. The states |00iand
|11ishould not be populated. Since only two states are
occupied, we can simplify the two-qubit model described
by the Hamiltonian in Eq. (1) to a single-qubit problem
with the two basis states |0i=|01iand |1i=|10i. In
this basis the Hamiltonian matrix reads
(hj|H2|j0i) = γσx+Uσz=Uγ
γU,(2)
whose eigenenergies are E1,2=±pU2+γ2.
The ME with nstroboscopic measurements is defined
by the evolution operator [15,19]
Mn=eiH2τ(P eiH2τ)n1, P =1− |jihj|=|j0ihj0|
(3)
with j, j0∈ {0,1}and j06=j, which can also be written
for n2 as
Mn=eiH2τ|j0i(hj0|eiH2τ|j0i)n2hj0|eiH2τ.(4)
Then, the FDR probability |φr,n|2=| hj|Mn|ji |2for
|ji→|jireads
(|hj|eiH2τ|ji|2n= 1
|hj0|eiH2τ|j0i|2n4|hj|eiH2τ|j0ihj0|eiH2τ|ji|2n2
(5)
and the FDT probability |φt,n|2=| hj0|Mn|ji |2for
|ji→|j0ireads
|hj0|eiH2τ|j0i|2(n1)|hj0|eiH2τ|ji|2.
For the Hamiltonian H2with U= 0 we get
|hj0|eiH2τ|j0i|2= cos2(γτ ) and |hj0|eiH2τ|ji|2=
sin2(γτ ). Similar but slightly more complex results are
obtained for the parameter c= cos pU2+γ2τin the
general case with U6= 0. Then, for U= 0 the distribu-
tion function |φr,n|2depends on c= cos (γτ) and reads
|φr,n|2=(c2n= 1
(1 c2)2c2(n2) n > 1(6)
for the FDR probability and for the FDT probability
|φt,n|2= (1 c2)c2(n1).(7)
Thus, the sum of the FDR probabilities for all n1 gives
1 and the mean FDR time is τhni. Subsequently, we will
call hnimean FDR time, assuming that it is implicitly
multiplied by the time step τ.
c2= 1 plays a special role because then the transition
|ji→|j0iis completely suppressed:
hni=X
n1
n|φr,n|2=(2c2<1
1c2= 1 .(8)
The corresponding results of the FDT probabilities are
X
n1
|φt,n|2=(0c2= 1
1c2<1(9)
hni=X
n1
n|φt,n|2=(0c2= 1
1/(1 c2)c2<1.(10)
These FDR/FDT results are obtained for an infinite
number of measurements. Since an experiment allows
only a finite number of measurements, the corresponding
mean FDR/FDT results for Nmeasurements are given
in the Supplemental Material. An important difference
for a finite number of measurements is that hniin Eq.
(10) vanishes rather than diverges for c21.
III. IMPLEMENTATION ON A QUANTUM
COMPUTER
A. Single-Qubit Implementation
General operators, such as the unitary evolution opera-
tor exp(iHτ ), must be constructed on a quantum com-
puter as a product of elementary gate operators. The
difficulty is that these gate operators do not commute.
3
However, if the Hamiltonian Hconsists of a sum of sim-
ple qubit operators we can employ time slicing or Trot-
terization [21]. In terms of exp(iH2τ) this means that
we divide the time τinto ktime slices ∆twith ∆t=τ/k,
which provides the approximation
eiH2τ(eσxteiUσzt)k.(11)
In the limit k→ ∞ the approximation becomes exact.
Therefore, for a good approximation the Trotter number
kmust be large. In the single-qubit case of H2we have
eσxt=cos (γt)isin (γt)
isin (γt) cos (γt)(12)
=Rx(2γt) (13)
and
eiUσzt=eiU t0
0eiUt=Rz(2Ut),(14)
such that the single-qubit unitary operator can be writ-
ten as
eiH2τ≈ {Rz(2Ut)·Rx(2γt)}k.(15)
The unitary evolution is followed by a projective mea-
surement in the computational basis, defined by the pro-
jectors P0=|0i h0|and P1=|1i h1|. To implement the
unitary operator in Eq. (15), followed by projective mea-
surements, we run the following quantum circuit (circuit
(1)) on the quantum device
k N
where kdenotes the number of Trotter steps and Nthe
number of measurements. The gates Rzand Rximple-
ment the rotations Rz(2Ut) and Rx(2γt) in Eq. (15),
respectively, and the initial state |jiis either |0ior |1i.
B. Two-Qubit Implementation
In analogy to the single-qubit case we approximate the
unitary time evolution for two qubits with the Hamilto-
nian
H2
2=γ(σxσx+σyσy) + Uσ0σz(16)
as
eiH2
2t(eσxσxteσyσyteiUσ0σzt)k.(17)
The single factors are written in the basis of |00i,..., |11i
as
eσxσxteσyσyt
=Rxx(2γt)·Ryy(2γt)
=
1 0 0 0
0 cos(2γt)isin(2γt) 0
0isin(2γt) cos(2γt) 0
0 0 0 1
(18)
and
ei(U·σ0σz·t)
=σ0Rz(2Ut)
=
eiUt000
0eiUt0 0
0 0 eiUt0
000eiUt
.(19)
The corresponding quantum circuit (circuit (2)) can be
visualized as
k N
where the gates Rz,Ryy, and Rxx implement the ro-
tations Rz(2Ut), Ryy (2γt), and Rxx(2γt), respec-
tively. The initial states |jiand |j0iare either |0ior |1i
and only the first qubit is projectively measured.
C. Error mitigation
In general, there are several sources of errors on cur-
rent quantum computing devices, e.g., amplitude damp-
ing, phase damping, depolarization, state preparation
and measurement errors. In this work, we focus on the
mitigation of the latter as we implement quantum cir-
cuits with up to 40 mid-circuit measurements and there-
fore anticipate that measurement errors have the most
significant impact on our experimental results.
Many readout-error mitigation schemes rely on clas-
sical post-processing techniques that involve measuring
a calibration matrix and applying this matrix to the
raw experimental data, which would render readout-
error mitigation inefficient and time-consuming in our
case. Furthermore, due to the relatively high number of
measurements, a regular updating of the measurement
calibration matrix would be necessary. Therefore, we
use a read-out error mitigation technique that is bet-
ter suited for a high number of mid-circuit measure-
ments. This scheme employs the framework of quantum
error correction and embeds the state after the appli-
cation of a unitary gate and before a measurement in
a non-local state of three entangled qubits, analogous
to the encoding in the three-qubit repetition code, as
depicted in the following quantum circuit (circuit (3)):
摘要:

MeasurementinducedquantumwalksonanIBMQuantumComputerSabineTornowResearchInstituteCODE,UniversitatderBundeswehrMunchen,Carl-Wery-Str.22,D-81739Munich,GermanyKlausZieglerInstitutfurPhysik,UniversitatAugsburg,D-86135Augsburg,Germany(Dated:October19,2022)Westudyaquantumwalkofasingleparticlethatissub...

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