Ab-initio tree-tensor-network digital twin for quantum computer benchmarking in 2D
2025-05-03
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Ab-initio tree-tensor-network digital twin for quantum computer benchmarking in 2D
Daniel Jaschke ∗,1, 2, 3 Alice Pagano ,1, 2, 3 Sebastian Weber ,4and Simone Montangero 1, 2, 3
1Institute for Complex Quantum Systems, Ulm University, Albert-Einstein-Allee 11, 89069 Ulm, Germany
2Dipartimento di Fisica e Astronomia "G. Galilei" & Padua Quantum Technologies
Research Center, Università degli Studi di Padova, Italy I-35131, Padova, Italy
3INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy∗
4Institute for Theoretical Physics III and Center for Integrated Quantum
Science and Technology, University of Stuttgart, 70550 Stuttgart, Germany
Large-scale numerical simulations of the Hamiltonian dynamics of a Noisy Intermediate Scale
Quantum (NISQ) computer – a digital twin – could play a major role in developing efficient and
scalable strategies for tuning quantum algorithms for specific hardware. Via a two-dimensional
tensor network digital twin of a Rydberg atom quantum computer, we demonstrate the feasibil-
ity of such a program. In particular, we quantify the effects of gate crosstalks induced by the
van der Waals interaction between Rydberg atoms: according to an 8×8 digital twin simulation
based on the current state-of-the-art experimental setups, the initial state of a five-qubit repetition
code can be prepared with a high fidelity, a first indicator for a compatibility with fault-tolerant
quantum computing. The preparation of a 64-qubit Greenberger-Horne-Zeilinger (GHZ) state with
about 700 gates yields a 99.9% fidelity in a closed system while achieving a speedup of 35% via
parallelization.
The present NISQ era of quantum computing poses
extreme experimental, theoretical, and engineering chal-
lenges for all promising quantum computing platforms,
being condensed matter or atomic-molecular-optical
based ones [1–4]. Indeed, identifying the best approaches,
engineering solutions, and optimizing strategies at the
physical, logical and algorithmic levels is necessary to
maximize the capability of NISQ computers and unlock
the fault-tolerant scalable era of general-purpose quan-
tum computing [5]. In the last two decades, these chal-
lenges have been mostly attacked at the level of few
qubits, with impressive developments in, e.g., qubits and
gate qualities [6–8]. However, the years ahead require
achieving scalability and that will only be possible by
understanding and characterizing the performances and
limitations of the existing building blocks while function-
ing as one. For example, high-fidelity implementations
will require taking into account also fast-decaying long-
range interactions. Moreover, to go beyond NISQ, deco-
herence effects shall be mitigated by reducing quantum
circuits depth while quantum error-correcting codes will
come at the price of additional gates: all this confront
the software stack with further challenges, e.g., to what
degree the gates on logical qubits can run in parallel.
Here, we develop an efficient digital twin of a two-
dimensional quantum processing unit (QPU) with access
to a variety of compelling features, e.g., additional lev-
els beyond the qubit states, long-range interactions, and
decoherence effects. These features of a large-scale digi-
tal twin of the QPU will be fundamental to support the
next decades of developments, e.g., comparable to the im-
pact that optimal control simulations had on the devel-
opment of high-fidelity single and two-qubit gates [8–10].
∗Corresponding author: daniel-1.jaschke@uni-ulm.de
Via tensor network methods [11–14], we perform two-
dimensional large-scale classical simulations of a quan-
tum computer running non-trivial quantum algorithms;
tensor network methods allow one to overcome the curse
of the exponentially increasing Hilbert space [15]. We
combine the digital twin with a customized compiler and
demonstrate how together they identify limiting factors
of current or future hardware. In this respect, the dif-
ferent topology and connectivity, e.g., 1-dimensional sys-
tems versus 2-dimensional ones, can lead to very distinct
results in terms of the scaling of algorithms. We thus
demonstrate how digital twins could guide the develop-
ment of future quantum algorithm compilers and tran-
spilers [16–18], specifically analyzing a Rydberg QPU in
two dimensions.
Rydberg atoms trapped in optical tweezers represent
one promising platform for realizing a quantum com-
puter [19–25]. The Rydberg architecture has been im-
pressively improved in various aspects over the last years
and the execution of quantum algorithms of increasing
complexity and circuit depth is in sight: recent Rydberg
experiments have demonstrated two-qubit gate fidelities
beyond 98% [26–29]. A key ingredient to increasing the
final computation quality and the achievable circuit com-
plexity is evidently the ability to run gates in parallel.
Parallel gate execution requires an independent paral-
lel control and addressing of each qubit, which has been
recently demonstrated [30–35]. Understanding if and
how an algorithm can be parallelized in the presence of
long-range interactions, different connectivity, and spu-
rious qubit crosstalks provides thus crucial insights to
attack the next quantum computing engineering chal-
lenges. The building blocks to experimentally implement
the protocol suggested in the following in this manuscript
are available, foremost the CZ has been realized for ex-
ample with 87Rb atoms [36].
To test the limits of the hardware on the digital twin,
arXiv:2210.03763v3 [quant-ph] 23 May 2024
2
we analyze an algorithm that can be easily verified via the
measurement statistics in an experiment, can be scaled
with system size, is compatible with NISQ devices from
the expected fidelity, and generates an amount of entan-
glement compatible with simulations on a classical com-
puter. The preparation of large GHZ states fulfills these
characteristics. Moreover, the GHZ state is becoming a
standard benchmark of the ability to control highly non-
classical properties of quantum hardware. Since the sem-
inal demonstration of 14-qubit GHZ states in a trapped
ion quantum computer [37], other platforms have ac-
cepted the challenge as well [38–40]. Recently, six-qubit
GHZ states have been realized in a Rydberg quantum
processor [41]. Therefore, this problem serves as an ex-
ample of how to use the digital twin from the problem
statement to the statistics of projective measurements
comparable to an actual experimental setup.
The digital twin allows one to establish the opti-
mal radius beyond which gates can run in parallel
without significant crosstalk on the 88Sr Rydberg plat-
form [36, 43, 44]. In particular, we characterize the trade-
off between higher parallelization versus smaller errors
due to crosstalk: on the one hand, a lower circuit depth
comes with higher error rates induced by the Rydberg in-
teraction; on the other hand, we pay for higher precision
gates with larger circuit depths and an increasing vulner-
ability to decoherence. As sketched in Fig. 1a), we com-
pile the circuit with a dedicated compiler RydberGHZ-C
targeting the GHZ state in a two-dimensional geometry.
Then, we use realistic and conservative parameters for
the simulation of an 8×8strontium-88 setup [44], for
example, taking into account long-range Rydberg inter-
actions. For a 64-qubit GHZ state, we obtain a state
infidelity of a 10−2level for the closed quantum system
and controllable crosstalk. The circuit depth is only 15%
above the theoretical minimum for a 2D square system
with nearest-neighbor connectivity and the same gate set.
Finally, we demonstrate an application of the digital twin
on parallel GHZ states generation, e.g., encountered in
the initial preparation of quantum error-correcting five-
qubit repetition code [45–47].
The numerical workhorse behind the ab-initio
Hamiltonian-based emulation of a parallel quantum com-
putation is a tree tensor network (TTN) simulating a
square lattice of qutrits taking into account the states
|0⟩,|1⟩, and the Rydberg state |r⟩[48]. Within the
family of tensor network algorithms [49–52], the TTN
is a powerful ansatz for two-dimensional systems. The
simulation of the system time evolution is implemented
by exploiting recent progress in tensor network meth-
ods [11–14]: in particular, the time-dependent variational
principle which supports the long-range interactions re-
quired for picturing crosstalk and the scheduling of par-
allel gates [11, 13]. On the one hand, the combination of
the many-body simulations with optimal control results
to include time-optimal gates and a compiler requires to
master and merge all the existing state-of-the-art build-
ing blocks. On the other hand, exactly this overarch-
ing approach distinguishes it from independently carried
out analysis by the insight that can be gained: with an
equivalent size of 101 qubits, the digital twins sets the
standard for emulating a QPU at the Hamiltonian level
on a classical computer.
The structure of the manuscript is the following: we
focus on the prospect of running quantum algorithms
in parallel on the Rydberg platform by constructing a
global GHZ state and preparing multiple GHZ states on
five qubits in Sec. I. A detailed description of the Ryd-
berg system including open quantum system effects fol-
lows in Sec. II. Afterward, we explain the technical as-
pects of the tensor network simulations in Sec. III and
the RydberGHZ-C designed for the parallel GHZ prepa-
ration in Sec. IV. We conclude with a brief summary and
outlook.
I. PARALLEL QUANTUM ALGORITHMS
The digital twin relies on an ab-initio Hamiltonian
description of the platform. We consider that there is
the potential to study all qubit platforms in the way
we demonstrate the simulations here for Rydberg atoms.
The most challenging numerical aspects, i.e., the two-
dimensional layout of superconducting hardware and the
long-range interactions of trapped ions, are combined in
the Rydberg platform. The two-dimensional structures
in superconducting QPUs are continuously scaled up and
achieve 127 qubits [53], which is already twice the size of
the 64-qubit system studied here. Although trapped ions
systems are one-dimensional, their strong long-range in-
teractions allow an all-to-all connectivity [54] which can
lead to a rapid growth of entanglement and the classical
resources needed for the digital twin. In the following,
we consider solely Rydberg systems, i.e., neutral atoms.
For the Rydberg platform, the physics required to un-
derstand the parallelization of the GHZ state preparation
can be summarized according to Fig. 1. We focus on the
crosstalk in a closed quantum system in this section and
discuss open system effects in Sec. II. Figure 1a) shows a
sketch of a 4×4setup of Rydberg atoms in optical tweez-
ers; the lattice constant of the grid aintroduces the first
relevant length scale. Within the Rydberg blockade ra-
dius rB, only a single atom can be excited to the Rydberg
state due to the van der Waals interactions. We work at
a fixed Rydberg blockade radius rB= 4.98µm=1.66ain
accordance with the experimental parameters proposed
in Ref. [44], which also provides the time-optimal CZ gate
used in the simulation here as well as the parameters for
a Strontium-88 setup; the pulse sequence for the time-
optimal CZ gate stems from optimal control. On the one
hand, the van der Waals interaction is giving rise to the
Rydberg blockade which is exploited to implement an
entangling gate between two atoms; on the other hand,
the van der Waals interaction leads to possible crosstalk
if multiple entangling gates act in parallel and atoms of
two different CZ gates interact via van der Waals inter-
3
a)
b)
Step 5 Step 10 Step 11 Step 15 Step 16 Step 17
Step 20 Step 21 Step 22 Step 23 Step 26 Legend:
RotX(φ)
RotZ(φ)
CZ(φ)
Two-state
readout
Count Count
Single-state
readout
Count
Count |0i
0
8
...
56
64
Count |1i
0
8
...
56
64
Probability
10−8
10−4
1
c) i) ii)
iii)
rG=√10a
Probability GHZ
10−10 10−910−810−710−610−510−410−310−2
Probability Non-GHZ
0.5
10−3
10−5
Probability
0.5
10−3
10−5
0 4 8 12
Count |0i
0.5
10−3
10−5
Probability
52 56 60 64 0.5
10−3
10−5
rG=rS=√16a
rG=√10a
Images from [42]
iv)
v)
FIG. 1. a) Rydberg quantum computer setup: We simulate a square grid of 88Sr Rydberg atoms trapped in optical tweezers. In
each step, a layer of parallel gates is applied: the controlled phase gates involve the strongly interacting Rydberg state |r⟩, thus
two of them applied simultaneously in close proximity introduce crosstalk errors. A tree tensor network (TTN) simulates the
Rydberg atoms as qutrits (states |0⟩,|1⟩, and |r⟩). The lasers, individually addressing the atoms, implement the single-qubit
and two-qubit gates (inset). b) Parallelization: Selected quantum algorithm layers of the GHZ state preparation in terms of
selected gates native to the Rydberg platform. Each square contains gates that are executed in parallel for a minimal distance
between CZ gates rG≥2a, where ais the lattice spacing. c) Experimental measurement schemes: Projective measurements
for a perfect GHZ state yield a 50% probability of counting either exactly 64 qubits in the |0⟩or |1⟩state for an 8×8 closed
quantum system. We can choose between a two-state readout scheme in i) or a single-state readout scheme in ii), which both
use an additional state. For rG=√10, we compare the measurement statistics of readout scheme i) in iii) to readout scheme
ii) in iv). We identify the states attributed to the GHZ state (in green) versus states introduced due to crosstalk (see color
bar). As a consequence of a remaining population in the Rydberg state, the sum of qubits measured in |0⟩and |1⟩does not
necessarily add up to one, showing the effect of the remaining population in the Rydberg state |r⟩. In comparison, rG=√16a
limits the crosstalk to an acceptable amount as shown for the single-state readout scheme (see v) and is defined as the safe
radius rSto run gates in parallel.
actions. Consider as an example atoms A and B being
the target of the first CZ gate and atoms C and D being
the target of a second CZ gate; crosstalk arises if A or
B interacts with atoms of the other gates, i.e., C and D,
while both gates are driven at the same time. In con-
trast to the van der Waals interaction used to implement
the CZ gate between A and B, additional atoms in the
Rydberg state like C and D corrupt the pulse sequence
and lead to what we refer to as crosstalk. Our compila-
tion strategy considers this condition and allows one to
specify the minimal radius rGwhich the RydberGHZ-C
and scheduler enforce between any two qubits participat-
ing in different entangling gates executed in parallel. We
focus on the radius rGwhile the other length scales are
kept constant. Finally, we are able to identify the ra-
dius rSwhere the crosstalk is negligible in comparison to
other errors and where the algorithm is safely executed
in parallel, i.e., we establish a criterion on the fidelity
of our state preparation. Figure 1b) presents the GHZ
preparation for rG= 2aon a 4×4square lattice for a
subset of native gates of the Rydberg platform which are
implemented for the digital twin. The radius rGchanges
the circuit depth Dand the fidelity via crosstalk, while
we keep the lattice constant aand the Rydberg blockade
radius rBfixed. In the following paragraphs, we analyze
these effects in detail.
4
Figure 1c) showcases a typical result enabled by the
digital twin simulation: the effects of crosstalk errors as
they become visible in projective measurements analog to
an experimental setup. The fact of simulating qutrits al-
lows one to explore different readout schemes either with
a readout of both qubit states, see i) in Fig. 1c), or a
single-state readout scheme where we choose the |0⟩state,
see ii). For each measurement of the |0⟩or |1⟩state, one
needs to transfer the corresponding state to an additional
state for readout, e.g., to the ground state of the optical
qubit. A single-state readout scheme, e.g., of state |0⟩,
avoids the additional overhead of transferring the second
state |1⟩also to state for the readout and measuring it.
The sum along the axis «Count |1⟩» of the histogram
iii) for the two-state readout leads to the histogram iv)
of the single-state readout. The sum of the probabilities
along one axis underlines how the readout schemes dif-
fer in information: counting only the number of qubits
in |0⟩, the state |11 . . . 11⟩originating in the GHZ state
is indistinguishable from the state |11 . . . 11r⟩, which has
an error due to a remaining population in the |r⟩state.
In the example of Fig. 1c), this fraction is below 10−5.
When comparing the probability bars representing errors
in the projective measurements, the impact of crosstalk
is evident as the bars differ by two orders of magnitude,
i.e., up to 10−3errors for rG=√10aversus 10−5for
rG=√16a. The probabilities for this set of states shown
in the histogram are extracted directly from the TTN by
sampling 1,000,000 projective measurements. For both
rG=√10aand rG=√16a, the samples cover at least
99.945% of the probability, i.e., a single measurement
appears with at most 0.055% probability not in the data
shown; these statistics can be directly compared to ex-
periments. Although simulations of projective measure-
ments are possible as shown in Fig. 1c), we concentrate in
the remainder on the infidelity which is more accessible
in its interpretation as a single number.
Figure 2 shows the change in the circuit depth D
and the infidelity Iof the algorithm for preparing a
global GHZ state as a function of the radius rG. The
fidelity of the algorithm Fis defined as the state fidelity
F=|⟨ψ(τ)|ψGHZ⟩|2at the end of the algorithm at time
τ; the infidelity is I= 1 −F. The total time of the com-
plete algorithm is τ=D·122ns, which is the product
of the circuit depth Dtimes the 122ns pulse time per
gate, and the circuit depth Ddepends itself on the com-
piler setting and system size. The circuit is generated
with the RydberGHZ-C compiler targeting directly the
Rydberg platform. On the one hand, the circuit depth
drops, e.g., from over 51 to 46 while decreasing the dis-
tance rGfrom √25ato √2afor the 8×8square lattice.
On the other hand, we observe that the infidelity changes
by more than two orders of magnitude while changing rG.
This change in infidelity is due to the Rydberg interac-
tion decaying with a power of six, especially for the 8×8
grid where we have the bigger number of CZ gates n. The
overall fidelity Fdepends on the number of CZ gates n,
therefore we define the average as F⊘=F1/n. For the
1 2 3 4
Distance rG/a
10−4
10−3
10−2
10−1
100
Indelity I
26
28
30
Circuit depth D
a)
12345
Distance rG/a
10−3
10−2
10−1
100
Indelity I
46
48
50
52
Circuit depth D
b)
Images from [42]
FIG. 2. Measuring the effect of controlled phase gates exe-
cuted in parallel in a closed quantum system. The infidelity
Idecreases towards larger circuit depth Dfor the GHZ state
preparation. a) For the 4×4grid, we identify a clear jump for
distances r≥2, which allows reducing the circuit depth by
more than six percent allowing an infidelity of a 10−3level.
b) For the 8×8grid, larger distances have to be considered to
go to a fidelity of the 10−2level at rG≥4a. A gain in circuit
depth of 35% is possible in comparison to a circuit without
parallel CZ gates.
largest values of rGshown in Fig. 2, the error is driven
by small numerical artifacts always remaining in an op-
timized pulse sequence, i.e., the value of F⊘is in good
agreement with the Bell state fidelity for a single gate of
the protocol for the closed system from Ref. [44].
We now choose the radius rG=rSfor a safely executed
algorithm as rS(L= 4) = √8aand rS(L= 8) = √16a.
In Sec. II on the Rydberg model, we show that the infi-
delity of the order of 10−3introduced in this scenario for
a4×4grid is below the largest errors of the order of 10−2
introduced by decay from the Rydberg state. The same
argument holds for an infidelity of the 10−2level for an
8×8system. In summary, Fig. 2 demonstrates that we
identify the errors originating from parallel CZ gates on
a Rydberg quantum computer; the ideal setting for the
system is rS(L= 4) = √8aand rS(L= 8) = √16a, which
leads to a tolerable loss in fidelity in comparison to a cir-
cuit serial in the CZ gates. We point out the reduction
of the circuit depth by 35% for the 8×8system from a
circuit depth 78 without any CZ gates in parallel to a cir-
cuit depth of 50, see Sec. IV for details. Meanwhile, the
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Ab-initiotree-tensor-networkdigitaltwinforquantumcomputerbenchmarkingin2DDanielJaschke∗,1,2,3AlicePagano,1,2,3SebastianWeber,4andSimoneMontangero1,2,31InstituteforComplexQuantumSystems,UlmUniversity,Albert-Einstein-Allee11,89069Ulm,Germany2DipartimentodiFisicaeAstronomia"G.Galilei"&PaduaQuantumTechn...
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