# Digitized-Counterdiabatic Quantum Algorithm for Protein Folding

Pranav Chandarana  Department of Physical Chemistry, University of the Basque
Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  EHU Quantum Center,
University of the Basque Country UPV/EHU, Barrio Sarriena, s/n, 48940 Leioa,
Biscay, Spain  Narendra N. Hegade<EMAIL_ADDRESS>Kipu
Quantum, Greifswalderstrasse 226, 10405 Berlin, Germany  International Center
of Quantum Artificial Intelligence for Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Iraitz
Montalban  Kipu Quantum, Greifswalderstrasse 226, 10405 Berlin, Germany
Department of Physics, University of the Basque Country UPV/EHU, Barrio
Sarriena, s/n, 48940 Leioa, Biscay, Spain  Enrique Solano
<EMAIL_ADDRESS>Kipu Quantum, Greifswalderstrasse 226, 10405
Berlin, Germany  International Center of Quantum Artificial Intelligence for
Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China
IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao,
Spain  Xi Chen<EMAIL_ADDRESS>Department of Physical Chemistry,
University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena,
s/n, 48940 Leioa, Biscay, Spain

###### Abstract

We propose a hybrid classical-quantum digitized-counterdiabatic algorithm to
tackle the protein folding problem on a tetrahedral lattice. Digitized-
counterdiabatic quantum computing is a paradigm developed to compress quantum
algorithms via the digitization of the counterdiabatic acceleration of a given
adiabatic quantum computation. Finding the lowest energy configuration of the
amino acid sequence is an NP-hard optimization problem that plays a prominent
role in chemistry, biology, and drug design. We outperform state-of-the-art
quantum algorithms using problem-inspired and hardware-efficient variational
quantum circuits. We apply our method to proteins with up to 9 amino acids,
using up to 17 qubits on quantum hardware. Specifically, we benchmark our
quantum algorithm with Quantinuum’s trapped ions, Google’s and IBM’s
superconducting circuits, obtaining high success probabilities with low-depth
circuits as required in the NISQ era.

## Introduction

Variational quantum algorithms (VQAs) have been proposed to solve problems
with noisy qubits in near-term quantum computers [1]. VQAs are hybrid
classical-quantum algorithms that optimize a cost function containing
information about the solution. The quantum part of a VQA consists of a
parameterized quantum circuit (PQC), also known as circuit ansatz, to produce
trial quantum states. The classical part consists of an optimization routine
that gives optimal parameters to solve the problem. The choice of PQC affects
the performance of the VQA to a great extent. These PQCs are broadly divided
into two categories: problem-inspired and hardware-efficient. Problem-inspired
ansatz utilizes the properties of the problem Hamiltonian to efficiently reach
the expected state. Hardware-efficient ansatz takes the information of the
device connections to reduce the noise due to deep circuits and
unimplementable connections. Some examples of problem-inspired ansatz are the
Quantum Approximate Optimization Algorithm (QAOA) [2], the unitary coupled-
cluster ansatz [3] or the Hamiltonian variational ansatz [4, 5]. On the other
hand, some noteworthy hardware-efficient ansätze can be found in Refs. [6, 7].

Implementing VQAs is challenging due to shot noise, measurement noise, and
others. It has also been shown that VQAs suffer from barren plateaus, where
the gradients vanish with increasing system size [8, 9, 10]. Generally, VQAs
with hardware-efficient ansätze suffer from this challenge due to their high
expressibility at larger depths. Hence, the use of the problem-inspired ansatz
is motivated. In problem-inspired ansatz, the limited search space results in
lower expressibility and higher trainability. That being said, problem-
inspired ansätze usually involve large circuit depths, so experimental
implementation becomes unfeasible on available noisy devices. Thus, a good
circuit ansatz has to be expressible so that it contains the solution, but not
too expressible that it becomes untrainable.

Recently, several works have reported the use of digitized-counterdiabatic
quantum computation (DCQC) to improve and compress state-of-the-art quantum
algorithms. These methods utilize counterdiabatic (CD) protocols to accelerate
given adiabatic quantum algorithms to generate many-body ground states [11]
and QAOA [12]. Furthermore, they have also shown drastic improvements in
industrial applications, like portfolio optimization [13] and integer
factorization [14]. These methods have some difficulties, like finding
suitable initial parameters and optimal CD terms. To solve these challenges, a
meta-learning technique was proposed recently to find suitable initial
parameters [15]. The choice of optimal CD terms may be tackled by machine
learning methods like reinforcement learning [16] and Monte-carlo tree search
[17]. CD protocols stem from the field of shortcuts to adiabaticity, which was
developed to accelerate the quantum adiabatic processes. Among many methods,
like fast forward [18, 19] and invariant-based engineering [20, 21], CD
driving [22, 23, 24] has been of prominent interest over the years for
studying many-body quantum systems. Apart from shortcuts to adiabaticity,
other quantum control protocols like quantum optimal control (QOC) are also
studied in the context of VQAs [25, 26]. VQAs have also been implemented to
find optimal control sequences [27].

In this article, we develop a hybrid classical-quantum digitized-
counterdiabatic algorithm to tackle a protein folding problem. It consists of
a PQC inspired by CD protocols and a classical optimization routine for
parameter optimization. Proteins are macromolecules consisting of a large
chain of amino acid residues and perform many vital functions in organisms
like DNA replication, catalyzing metabolic reactions, and more. Knowledge of
how proteins fold is crucial in understanding enzymes. The mechanics of
folding may unravel remedies for diseases like Alzheimer’s, Huntington’s, and
Parkinson’s that are induced due to misfolding of proteins. With the
combinatorially increasing solution space, protein folding problem is highly
complex for classical computation, which suggests the potential use of quantum
computers.

Generally, protein folding is modeled by suitable 2D or 3D lattices, while the
amino acids are allowed to be placed such that the interaction energy is
minimized. By proper encoding schemes, this problem can be converted into a
problem Hamiltonian whose ground state shows the configuration of the
concerned protein in the given lattice. Over the last decade, numerous
attempts have been made to tackle this problem with quantum computing. For
instance, Perdomo et al. [28] studied a hydrophobic polar (HP) model with a 2D
lattice and later also studied a coarse-grained model (3D lattice) with
quantum annealing [29]. Babbush et al. [30] also studied the protein folding
problem with turn encoding in quantum adiabatic algorithms. Babej et al. [31]
studied this problem using the quantum alternating operator ansatz [32].
Recently, a resource-efficient version of the same problem was considered by
Anton Robert et al. [6] with an experimental demonstration on IBM
superconducting device.

To address this problem, we propose a hybrid digitized-counterdiabatic
algorithm that includes a PQC we call “CD-inspired” ansatz. While being
problem-inspired, this ansatz is also hardware implementable and has a
parameterization that scales as $\mathcal{O}(N^{2})$, where $N$ is the number
of qubits. We benchmark the performance against state-of-the-art problem-
inspired ansatz as well as hardware-efficient ansatz. In addition, we also
perform experiments with system sizes up to 17 qubits using several quantum
hardware platforms with different connectivity and native gates.

Figure 1: A Schematic diagram of different types of ansatz with $p=1$ layer.
In (a) hardware-efficient ansatz (HEA), which has no information from the
problem, (b) CD-inspired ansatz proposed in this work, and (c) QAOA, which is
a problem-inspired ansatz. Horizontal lines show qubit registers that are
initialized in $\ket{+}^{\otimes N}$ state. In HEA, there are parameterized
$Y$ rotations followed by nearest-neighbor entangling gates, then again
parameterized $Y$ rotations. In QAOA, we have Hamiltonian term $U_{c}(\gamma)$
and mixer term $U_{b}(\beta)$. And, for the CD-inspired ansatz, we have
parameterized $Y$ rotations followed by $YZ(\theta)$ rotations. In all cases,
cost function $C$ (Eq. (3)) is computed, then the parameters are updated using
gradient-based classical optimizers until $C$ is minimized. As we go from left
to right, the implementation difficulty increases.

## Results

CD-inspired ansatz. In this section, we will discuss the construction of the
CD-inspired ansatz for the algorithm and study its performance. We begin by
considering quantum adiabatic evolution with counterdiabatic protocol
Hamiltonian $H_{cd}$ given by

$H_{cd}(t)=(1-\lambda(t))H_{mixer}+\lambda(t)H+\dot{\lambda}(t)A_{\lambda},$
(1)

where $H_{mixer}$ is a Hamiltonian whose ground state is easy to prepare,
$\lambda(t)$ is a scheduling function with boundary conditions $\lambda(0)=0$,
$\lambda(T)=1$, $T$ is the total evolution time, and $A_{\lambda}$ is the
approximate CD term, calculated by using the nested commutator (NC) method
(See Methods .1). Lower orders of $A_{\lambda}$ give the approximate CD terms
while $l\to\infty$ will give the exact CD term. While working in the adiabatic
regime, the scheduling function $\lambda(t)$ should be slow enough to satisfy
the adiabatic theorem to reach the ground state of the target Hamiltonian with
high probability. However, with the CD term, this condition is lifted as the
non-adiabatic transitions can be suppressed [33] and at
$|~{}\dot{\lambda}~{}|\to 0$, we retrieve the adiabatic Hamiltonian. Now
consider a certain $\lambda(t)$ that satisfies
$|~{}\dot{\lambda}(t)~{}|\gg|~{}\lambda(t)~{}|$ . For this scenario, the
evolution will almost be non-adiabatic and most of the contribution will be
from $A_{\lambda}$ term. In theory, this evolution should also be successful
but it will require the calculation of the exact CD term which is a
challenging task as information on all the spectral properties of the
Hamiltonian becomes necessary.

In DC-QAOA [12], the Eq. (1) is digitized with approximate CD terms, to get
faster evolution. Instead of using actual scheduling functions, we take the
aid of classical optimization routines to optimize trotter evolution to reach
to the ground state. Under the assumption that there exists a scenario as
mentioned above, we can get rid of the first two terms of Eq. (1) and only the
CD terms can be considered. This is the intuitive motivation behind the CD-
inspired ansatz: Instead of implementing all the evolution, we implement only
the contributions from the digitized CD term as a parameterized circuit and
allow the classical optimization to take care of the evolution to lead to the
ground state. Thus, the CD-inspired ansatz will have the form

$U_{cd}(\theta)=e^{-i\theta A},$ (2)

where $A\in A_{\lambda}$ and $A_{\lambda}$ is a set of all the terms computed
from the NC method. This is advantageous for VQAs in the sense that this
condition gets rid of most of the terms from the ansatz which makes it
implementable in the near-term devices and as these algorithms aim to find
approximate solutions, this ansatz should lead to good solutions if a suitable
optimization strategy is used. This claim can also be backed by QOC theory
where the dynamical Lie algebra of non-commuting control Hamiltonians can be
computed in order to understand the performance of the ansatz [34]. However,
as the terms in the CD pool operators, $A_{\lambda}$ are calculated both by
executing only odd commutations (See Methods .1), it will yield a relatively
compact pool of operators. Thus, the NC method provides us with a way to
truncate the full expansion and then select the appropriate terms from the
reduced pool of operators.

Apart from that, an important task is to choose the parameterization of the
ansatz. There are several ways to accomplish this, for instance by studying
the symmetries of the problem Hamiltonian [35]. Given we are relying on few
terms from the NC method, it is beneficial that each term has its own free
parameter, increasing the degrees of freedom of the ansatz.

Summing up, the algorithm will consist of a PQC initialized in the ground
state of $H_{mixer}$. Then we choose a pool of operators obtained from Eq. (2)
to construct the ansatz, where each term has its free parameter to be
optimized by a classical optimizer. The number of free parameters will depend
on the number of interaction terms in the Hamiltonian. A schematic diagram of
hardware-efficient ansatz, CD-inspired ansatz, and problem-inspired ansatz is
shown in Fig. 1.

Figure 2: Success Probability as a function of 20 randomly initialized
instances for $6$, $7$, and $8$ amino acid proteins with $N=6$, $N=9$, and
$N=13$ qubits respectively using the CD-inspired ansatz with $p=1$ layer,
maximum 500 iteration steps and gradient-based optimizers Adam and Adagrad
were used for classical optimizations.

Performance analysis. In this section, we study the application of the CD-
inspired ansatz to various proteins with different numbers of amino acids.
These include the amyloid-beta peptide sequence (KLVFFA) which translates to a
6-qubit system, the Neuropeptide-alpha bag cell (APRLRFY), which translates to
a 9-qubit system, cyclic peptide inhibitor (AVDINNNA) which translates to a
13-qubit system and Oxytocin (CYIQNCPLG) which translates to a 17-qubit
system. Each of the letters represents an amino acid, for example, A: Alanine,
C: Cysteine, D: Aspartic Acid, among others, with the Miyazawa and Jernigan
(MJ) interactions [36].

In each case, we generate a 5-local Ising Hamiltonian (See Methods .2)
intending to reach the ground-state, that shows the protein configuration by
minimizing the expectation value. By considering the NC commutator in Eq. (7)
with $l=2$, we obtain a set of operators with increasing locality. We truncate
these terms to up to two-body terms, which result in
$U_{cd}=\\{Y,YZ,ZY,XY,YX\\}$. Here, each of the terms shows the exponentiation
of the corresponding Pauli terms, for example
$XY=e^{-i\sum_{i,j}J_{ij}\sigma_{i}^{x}\sigma_{j}^{y}}$. Out of these, we
select $\\{Y,YZ\\}$ where $Y=e^{-i\sum_{m}\sigma_{m}^{y}}$ with $m=0,...,N$,
and $YZ=e^{-i\sum_{i,j}J_{ij}\sigma_{i}^{y}\sigma_{j}^{z}}$ where $(i,j)$
correspond to two-body interaction sites of the Hamiltonian with coefficients
$J_{ij}$ (Fig. 1(b)). The number of two-body terms $N_{2loc}\leq N(N-1)/2$
make this ansatz hardware-efficient, in the sense that its experimental
implementation is much more feasible. Regarding the optimizable parameters,
each of the gates that are applied has its free parameter. Hence, the number
of parameters per layer is $R=N_{2loc}+N$ so the parameter scaling is
$\mathcal{O}(N^{2})$. In Appendix A, we have shown the parameterization as a
function of the system size for various ansatz.

For the classical optimization part, we implement stochastic gradient-descent-
based optimizers, called Adam [37] and Adagrad [38], where the gradients are
computed using the parameter-shift rule [39]. For each protein, we run the
algorithm 20 times with random initial parameters for the $p=1$ layer. To
quantify the performance of the algorithm, we use success probability as a
metric, showing the probability of getting the ground state at the end of the
algorithm. We also study the expectation values as a function of the iteration
steps to understand the convergence.

Figure 3: Energy as a function of iterations for $N=13$ qubit protein
AVDINNNA. Results show simulator data for 100 iterations using Adam optimizer
comparing QAOA and CD-inspired ansatz. Dashed lines display the average energy
of 20 instances considered. Crosses show the energy variation of the best
instance. Shaded regions indicate the standard deviation. The green line shows
the exact ground state energy.

To benchmark the performance, we implemented $p=1$ algorithm as a function of
the number of instances. For each of them, the ansatz was randomly initialized
and the success probabilities are shown in Fig. 2. The ansatz performs
extremely well for all the cases we investigate but with increasing system
size, the number of successful instances decreases. This is due to the fact
that the solution space increases drastically with system size making it
difficult to find a global solution. For the numerical simulations, we set the
number of iterations to 500 with the tolerance of $10^{-6}$, in the sense that
the iterations will stop once this tolerance is reached. The ‘unsuccessful’
instances show almost zero success probability, this behavior can be
attributed to the fact that the spectrum of energy eigenstates is densely
packed and has many degenerate excited states, so there is a finite
possibility of the output state being stuck at the lower excited states with
high probability. These results were computed with the intent to check how
efficiently the algorithm can reach the exact ground state. However, the
hybrid classical-quantum algorithms usually aim at finding approximate
solutions near the ground state. The observed trend of decreasing successful
instances with increased system size shows that scaling to a higher systems
size might require good initialization techniques to get to the exact ground
state.

To evaluate CD-inspired ansatz against state-of-the-art quantum algorithms, we
begin by doing a performance comparison with a well-known problem-inspired
ansatz: QAOA. To do so, we studied the convergence of 20 instances with $p=1$
layer, each having random initialization, with $N=13$ qubits protein AVDINNNA
and the results are shown in Fig. 3. Average energy with standard deviation
and the best instance (in terms of convergence) is plotted as a function of
iteration steps and compared with the exact ground state energy. Indeed, the
performance of the CD-inspired ansatz is much better than QAOA. This
performance enhancement can be attributed to the difference in the optimizable
parameters for both algorithms. Only two parameters result in low
expressibility, and to counter this challenge, a high $p$ ansatz is required.
Taking into consideration that QAOA ansatz at $p=1$ already contains five-body
terms, increasing layers would serve to be problematic as the two-qubit gate
errors will start to accumulate and the circuit depth will also increase,
leading to decoherence. Apart from this, the standard deviation of the average
energy of CD-inspired ansatz is initially high but it reduces as the iteration
steps increase. Hence, the algorithm will lead to states close to the ground
state independent of the initial parameters chosen. Lastly, the algorithm
reaches the approximate solutions in a relatively low number of iteration
steps. This behavior is useful in the hardware setting as it will reduce the
run-time of the algorithm.

Figure 4: Energy as a function of iterations for $N=17$ qubits system
comparing CD-inspired ansatz and hardware-efficient ansatz. Numerical
simulations were performed for 100 iterations with the Adam optimizer. Dashed
lines show the average energy of the 20 instances considered and Crosses show
the best instance. The inset plot shows the minimum energy achieved during the
optimization of the best instance. The green line shows the exact ground
state.

After comparing to a problem-inspired ansatz, we compare CD-inspired ansatz
with a hardware-efficient ansatz (HEA). To do so, we implemented a circuit
with parameterized $Y$ rotation applied to all qubits followed by CNOTs to the
cyclic nearest neighbors for entanglement, followed by parameterized $Y$
rotation applied to all qubits (Fig. 1(a)). We study a nine amino-acid protein
CYIQNCPLG that translates to a $N=17$ qubits system by comparing both average
energy and best instance for $p=1$ algorithm and results are shown in Fig. 4.
The inset plot shows the minimum energy obtained during the optimization of
the best instance. We observe that the minimum with the best CD-inspired
ansatz instance is much closer to the exact ground-state energy as compared to
HEA. Regarding the convergence with iteration steps, the HEA has a relatively
smoother convergence as compared to the CD-inspired ansatz. As far as the
circuit evaluations are concerned, the parameter-shift method will require 2
evaluations per parameter to estimate the gradients, hence the CD-inspired
ansatz will take $2(N_{2loc}+N)$ while HEA will take $2N$ circuit evaluations.
This is a challenge since, at larger system sizes, these would increase
drastically. To circumvent this, different methods to compute the gradients
can be adopted. For instance, Ref. [40] studies a method that can compute the
gradients with only two circuit evaluations; independent of the number of
circuit parameters. Besides this, the energy difference between the ground
state energy $E_{GS}$ and the minimum energy obtained from the best CD-
inspired ansatz instance $E_{cd}$ is $E_{cd}-E_{GS}\approx 0.54$. Thus, even
at a large system size, we get solutions very close to the ground state. With
the CD-inspired ansatz, we observed that for a system size as high as $N=17$
qubits, the ansatz finds it hard to converge to some steady value at 100
iterations which might be due to the classical optimization routines or lower
iteration steps. This also means that near the ground state, the energy
landscape is extremely featured which makes it harder for the local optimizer
to converge to a particular energy value.

Figure 5: Energy as a function of iterations for $N=9$ qubits with noise model
for ibmq_guadalupe. Numerical simulations were performed for 100 iterations
with Adagrad optimizer for 5 instances. Blue dots show the average of these
instances and the red-shaded region shows the standard deviation. The inset
shows the last 10 iteration steps. The green line shows the exact ground state
energy.

In order to study the performance of the algorithm against noise, we
considered a model corresponding to ibmq_guadalupe device from IBM. This model
uses the actual backend parameters to mimic noise with some exemptions,
namely, the cross-talk errors and leakage errors are not included [41]. Noisy
simulations are key to understanding how the energy landscape changes under
the effect of noise. In Fig. 5, we take the optimal initial parameters (of
ideal simulation) of a $N=9$ qubit system APRLRFY with $p=1$ and perform the
algorithm against the noise model five times with Adagrad optimizer and plot
the average energy as a function of iterations. The aim is to check how much
noise changes the convergence of the optimal algorithm.

We can observe that even in the presence of noise, the ansatz does work
considerably well. Having said that, the final convergence energy is lifted
considerably as compared to the ground state energy and the algorithm is not
exactly converging. There are still fluctuations as we reach the end (refer
inset Fig. 5), which is natural due to existing noise. This gives an intuition
about the energy landscape when CD-inspired ansatz is implemented. From Fig.
5, it is evident that the energy landscape with CD-inspired ansatz and
gradient-based optimizers is smooth enough to take the state near to the
ground state but there exist lots of local minima near the exact ground state.
These landscape features make it easier for the classical optimizer to get
near to the ground state but to reach exactly to the ground state becomes
difficult with increasing system size. This fact is also evident from Fig. 2,
as with increasing system size the number of successful instances decreases.
Hence, at higher system sizes, there might be a requirement for global
optimizations to efficiently reach the ground state with a high success
probability. Also, these noisy simulations were run without any error
mitigation techniques. The involvement of error mitigation techniques is
crucial at the time of experimental implementation. There has been significant
attention in studying error mitigation techniques [42, 43, 44, 45, 46, 47,
48]. Nevertheless, we still are able to reach considerable approximate
solutions while performing the algorithm under noise.

With the advantages of implementing this ansatz, several challenges also need
to be addressed. First, the choice of suitable CD terms from the pool of NC
commutation operators in this work is heuristic but clever techniques need to
be developed for choosing the dominant CD term that performs the best
depending on the problem. It can also be seen that, when the ansatz does not
perform well, the ground state success probability is close to zero. This
means that when we scale to higher system sizes, it might be difficult to find
the exact ground state as the size of the Hilbert space will be huge. Lastly,
although we have successfully reduced the circuit depth, there will still be a
requirement for additional circuit optimization strategies to improve the
results while performing experiments on real hardware, which we show in the
next section for several cloud quantum computing platforms.

Figure 6: Output probability distribution of $N=13$ AVDINNNA protein and
$N=17$ CYIQNCPLG protein on a trapped-ion system: Quantinuum system H1 with
1000 shots. (a) show the $N=13$ case and (b) show the $N=17$ case. (a1) and
(b1) show the graph corresponding to two-body interactions implemented in the
CD-inspired ansatz. Blue edges show the present two-body connections while the
purple edges show the connections that are absent. (a2) shows the optimal
protein configuration with a dotted green line depicting the connection of the
nearest-neighbor interaction. (b2) show the optimal configuration of protein
obtained from exactly solving the problem whereas (b3) shows the protein
configuration obtained in the experiment. In (b3) amino acids ‘P’ and ‘G’ are
overlapping.

Experimental implementations. In this section, we implement the proposed CD-
inspired ansatz on different available noisy hardware and emulators,
specifically on trapped-ions and superconducting systems. Having various
native gate sets and connectivity, all these devices pose different challenges
that require to be dealt with to get appreciable results. Details about the
hardware, circuit optimization, and error mitigation techniques are given in
Appendix C.

Figure 7: Output probability distribution of $N=9$ qubits by implementing the
optimal circuit on (a) IBM ibmq_guadalupe where the experiment was performed
with 8192 shots and (b) Google’s quantum virtual machine rainbow [49] where
the experiment was performed with 10000 shots. Dark-colored bars show the
ground-state probability of the physical qubits whereas light-colored bars
show the rest of the distribution (a1) and (b1) show the hardware topology and
selected qubits are shown using red color. (a2) and (b2) both show the optimal
protein configurations with the nearest neighbor connection between ‘A’ and
‘F’ shown by a dotted green line.

####  .0.1 Quantinuum Trapped-ions

We implemented two systems, 8 amino acid protein AVDINNNA ($N=13$ qubits) and
9 amino acid protein CYIQNCPLG ($N=17$ qubits) with $p=1$ layers on the
Quantinuum H1-1 [50] device. For the trapped-ions system, all the qubits are
identical and errors depend upon the interaction zones, so the selection of
qubits becomes trivial since we can choose any. In both cases, the system was
initialized in the $\ket{+}^{\otimes N}$ state by applying Hadamard gates to
all the qubits. Following that, we implement parameterized
$R_{y}({\theta_{i}})$ rotations on all qubits, and the interaction terms
$YZ(\theta)$ were constructed by the native $ZZ(\theta)$ interaction. This is
done by applying two rotations, $YZ^{ij}(\theta_{m})\equiv
R_{x}^{i}(\pi/2)~{}ZZ^{ij}(J_{ij}\theta_{m})~{}R_{x}^{i}(-\pi/2)$ (refer Fig.
10(a)). For $N=13$ system, the Hamiltonian we considered had $N_{2loc}=52$
two-body interactions and for system with $N=17$ qubits, $N_{2loc}=80$ two-
body interactions were present. Graphs in Fig. 6(a1) and Fig. 6(b1) show the
full connectivity with blue edges showing the present interactions and purple
edges showing the interactions that are absent. When dealing with the real
hardware, gates are applied in a parallel manner to reduce the circuit depth
but this comes at a cost of performing multiple operations at the same time.
In this system, there is a limit on how many parallel operations can be
performed efficiently, so we apply the gates such that they do not exceed 5
operations at a time. Due to the limited hardware access, we ran the
optimization part on a local simulator with Adam optimizer and 500 iteration
steps and implemented the circuit obtained with the optimal parameters on the
real hardware.

In Fig. 6, we show the probability distribution from the real hardware for
both systems. Both experiments were performed with 1000 shots. For $N=13$
qubits, we achieve around 85% probability of getting the ground state shown in
Fig. 6(a), while Fig. 6(a2) shows the folded protein in 3D. The nearest-
neighbor amino acid connection is ‘V’–‘N’, which gives us $q_{2,7}=1$. The
optimal turn sequence is $t_{expt}=[\bar{1},0,\bar{1},2,\bar{0},1,\bar{0}]$.
On the other hand, Fig. 6(b) shows the $N=17$ qubits case with around 79%
probability of the output state achieved during the experiment which is the
4th excited state of the Hamiltonian. As the system size is large and we are
implementing only $p=1$ layered ansatz, the classical optimization routine
gets close to the ground state energy but still reaches local minima.
Fig.6(b2) shows the protein configuration with the ground state, while
Fig.6(b3) is the protein configuration with the fourth excited state. The
optimal turn sequence is $t_{opt}=[\bar{1},0,\bar{3},1,\bar{0},2,\bar{1},3]$
and the turn sequence we get while experimenting is
$t_{expt}=[\bar{1},0,\bar{3},2,\bar{0},3,\bar{1},1]$. We can notice that the
last turn taken by the amino acid is non-optimal which leads to overlap at the
second last lattice point. For optimal configuration, the nearest-neighbor
connections are ‘C’–‘C’ and ‘Y’–‘G’, and hence $q_{1,6}=1$ and $q_{2,9}=1$.
However, with the 4th excited state the connections are ‘Y’–‘G’ and ‘Y’–‘P’,
hence $q_{2,9}=1$ and $q_{2,7}=1$. So, for the fourth excited state, there is
one connection and several turns that differ from the exact ground state. In
terms of the energy difference, both these configurations differ by a factor
of 0.54. This is close to the actual ground state and, thus, the algorithm is
likely to jump to the excited state. Implementation of our hybrid quantum
algorithm results in high success probability. This is due to the all-to-all
connectivity and the fact that the two-body interactions in the circuit can be
applied effectively with native gates.

####  .0.2 IBM Superconducting chip

We implemented the 7 amino acid protein APRLRFY on 16-qubit ibmq_guadalupe
[51] device with $p=1$ layer of the ansatz. To choose 9 out of the 16
available qubits, we apply a subgraph isomorphism algorithm [52] to get the
best layout possible. As usual, we start with the $\ket{+}^{\otimes N}$ state
by applying the Hadamard gate to all the qubits and then apply
$R_{y}(\theta_{i})$. The two-qubit gate decomposition in terms of the two-
qubit native gate is given by $YZ^{ij}(\theta_{m})\equiv
R_{z}^{i}(-\pi/2)~{}CR^{ji}(J_{ij}\theta_{m})~{}R_{z}^{i}(\pi/2)$ (refer Fig.
10(b)). We arrange these gates in parallel before the optimization such that
the resultant circuit is as shallow as possible. We ran the optimization with
this circuit on a local simulator and obtained the optimal parameters
$\theta_{opt}$ as before.

We show the resultant probability distribution in Fig. 7(a) with 8192 shots.
We can achieve a ground state probability of around 20%. Apart from choosing
the best layout, we implemented several additional strategies like SWAP
strategy for achieving the desired connectivity [53], native gate and pulse
optimization strategies [54], dynamical decoupling [55], and finally
measurement error mitigation techniques [56]. Detailed discussions about these
techniques are given in Appendix C. After the circuit optimization and error
mitigation, the success probability is enhanced from around 8% to 20%. Fig.
7(a2) show the configuration of the protein. The nearest neighbor connection
is between ‘A’–‘F’, and hence $q_{1,6}=1$. The optimal turn sequence is given
by $t_{expt}=[\bar{1},0,\bar{3},1,\bar{0},1]$. This configuration is one of
the two most stable (the other one is shown in Fig. 7(b2)) configurations.
Scaling to higher qubits for IBM devices is a major challenge due to the
increased requirement of SWAP gates and finite coherence times of the qubits.
Hence, CD-inspired ansatz is hardware-implementable but still requires
improvements to make it scalable for hardware with low connectivity.

####  .0.3 Google QVM

We implemented a 7 amino acid protein APRLRFY ($N=9$) with $p=1$ layers of the
ansatz on quantum virtual machine rainbow. Since we opted for a 2D grid
connectivity, we selected one out of three possible $3\times 3$ grids with low
error rates to map our problem. The native entangling gate $CZ$ was selected
and $YZ(\theta)$ interactions were implemented by first creating
$ZZ^{ij}(\theta_{m})\equiv
H^{j}CZ^{ij}H^{j}~{}R_{z}^{j}(J_{ij}\theta_{m})~{}H^{j}CZ^{ij}H^{j}$. From
here, $YZ^{ij}(\theta_{m})\equiv R_{x}(\pi/2)ZZ^{ij}(\theta_{m})R_{x}(-\pi/2)$
(Fig. 10(c)). In this implementation, $N_{2loc}=25$ two-body interactions were
present in the Hamiltonian.

After performing the circuit optimization (Appendix C), the probability
distribution of the ground state with 10000 shots is shown in Fig. 7(b).
Around 48% success probability was achieved with the optimized circuit. This
behavior is mainly due to the circuit decomposition and the connectivity of
the system. Also, as this is a noisy emulator, the real hardware is expected
to show a lower success probability. The folded protein is shown in Fig.
7(b2). The nearest neighbor connection is between ‘A’–‘F’ which is shown by a
green dotted line which means that interaction qubit $q_{1,6}=1$. The optimal
turn sequence is given by $t_{expt}=[\bar{1},0,\bar{3},1,\bar{0},2]$ . We can
observe that as compared to the previous case, the change in this sequence is
only the last turn. Both these cases are optimal and thus the Hamiltonian we
considered for APRLRFY protein has a doubly degenerate ground state.

## Discussions

We proposed a hybrid digitized-counterdiabatic quantum algorithm to
investigate the protein folding problem. The parameterized quantum circuit
associated with this algorithm is inspired by counterdiabatic protocols, has
$\mathcal{O}(N^{2})$ parameterization, and consists of only one-qubit and two-
qubit gates. We applied this algorithm to a tetrahedral lattice protein
folding problem, encoded in a 5-local Ising Hamiltonian where the ground state
contains the corresponding protein configuration. We study various proteins
with increasing amino acid chains and show that the proposed algorithm has an
excellent performance in terms of convergence and circuit depth. These results
outperform previously utilized quantum algorithms, enhancing the experimental
feasibility with various circuit optimization strategies. We prove that claim
by implementing up to 9 amino acid proteins with 17 qubits on several quantum
hardware like trapped-ions and superconducting systems and achieving high
success probabilities.

This work paves the way for implementing problem-inspired ansatz to industrial
use cases in the current NISQ era by utilizing digitized counterdiabatic
protocols. We believe this quantum algorithm can be extended to other relevant
applications. There is also a connection between counterdiabatic protocols
with adiabatic gauge potentials and dynamical Lie algebra. As mentioned
before, some challenges need to be addressed, for instance, the sensitivity
toward the initial parameters and the choice of the appropriate CD terms from
the adiabatic gauge pool. Using various modifications like global optimizers
would be interesting, as these algorithms find it difficult to reach the exact
ground state at high system sizes. This is crucial as the protein structure
changes significantly if we end up in an excited state. This also motivates
the use of purely digitized-counterdiabatic quantum algorithms where the task
is to find the exact ground state [11]. We also emphasize that using digital-
analog quantum computing with digitized-counterdiabatic quantum computing can
also be crucial. We believe this work advances quantum computing, bringing us
one step closer to practical quantum advantage, which would have to challenge
success in classical computing including the recent AlphaFold achievements
[57].

## Methods

###  .1 Variational quantum algorithms and counterdiabaticity

In VQA, a circuit ansatz and a classical optimizer combine to solve an
optimization problem. The task of the circuit ansatz is to generate quantum
states based on the optimal parameters provided by the classical routines. The
classical parts can be done by gradient-based optimizers (like Adam or
Adagrad) or gradient-free optimizers (like COBYLA or POWELL). The goal is to
minimize a cost function $C$ that can take various forms depending upon the
problem. But in our case, we will use the expectation value of the problem
Hamiltonian $H$ given by

$C(\vec{\theta})=\bra{\psi(\vec{\theta})}H\ket{\psi(\vec{\theta})},$ (3)

where $\vec{\theta}=\\{\theta_{1},\theta_{2},...\\}$ shows the parameters
associated with the circuit ansatz. As mentioned earlier, both problem-
inspired ansatz and hardware-efficient ansatz have their advantages and
disadvantages. Despite the promising performance of the problem-inspired
ansatz, the current hardware experience several bottlenecks like limited qubit
connectivity, imperfect implementation of gates, limited coherence times, etc.
This makes the implementation impractical, so that hardware-efficient ansatz
is preferred. Generally, the hardware-efficient ansatz is of the form

$U(\vec{\theta})=\prod_{k=1}^{p}U_{k}(\vec{\theta}_{k})W_{k},$ (4)

where $\vec{\theta}_{k}$ are the optimizable parameters and $p$ is the number
of layers. $U_{k}=exp[-i\theta_{k}V_{k}]$, where $V_{k}$ is a Hermitian
operator. $W_{k}$ are non-parameterized gates usually consisting of 2-qubit
connecting gates like $CNOT$s and $U_{k}$ are parameterized single qubit
rotations.

On the other hand, the problem-inspired ansatz use evolutions of the form

$U(\theta)=e^{-i\hat{g}t},$ (5)

where $\hat{g}$ is a Hermitian operator and $t$ is a parameter. These
$\hat{g}$ are derived from the system of interest. For instance, in QAOA [2],
$\hat{g}=H$ corresponds to the problem Hamiltonian, thus Eq. (5) resembles the
trotterized-time evolution. In QAOA, the quantum circuit consists of two
unitaries: Hamiltonian term $U_{c}(\gamma)$ and mixing term $U_{b}(\beta)$
applied $p$-times to the initial state $\ket{\psi_{0}}$. Here,
$(\gamma,\beta)$ are the parameters to optimize by the classical optimizer.
Along these lines, the evolution looks like

$\ket{\psi_{f}}=U_{b}{(\beta_{p})}U_{c}{(\gamma_{p})}U_{b}{(\beta_{p-1})}U_{c}{(\gamma_{p-1})}\dots
U_{b}{(\beta_{1})}U_{c}{(\gamma_{1})}\ket{\psi_{0}},$ (6)

where $\ket{\psi_{f}}$ shows the output state, while generally
$U_{b}(\beta)=e^{-i\beta\sum_{i}\sigma_{i}^{x}}$ and
$U_{c}(\gamma)=e^{-i\gamma H}$. Mixer term $U_{b}(\beta)$ can take several
other forms as well [32]. The aim is to minimize the cost function given by
Eq. (3). QAOA directly relates to the quantum adiabatic evolution [58] hence
it has believed to be a successful algorithm at large $p$ layers due to the
adiabatic theorem. That being said, implementing circuits with large $p$
results in high circuit depths, not feasible for current near-term devices.
Many adaptations to QAOA have been reported [32, 59, 60]. Among them, CD
protocol has been of interest recently [61, 16, 12], from which the newly
proposed digitized-counterdiabatic QAOA (DC-QAOA) reports that the addition of
CD terms to the usual QAOA ansatz gives significant improvements in the
performance of QAOA. Finding CD terms is a critical task and is done by
obtaining approximate CD terms by the adiabatic gauge potentials using the
nested commutator (NC) method [62]. In the considered NC method, the
approximate CD terms are given by $A_{\lambda}^{(l)}$ which can be calculated
as

$A_{\lambda}^{(l)}=i\sum_{k=1}^{l}\alpha_{k}(t)\underbrace{[H_{a},[H_{a},......[H_{a},}_{2k-1}\partial_{\lambda}H_{a}]]],$
(7)

where $l$ shows the order of expansion and
$H_{a}(\lambda)=(1-\lambda(t))H_{mixer}+\lambda(t)H$. The CD term is then
digitized and the coefficient $\alpha$ is considered as an additional free
parameter along with $(\beta,\gamma)$ to increase the ansatz expressibility.
Summing up, DC-QAOA uses three unitary terms iteratively $p$-times:
Hamiltonian term, mixer term, and CD term to minimize the cost function more
effectively. However, we have to pay the price in terms of increased circuit
depth per layer. To upgrade this method, we propose a CD-inspired circuit
ansatz that is also hardware implementable. Hence, it partakes the advantages
of both problem-inspired and hardware-efficient ansatz (see Results).

###  .2 Protein folding

Figure 8: Schematic diagram showing the tetrahedral lattice structures. (a)
shows the lattice $\mathcal{P}$ and (b) shows the inverted lattice
$\mathcal{Q}$. The turns taken by the amino acids will be one of the four
directions ($t=0,1,2,3$ or $\bar{0},\bar{1},\bar{2},\bar{3}$) and both
lattices are switched at each turn. (c) shows the mesh made of $\mathcal{P}$
(shown in red) and $\mathcal{Q}$ (shown in blue), while the green line shows a
schematic of the turns taken by an arbitrary protein in 3D.

In this section, we describe the current quantum scenario regarding protein
folding, tackled as a lattice problem where amino acids are sequentially added
to a given lattice such that the total conformation energy is minimum. The
lattice can be 2D (plane) or 3D (cubic or tetrahedral). The complexity
increases drastically while going from 2D to 3D because of the highly
increasing number of possible protein configurations. After selecting the
lattice structure, the next important step is to choose the type of encoding.
The widely studied encoding types are position encoding, where the lattice
coordinates are encoded as qubits, and turn encoding, where the turn direction
is encoded as qubits. In position encoding, the solution bitstring will
represent the coordinates. Here, the respective amino acids should be placed
for energy minimization, and, for turn encoding, the solution contains
information about the turns taken by each amino acid sequentially to result in
minimum energy configuration. The difficulty and the form of the Hamiltonian
are highly dependent on the type of encoding chosen. Another vital task is to
assign amino acids’ interaction energies. There are two types of interactions,
namely, the Hydrophobic model (HP), where the interaction coefficients are
limited to binary values [28], and the Miyazawa and Jernigan (MJ) interaction,
where the coefficients are arbitrary depending upon the contact of amino acids
[36]. Depending upon the structure, we have to define some constraints that
will help us avoid configurations that are not allowed, like stacking
different amino acids on the same lattice point.

In this work, we consider the 3D tetrahedral lattice model introduced in Ref.
[6] with the turn encoding and MJ interactions. The qubits are encoded to
represent the turn $t_{i}$ taken by the $(i+1)^{th}$ amino acid after $i^{th}$
amino acid. There are two sets of lattice $\mathcal{P}$ and $\mathcal{Q}$
exactly inverse to each other where the possible four turns are shown by
$t=0,1,2,3$ for $\mathcal{P}$ (and $\bar{0},\bar{1},\bar{2},\bar{3}$ for
$\mathcal{Q})$ and each of the numbers shows the different direction of the
tetrahedron shown in Fig. 8. We devote two qubits to encode each turn so that
the turns are given by $t_{j}=q_{j}q_{j+1}$ which scales as $2(N_{a}-3)$ where
$N_{a}$ is the number of amino acids considered. $\mathcal{P}$ and
$\mathcal{Q}$ are changed at every turn so that even turns can be shown by
$\mathcal{P}$ and odd turns can be shown by $\mathcal{Q}$. An example of the
turn bitstring will be of the form $t=[\bar{1},0,\bar{3},...]$. The schematic
diagram is shown in Fig. 8. The initial two turns can be fixed to $t_{1}=1$
and $t_{2}=\bar{0}$ due to the symmetry of space we consider. In addition, one
more qubit can also be saved due to space symmetry if no side chains are
considered. Hence, the conformation qubits $Q_{c}$ will look like

$Q_{c}=[00][01][q_{5}1][q_{7}q_{8}].....[q_{2(N-1)-1}q_{2(N-1)}]\,,$ (8)

where $q_{6}=1$ due to spatial symmetry. To keep track of the turns, a
function $g_{m}$ where $m={0,1,2,3}$ is constructed and this function will
return 1 if the axis $m$ is selected at $i^{th}$ turn. The shortest distance
between any two beads can be found by keeping track of the number of turns the
beads had taken with lattices $\mathcal{P}$ and $\mathcal{Q}$. As far as the
constraints are concerned, there are two types, namely, growth constraints
that penalize unwanted conformations and chirality constraints that enforce
correct values. To impose these, two terms $H_{gc}(Q_{c})$ and $H_{ch}(Q_{c})$
are added to the problem Hamiltonian with positive Lagrange multipliers
$(\theta_{gc},\theta_{ch})$. Details about how to create these functions are
given in Ref. [6]. Along with $Q_{c}$, a set of qubits $Q_{in}$ are included
that takes account of the interactions between the nearest-neighbor beads in
the protein chain. $Q_{in}=q_{i,j}$ are a set of two-indices qubits that have
information about nearest neighbor contact with $i^{th}$ and $j^{th}$ bead. If
the contact occurs, the energy $e_{i,j}$ is applied to the Hamiltonian
$H_{in}$. We consider the nearest neighbor interactions, where the contact is
considered if the distance between two non-consecutive beads is unity. Thus
the total qubits $Q_{tot}=\\{Q_{c},Q_{in}\\}$ and the Hamiltonian $H$ is given
by

$H(Q_{tot})=H_{gc}(Q_{c})+H_{ch}(Q_{c})+H_{in}(Q_{in}).$ (9)

The conversion of $H(Q_{tot})$ into an Ising Hamiltonian $H$ will result in

$\displaystyle H$
$\displaystyle=\sum_{i}h_{i}\sigma_{z}^{i}+\sum_{ij}J_{ij}\sigma_{z}^{i}\sigma_{z}^{j}+\sum_{ijk}K_{ijk}\sigma_{z}^{i}\sigma_{z}^{j}\sigma_{z}^{k}$
(10)
$\displaystyle+\sum_{ijkl}L_{ijkl}\sigma_{z}^{i}\sigma_{z}^{j}\sigma_{z}^{k}\sigma_{z}^{l}+\sum_{ijklm}M_{ijklm}\sigma_{z}^{i}\sigma_{z}^{j}\sigma_{z}^{k}\sigma_{z}^{l}\sigma_{z}^{m},$

where the indices and the coefficients depend on specific proteins. Hence, $H$
is a 5-local Ising Hamiltonian whose ground state shows the optimal turn
sequence for the folded protein. The specific reason for the selection of this
model is that this model has a tetrahedral lattice that captures many physical
and chemical properties. This model uses a lower number of qubits for a
specified amino-acid chain as compared to other methods, but this comes at the
cost of increasing the locality of the Hamiltonian. In Results, we show that
this increased locality does not affect our CD-inspired ansatz, and we can get
to the optimal solutions by using only 2-local terms in the PQC. This happens
in contrast to QAOA, where the circuit ansatz will require 5-local terms,
which makes it challenging to implement on a real device.

###### Acknowledgements.

We acknowledge the Azure quantum credits program for providing access to the
Quantinuum trapped-ions systems. P.C. acknowledges Mikel Garcia de Andoin and
Martin Larocca for useful discussions. This work is supported by EU FET Open
Grant EPIQUS (899368), QUANTEK project (KK-2021/00070), the Basque Government
through Grant No. IT1470-22, the project grant PID2021-126273NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ERDF
Invest in your Future”, NSFC (12075145), and STCSM (2019SHZDZX01-ZX04). X.C.
acknowledges ayudas para contratos Ramón y Cajal–2015-2020 (RYC-2017-22482).

##  Appendix A Parameter Scaling

In Fig. 9, we show how the parameters of $p=1$ CD-inspired ansatz of the
protein folding problem vary as a function of system size as compared to both
HEA and the case where all the possible two-body interactions are present.

Figure 9: Number of optimizable parameters as a function of system size for
various ansatz. The blue line shows hardware-efficient ansatz
parameterization, the red line shows CD-inspired ansatz parameterization, and
the green line shows the CD-inspired ansatz parameterization where all-to-all
two-body interactions are present contrary to the protein folding problem.

##  Appendix B Native gate decomposition

The native gate decomposition of $YZ(\theta)$ interaction concerning various
hardware is shown in Fig. 10. Fig. 10(a) shows the decomposition with respect
to the $ZZ(\theta)$ interaction native to the Quantinuum trapped-ions
hardware, Fig. 10(b) shows the decomposition in terms of $CR(\theta)$ gate
native to IBM superconducting chip, and Fig. 10(c) show the decomposition in
terms of $CZ$ gates native to the Google superconducting hardware.

Figure 10: Native gate decomposition of $YZ(\theta)$ gate in real hardware for
qubits $\ket{i}$ and $\ket{j}$. (a) shows the decomposition for Quantinuum
tapped-ions hardware (native gate $ZZ(\theta)$), (b) shows the decomposition
for IBM superconducting hardware (native gate $CR(\theta)$) and (c) shows the
decomposition for Google QVM (native gate $CZ$).

##  Appendix C Experimental details

###  C.1 Quantinuum Trapped-ions

Hardware details. The Quantinuum H1-1 [50] is an all-to-all connected trapped-
ion system with 20 qubits. There are multiple interaction zones where these
physical qubits can move and the quantum operations are applied using lasers.
Due to the existence of these zones, this device can perform parallel
operations effectively. All-to-all connectivity is achieved by the physical
rearrangement of qubits to perform all the two-qubit interactions. All the
qubits are identical but the errors arise due to the quantum operations
depending upon the locality of those qubits. Typical one-qubit gate infidelity
is $4\times 10^{-5}$, while the two-qubit gate infidelity is $3\times 10^{-3}$
and the native gate-set are the two-qubit $ZZ(\theta)$ gates.

Circuit optimization and Error mitigation. Minimizing two-qubit gate error is
an essential part of the experimental implementation. Due to all-to-all
connectivity and suitable two-qubit native gate, we already have an
implementable circuit. However, intending to obtain the best results, we
strategically removed several $YZ(\theta)$ gates by analyzing the angles
associated with them. Rotations with angles near zero do not contribute highly
to the circuit but their gate errors do accumulate. Thus these gates can be
avoided to minimize two-qubit gate errors. This should be done carefully
keeping in mind the error rates and change in fidelity. The final $ZZ(\theta)$
gate count at the time of implementation was reduced to 35 for $N=13$ qubits
case and to 70 for $N=17$ qubits case.

###  C.2 Google QVM

Hardware details. The hardware from Google has Sycamore architecture with 53
transmon qubits arranged in a 2D grid. Hence, each physical qubit is connected
to at most 4 other qubits. The single-qubit gates are executed by microwave
pulses of a fixed frequency and two-qubit gates can be executed by bringing
the nearest qubits on resonance and then turning on a coupling. The two-qubit
native gates are the $CZ$, $\sqrt{iSWAP}$, and Sycamore interactions. Typical
two-qubit $\sqrt{iSWAP}$ implementation error is 1.4% per gate when applied in
parallel and the typical single-qubit gate error is 0.1% per gate. More
information can be found in Ref. [63]. In our experiment, we utilized the
quantum virtual machine (QVM) offered by Google. Google offers two QVMs-
rainbow and weber\- where a noise model is implemented that closely mimics the
actual noise of the hardware. In this work, we utilize QVM rainbow, a 23-qubit
device with square-grid lattice connectivity (Fig. 7(b1)).

Circuit optimization and error mitigation. As the connectivity is not all-to-
all, a SWAP strategy is required to implement the circuit. We apply a SWAP
strategy based on Ref. [53] that implements the circuit by using 11 SWAP gates
as all the two-qubit interactions are not present in the circuit. In order to
get the best results, we optimized the ‘moments’ of the implemented circuit. A
moment is defined as a set of operations, acting on different qubits such that
all of them can be applied at a single abstract time slice. Essentially, the
number of moments should be minimized to reduce the circuit execution time. In
other words, we arranged the circuit such that the number of operations that
can be applied at a single time is maximized. There are several ways to
achieve this, for example, aligning the circuit to the left where the maximum
possible number of operations are arranged from the start of the circuit. For
our implementation, we optimized the circuit where all the operations are
aligned into similar categories, that is, all the one-qubit and two-qubit
operations are aligned in separate moments. This method is adopted to reduce
the qubit idling since if these operations are isolated, the other qubits will
be idle, leading to errors. Since we have only used the QVM, we limit
ourselves to the circuit optimizations mentioned above. However, for the real
device, there might be a need for further circuit optimizations. This can be
done by adopting strategies like dropping moments that have a lower impact on
the circuit results and adding dynamical decoupling to resist the qubit
idling. Testing combinations of native gates to generate $YZ(\theta)$
interactions more effectively can also prove to be effective. For example,
this is shown in Ref. [64] where they use the Sycamore gates to generate the
combination of $ZZ-SWAP$ interactions.

###  C.3 IBM superconducting chips

Hardware details. IBM systems consist of fixed-frequency transmon qubits and
the gate operations are applied with microwave pulses [65]. The connectivity
is heavy-hex and for our implementation, we chose qubits with linear
connectivity (Fig. 7(a1)). The two-qubit native gate for this device is the
cross-resonance gate
$R^{ij}_{zx}(\theta)=e^{-i\frac{\theta}{2}\sigma_{i}^{z}\sigma_{j}^{x}}$ also
known as $CR(\theta)$ gate. The average single-qubit gate error is around
0.05% and the average two-qubit $CNOT$ gate error is around 1.4%. Error map of
the ibmq_guadalupe device when the experiment was performed is shown in Fig.
11. This error map shows the connectivity of the qubits, the Hadamard gate
error of each of the qubits, the $CNOT$ error of all the present connections,
and the read-out errors of the qubits. The qubits shown within the red box
were selected for the experiment.

Figure 11: Error map of IBM’s 16 qubit ibmq_guadalupe device when the
experiment was performed. The experiment was performed using the qubits
highlighted in red color.

Circuit optimization and error mitigation.

Best layout- Qubit selection is a crucial task while implementing the circuit
as each qubit possesses its own single-qubit gate errors and two-qubit gate
errors with the connecting qubits. To select the best qubits from the device,
we implement a sub-graph isomorphism algorithm that selects 9 qubits out of 16
that minimize the expected gate errors [52].

SWAP strategies- IBM’s heavy-hex connectivity scheme means there only exists
nearest neighbor coupling. As we have a linear connection of qubits, we need a
good SWAP strategy to implement our circuit using the least SWAP gates
possible. To achieve this, we implement the strategy performed in Ref. [53].
As some of the two-qubit interactions are missing, we could cover our
connectivity by applying 30 SWAP gates for the case involving $N=9$ qubits.

Native gate compilation- SWAP, $CNOT$, and $ZZ(\theta)$ gates are not native
to IBM’s chips, being translated to the native $CR(\theta)$ gate before these
circuits are executed. Ref. [54] showed how by being hardware aware, these
native gates can be translated beforehand so by the annihilation of commuting
gates we can reduce the depth of the final logical circuit. By being hardware-
specific in this regime, we could also calibrate the $CR(\theta)$ gates so
that no other translation is done. Therefore, the actual physical coupling
represents the intended interaction gate between two qubits. Hence, the final
pulse schedule gets reduced to its minimal expression according to the
specifications of the hardware while preserving the intended structure for the
ansatz. Once the circuit is reduced to its minimum expression, there are still
some techniques we can apply for noise mitigation. Some are encoded within our
pulse definition and others by statistically correcting the systematic error
upon measurement.

Dynamical decoupling- Dynamical decoupling techniques introduce different gate
schemes to minimize the noise generated by the idling effect of the qubits
while other longer-time two-qubit gates are being applied [46]. They observed
that if idling time can be used up by applying gates (that will not change the
final result), the effect of noise can be reduced significantly. A well-known
scheme is the $XY4$ scheme which applies a sequence of four rotation
operations on the $X$ and $Y$ axis. Following a symmetrized version of the
$XY4$ scheme as in Ref. [55], the $XY8$ scheme applies these rotations eight
times instead of four. For our experiment runs, we use the $XY8$ sequences
shown by

$X_{\pi}\xrightarrow{}Y_{\pi}\xrightarrow{}...\xrightarrow{}X_{-\pi}\xrightarrow{}Y_{-\pi}.$
(11)

Here, ${-\pi}$ and ${\pi}$ reflect the opposing pulse amplitude being applied
on each axis. Thus, the result of the gate application renders the $I$ and by
the active involvement, low-frequency noise is generated while idling gets
reduced which results in an overall reduction in the noise.

Measurement Error Mitigation- Finally, readout error can be seen as
systematically and consistent for a given chip, always representing a similar
error distribution along the usage of a chip. By compensating for this final
measurement error one could boost the final results to their statistically
corrected error-less version. Usually, this is done by an approach where for a
given ideally simulated probability distribution $\vec{p}_{ideal}$, upon the
measurement on a real device would yield a noisy distribution
$\vec{p}_{noisy}$ such that

$\vec{p}_{noisy}=\mathbf{A}\vec{p}_{ideal}.$ (12)

Here, $\mathbf{A}$ represents a matrix called the assignment matrix that has
information about the noise. The probability can be enhanced by finding
$\mathbf{A^{-1}}$ and thus finding the error-mitigated quasi-probabilities.
However, as the system sizes increase, the calculation of this matrix becomes
difficult. In Ref. [56], an approach is proposed where, instead of calculating
the full matrix, a reduced matrix is calculated. This is done by considering
the qubits needed for the experiment runs instead of considering the full
space, hence reducing the computational overhead. In our implementation, we
use this technique to perform measurement error mitigation.

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# Highlights of EPS HEP 2019

Department of Physics & Astronomy, University College London
E-mail

###### Abstract:

An opinionated and informal recap of highlights from the EPS HEP 2019
conference in Ghent, including some aspects of flavour physics, neutrinos,
high-density QCD, astrophysics and energy frontier collider physics, and some
thoughts about the future.

##  1 Introduction

The purpose of this write-up, as with the talk, is not to be definitive but to
give an editorialised point of view on what was a packed and exciting meeting.
As befits such a contribution, references will mostly be to other (mostly
plenary) talks at the conference, and I will not include figures, on the
assumption these will be included in the original contributions. Although it
is a necessarily personal selection, I will not dwell on a satisfying semi-
final and a stunning final from the cricket world cup, which took place during
the conference, even though they will always be associated with Ghent for me
from now on. I will say though that it was a privilege to be asked to give
this talk by the European Physical Society, especially bearing in mind the
turbulent politics between the UK and most of the rest of Europe at present.

For the rest of the contribution, I will stick to the physics. There was a lot
of it.

##  2 Flavour Physics

Impressive advances were reported in flavour physics, with Katharina Müller
presenting the first observation of charge-parity violation in charm decays,
by LHCb [1]. The time-integrated CP asymmetry is measured in neutral charmed
hadron ($D^{0}/\bar{D}^{0}$) decays to both $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$.
A combination of the new LHC Run 2 result with the Run 1 result gives a
5.3$\sigma$ difference of the measured value from zero, indicating dominantly
direct CP violation and roughly compatible with the Standard Model (SM). The
uncertainties in the SM prediction are greater than those in the data.

Müller also presented an updated combination of measurements of the unitarity
triangle angle $\gamma$, incuding a new measurement in the $B^{0}\rightarrow
DK^{*0}(D\rightarrow K\pi,KK,\pi\pi)~{}$[2]. There are some tensions, at the
2$\sigma$ level, between different measurments, worth mentioning to emphasise
the importance of measuring this angle in several different ways, since new
physics may manifest itself differently depending upon the process. Updates of
$B_{s}$ mixing from LHCb and ATLAS were also presented, using integrated
luminosities of 4.9-1 and 99.7 fb-1 respectively. These results were also
discussed by Johannes Albrecht. The main focus in his talk however was on
semi-leptonic decays of $B$ hadrons to Kaons and either $e^{+}e^{-}$ or
$\mu^{+}\mu^{-}$, and in particular the ratios and angular variables, where
the consistency with the SM is at level of around 2.5$\sigma$, a level which
has not changed with the increased precision from including part of the Run 2
data. While no individual measurement is compelling as evidence for physics
beyond the SM (BSM), taken together (and as pointed out by Marco Nardecchia in
his summary of these and other comparisons) there is a suggestive pattern
which is readily accommodated by extensions to the SM. This is clearly an
interesting space to watch, especially as more LHC luminosity is included and,
as we heard from Francesco Forti, Belle II data are on the horizon, with the
accelerator currently on its commissioning track to high luminosities and the
detector already recording collisions.

##  3 Neutrinos

That we are now into an era of precision, high statistics neutrino physics was
amply illustrated by the dataset of atmospheric neutrino events collected by
SuperKamiodande [3] and now used in global fits, with a “postage stamp” style
slide shown by Gioacchino Ranucci containing 50,000 events split into 19
different analysis samples, each with enough statistics to allow detailed
exploration of the event characteristics. As discussed by Francesca Di
Lodovico and Sylvia Pasquale, data principally from T2K and No$\nu$a are
beginning to constrain both the neutrino mass hierarchy (with the normal
hierarchy marginaly favoured) and the CP violating phase in the PMNS matrix
(with a non-zero value marginally favoured). Reactor experiments are also
making major contributions to characterising the physics of the neutrino
sector, particularly with their measurements of the mixing angle
$\theta_{13}$.

##  4 New “Standard Model” Physics

When we discuss the search for “new physics”, it grates on me sometimes, since
as we probe above the electroweak symmetry breaking scale with ATLAS and CMS,
everything we measure is in a sense new. We are measuring what are sometimes
previously unseen processes, and always in a region of physics previously
unexplored. Every time we do so and find agreement with SM, we are validating
our ideas about the fundamental constituents and forces in a qualitatively new
regime. By the standards of most fields, this is new physics.

Several examples were shown by Wolfgang Adam, Lucia Di Ciaccio and Andreas
Hoecker, with (to give just a few examples) impressive results on diboson
production, top cross sections and asymmetry (including new calculations of
NNLO spin correlations), prompt photons (now out to 2 TeV in transverse
momentum) and a dilepton+photon measurements with the full Run 2 data set from
ATLAS. As Giulia Zanderighi made clear, for the full exploitation of these
measurements our ability to make exclusive calculations, implementing the
kinematic cuts which allow comparisons to data in the fiducial phase space of
the measurement, is essential and is an area of intense effort and much
progress.

The new regimes probed, and the new precision achieved with QCD theory, means
that, as also pointed out by Zanderighi, there is now a need for both QCD and
electroweak higher order corrections to be incorporated together in
calculations. This need will become more urgent throughout the high luminosity
LHC (HL-LHC) era. An early step in this direction was in fact shown in a
parallel session by German Sborlini, in the form of a coherent QED and QCD
higher-order calculation of Drell-Yan dilepton production [4]. It seems from
this that a multiplicative rather than additive combination of the corrections
is a better approximation to the fully coherent result.

At the time, I thought I understood something about elliptical Feynman
diagrams from the fascinating talk by Claude Duhr. This understanding now
escapes me, but it seems there is an interesting and useful advance on the
boundaries between physics and mathematics there.

As predicted, jet substructure analysis has proved itself “useful more
generally in the identification of hadronically decaying massive particles
which have energy large compared to their mass” [5], and highly boosted
$W,Z,H$ and top are now an important feature of physics at the LHC. Several
new results were shown, including a very direct technique from CMS of
measuring the top mass by measuring the mass of a “fat” jet containing three
subjets from the decay of a top quark. There are interesting questions about
how these techniques will change, and what demands they will place on the
detectors, if even higher boosts are available at future colliders, and when
also the scales are such that these particles will need to be treated as
partons in the initial and final state showers. Present theoretical tools are
likely to require substantial development for us even to make a reasonable
estimate.

##  5 High Density QCD

Jet substructure also featured as a powerful tool in heavy ion physics and the
study of high-density QCD. An array of impressive new results was shown by
Marco Van Leeuwen, Dennis Perepelitsa and Carlos Albert Salgado Lopez. The
impact of the medium on QCD splitting is studied by observing differences
between the subjet momenta in proton-proton and heavy ion collisions [6, 7],
with interesting effects seen which should help characterise the medium.
However, such comparisions must be done with care, given the obersvation of
collective effects even in proton-proton collisions [8], a system thought,
before LHC data, to be too small to manifest such a thing. As Jan Fiete
Grosse-Oetringhaus strikingly put it, this challenges two paradigms at once.
What is the smallest system in which the heavy-ion “standard model” remains
valid? And can the standard tools for proton-proton physics remain standard?
Traditional high-energy physics and traditional heavy ion studies, often
sociologically and scientifically rather distant, must grow together. The
underlying QCD is the same theory, and the aim must be to demonstrate that
unified description for $ee$, $ep$, $pp$ and $AA$ is feasible, or show that
different mechanisms are justified and how they arise.

##  6 The Higgs

Even if you don’t buy my claim that most of the measurements from the LHC are
new physics, perhaps you will agree with Roberto Salerno’s statement that
Higgs physics, at least, is really new physics. Christoph Grojean in the ECFA
session on Sunday gave a powerful reminder, in the context of proposals for
future colliders [9], of just how special the Higgs boson is. It can be seen
as a new force, of a different nature to the gauge interactions known so far.
There is no underlying local symmetry, no quantised charge. It is deeply
connected to the vacuum structure of space-time. The up and down-quark Yukawa
couplings determine the relationship between the proton and neutron masses,
and thus the stability of nuclei. The electron Yukawa controls the size of the
atoms (and thus the size of the Universe?). The top quark Yukawa decides (in
part) the stability of the electroweak vacuum. The Higgs self-coupling
controls the (thermo)dynamics of the EW phase transition, and therefore might
be responsible for the dominance of matter over antimatter in the Universe.
The Higgs Boson really is special.

Salerno, Adam and Hoecker showed a plethora of new results, many using the
full LHC Run 2 dataset. ATLAS and CMS have established the existence of Higgs
couplings to the third generation charged fermions, as well as the gauge
bosons. New studies were shown on the pursuit of the second generation — muons
and charm. The limits are still factors above the SM, but are getting close
enough to have interesting sensitivity to any upward deviations, and more
precision is promised. Differential cross sections for Higgs production and
decay test whether is it pointlike, whether its couplings evolve as expected
with (for example) transverse momentum. Direct measurement of the Higgs self
coupling requires much more data, with observation possible with the full HL-
LHC, and any precision requiring a future hadron collider, just as a
measurement of the total Higgs width with a degree of model-independence
requires a lepton collider.

Finally on the Higgs, Francesco Riva detailed how intertwined the Higgs sector
is with many of the processes measured at the LHC – dibosons, top and more.
This implies that when we measure such things, we are, within the context of
the SM, making powerful consistency tests of the parameters of the Higgs
sector – or doing Higgs physics without a Higgs, as he styles it. So, if you
agreed that Higgs physics is new physics, you now have to accept my earlier
claim that so are many of the other measurements ATLAS and CMS are producing,
even if no trace of BSM physics has yet shown itself.

##  7 Dark Matter, Astroparticle Physics and Cosmology

The progress in searching for Dark Matter was described by Igor Garcia
Irastorza, Carlos de los Heros and Kfir Blum. Amongst the items which were
fresh news to me was the fact that the famous picture of the black hole at the
centre of the giant elliptical galaxy M87 [10] actually excludes some ultra-
light Dark Matter models, in which such black holes do not form. This is
partly I think a reflection of a resurgence in model building and exploration,
now the WIMP miracle is (at best) delayed. This is also reflected in the
increased interest in axion searches as described by Irastorza.

We also learned that the so-called “neutrino floor”, the point in sensitivity
at which detectors looking directly for Dark Matter scattering off normal
matter start seeing solar neutrino interactions, is not a hard floor, “more of
a swampland” in the words of de los Heros. Anisotropic detectors, directional
detection and time modulation offering possibilities for sinking below it.

The anomalously large number of positrons seen by AMS (shown by Barbara De
Lotto) remains intriguing – something interesting is going on there, whether
it is Dark Matter-related or not. And the IceCube map of the neutrino sky
shown by Elisa Bernardini offers a truly new view of the Universe, with the
obvious potential for identifying point sources perhaps from Dark Matter
annihilation, amongst other things.

Our newest messenger, gravitational waves, continues to surprise and excite.
One surprise to me was the fact that the “chirp” pattern of a neutron star
merger, as shown by Patricia Schmidt, has – via tidal distortion –
implications for the equations of state of high density QCD, as discussed by
Carlos Albert Salgado Lopez. The prospect of the Einstein telescope, which
with its sensitivity to lower frequencies should allow impending mergers to be
spotted earlier and thus observed in more channels, is mouth-watering.

##  8 The Future, and Beyond the Standard Model

And so to something on future prospects. Much of this hangs on development of
accelerator technology, of which we heard summaries from Catarina Biscari in
the ECFA session and Ralph Assman in the final session of the plenaries.
Highlights included operational crab cavities for protons in the SPS (shown by
Olivier Bruning in the parallel sessions), 2 GeV electron acceleration in the
AWAKE proton-driven plasma wakefield experiment, a dipole magnet demonstrator
so far tested at FNAL to 14 Tesla with an end goal of 15 T, and work on Nb3Sn
wire toward 16 T from companies engaged in the Future Circular Collider
project. The success (and speed) of such developments will be critical for the
long-term future of the field.

Nearer in time, we have LHC Run 3 and the HL-LHC era approaching. Isabell
Melzer-Pellmann and Marie-Helene Genest showed the results of many ingenious
and important searches for BSM physics, none of which have revealed anything
other than increasingly stringent limits. This is itself is great progress and
is having a profound impact on the theoretical field, as was described by
Giuliano Panico and alluded to several times already in this contribution.
Over the next few years, I believe we need, and will see, a profound change of
approach here.

In my opinion, while there is still important scope for novel signatures not
yet covered (for example, exotic long-lived particles, disappearing tracks,
non-standard jets, and probably other things still to be dreamt up) the main
emphasis of the experiments should shift toward making precise, model-
independent measurements which can be confronted with increasingly precise and
exclusive SM predictions. It is such confrontation in my view that gives us
the best chance of establishing whether or not we face a desert above the
electroweak symmetry-breaking scale, as far as BSM physics is concerned. Such
measurements, including those of the Higgs, seem likely to me to provide the
main legacy of the LHC, an exacting challenge laid down to future model
builders.

A clearer idea of exactly what is being measured, in terms of separating
theoretical interpretation from measurement, is needed. This implies not only
measuring in fiducial regions reflecting the detector acceptance, but also in
what is considered background and what is considered signal. An example: if we
are aiming to measure $WW$ scattering, and we see an event with two isolated
leptons and missing energy in the detector, should we really be subtracting it
because our Monte Carlo says it came from top quarks? What about off-shell
tops? It is often better to measure a final state, and do any subtraction
later as part of the interpretation. This throws down a challenge to the
theory to calculate the final states that are actually measured, of course, as
discussed above.

When it comes to interpretation of searches too, we need to on the one hand
provide the data in such a way that a search for one model may be readily
reinterpreted in terms of others with similar signatures, and on the other
hand we should be moving away from simplified benchmark models toward making
more general statements about whole classes of theory. An excellent example of
the latter approach was given by Peter Athron in his parallel session talk
[11], where the GAMBIT system is used to show that rumours of the death of the
MSSM may be exaggerated. Not only are there still allowed parameter points for
any and all values of neutralino and chargino mass (even though swathes of the
multidimensional parameter space are indeed ruled out by LHC data) but there
are even some regions of SUSY parameter space which are marginally favoured
over the SM.

I will take the opportunity to highlight a few other techniques and ideas of
growing importance.

The potential of machine learning, long extant around the periphery of the
field, is now being realised in many different particle and astroparticle
physics applications, and has much more to offer if used judiciously. Jet
substructure has already been mentioned and hardly counts as new, but will in
my view continue to grow in importance, including in the evaluation of the
potential of future colliders. The same goes for the theoretical advances
needed to calculate such variables incorporating both QCD and electroweak
higher-order corrections.

And finally, the field of particle physics really must get to grips with Open
Data. The culture in astrophysics should be an example; the fact that LIGO
data from the first gravitational wave observations is already public is
immensely impressive, when the Higgs discovery data are still behind
collaboration firewalls. Of course, the technical issues are different.
Release of data is not without effort, and brings challenges. But in the final
analysis, data produced by our experiments is part of the store of human
knowledge, and is not ours to keep to ourselves in the long term. On the LHC
side, CMS have led the way in opening a subset of early LHC data for physics
analysis, some of which has been used by theorists to develop new results, and
as shown in a parallel session by Matthias Schott, a member of ATLAS, to
perform open cross-checks, new measurements and comparisons. This shows
opening up our data is technically possible, and useful, for collider
experiments. However, such activity – especially by a member of a rival
collaboration – causes concern to some, and may even be discouraged or seen as
a threat by the collaborations. This would be a mistake. While there are, for
example, legitimate concerns about shortage of effort within the
collaborations, attempting to control what colleagues do with public data is
not the way to address those concerns, and in my opinion crosses an
unacceptable line in terms of academic freedom. Open data, and its analysis,
will in the end be good for science and innovation, and should be at least
tolerated by even the most sceptical. External pressures are pushing the field
in this direction anyway. We should go enthusiastically, not dragging our
feet.

##  9 Conclusion

With the ongoing results from the LHC experiments, Belle II arriving, HyperK
and DUNE on the horizon, continuing improvement in theoretical and experiment
techniques and a deepening engagement between particle physics, astrophysics
and nuclear physics, this week has provided an exciting snapshot of a field
still digesting the implications of the results of the last decade or so. I
did not mention charged-lepton flavour measurements, or experiments probing
the neutrino mass directly (its value, and whether it is Majorana or Dirac in
nature), but there are also exciting prospects there. The main impression I
took away from the conference is that while we need to pay attention to the
far future, and it contains many uncertainties, the next few years promise a
great richness of data, and therefore possibility, which should inspire and
excite us all. But there is still much work to do to make this a reality.

I apologise to anyone I didn’t represent well, or at all, here. I already
thanked the organisers for the invitation. I would also like to thank them and
the local organisers, colleagues, and the city of Ghent and its brewers, for
hosting and attending a superb conference.

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