arXiv subjects

Andreas Bärtschi

Publications and source records attributed to Andreas Bärtschi.

At least 19 recordsLinked to original sources

arXiv2026

Potential Applications of Quantum Computing at Los Alamos National Laboratory

The emergence of quantum computing technology over the last decade indicates the potential for a transformational impact in the study of quantum mechanical systems. It is natural to presume that such computing technologies would be valuable to large scientific institutions, such as United States national laboratories. However, detailed descriptions of what these institutions would like to use these computers for are limited. To help provide some initial insights into this topic, this report develops detailed use cases of how quantum computing technology could be utilized to enhance a variety of quantum physics research activities at Los Alamos National Laboratory, including quantum magnetic materials, high-temperature superconductivity and nuclear astrophysics simulations. The report discusses how current high-performance computers are used for scientific discovery today and develops detailed descriptions of the types of quantum physics simulations that Los Alamos National Laboratory scientists would like to conduct, if a sufficient computing technology became available. While the report strives to highlight the breadth of potential application areas for quantum computation, this investigation has also indicated that many more use cases exist at Los Alamos National Laboratory, which could be documented in similar detail with sufficient time and effort.

quant-ph↗

arXiv2026

Snapshot-QAOA: Extending QAOA to Quantum Hamiltonian Simulation

We present Snapshot-QAOA, a variation of the Quantum Approximate Optimization Algorithm (QAOA) that finds approximate minimum energy eigenstates of a large set of quantum Hamiltonians (i.e. Hamiltonians with non-diagonal terms). Traditionally, QAOA targets the task of approximately solving combinatorial optimization problems; Snapshot-QAOA enables a significant expansion of the use case space for QAOA to more general quantum Hamiltonians, where the goal is to approximate the ground-state. Such ground-state finding is a common challenge in quantum chemistry and material science applications. Snapshot-QAOA retains desirable variational-algorithm qualities of QAOA, in particular small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the QEC-era Quantum Phase Estimation (QPE) algorithm. Our fundamental approach is inspired by the idea of Trotterization of a continuous-time linear adiabatic anneal schedule, which for sufficiently large QAOA depth gives very good performance. Snapshot-QAOA restricts the QAOA evolution to not phasing out the mixing Hamiltonian completely at the end of the evolution, instead evolving only a partial typical linear QAOA schedule, thus creating a type of snapshot of the typical QAOA evolution. As a test case, we simulate Snapshot-QAOA on a 16 qubit J1-J2 frustrated square transverse field Ising model with periodic boundary conditions.

quant-ph↗

arXiv2026

Evaluating the Limits of QAOA Parameter Transfer at High-Rounds on Sparse Ising Models With Geometrically Local Cubic Terms

The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters. Parameter transfer is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable non-variational learning method for QAOA angle finding. We systematically study QAOA parameter transferability from small problems (16, 27 qubits) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher order terms using the Julia based QAOA simulation tool JuliQAOA to perform classical angle finding for up to 49 QAOA layers. Parameter transfer of the fixed angles is validated using a combination of full statevector, Projected Entangled Pair States, Matrix Product State, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to unseen problem instances does not in general provide monotonically improving performance as a function of p - there are many cases where the performance temporarily decreases as a function of p - but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100+ qubit instances. We also sample the hardware-compatible Ising models using the ensemble of fixed QAOA angles on several superconducting qubit IBM Quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM NISQ processors up to p=5 on ibm_fez, p=9 on ibm_torino, and p=10 on ibm_pittsburgh.

quant-ph↗

arXiv2025

Quantum Data Learning of Topological-to-Ferromagnetic Phase Transitions in the 2+1D Toric Code Loop Gas Model

Quantum data learning (QDL) provides a framework for extracting physical insights directly from quantum states, bypassing the need for any identification of the classical observable of the theory. A central challenge in many-body physics is that the identity of quantum phases, especially those with topological order, are often inaccessible through local observables or simple symmetry-breaking diagnostics. Here, we apply QDL techniques to the 2+1-dimensional toric-code loop-gas model in a magnetic field. Ground states are generated across multiple lattice sizes using a parametrized loop-gas circuit (PLGC) with a variational quantum-eigensolver (VQE) approach. We then train a quantum convolutional neural network (QCNN) across the full field-parameter range to perform phase classification and capture the overall phase structure. We also employ a physics-aware training protocol that excludes the near-critical region (0.2 <= x <= 0.4)) around (x_c = 0.25), the phase-transition point estimated by quantum Monte Carlo, reserving this window for testing to evaluate the ability of the model to learn the phase transition. In parallel, we implement an unsupervised quantum k-means method based on state overlaps, which partitions the dataset into two phases without prior labeling. Our supervised QDL approach recovers the phase structure and accurately locates the phase transition, in close agreement with previously reported values; the unsupervised QDL approach recovers the phase structure and locates the phase transition with a small offset as expected in finite volumes; both QDL methods outperform classical alternatives. These findings establish QDL as an effective framework for characterizing topological quantum matter, studying finite volume effects, and probing phase diagrams of higher-dimensional systems.

quant-ph↗

arXiv2025

Variational Quantum Simulations of a Two-Dimensional Frustrated Transverse-Field Ising Model on a Trapped-Ion Quantum Computer

Quantum computers are an ideal platform to study the ground state properties of strongly correlated systems due to the limitation of classical computing techniques particularly for systems exhibiting quantum phase transitions. While the error rates of Noisy Intermediate-Scale Quantum (NISQ) computers are still high, simulating strongly correlated systems on such devices and extracting information of possible phases may be within reach. The frustrated transverse-field Ising model (TFIM) is such a system with multiple ordered magnetic phases. In this study, we simulate a two-dimensional frustrated TFIM with next-nearest-neighbor spin-exchange interactions at zero temperature. The competition between the nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic coupling gives rise to frustration in the system. Moreover, the presence of quantum fluctuations makes the ground-state phase profile even richer. We use the Variational Quantum Eigensolver (VQE) to compute the phases on a square lattice with periodic boundary conditions for a system of 16 sites (qubits). The trained VQE circuits are compared to exact diagonalization, allowing us to extract error measures of VQE. We focus on the ground-state phase transitions of this model, where VQE succeeds in finding the dominant magnetic phases. The optimized VQE circuits are then executed on the Quantinuum H1-1 trapped-ion quantum computer without using any error mitigation techniques. Our experiments show near perfect recovery of the magnetic phases of the frustrated model through ground-state energy, the energy derivative, and the spin correlation functions. Thus, we show that the trapped-ion quantum processor is able to achieve reliable simulations of a strongly correlated system within the limitations of the VQE approach.

quant-ph↗

arXiv2025

Classical Combinatorial Optimization Scaling for Random Ising Models on 2D Heavy-Hex Graphs

Motivated by near term quantum computing hardware limitations, combinatorial optimization problems that can be addressed by current quantum algorithms and noisy hardware with little or no overhead are used to probe capabilities of quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). In this study, a specific class of near term quantum computing hardware defined combinatorial optimization problems, Ising models on heavy-hex graphs both with and without geometrically local cubic terms, are examined for their classical computational hardness via empirical computation time scaling quantification. Specifically the Time-to-Solution metric using the classical heuristic simulated annealing is measured for finding optimal variable assignments (ground states), as well as the time required for the optimization software Gurobi to find an optimal variable assignment. Because of the sparsity of these Ising models, the classical algorithms are able to find optimal solutions efficiently even for large instances (i.e. $100,000$ variables). The Ising models both with and without geometrically local cubic terms exhibit average-case linear-time or weakly quadratic scaling when solved exactly using Gurobi, and the Ising models with no cubic terms show evidence of exponential-time Time-to-Solution scaling when sampled using simulated annealing. These findings point to the necessity of developing and testing more complex, namely more densely connected, optimization problems in order for quantum computing to ever have a practical advantage over classical computing. Our results are another illustration that different classical algorithms can indeed have exponentially different running times, thus making the identification of the best practical classical technique important in any quantum computing vs. classical computing comparison.

math.OC↗

arXiv2025

Lower bounds on the number of rounds of the quantum approximate optimization algorithm required for guaranteed approximation ratios

The quantum approximate optimization algorithm, also known in its generalization as the quantum alternating operator ansatz, (QAOA) is a heuristic hybrid quantum-classical algorithm for finding high-quality approximate solutions to combinatorial optimization problems, such as maximum satisfiability. While the QAOA is well studied, theoretical results as to its runtime or approximation ratio guarantees are still relatively sparse. We provide some of the first lower bounds for the number of rounds (the dominant component of QAOA runtimes) required for the QAOA. For our main result, we (i) leverage a connection between quantum annealing times and the angles of the QAOA to derive a lower bound on the number of rounds of the QAOA with respect to the guaranteed approximation ratio. We apply and calculate this bound with Grover-style mixing unitaries and (ii) show that this type of QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratios for most problems. We also (iii) show that the bound depends only on the statistical values of the objective functions, and when the problem can be modeled as a $k$-local Hamiltonian, can be easily estimated from the coefficients of the Hamiltonians. For the conventional transverse-field mixer, (iv) our framework gives a trivial lower bound to all bounded-occurrence local cost problems and for all strictly $k$-local cost Hamiltonians matching known results that constant approximation ratio is obtainable with a constant-round QAOA for a few optimization problems from these classes. Using our proof framework, (v) we recover the Grover lower bound for unstructured search and, with small modification, show that our bound applies to any QAOA-style search protocol that starts in the ground state of the mixing unitaries.

quant-ph↗

arXiv2025

Quantum Graph Transformer for NLP Sentiment Classification

Quantum machine learning is a promising direction for building more efficient and expressive models, particularly in domains where understanding complex, structured data is critical. We present the Quantum Graph Transformer (QGT), a hybrid graph-based architecture that integrates a quantum self-attention mechanism into the message-passing framework for structured language modeling. The attention mechanism is implemented using parameterized quantum circuits (PQCs), which enable the model to capture rich contextual relationships while significantly reducing the number of trainable parameters compared to classical attention mechanisms. We evaluate QGT on five sentiment classification benchmarks. Experimental results show that QGT consistently achieves higher or comparable accuracy than existing quantum natural language processing (QNLP) models, including both attention-based and non-attention-based approaches. When compared with an equivalent classical graph transformer, QGT yields an average accuracy improvement of 5.42% on real-world datasets and 4.76% on synthetic datasets. Additionally, QGT demonstrates improved sample efficiency, requiring nearly 50% fewer labeled samples to reach comparable performance on the Yelp dataset. These results highlight the potential of graph-based QNLP techniques for advancing efficient and scalable language understanding.

cs.CL↗

arXiv2025

Scalable Experimental Bounds for Entangled Quantum State Fidelities

Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as $O(3^N)$ for $N$-qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma and others [PhysRevA.74.052302] established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke States and GHZ States. These bounds must still be tight enough for larger states to provide reasonable estimations on NISQ devices. For the first time and more than 15 years after the theoretical introduction, we report meaningful lower bounds for the state preparation fidelity of all Dicke States up to $N=10$ and all GHZ states up to $N=20$ on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting systems for much smaller states. Furthermore, we provide evidence that for large Dicke States $D^N_{N/2}$, we may resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path forward to benchmarking entanglement as NISQ devices improve in size and quality.

quant-ph↗

arXiv2024

Challenges and Opportunities in Quantum Optimization

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

quant-ph↗

arXiv2024

Scaling Whole-Chip QAOA for Higher-Order Ising Spin Glass Models on Heavy-Hex Graphs

We show through numerical simulation that the Quantum Approximate Optimization Algorithm (QAOA) for higher-order, random-coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from $16$ up to $127$ qubits for $p=1$ up to $p=5$, which allows for straight-forward transfer learning of QAOA angles on instance sizes where exhaustive grid-search is prohibitive even for $p>1$. We use Matrix Product State (MPS) simulation at different bond dimensions to obtain confidence in these results, and we obtain the optimal solutions to these combinatorial optimization problems using CPLEX. In order to assess the ability of current noisy quantum hardware to exploit such parameter concentration, we execute short-depth QAOA circuits (with a CNOT depth of 6 per $p$, resulting in circuits which contain $1420$ two qubit gates for $127$ qubit $p=5$ QAOA) on $100$ higher-order (cubic term) Ising models on IBM quantum superconducting processors with $16, 27, 127$ qubits using QAOA angles learned from a single $16$-qubit instance. We show that (i) the best quantum processors generally find lower energy solutions up to $p=3$ for 27 qubit systems and up to $p=2$ for 127 qubit systems and are overcome by noise at higher values of $p$, (ii) the best quantum processors find mean energies that are about a factor of two off from the noise-free numerical simulation results. Additional insights from our experiments are that large performance differences exist among different quantum processors even of the same generation and that dynamical decoupling significantly improve performance for some, but decrease performance for other quantum processors. Lastly we show $p=1$ QAOA angle mean energy landscapes computed using up to a $414$ qubit quantum computer, showing that the mean QAOA energy landscapes remain very similar as the problem size changes.

quant-ph↗

arXiv2024

Simulating Heavy-Hex Transverse Field Ising Model Magnetization Dynamics Using Programmable Quantum Annealers

Recently, a Hamiltonian dynamics simulation was performed on a kicked ferromagnetic 2D transverse field Ising model with a connectivity graph native to the 127 qubit heavy-hex IBM Quantum architecture using ZNE quantum error mitigation. We demonstrate that one of the observables in this Trotterized Hamiltonian dynamics simulation, namely magnetization, can be efficiently simulated on current superconducting qubit-based programmable quantum annealing computers. We show this using two distinct methods: reverse quantum annealing and h-gain state encoding. This simulation is possible because the 127 qubit heavy-hex connectivity graph can be natively embedded onto the D-Wave Pegasus quantum annealer hardware graph and because there exists a direct equivalence between the energy scales of the two types of quantum computers. We derive equivalent anneal pauses in order to simulate the Trotterized quantum circuit dynamics for varying Rx rotations $θ_h \in (0, \fracπ{2}]$, using quantum annealing processors. Multiple disjoint instances of the Ising model of interest can be embedded onto the D-Wave Pegasus hardware graph, allowing for parallel quantum annealing. We report equivalent magnetization dynamics using quantum annealing for time steps of 20, 50 up to 10,000, which we find are consistent with exact classical 27 qubit heavy-hex Trotterized circuit magnetization dynamics, and we observe reasonable, albeit noisy, agreement with the existing simulations for single site magnetization at 20 Trotter steps. The quantum annealers are able to simulate equivalent magnetization dynamics for thousands of time steps, significantly out of the computational reach of the digital quantum computers on which the original Hamiltonian dynamics simulations were performed.

quant-ph↗

arXiv2024

Trainability Barriers in Low-Depth QAOA Landscapes

The Quantum Alternating Operator Ansatz (QAOA) is a prominent variational quantum algorithm for solving combinatorial optimization problems. Its effectiveness depends on identifying input parameters that yield high-quality solutions. However, understanding the complexity of training QAOA remains an under-explored area. Previous results have given analytical performance guarantees for a small, fixed number of parameters. At the opposite end of the spectrum, barren plateaus are likely to emerge at $Ω(n)$ parameters for $n$ qubits. In this work, we study the difficulty of training in the intermediate regime, which is the focus of most current numerical studies and near-term hardware implementations. Through extensive numerical analysis of the quality and quantity of local minima, we argue that QAOA landscapes can exhibit a superpolynomial growth in the number of low-quality local minima even when the number of parameters scales logarithmically with $n$. This means that the common technique of gradient descent from randomly initialized parameters is doomed to fail beyond small $n$, and emphasizes the need for good initial guesses of the optimal parameters.

quant-ph↗

arXiv2024

Hierarchical Multigrid Ansatz for Variational Quantum Algorithms

Quantum computing is an emerging topic in engineering that promises to enhance supercomputing using fundamental physics. In the near term, the best candidate algorithms for achieving this advantage are variational quantum algorithms (VQAs). We design and numerically evaluate a novel ansatz for VQAs, focusing in particular on the variational quantum eigensolver (VQE). As our ansatz is inspired by classical multigrid hierarchy methods, we call it "multigrid" ansatz. The multigrid ansatz creates a parameterized quantum circuit for a quantum problem on $n$ qubits by successively building and optimizing circuits for smaller qubit counts $j < n$, reusing optimized parameter values as initial solutions to next level hierarchy at $j+1$. We show through numerical simulation that the multigrid ansatz outperforms the standard hardware-efficient ansatz in terms of solution quality for the Laplacian eigensolver as well as for a large class of combinatorial optimization problems with specific examples for MaxCut and Maximum $k$-Satisfiability. Our studies establish the multi-grid ansatz as a viable candidate for many VQAs and in particular present a promising alternative to the QAOA approach for combinatorial optimization problems.

quant-ph↗

arXiv2024

Quantum Approximate Optimization: A Computational Intelligence Perspective

Quantum computing is an emerging field on the multidisciplinary interface between physics, engineering, and computer science with the potential to make a large impact on computational intelligence (CI). The aim of this paper is to introduce quantum approximate optimization methods to the CI community because of direct relevance to solving combinatorial problems. We introduce quantum computing and variational quantum algorithms (VQAs). VQAs are an effective method for the near-term implementation of quantum solutions on noisy intermediate-scale quantum (NISQ) devices with less reliable qubits and early-stage error correction. Then, we explain Farhi et al.'s quantum approximate optimization algorithm (Farhi's QAOA, to prevent confusion). This VQA is generalized by Hadfield et al. to the quantum alternating operator ansatz (QAOA), which is a nature-inspired (particularly, adiabatic) quantum metaheuristic for approximately solving combinatorial optimization problems on gate-based quantum computers. We discuss connections of QAOA to relevant domains, such as computational learning theory and genetic algorithms, discussing current techniques and known results regarding hybrid quantum-classical intelligence systems. We present a schematic of how QAOA is constructed, and also discuss how CI techniques can be used to improve QAOA. We conclude with QAOA implementations for the well-known maximum cut, maximum bisection, and traveling salesperson problems, which can serve as templates for CI practitioners interested in using QAOA.

quant-ph↗

arXiv2024

Increasing the Measured Effective Quantum Volume with Zero Noise Extrapolation

Quantum Volume is a full-stack benchmark for near-term quantum computers. It quantifies the largest size of a square circuit which can be executed on the target device with reasonable fidelity. Error mitigation is a set of techniques intended to remove the effects of noise present in the computation of noisy quantum computers when computing an expectation value of interest. Effective quantum volume is a proposed metric that applies error mitigation to the quantum volume protocol in order to evaluate the effectiveness not only of the target device but also of the error mitigation algorithm. Digital Zero-Noise Extrapolation (ZNE) is an error mitigation technique that estimates the noiseless expectation value using circuit folding to amplify errors by known scale factors and extrapolating to the zero-noise limit. Here we demonstrate that ZNE, with global and local unitary folding with fractional scale factors, in conjunction with dynamical decoupling, can increase the effective quantum volume over the vendor-measured quantum volume. Specifically, we measure the effective quantum volume of four IBM Quantum superconducting processor units, obtaining values that are larger than the vendor-measured quantum volume on each device. This is the first such increase reported.

quant-ph↗

arXiv2024

Graph Neural Networks for Parameterized Quantum Circuits Expressibility Estimation

Parameterized quantum circuits (PQCs) are fundamental to quantum machine learning (QML), quantum optimization, and variational quantum algorithms (VQAs). The expressibility of PQCs is a measure that determines their capability to harness the full potential of the quantum state space. It is thus a crucial guidepost to know when selecting a particular PQC ansatz. However, the existing technique for expressibility computation through statistical estimation requires a large number of samples, which poses significant challenges due to time and computational resource constraints. This paper introduces a novel approach for expressibility estimation of PQCs using Graph Neural Networks (GNNs). We demonstrate the predictive power of our GNN model with a dataset consisting of 25,000 samples from the noiseless IBM QASM Simulator and 12,000 samples from three distinct noisy quantum backends. The model accurately estimates expressibility, with root mean square errors (RMSE) of 0.05 and 0.06 for the noiseless and noisy backends, respectively. We compare our model's predictions with reference circuits [Sim and others, QuTe'2019] and IBM Qiskit's hardware-efficient ansatz sets to further evaluate our model's performance. Our experimental evaluation in noiseless and noisy scenarios reveals a close alignment with ground truth expressibility values, highlighting the model's efficacy. Moreover, our model exhibits promising extrapolation capabilities, predicting expressibility values with low RMSE for out-of-range qubit circuits trained solely on only up to 5-qubit circuit sets. This work thus provides a reliable means of efficiently evaluating the expressibility of diverse PQCs on noiseless simulators and hardware.

quant-ph↗

arXiv2024

Probing Quantum Telecloning on Superconducting Quantum Processors

Quantum information can not be perfectly cloned, but approximate copies of quantum information can be generated. Quantum telecloning combines approximate quantum cloning, more typically referred as quantum cloning, and quantum teleportation. Quantum telecloning allows approximate copies of quantum information to be constructed by separate parties, using the classical results of a Bell measurement made on a prepared quantum telecloning state. Quantum telecloning can be implemented as a circuit on quantum computers using a classical co-processor to compute classical feed forward instructions using if statements based on the results of a mid-circuit Bell measurement in real time. We present universal, symmetric, optimal $1 \rightarrow M$ telecloning circuits, and experimentally demonstrate these quantum telecloning circuits for $M=2$ up to $M=10$, natively executed with real time classical control systems on IBM Quantum superconducting processors, known as dynamic circuits. We perform the cloning procedure on many different message states across the Bloch sphere, on $7$ IBM Quantum processors, optionally using the error suppression technique X-X sequence digital dynamical decoupling. Two circuit optimizations are utilized, one which removes ancilla qubits for $M=2, 3$, and one which reduces the total number of gates in the circuit but still uses ancilla qubits. Parallel single qubit tomography with MLE density matrix reconstruction is used in order to compute the mixed state density matrices of the clone qubits, and clone quality is measured using quantum fidelity. These results present one of the largest and most comprehensive NISQ computer experimental analyses on (single qubit) quantum telecloning to date. The clone fidelity sharply decreases to $0.5$ for $M > 5$, but for $M=2$ we are able to achieve a mean clone fidelity of up to $0.79$ using dynamical decoupling.

quant-ph↗