Combinatorial optimization problems have wide-ranging applications in industry and academia. Quantum computers may help solve them by sampling from carefully prepared Ansatz quantum circuits. However, current quantum computers are limited by their qubit count, connectivity, and noise. This is particularly restrictive when considering optimization problems beyond the quadratic order. Here, we introduce Ansatze based on an approximate quadratization of high-order Hamiltonians which do not incur a qubit overhead. The price paid is a loss in the quality of the noiseless solution. Crucially, this approximation yields shallower Ansatze which are more robust to noise than the standard QAOA one. We show this through simulations of systems of 8 to 16 qubits with variable noise strengths. Furthermore, we also propose a noise-aware Ansatz design method for quadratic optimization problems. This method implements only part of the corresponding Hamiltonian by limiting the number of layers of SWAP gates in the Ansatz. We find that for both problem types, under noise, our approximate implementation of the full problem structure can significantly enhance the solution quality. Our work opens a path to enhance the solution quality that approximate quantum optimization achieves on noisy hardware.

Error mitigation has enabled quantum computing applications with more than a hundred qubits and deep circuits. Many error mitigation methods are noise-aware, relying on a faithful characterization of the noise channels of the hardware. However, fundamental limitations lead to unlearnable degrees of freedom of the underlying noise models when considering qubits. Here, we show how to leverage noncomputational states as an additional resource to learn state-preparation errors in superconducting qubits. This allows one to fully constrain the noise models. We can thus independently and accurately mitigate state-preparation errors, gate errors, and measurement errors. Our proposed method is also applicable to dynamic circuits with midcircuit measurements. This work opens the door to improved error mitigation for measurements, both at the end of the circuit and midcircuit.

Through recent progress in hardware development, quantum computers have advanced to the point where benchmarking of (heuristic) quantum algorithms at scale is within reach. Particularly in combinatorial optimization - where most algorithms are heuristics - it is key to empirically analyze their performance on hardware and track progress towards quantum advantage. To this extent, we present ten optimization problem classes that are difficult for existing classical algorithms and can (mostly) be linked to practically relevant applications, with the goal to enable systematic, fair, and comparable benchmarks for quantum optimization methods. Further, we introduce the Quantum Optimization Benchmarking Library (QOBLIB) where the problem instances and solution track records can be found. The individual properties of the problem classes vary in terms of objective and variable type, coefficient ranges, and density. Crucially, they all become challenging for established classical methods already at system sizes ranging from less than 100 to, at most, an order of 100,000 decision variables, allowing to approach them with today's quantum computers. We reference the results from state-of-the-art solvers for instances from all problem classes and demonstrate exemplary baseline results obtained with quantum solvers for selected problems. The baseline results illustrate a standardized form to present benchmarking solutions, which has been designed to ensure comparability of the used methods, reproducibility of the respective results, and trackability of algorithmic and hardware improvements over time. We encourage the optimization community to explore the performance of available classical or quantum algorithms and hardware platforms with the benchmarking problem instances presented in this work toward demonstrating quantum advantage in optimization.

The goal of multi-objective optimization is to understand optimal trade-offs between competing objective functions by finding the Pareto front, that is, the set of all Pareto-optimal solutions, where no objective can be improved without degrading another one. Multi-objective optimization can be challenging classically, even if the corresponding single-objective optimization problems are efficiently solvable. Thus, multi-objective optimization represents a compelling problem class to analyze with quantum computers. Here we use a low-depth quantum approximate optimization algorithm to approximate the optimal Pareto front of certain multi-objective weighted maximum-cut problems. We demonstrate its performance on an IBM Quantum computer, as well as with matrix product state numerical simulation, and show its potential to outperform classical approaches.

Current quantum computers suffer from noise that stems from interactions between the quantum system that constitutes the quantum device and its environment. These interactions can be suppressed through dynamical decoupling to reduce computational errors. However, the performance of dynamical decoupling depends on the type of the system-environment interactions that are present, which often lack an accurate model in quantum devices. We show that the performance of dynamical decoupling can be improved by optimizing its rotational gates to tailor them to the quantum hardware. We find that compared to canonical decoupling sequences, such as CPMG and XY4, the optimized dynamical decoupling sequences yield the best performance in suppressing noise in superconducting qubits. Our work thus enhances existing error suppression methods which helps increase circuit depth and result quality on noisy hardware.

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage-namely, a significant (in some cases exponential) speedup of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small-scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models that are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this Roadmap paper, led by CERN, DESY, and IBM, we provide the status of high-energy physics quantum computations and give examples of theoretical and experimental target benchmark applications, which can be addressed in the near future. Having in mind hardware with about 100 qubits capable of executing several thousand two-qubit gates, where possible, we also provide resource estimates for the examples given using error-mitigated quantum computing. The ultimate declared goal of this task force is therefore to trigger further research in the high-energy physics community to develop interesting use cases for demonstrations on near-term quantum computers.

Variational quantum algorithms offer fascinating prospects for the solution of combinatorial optimization problems using digital quantum computers. However, the achievable performance in such algorithms and the role of quantum correlations therein remain unclear. Here, we shed light on this open issue by establishing a tight connection to the seemingly unrelated field of quantum metrology: Metrological applications employ quantum states of spin-ensembles with a reduced variance to achieve an increased sensitivity, and we cast the generation of such squeezed states in the form of finding optimal solutions to a combinatorial MaxCut problem with an increased precision. By solving this optimization problem with a quantum approximate optimization algorithm (QAOA), we show numerically as well as on an IBM quantum chip how highly squeezed states are generated in a systematic procedure that can be adapted to a wide variety of quantum machines. Moreover, squeezing tailored for the QAOA of the MaxCut permits us to propose a figure of merit for future hardware benchmarks.

Quantum computation promises to advance a wide range of computational tasks. However, current quantum hardware suffers from noise and is too small for error correction. Thus, accurately utilizing noisy quantum computers strongly relies on noise characterization, mitigation, and suppression. Crucially, these methods must also be efficient in terms of their classical and quantum overhead. Here, we efficiently characterize and mitigate crosstalk noise, which is a severe error source in, e.g., cross-resonance based superconducting quantum processors. For crosstalk characterization, we develop a simplified measurement experiment. Furthermore, we analyze the problem of optimal experiment scheduling and solve it for common hardware architectures. After characterization, we mitigate noise in quantum circuits by a noise-aware qubit routing algorithm. Our integer programming algorithm extends previous work on optimized qubit routing by swap insertion. We incorporate the measured crosstalk errors in addition to other, more easily accessible noise data in the objective function. Furthermore, we strengthen the underlying integer linear model by proving a convex hull result about an associated class of polytopes, which has applications beyond this work. We evaluate the proposed method by characterizing crosstalk noise for a complete 27 qubit chip and leverage the resulting data to improve the approximation ratio of the Quantum Approximate Optimization Algorithm by up to 10 % compared to other established noise-aware routing methods. Our work clearly demonstrates the gains of including noise data when mapping abstract quantum circuits to hardware native ones.

Quantum computers are increasing in size and quality, but are still very noisy. Error mitigation extends the size of the quantum circuits that noisy devices can meaningfully execute. However, state-of-the-art error mitigation methods are hard to implement and the limited qubit connectivity in superconducting qubit devices restricts most applications to the hardware's native topology. Here we show a quantum approximate optimization algorithm (QAOA) on non-planar random regular graphs with up to 40 nodes enabled by a machine learning-based error mitigation. We use a swap network with careful decision-variable-to-qubit mapping and a feed-forward neural network to demonstrate optimization of a depth-two QAOA on up to 40 qubits. We observe a meaningful parameter optimization for the largest graph which requires running quantum circuits with 958 two-qubit gates. Our work emphasizes the need to mitigate samples, and not only expectation values, in quantum approximate optimization. These results are a step towards executing quantum approximate optimization at a scale that is not classically simulable. Reaching such system sizes is key to properly understanding the true potential of heuristic algorithms like QAOA.

The real-time simulation of large many-body quantum systems is a formidable task, that may only be achievable with a genuine quantum computational platform. Currently, quantum hardware with a number of qubits sufficient to make classical emulation challenging is available. This condition is necessary for the pursuit of a so-called quantum advantage, but it also makes verifying the results very difficult. In this paper, we flip the perspective and utilize known theoretical results about many-body quantum critical dynamics to qualitatively benchmark digital quantum hardware and various error mitigation techniques. In particular, we benchmark against known universal scaling laws in the Hamiltonian simulation of a time-dependent transverse-field Ising Hamiltonian of up to 133 qubits. Incorporating only basic error mitigation and suppression methods, our study shows reliable control up to a two-qubit gate depth of 28, featuring a maximum of 1396 two-qubit gates, before noise becomes prevalent. These results are transferable to applications such as Hamiltonian simulation, variational algorithms, optimization, or quantum machine learning. We demonstrate this on the example of digitized quantum annealing for optimization and identify an optimal working point in terms of both circuit depth and time step on a 133-site optimization problem.

Current quantum computers suffer from noise that stems from interactions between the quantum system that constitutes the quantum device and its environment. These interactions can be suppressed through dynamical decoupling to reduce computational errors. However, the performance of dynamical decoupling depends on the type of system-environment interactions that are present, which often lack an accurate model in quantum devices. We show that the performance of dynamical decoupling can be improved by optimizing its rotational gates to tailor them to the quantum hardware. We find that, compared to canonical decoupling sequences, such as CPMG, XY4, and UR6, the optimized dynamical decoupling sequences yield the best performance in suppressing noise in superconducting qubits. Our work thus enhances existing error suppression methods, which helps increase the circuit depth and result quality on noisy hardware.

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.

Quantum computing may solve complex problems in areas as diverse as natural sciences, optimization, machine learning, and finance. I will briefly introduce the IBM Quantum roadmap and explore some of the applications of quantum computing. Full-text article not available; see video presentation

In this paper, we explore the impact of noise on quantum computing, particularly focusing on the challenges when sampling bit strings from noisy quantum computers as well as the implications for optimization and machine learning applications. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the Conditional Value at Risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on a real quantum computer involving up to 127 qubits. The results show a strong alignment with theoretical predictions.

Quantum computers process information with the laws of quantum mechanics. Current quantum hardware is noisy, can only store information for a short time and is limited to a few quantum bits, that is, qubits, typically arranged in a planar connectivity1. However, many applications of quantum computing require more connectivity than the planar lattice offered by the hardware on more qubits than is available on a single quantum processing unit (QPU). The community hopes to tackle these limitations by connecting QPUs using classical communication, which has not yet been proven experimentally. Here we experimentally realize error-mitigated dynamic circuits and circuit cutting to create quantum states requiring periodic connectivity using up to 142 qubits spanning two QPUs with 127 qubits each connected in real time with a classical link. In a dynamic circuit, quantum gates can be classically controlled by the outcomes of mid-circuit measurements within run-time, that is, within a fraction of the coherence time of the qubits. Our real-time classical link enables us to apply a quantum gate on one QPU conditioned on the outcome of a measurement on another QPU. Furthermore, the error-mitigated control flow enhances qubit connectivity and the instruction set of the hardware thus increasing the versatility of our quantum computers. Our work demonstrates that we can use several quantum processors as one with error-mitigated dynamic circuits enabled by a real-time classical link. A 142-qubit processor can be realized by connecting two smaller quantum processors using classical communications and circuit cutting.

Most quantum circuits require SWAP gate insertion to run on quantum hardware with limited qubit connectivity. A promising SWAP gate insertion method for blocks of commuting two-qubit gates is a predetermined swap strategy which applies layers of SWAP gates simultaneously executable on the coupling map. A good initial mapping for the swap strategy reduces the number of required swap gates. However, even when a circuit consists of commuting gates, e.g., as in the Quantum Approximate Optimization Algorithm (QAOA) or trotterized simulations of Ising Hamiltonians, finding a good initial mapping is a hard problem. We present a SAT-based approach to find good initial mappings for circuits with commuting gates transpiled to the hardware with swap strategies. Our method achieves a 65% reduction in gate count for random three-regular graphs with 500 nodes. In addition, we present a heuristic approach that combines the SAT formulation with a clustering algorithm to reduce large problems to a manageable size. This approach reduces the number of swap layers by 25% compared to both a trivial and random initial mapping for a random three-regular graph with 1000 nodes. Good initial mappings will therefore enable the study of quantum algorithms, such as QAOA and Ising Hamiltonian simulation applied to sparse problems, on noisy quantum hardware with several hundreds of qubits.

QiskitDynamics is an open-source Python library for numerically simulating the time dynamics of finite-dimensional quantum systems.The goal of the package is to provide flexible configuration of the numerical methods used for simulation: general tools for transforming models of quantum systems for more efficient simulation (rotating frames and the rotating wave approximation), choice of array representations (dense vs. sparse, and different array libraries), and access to different types of underlying solvers (standard ODE vs. geometric solvers).The package also contains advanced functionality for computing time-dependent perturbation theory expressions used in robust quantum control optimization (Haas et al.,

Qiskit Experiments is a Python package for designing and running quantum computing experiments with a focus on calibration and characterization of quantum devices.It consists of a general purpose experiments framework that can be used by researchers to rapidly implement new experiments and a library of common experiments for calibration, characterization, and verification of quantum devices.

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage, namely a significant (in some cases exponential) speed-up of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models which are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this roadmap paper, led by CERN, DESY and IBM, we provide the status of high-energy physics quantum computations and give examples for theoretical and experimental target benchmark applications, which can be addressed in the near future. Having the IBM 100 x 100 challenge in mind, where possible, we also provide resource estimates for the examples given using error mitigated quantum computing.

Simulating the time-evolution of a Hamiltonian is one of the most promising applications of quantum computers. Multi-Product Formulas (MPFs) are well suited to replace standard product formulas since they scale better with respect to time and approximation errors. Hamiltonian simulation with MPFs was first proposed in a fully quantum setting using a linear combination of unitaries. Here, we analyze and demonstrate a hybrid quantum-classical approach to MPFs that classically combines expectation values evaluated with a quantum computer. This has the same approximation bounds as the fully quantum MPFs, but, in contrast, requires no additional qubits, no controlled operations, and is not probabilistic. We show how to design MPFs that do not amplify the hardware and sampling errors, and demonstrate their performance. In particular, we illustrate the potential of our work by theoretically analyzing the benefits when applied to a classically intractable spin-boson model, and by computing the dynamics of the transverse field Ising model using a classical simulator as well as quantum hardware. We observe an error reduction of up to an order of magnitude when compared to a product formula approach by suppressing hardware noise with Pauli Twirling, pulse efficient transpilation, and a novel zero-noise extrapolation based on scaled cross-resonance pulses. The MPF methodology reduces the circuit depth and may therefore represent an important step towards quantum advantage for Hamiltonian simulation on noisy hardware.