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Present an outline of important aspects for benchmarking quantum algorithms, in particular quantum heuristics. Using quantum approximate optimization algorithm as an example, we demonstrate how the choice of cost Hamiltonian, of mixer, and of initial states can affect the performance. We also discuss how realistic aspects of near-term quantum hardware will influence the algorithm performance.
The computational capabilities of near-term quantum computers are limited by the noisy execution of gate operations and a limited number of physical qubits. Hybrid variational algorithms are well-suited to near-term quantum devices because they allow for a wide range of tradeoffs between the amount of quantum and classical resources used to solve a problem. This paper investigates tradeoffs available at both the algorithmic and hardware levels by studying a specific case – applying the Quantum Approximate Optimization Algorithm (QAOA) to instances of the Maximum Independent Set (MIS) problem. We consider three variants of the QAOA which offer different tradeoffs at the algorithmic level in terms of their required number of classical parameters, quantum gates, and iterations of classical optimization needed. Since MIS is a constrained combinatorial optimization problem, the QAOA must respect the problem constraints. This can be accomplished by using many multi-controlled gate operations which must be decomposed into gates executable by the target hardware. We study the tradeoffs available at this hardware level, combining the gate fidelities and decomposition efficiencies of different native gate sets into a single metric called the gate decomposition cost .
Efficient preparation of arbitrary entangled quantum states is crucial for quantum computation. This is particularly important for noisy intermediate-scale quantum simulators relying on variational hybrid quantum-classical algorithms. To that end, we propose symmetry-conserving modified quantum approximate optimization algorithm (SCom-QAOA) circuits. The depths of these circuits depend not only on the desired fidelity to the target state but also on the amount of entanglement the state contains. The parameters of the SCom-QAOA circuits are optimized using the quantum natural gradient method based on the Fubini-Study metric. The SCom-QAOA circuit transforms an unentangled state into a ground state of a gapped one-dimensional Hamiltonian with a circuit depth that depends not on the system size but rather on the finite correlation length. In contrast, the circuit depth grows proportionally to the system size for preparing low-lying states of critical one-dimensional systems. Even in the latter case, SCom-QAOA circuits with depth less than the system size were sufficient to generate states with fidelity in excess of 99%, which is relevant for near-term applications. The proposed scheme enlarges the set of the initial states accessible for variational quantum algorithms and widens the scope of investigation of nonequilibrium phenomena in quantum simulators. Published by the American Physical Society 2024
We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the Max-Cut problem. Our main focus is on compilation with trapped-ion simulators using Pauli-X operations and all-to-all Ising Hamiltonian HIsing evolution generated by Molmer-Sorensen or optical dipole force interactions, though some of our results also apply to standard gate-based compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity. For quantum hardware that uses edge-by-edge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trapped-ion quantum simulators implementing all-to-all HIsing pulses, we show that for a (1−ϵ) factor loss in the Max-Cut approximation (ϵ>0), our compilations improve the (worst-case) number of HIsing pulses from O(n2) to O(nlog(n/ϵ)) and the (worst-case) number of Pauli-X bit flips from O(n2) to O(nlog(n/ϵ)ϵ2) for n-node graphs. This is an asymptotic improvement for any constant ϵ>0. We demonstrate that significant improvements to the approximation ratio are obtained using decomposition in simulated trapped-ion experiments with dephasing noise. We further present a generic argument showing that sparsification results in an exponentially improved circuit fidelity lower bound in digital computing schemes based on one- and two-qubit gates, which are relevant to a wide variety of hardwares such as superconducting qubits and certain neutral atom or trapped ion setups, and more sophisticated noise models. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.
Simulating quantum circuits is essential for validating quantum algorithms. However, no single simulator consistently performs best - efficiency depends on circuit structure, entanglement, and depth. In this work, we integrate Qiskit-Aer (state-vector and matrix product state) and QTensor, a tree-tensor-network based simulator, into the Quantum Framework (QFw), a modular platform that supports multiple quantum backends via a unified interface. We also enable distributed quantum approximate optimization algorithm (DQAOA) application compatibility with QFw, allowing sub-problems to be solved in parallel at scale. We then benchmark DQAOA and TFIM (transverse field Ising model) circuits across supported simulators, showing how performance varies significantly with problem type. All simulations are deployed on the Frontier supercomputer using QFw's MPI-based orchestration for distributed, multinode execution. These results underscore the need for simulatoragnostic infrastructure to enable systematic evaluation and highperformance scaling of quantum workloads. QFw provides a practical and extensible path toward reproducible quantum algorithm development across diverse application domains.
Here we investigate the use of deep multiscale entanglement renormalization ansatz (DMERA) circuits as a variational ansatz. We use the exactly solvable one-dimensional critical transverse-field Ising model as a test bed. Numerically exact simulation of the quantum circuit ansatz can in this case be carried out to hundreds of qubits by exploiting efficient classical algorithms for simulating matchgate circuits. We find that, for this system, the DMERA strongly outperforms a standard quantum approximate optimization algorithm (QAOA)–style ansatz, and that a major source of systematic error in correlation functions approximated using the DMERA is the breaking of the translational and Kramers-Wannier symmetries of the transverse-field Ising model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging, without incurring additional cost in qubits or circuit depth. Here, we propose that this technique for mitigating systematic error could be applied to noisy intermediate-scale quantum (NISQ) simulations of physical systems with other symmetries.
Abstract We show 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 computationally efficient parameter transfer of QAOA angles. Matrix product state (MPS) simulation is used to compute noise-free QAOA performance. Hardware-compatible short-depth QAOA circuits are executed on ensembles of 100 higher-order Ising models on noisy IBM quantum superconducting processors with 16, 27, and 127 qubits using QAOA angles learned from a single 16-qubit instance using the JuliQAOA tool. We show that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution. We show that p = 1 QAOA energy landscapes remain very similar as the problem size increases using NISQ hardware gridsearches with up to a 414 qubit processor.
Abstract The SAT problem is a prototypical NP-complete problem of fundamental importance in computational complexity theory with many applications in science and engineering; as such, it has long served as an essential benchmark for classical and quantum algorithms. This study shows numerical evidence for a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random sampling for finding all solutions to 3-SAT (All-SAT) and Max-SAT problems. G-QAOA is less resource-intensive and more adaptable for these problems than Grover’s algorithm, and it surpasses conventional QAOA in its ability to sample all solutions. We show these benefits by classical simulations of many-round G-QAOA on thousands of random 3-SAT instances. We also observe G-QAOA advantages on the IonQ Aria quantum computer for small instances, finding that current hardware suffices to determine and sample all solutions. Interestingly, a single-angle-pair constraint that uses the same pair of angles at each G-QAOA round greatly reduces the classical computational overhead of optimizing the G-QAOA angles while preserving its quadratic speedup. We also find parameter clustering of the angles. The single-angle-pair protocol and parameter clustering significantly reduce obstacles to classical optimization of the G-QAOA angles.
We present a comparison of different quantum state preparation algorithms and their overall efficiency for the Schwinger model with a theta term. While adiabatic state preparation is proved to be effective, in practice it leads to large gate counts to prepare the ground state. The quantum approximate optimization algorithm (QAOA) provides excellent results while keeping the counts small by design, at the cost of an expensive classical minimization process. We introduce a “blocked” modification of the Schwinger Hamiltonian to be used in the QAOA that further decreases the length of the algorithms as the size of the problem is increased. The rodeo algorithm (RA) provides a powerful tool to efficiently prepare any eigenstate of the Hamiltonian, as long as its overlap with the initial guess is large enough. We obtain the best results when combining the blocked QAOA ansatz and the RA, as this provides an excellent initial state with a relatively short algorithm without the need to perform any classical steps for large problem sizes. Published by the American Physical Society 2025
We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, ADAPT-Clifford, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an N -vertex graph by building a depth O ( N ) Clifford circuit. The algorithm has runtime complexity O ( N 2 ) and O ( N 3 ) for sparse and dense graphs, respectively, and space complexity O ( N 2 ) , with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to ∼ 94 % of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW. Published by the American Physical Society 2024
Current NISQ devices are prone to errors. In order to be used for practical applications or achieve fault-tolerant thresholds, strategies to suppress error rates will be needed to maximize the potential of noisy devices. Dynamical decoupling (DD) is one such strategy for suppressing — or at least alleviating — the effects of decoherence, in which sequences of pulses are applied to qubits to decouple their interaction with the environment. Through experimental runs performed on several Rigetti quantum computing units (QPUs), we first demonstrate that DD is capable of improving coherence times for isolated qubits, as well as suppressing errors caused by the ZZ coupling between pairs of qubits. Extending this framework to cycles containing2-qubit gates, we show that DD can be inserted to decouple qubits from crosstalk occurring during neighboring 2-qubit gates, and demonstrate the efficacy of this procedure on quantum approximate optimization algorithm (QAOA) circuits. We also explore the usage of tailored DD sequences for the suppression of characterized error channels. We are grateful for support from the NASA Ames Research Center and from the DARPA ONISQ program under interagency agreement IAA 8839,Annex 114. HYH is supported by the USRA Feynman QuantumAcademy funded by the NAMS R&D Student Program and a UCHellman Fellowship. JS, ZGI and ZW are supported by USRA NASAAcademic Mission Service (NNA16BD14C).
Quantum computing has emerged as a powerful computational paradigm capable of solving problems beyond the reach of classical computers. However, today’s quantum computers are noisy, posing challenges to obtaining accurate results. Here, we explore the impact of noise on quantum computing, focusing on the challenges in sampling bit strings from noisy quantum computers and the implications for optimization and machine learning. 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 real quantum computers involving up to 127 qubits. The results show strong alignment with theoretical predictions.
Many of the most challenging computational problems arising in practical applications are tackled by heuristic algorithms which have not been rigorously proven to outperform other approaches but rather have been empirically demonstrated to be effective. While quantum heuristics have been proposed since the early days of quantum computing, true empirical evaluation of the real-world performance of these algorithms is only becoming possible now as increasingly powerful quantum gate-model devices continue to come online.In this talk, I will give an overview of the NASA QuAIL team's ongoing investigation into quantum gate-model heuristic algorithms for exact and approximate optimization. In particular, we consider the performance of the Quantum Approximate Optimization Algorithm on NP-hard optimization problems, and describe algorithm parameter setting strategies for real-world quantum hardware. We then show a generalization of QAOA circuits, the Quantum Alternating Operator Ansatz, especially suitable for low-resource implementations of QAOA for problems with hard (feasibility) constraints. The talk will conclude with a discussion of research challenges, particularly for optimization and sampling applications of QAOA, and the potential of more general quantum heuristics to give advantages over classical computers.
Constraint satisfaction problems are an important area of computer science. Many of these problems are in the complexity class NP which is exponentially hard for all known methods, both for worst cases and often typical. Fundamentally, the lack of any guided local minimum escape method ensures the hardness of both exact and approximate optimization classically, but the intuitive mechanism for approximation hardness in quantum algorithms based on Hamiltonian time evolution is poorly understood. We explore this question using the prototypically hard MAX-3-XORSAT problem class. We conclude that the mechanisms for quantum exact and approximation hardness are fundamentally distinct. We qualitatively identify why traditional methods such as quantum adiabatic optimization are not good approximation algorithms. We propose a new spectral folding optimization method that does not suffer from these issues and study it analytically and numerically. We consider random rank-3 hypergraphs including extremal planted solution instances, where the ground state satisfies an anomalously high fraction of constraints compared to truly random problems. We show that, if we define the energy to be $E = N_{unsat}-N_{sat}$, then spectrally folded quantum optimization will return states with energy $E \leq A E_{GS}$ (where $E_{GS}$ is the ground state energy) in polynomial time, where conservatively, $A \simeq 0.6$. We thoroughly benchmark variations of spectrally folded quantum optimization for random classically approximation-hard (planted solution) instances in simulation, and find performance consistent with this prediction. We do not claim that this approximation guarantee holds for all possible hypergraphs, though our algorithm's mechanism can likely generalize widely. These results suggest that quantum computers are more powerful for approximate optimization than had been previously assumed.
Approximate quantum error correction (AQEC) not only dictates the performance of discrete- and continuous-variable quantum error correction codes but also serves as a unifying framework across various physical disciplines. Identifying the optimal recovery channel to maximize the entanglement fidelity via standard semidefinite programming is computationally bottlenecked by the exponentially growing number of Kraus operators with system size, rendering large-scale optimization prohibitive. While analytical near-optimal maps exist, they typically work only when the Knill-Laflamme conditions are nearly satisfied. In this Letter, we establish an efficient framework by leveraging the duality between recovery and environment decoupling. This framework yields a tighter analytical lower bound on entanglement fidelity than the conventional limit set by the transpose channel. Furthermore, by exploiting the decayed weights of noise Kraus operators, we introduce a framework based on principal component analysis to reduce the dimension. In thermal loss channels where the weights decay exponentially, our approach achieves a 33-fold computational speedup while maintaining rigorous accuracy. Our approach enables high-precision optimization for AQEC codes that were previously intractable due to the curse of dimensionality.
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We present a fast and Bayes-optimal-approximating tensor network decoder for planar quantum LDPC codes based on the tensor renormalization group algorithm, originally proposed by Levin, and Nave. By precomputing the renormalization group flow for the null syndrome, we need only recompute tensor contractions in the causal cone of the measured syndrome at the time of decoding. This allows us to achieve an overall runtime complexity of ($pnχ^6$) where p is the depolarizing noise rate, and χ is the cutoff value used to control singular value decomposition approximations used in the algorithm. We apply our decoder to the surface code in the code capacity noise model and compare its performance to the original matrix product state (MPS) tensor network decoder introduced by Bravyi, Suchara, and Vargo. The MPS decoder has a p-independent runtime complexity of $\mathcal{O}(nχ^3)$ resulting in significantly slower decoding times compared to our algorithm in the low-p regime.
Unitary coupled cluster theory (UCC) is a common wave function ansatz for quantum simulation of molecular electronic structure using the variational quantum eigenvalue solver (VQE). Even for small molecules using a double-ζ basis, the number of variational parameters required to minimize the electronic energy (i.e., optimize the circuit) is large and beyond the reach of current quantum computers. For example, a circuit simulating C2 using the UCCSD ansatz and the cc-pVDZ basis set with frozen-core will require over 10,000 variational parameters and a Hilbert space of over 10^8 determinants. To make progress on simulating such molecular systems on near-term quantum computers, we explore how much of the optimization can be approximately prepared with classical simulation while reducing the number of optimization steps performed on a quantum device. Recently, Chen, Cheng, and Freericks [J. Chem. Theory Comput. 2021, 17, 841-847] presented an algorithm for the factorized form of the UCC ansatz that allows for efficient UCC optimizations on classical hardware. We flip the algorithm around and use it to prepare approximate quantum circuits for systems that require a large number of qubits to represent. We will present results from our implementation and discuss strategies for incorporating this implementation for algorithms involving near-term quantum computers.