Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices

Zhou, Leo; Wang, Shengtao; Choi, Soonwon; Pichler, Hannes; Lukin, Mikhail D.

doi:10.1103/physrevx.10.021067

Cited by 583 publications

(630 citation statements)

References 55 publications

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“…For all instances, the parameters were initialized such that they realize a simple linear interpolation between H P and H B -i.e., for odd j (parameters of H P terms) we have that θ j = j/d while for even j (parameters of H B terms) we have that θ j = 1 − j/d. This initialization is informed by recent studies which identified global optima for MaxCut problems, which were typically close to linear interpolation solutions [46]. The optimization was carried out using the Adam optimizer, with initial learning rate α = 0.001 and hyper-parameters β 1 = 0.8 and β 2 = 0.999.…”

Section: Stochastic Gradient Descent For Qaoamentioning

confidence: 99%

Stochastic gradient descent for hybrid quantum-classical optimization

Sweke

Wilde

Meyer

et al. 2020

Quantum

197

123

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Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.

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Section: Stochastic Gradient Descent For Qaoamentioning

confidence: 99%

Stochastic gradient descent for hybrid quantum-classical optimization

Sweke

Wilde

Meyer

et al. 2020

Quantum

197

123

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show abstract

“…Moreover, there is a significant interest in solving optimization problems on quantum computers, in particular with the quantum approximate optimization algorithm (QAOA) [11][12][13]. This variational algorithm has been used to study a range of discrete [11,[14][15][16] and continuous [17] optimization problems, and may have applications for unstructured search [18]. While there is currently no proof that it can provide an asymptotic quantum advantage, QAOA is an emerging approach for benchmarking quantum devices and is a candidate for demonstrating a practical quantum speed up on near-term NISQ devices.…”

Section: Introductionmentioning

confidence: 99%

Improving the Performance of Deep Quantum Optimization Algorithms with Continuous Gate Sets

et al. 2020

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Variational quantum algorithms are believed to be promising for solving computationally hard problems on noisy intermediate-scale quantum (NISQ) systems. Gaining computational power from these algorithms critically relies on the mitigation of errors during their execution, which for coherence-limited operations is achievable by reducing the gate count. Here, we demonstrate an improvement of up to a factor of 3 in algorithmic performance for the quantum approximate optimization algorithm (QAOA) as measured by the success probability, by implementing a continuous hardware-efficient gate set using superconducting quantum circuits. This gate set allows us to perform the phase separation step in QAOA with a single physical gate for each pair of qubits instead of decomposing it into two CZ gates and single-qubit gates. With this reduced number of physical gates, which scales with the number of layers employed in the algorithm, we experimentally investigate the circuit-depth-dependent performance of QAOA applied to exact-cover problem instances mapped onto three and seven qubits, using up to a total of 399 operations and up to nine layers. Our results demonstrate that the use of continuous gate sets may be a key component in extending the impact of near-term quantum computers.

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“…Quantum annealing (QA) [2][3][4][5][6], alias adiabatic quantum computation (AQC) [7,8], is a promising quantum algorithm implemented [9] in present noisy intermediate-scale quantum devices [10]. More recently, the quantum approximate optimization algorithm (QAOA) [11]-a hybrid quantum-classical variational optimization scheme [12]-has gained momentum [13][14][15][16] and has been successfully realized in several experimental platforms [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…QAOA consists of a classical minimization in the 2P-dimensional energy landscape, which is in general not a trivial task [21], because local optimizations tend to get trapped into one of the many local minima, producing irregular parameters (γ * , β * ), hard to implement and sensitive to noise. To obtain stable and regular schedules (γ * , β * ), easily generalized to different values of P and implemented experimentally, iterative procedures should be employed [14,16,17]. For quantum Ising chains, smooth regular optimal parameters can be found [14], which are adiabatic in a digitized-QA/AQC [22] context.…”

Section: Introductionmentioning

confidence: 99%

Reinforcement-learning-assisted quantum optimization

Wauters

Panizon

Mbeng

et al. 2020

Phys. Rev. Research

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We propose a reinforcement learning (RL) scheme for feedback quantum control within the quantum approximate optimization algorithm (QAOA). We reformulate the QAOA variational minimization as a learning task, where an RL agent chooses the control parameters for the unitaries, given partial information on the system. Such an RL scheme finds a policy converging to the optimal adiabatic solution of the quantum Ising chain that can also be successfully transferred between systems with different sizes, even in the presence of disorder. This allows for immediate experimental verification of our proposal on more complicated models: the RL agent is trained on a small control system, simulated on classical hardware, and then tested on a larger physical sample.

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Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices

Cited by 583 publications

References 55 publications

Stochastic gradient descent for hybrid quantum-classical optimization

Stochastic gradient descent for hybrid quantum-classical optimization

Improving the Performance of Deep Quantum Optimization Algorithms with Continuous Gate Sets

Reinforcement-learning-assisted quantum optimization

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