Variational quantum algorithms are a promising candidate for execution on near-term quan- tum computers, and a number of experimental demonstrations of these algorithms have al- ready been performed. These algorithms rely on a classical optimization subroutine, and hence the efficiency of these algorithms can be limited by the performance of these opti- mizers. Here, we saw that to accurately assess the performance of these optimizers, it is crucial to develop a good cost model, and tune available hyperparameters to operational specifications.
Given the unique considerations of quantum systems, we developed two new surrogate model-based optimizers, MGD and MPG, to fill some of the gaps of previous methods. We numerically compared their performance with other popular alternatives, and found it advantageous in several realistic settings. We also probed how the cost model and presence of errors can significantly impact the choice of optimizer in a practical setting.
Now that quantum computers are coming online, accessing superconducting qubits through a cloud interface is an important scenario to consider. The latency of communicat- ing over the Internet can cause large increases in running times, but this can be mitigated by circuit batching, though the cost savings depends on the optimizer.
We also observed that inherently stochastic optimizers, such as MPG, MGD and SPSA, were more robust to variations in problems or setting once properly tuned. This extended to situations where finite gate or circuit noise was present. In contrast, while it was sometimes possible to make deterministic optimizers competitive through careful tuning, these tunings were fragile with respect to small variations in the problem or the introduction of noise. Overall, MPG and MGD’s tolerance of noise, ability to take advantage of circuit batching, and good overall performance make them good candidates for actual experiments, but the best optimizer can depend on the processor’s wall-clock model, level of noise, number of
parameters, or the specific circuit ansatz.
In this work, we have shown how practical considerations can significantly affect the calculus of choosing an optimizer for running variational algorithms. Future work will develop more accurate noise and cost models, and further development of optimizers can take these unique considerations into account.
Code Availability
Implementations of Model Gradient Descent and Model Policy Gradient are available at https://github.com/quantumlib/ReCirq.
CHAPTER 3
Quantum Approximate Optimization on a
Superconducting Qubit Processor
3.1
Introduction
The Google Sycamore superconducting qubit platform has been used to demonstrate com- putational capabilities surpassing those of classical supercomputers for certain sampling tasks [8]. However, it remains to be seen whether such processors will be able to achieve a similar computational advantage for problems of practical interest. Along with quantum chemistry [75,76], machine learning [77], and simulation of physical systems [3], discrete optimization has been widely anticipated as a promising area of application for quantum computers.
Beginning with a focus on quantum annealing [78] and adiabatic quantum computing [79], the possibility of quantum enhanced optimization has driven much interest in quantum technologies over the years. This is because faster optimization could prove transforma- tive for diverse areas such as logistics, finance, machine learning, and more. Such discrete optimization problems can be expressed as the minimization of a quadratic function of bi- nary variables [80, 81], and one can visualize these cost functions as graphs with binary variables as nodes and (weighted) edges connecting bits whose (weighted) products sum to the total cost function value. For most industrially-relevant problems, these graphs are non-planar and many ancilla would be required to embed them in (quasi-)planar graphs matching the qubit connectivity of most hardware platforms [82]. This limits the appli- cability of scalable architectures for quantum annealing [83] and corresponds to increased circuit complexity in digital quantum algorithms for optimization such as QAOA.
The quantum approximate optimization algorithm (QAOA) is the most studied gate model approach for optimization using near-term devices [9]. While the prospects for achieving quantum advantage with QAOA remain unclear, QAOA prescribes a simple
Hardware Grid Three-Regular MaxCut Sherrington-Kirkpatrick Model d.
a. b. c.
Figure 3.1: We studied three families of optimization problems: a. Hardware Grid prob- lems with a graph matching the hardware connectivity of the 23 qubits used in this ex- periment. b. MaxCut on random 3-regular graphs, with the largest instance depicted (22 qubits). c. The fully-connected Sherrington-Kirkpatrick (SK) model shown at the largest size (17 qubits). d. QAOA uses p applications of problem and driver unitaries to approxi- mate solutions to optimization problems. The parameters γ and β are shared among qubits in a layer but different for each of the p layers.
paradigm for optimization which makes it amenable to both analytical results and imple- mentation on current processors [84, 85, 86, 87, 88, 89, 90, 91, 92]. For these reasons, QAOA has also become popular as a system-level benchmark of quantum hardware. This work builds on prior experimental demonstrations of QAOA on superconducting qubits [24,93,94,95], ion traps [27], and photonics systems [96].
We are able to experimentally resolve, for the first time, increased performance with greater QAOA depth and apply QAOA to cost functions on graphs that deviate significantly from our hardware connectivity. Owing to the low error rates of the Sycamore platform, the trade-off between the theoretical increase in quality of solutions with increasing QAOA depth and additional noise is apparent for hardware-native problems. We also apply the al- gorithm to non-native graph problems with their necessary compilation overhead and study the scaling of solution quality and problem size. Our results reveal that the performance of QAOA is qualitatively different when applied to hardware native graphs versus more complex graphs, highlighting the challenge of scaling QAOA to problems of industrial importance.
For this study, we used a “Sycamore” quantum processor which consists of a two- dimensional array of 54 transmon qubits [8]. Each qubit is tunably coupled to four nearest neighbors in a rectangular lattice. In this case, all device calibration was fully automated and data was collected using a cloud interface to the platform programmed using Cirq [97]. Our experiment was restricted to 23 physical qubits of the larger Sycamore device, arranged in a topology depicted in Figure3.1a.