Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor (2004.04197v3)

Abstract: We demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the (planar) connectivity graph of our hardware; however, we also apply the QAOA to the Sherrington-Kirkpatrick model and MaxCut, both high dimensional graph problems for which the QAOA requires significant compilation. Experimental scans of the QAOA energy landscape show good agreement with theory across even the largest instances studied (23 qubits) and we are able to perform variational optimization successfully. For problems defined on our hardware graph we obtain an approximation ratio that is independent of problem size and observe, for the first time, that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size but still provides an advantage over random guessing for circuits involving several thousand gates. This behavior highlights the challenge of using near-term quantum computers to optimize problems on graphs differing from hardware connectivity. As these graphs are more representative of real world instances, our results advocate for more emphasis on such problems in the developing tradition of using the QAOA as a holistic, device-level benchmark of quantum processors.

• The paper demonstrates that QAOA achieves stable approximation ratios on native grid problems up to 23 qubits using the Sycamore processor.
• It employs swap networks to compile non-planar problems like the SK model, revealing challenges from increased circuit depth and noise.
• The study highlights the need for improved error mitigation and compilation techniques to fully realize quantum advantages on near-term devices.

Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor

The paper "Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor" addresses the application of Google's Sycamore superconducting quantum processor to combinatorial optimization problems via the Quantum Approximate Optimization Algorithm (QAOA). The paper encompasses experiments involving problems naturally suited to the hardware topology and those necessitating compilation on graphs of higher complexity than the device's planar connectivity, such as the Sherrington-Kirkpatrick (SK) model and MaxCut problems.

Experimental Methodology

Hardware Graphs and Compilation: The experiments utilize the Sycamore quantum processor, which features a 2D array of transmon qubits. The paper focuses on both hardware-native problems, directly mapped onto the processor's grid, and compiled problems where logical qubit interactions do not align with hardware connectivity. For non-native problems such as the SK model, which has a fully connected graph, swap networks facilitate the necessary qubit interactions through layers of two-qubit operations, leading to extensive circuit depths proportional to the problem size.

Quantum Approximate Optimization Algorithm (QAOA): The QAOA's variational approach minimizes a problem-specific cost function by iterating between problem and driver unitaries, parameterized by $\gamma$ and $\beta$. For increasing depth $p$, this iterated structure allows exploration of more complex solution spaces. The paper examines QAOA for several graph structures, providing insights into the algorithm's resilience and limitations on near-term quantum hardware.

Results and Analysis

The primary achievement is the demonstration that QAOA, when applied to hardware-native grid problems, yields results with an approximation ratio that does not degrade with scaling problem size up to 23 qubits. This behavior is attributed to the local nature of errors that do not propagate across the problem graph, thereby maintaining the solution quality. As QAOA circuit depth $p$ increases, experimental results reveal improved solution quality due to an enhanced optimization landscape, up to a practical limit where additional depth introduces excessive noise.

Conversely, the paper of non-native, compiled problems—particularly the SK model—illustrates a notable decrease in performance as problem size and compilation overhead increase, indicating a significant challenge for near-term devices in moving beyond hardware constraints. The experimental results, when interpolated with theoretical predictions, suggest that while performance decreases with problem size, the system maintains a quantum advantage over random guessing for deep circuits.

Implications and Future Directions

From a theoretical standpoint, this research demonstrates the complexity and variability of benchmarking quantum processors using QAOA. The findings underscore the difficulty of leveraging near-term devices for non-native, compiled problem instances while maintaining quantum benefits. These challenges are inherent due to the lack of direct mapping between complex problem graphs and hardware topology.

Practically, the results necessitate continued exploration of error mitigation techniques and compiler optimizations to better align problem Hamiltonians with hardware capabilities. The paper advocates for increased focus on non-planar graph problems, as they represent the nature of most real-world applications, offering a more holistic benchmarking approach that aligns closely with potential industrial uses.

Future research directions should involve advancing fault-tolerance through error correction and exploring hardware architectures with flexible connectivity or enhanced error rates. Additionally, the continued development of compilation techniques that minimize depth and gate count will be vital in improving QAOA's applicability to larger, more intricate optimization problems.

This paper solidifies QAOA's role in evaluating quantum processors, providing significant insights into the capabilities and limitations of current quantum technologies in addressing complex optimization problems beyond their native problem graphs.