[Paper Review] Quantum Algorithms for Fixed Qubit Architectures

Edward Farhi, Jared V. Goldstone+2 more|arXiv (Cornell University)|Mar 17, 2017

Quantum Computing Algorithms and Architecture31 references90 citations

TL;DR

The paper proposes a variational, hardware-efficient quantum algorithm on fixed qubit layouts to solve combinatorial optimization tasks without full error correction, demonstrating a baseline 0.5293 approximation for 3-regular MaxCut and exploring parameter-rich, grid-based implementations.

ABSTRACT

Gate model quantum computers with too many qubits to be simulated by available classical computers are about to arrive. We present a strategy for programming these devices without error correction or compilation. This means that the number of logical qubits is the same as the number of qubits on the device. The hardware determines which pairs of qubits can be addressed by unitary operators. The goal is to build quantum states that solve computational problems such as maximizing a combinatorial objective function or minimizing a Hamiltonian. These problems may not fit naturally on the physical layout of the qubits. Our algorithms use a sequence of parameterized unitaries that sit on the qubit layout to produce quantum states depending on those parameters. Measurements of the objective function (or Hamiltonian) guide the choice of new parameters with the goal of moving the objective function up (or lowering the energy). As an example we consider finding approximate solutions to MaxCut on 3-regular graphs whereas the hardware is physical qubits laid out on a rectangular grid. We prove that the lowest depth version of the Quantum Approximate Optimization Algorithm will achieve an approximation ratio of at least 0.5293 on all large enough instances which beats random guessing (0.5). We open up the algorithm to have different parameters for each single qubit $X$ rotation and for each $ZZ$ interaction associated with the nearest neighbor interactions on the grid. Small numerical experiments indicate that an enveloping classical algorithm can be used to find the parameters which sit on the grid to optimize an objective function with a different connectivity. We discuss strategies for finding good parameters but offer no evidence yet that the proposed approach can beat the best classical algorithms. Ultimately the strength of this approach will be determined by running on actual hardware.

Motivation & Objective

• Develop a strategy to program gate-model quantum computers with fixed qubit architecture without error correction or compilation.
• Use hardware-native two-qubit gates to build parameterized circuits that optimize a classical objective or a quantum Hamiltonian.
• Demonstrate that shallow, grid-embedded quantum circuits can achieve nontrivial approximation ratios for MaxCut on 3-regular graphs.
• Investigate how increasing parameter freedom and connectivity affects performance and potential classical speedups.

Proposed method

• Represent the objective as a sum of two-qubit terms and implement unitaries on hardware-available qubit pairs.
• Use a variational circuit with layers of single-qubit X-rotations and ZZ-type two-qubit interactions aligned to the hardware graph.
• Analyze a lowest-depth QAOA variant on a square grid to bound approximation ratios for MaxCut on 3-regular graphs.
• Introduce extra parameters by allowing distinct angles for each single-qubit and each two-qubit gate.
• Show that with grid-embedded unitaries one can bound the approximation ratio to at least 0.5293 for large instances under suitable embeddings.
• Provide numerical experiments illustrating parameter-rich grid implementations and warm-start strategies.

Experimental results

Research questions

• RQ1 Can fixed-architecture, gate-model quantum computers approximate combinatorial optimization problems without error correction?
• RQ2 What is the baseline performance (approximation ratio) achievable by shallow, hardware-native quantum circuits for MaxCut on 3-regular graphs when mapped to a grid layout?
• RQ3 Does increasing the number of tunable parameters (per-qubit and per-edge) improve performance, and is grid-based wiring viable for unrelated problem graphs?
• RQ4 Can warm-start classical solutions improve quantum optimization performance on fixed architectures?
• RQ5 To what extent can a grid-based QAOA framework beat random guessing and approach or surpass classical baselines?

Key findings

• The lowest-depth version of the Quantum Approximate Optimization Algorithm (QAOA) on a fixed grid achieves an approximation ratio of at least 0.5293 for all large enough 3-regular MaxCut instances.
• The approximation ratio bound is derived by analyzing edge-case configurations and embedding the graph onto the grid with a greedy pairing that yields m1 ≥ m/3, with m = 3n/2 edges.
• At 16 bits, a grid-embedded QAOA with p=4 (grid 4x4) achieved an approximation ratio of 0.6424 after vertex-to-grid assignment and angle optimization, compared to 0.7519 for the original graph at p=1.
• Expanding the parameter space to allow distinct angles for each ZZ gate and each X rotation can be beneficial, as shown by numerical experiments with increased parameter freedom.
• Warm-start experiments indicate that initializing parameters to reproduce near-optimal classical solutions can facilitate climbing to higher objective values, suggesting potential practical gains.