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arXiv:1904.09314 · Physical Review A

XY mixers: Analytical and numerical results for the quantum alternating operator ansatz

Zhihui Wang, Nicholas C. Rubin, Jason Dominy, Eleanor Gilbert Rieffel

Published Jan 14, 2020• 189 citations
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Abstract

The quantum alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using $XY$ Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that, for an integer variable admitting $\ensuremath{\kappa}$ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\ensuremath{\kappa})$. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$ mixers is borne out numerically, demonstrating a significant improvement over the general $X$ mixer, and moreover the generalized $W$ state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.

DOI
10.1103/physreva.101.012320

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The quantum alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using XYXYXY Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the XYXYXY model, we demonstrate that, for an integer variable admitting \ensuremathκ\ensuremath{\kappa}\ensuremathκ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth O(\ensuremathκ)O(\ensuremath{\kappa})O(\ensuremathκ). We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using XYXYXY mixers is borne out numerically, demonstrating a significant improvement over the general XXX mixer, and moreover the generalized WWW state yields better performance than easier-to-generate classical initial states when XYXYXY mixers are used.
Quantum Approximate Optimization Algorithm, further generalized as Quantum Alternating Operator Ansatz (QAOA), is a family of algorithms for combinatorial optimization problems. It is a leading candidate to run on emerging universal quantum computers to gain insight into quantum heuristics. In constrained optimization, penalties are often introduced so that the ground state of the cost Hamiltonian encodes the solution (a standard practice in quantum annealing). An alternative is to choose a mixing Hamiltonian such that the constraint corresponds to a constant of motion and the quantum evolution stays in the feasible subspace. Better performance of the algorithm is speculated due to a much smaller search space. We consider problems with a constant Hamming weight as the constraint. We also compare different methods of generating the generalized W-state, which serves as a natural initial state for the Hamming-weight constraint. Using graph-coloring as an example, we compare the performance of using XY model as a mixer that preserves the Hamming weight with the performance of adding a penalty term in the cost Hamiltonian.
Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by 5.30%5.30\%5.30% on ER random graphs and 5.41%5.41\%5.41% on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only 15.30%15.30\%15.30% and 25.18%25.18\%25.18% of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.
The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most of the literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints. We generalize the “XY”-mixer designed to preserve the subspace of “one-hot” states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of the number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for an “XY”-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.
ABSTRACT Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA‐QAOA+. Unlike QAOA+, AMA‐QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA‐QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA‐QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by on Erdős–Rényi random graphs and on 3‐regular graphs. Moreover, AMA‐QAOA+ significantly reduces the CNOT gate consumption, requiring only and of the CNOT gates used by QAOA+ on Erdős–Rényi and 3‐regular random graphs, respectively. These results demonstrate that AMA‐QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource‐efficient quantum circuits.
The Quantum Alternating Operator Ansatz is a generalization of the Quantum Approximate Optimization Algorithm (QAOA) designed for finding approximate solutions to combinatorial optimization problems with hard constraints. In this paper, we study Maximum k-Vertex Cover under this ansatz due to its modest complexity, while still being more complex than the well studied problems of Max-Cut and Max E3-LIN2. Our approach includes (i) a performance comparison between easy-to-prepare classical states and Dicke states as starting states, (ii) a performance comparison between two XY-Hamiltonian mixing operators: the ring mixer and the complete graph mixer, (iii) an analysis of the distribution of solutions via Monte Carlo sampling, and (iv) the exploration of efficient angle selection strategies. Our results are: (i) Dicke states improve performance compared to easy-to-prepare classical states, (ii) an upper bound on the simulation of the complete graph mixer, (iii) the complete graph mixer improves performance relative to the ring mixer, (iv) numerical results indicating the standard deviation of the distribution of solutions decreases exponentially inppp(the number of rounds in the algorithm), requiring an exponential number of random samples to find a better solution in the next round, and (v) a correlation of angle parameters which exhibit high quality solutions that behave similarly to a discretized version of the Quantum Adiabatic Algorithm.
The Quantum Alternating Operator Ansatz (QAOA) is a promising quantum heuristic to combinatorial optimization, designed for the Noisy Intermediate-Scale Quantum (NISQ) era. A key quantum operator of the algorithm is the mixing, which introduces quantum interference. This work investigates analytically the performance of QAOA with the Grover mixer, a mixing operator inspired by Grover’s algorithm. A critical feature of QAOA with Grover mixer is that it depends only on the probability distribution of the solutions, ignoring the structure of the problem. Due to this, it was conjectured that QAOA with this mixer is bounded to the quadratic speed-up of Grover’s algorithm over the classical brute force on the unstructured search problem. On the other hand, this feature allows the development of a statistical approach, which we explore in this work. For a variant of QAOA called Grover mixing QAOA (GM-QAOA), the analysis leads to an expression for the expectation value whose number of terms depends exponentially on the number of layers of the QAOA. In contrast, for Grover mixer Threshold QAOA (GM-Th-QAOA), a simpler variant that has an intimate relationship with Grover’s algorithm, we obtain a constant-time expression, which allows us to get bounds for different performance metrics, such as the statistical quantities of standard score and quantile. Subsequently, we generalized QAOA with the Grover mixer in a framework we called Grover-based QAOA. By using a contradiction argument with the optimality of Grover’s algorithm, we extend all the bounds of GM-Th-QAOA to the general context of Grover-based QAOA. As a result, we get the main contribution of this work: the proof of the aforementioned conjecture on the performance of QAOA with Grover mixer. We apply our bounds to the class of complete bipartite graphs on the Max-Cut problem, demonstrating that we cannot obtain a polynomial-time algorithm with Grover-based QAOA for these instances, showing that the performance limitation of the Grover mixer can be very severe. Our results suggest that achieving competitive performance with QAOA requires a mixing operator capable of leveraging the structure of combinatorial optimization problems.