The quantum approximate optimization algorithm~(QAOA) first proposed by Farhi et al. promises near-term applications based on its simplicity, universality, and provable optimality. A depth-p QAOA consists of p interleaved unitary transformations induced by two mutually non-commuting Hamiltonians. A long-standing question concerning the performance of QAOA is the dependence of its success probability as a function of circuit depth p. We make initial progress by analyzing the success probability of QAOA for realizing state transfer in a one-dimensional qubit chain using two-qubit XY Hamiltonians and single-qubit Hamiltonians. We provide analytic state transfer success probability dependencies on p in both low and large p limits by leveraging the unique spectral property of the XY Hamiltonian. We support our proof under a given QAOA ansatz with numerical optimizations of QAOA for up to \(N\)=20 qubits. We show that the optimized QAOA can achieve the well-known quadratic speedup, Grover speedup, over the classical alternatives. Treating QAOA optimization as a quantum control problem, we also provide numerical evidence of how the circuit depth determines the controllability of the QAOA ansatz.
The quantum approximate optimization algorithm~(QAOA) first proposed by Farhi et al. promises near-term applications based on its simplicity, universality, and provable optimality. A depth-p QAOA consists of p interleaved unitary transformations induced by two mutually non-commuting Hamiltonians. A long-standing question concerning the performance of QAOA is the dependence of its success probability as a function of circuit depth p. We make initial progress by analyzing the success probability of QAOA for realizing state transfer in a one-dimensional qubit chain using two-qubit XY Hamiltonians and single-qubit Hamiltonians. We provide analytic state transfer success probability dependencies on p in both low and large p limits by leveraging the unique spectral property of the XY Hamiltonian. We support our proof under a given QAOA ansatz with numerical optimizations of QAOA for up to (N)=20 qubits. We show that the optimized QAOA can achieve the well-known quadratic speedup, Grover speedup, over the classical alternatives. Treating QAOA optimization as a quantum control problem, we also provide numerical evidence of how the circuit depth determines the controllability of the QAOA ansatz.
The quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm intending to find the ground state of a target Hamiltonian. Theoretically, QAOA can obtain the approximate solution if the quantum circuit is deep enough. Actually, the performance of QAOA decreases practically if the quantum circuit is deep since near-term devices are not noise free, and the errors caused by noise accumulate as the quantum circuit increases. In order to reduce the depth of quantum circuits, we propose an ansatz dubbed as ``Shortcuts to QAOA'' (S-QAOA), S-QAOA provides shortcuts to the ground state of the target Hamiltonian by including more two-body interactions and releasing the parameter freedoms. To be specific, besides the existing ZZ interaction in the QAOA ansatz, other two-body interactions are introduced in the S-QAOA ansatz such that the approximate solutions could be obtained with smaller circuit depth. Considering the Max-Cut problem and the Sherrington-Kirkpatrick model, numerically computation shows the YY interaction has the best performance. The reason for this might arise from the counterdiabatic effect generated by YY interaction. On top of this, we release the freedom of parameters of two-body interactions, which a priori do not necessarily have to be fully identical, and numerical results show that it is worth paying the extra cost of having more parameter freedom since one has a greater improvement on success rate.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.
Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires 2mp CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate n−1 CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by ≃84.8% for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with ≃5%; fewer SWAP gates for Erdős-Renyi graphs.
The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for solving NP hard combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) hardware. However, its performance is critically dependent on the selection of the circuit depth a parameter that must be specified a priori without clear guidance. In this paper, we introduce a variant of QAOA called dynamic depth Quantum Approximate Optimization Algorithm (DDQAOA) that resolves the challenge of pre selecting a fixed circuit depth. Our method adaptively expands circuit depth, starting from p = 1 and progressing up to p = 10, by transferring learned parameters to deeper circuits based on convergence criteria. We tested this approach on 100 instances of the Constrained Shortest Path Problem (CSPP) at 10 qubit and 16 qubit scales. Our DDQAOA achieved superior approximation ratios and success probabilities with fewer CNOT gate evaluations than the standard QAOA for p = 3, 5, 10, and 15. In particular, while standard QAOA at p = 15 achieved results close to our approach, it used 217% and 159.3% more CNOT gates for 10 qubit and 16 qubit instances, respectively. This demonstrates the performance and practical applicability of DDQAOA to solve combinatorial optimization problems on near term devices.
Farhi et al. recently proposed a class of quantum algorithms, the quantum approximate optimization algorithm (QAOA), for approximately solving combinatorial optimization problems (E. Farhi et al., arXiv:1411.4028; arXiv:1412.6062; arXiv:1602.07674). A level-p QAOA circuit consists of p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here we analytically and numerically study parameter setting for the QAOA applied to MaxCut. For the level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the ``ring of disagrees,'' or the one-dimensional antiferromagnetic ring, we provide an analysis for an arbitrarily high level. Using a fermionic representation, the evolution of the system under the QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of the QAOA for any p. It also greatly simplifies the numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional submanifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
The quantum approximate optimization algorithm (QAOA) is promising for achieving quantum computational advantage with near-term quantum devices. It was numerically shown that the QAOA cost functions exhibit a phenomenon called reachability deficit (RD), where the success probability cannot reach unity until the circuit depth exceeds a certain critical value. However, an in-depth theoretical understanding of RD remains lacking. Here we focus on a variational variant of Grover search on multiple marked solutions as a prototype for analyzing the RD problem, where we further relax the criterion of reachability by tolerating a certain probability of failure. Specifically, we obtain a general analytical expression relating the critical depth of the quantum circuit to the solution density. In the dilute limit, the critical depth is consistent with the Grover bound, exhibiting a robust quadratic scaling that is insensitive to the failure probability. Moreover, we also find that the projective mixing Hamiltonian performs significantly better than the traditional mixing Hamiltonian in the QAOA, although it is less favorable in terms of physical implementation. However, by taking into account two-body interactions in the mixing Hamiltonian, the performance becomes on par with the projective mixing Hamiltonian at the cost of O(n2) additional terms. These results represent a simplified but insightful model of the QAOA, fully addressing the dependence of required circuit depth over the ground-state degeneracy.