The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.
The hybrid quantum-classical algorithm is actively examined as a technique applicable even to intermediate-scale quantum computers. To execute this algorithm, the hardware efficient ansatz is often used, thanks to its implementability and expressibility; however, this ansatz has a critical issue in its trainability in the sense that it generically suffers from the so-called gradient vanishing problem. This issue can be resolved by limiting the circuit to the class of shallow alternating layered ansatz. However, even though the high trainability of this ansatz is proved, it is still unclear whe…
This paper investigates the experimental performance of a discrete portfolio optimization problem relevant to the financial services industry on the gate-model of quantum computing. We implement and evaluate a portfolio rebalancing use case on an idealized simulator of a gate-model quantum computer. The characteristics of this exemplar application include trading in discrete lots, non-linear trading costs, and the investment constraint. We design a novel problem encoding and hard constraint mixers for the Quantum Alternating Operator Ansatz, and compare to its predecessor the Quantum Approxima…
We present a new hybrid, local search algorithm for quantum approximate optimization of constrained combinatorial optimization problems. We focus on the Maximum Independent Set problem and demonstrate the ability of quantum local search to solve large problem instances on quantum devices with few qubits. This hybrid algorithm iteratively finds independent sets over carefully constructed neighborhoods and combines these solutions to obtain a global solution. We study the performance of this algorithm on 3-regular, Community, and Erdős-Rényi graphs with up to 100 nodes.
The maximum independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, $K_{2,3}$, and $K_{3,3}$ graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs it …
Gate-based universal quantum computers form a rapidly evolving field of quantum computing hardware technology. In previous work, we presented a quantum algorithm for lattice protein folding on a cubic lattice, tailored for quantum annealers. In this paper, we introduce a novel approach for solving the lattice protein folding problem on universal gate-based quantum computing architectures. Lattice protein models are coarse-grained representations of proteins that have been used extensively over the past thirty years to examine the principles of protein folding and design.These models can be use…
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 mix…
Variational quantum algorithms (VQAs) are the quantum analog of classical neural networks (NNs). A VQA consists of a parameterized quantum circuit (PQC) which is composed of multiple layers of ansatzes (simpler PQCs, which are an analogy of NN layers) that differ only in selections of parameters. Previous work has identified the alternating layered ansatz as potentially a new standard ansatz in near-term quantum computing. Indeed, shallow alternating layered VQAs are easy to implement and have been shown to be both trainable and expressive. In this work, we introduce a training algorithm with …
This paper establishes the tractability of finding the optimal Nash equilibrium, as well as the optimal social solution, to a discrete congestion game using a gate-model quantum computer. The game is of the type originally posited by Rosenthal in the 1970's. To find the optimal Nash equilibrium, we formulate an optimization problem encoding based on potential functions and path selection constraints, and solve it using the Quantum Alternating Operator Ansatz. We compare this formulation to its predecessor, the Quantum Approximate Optimization Algorithm. We implement our solution on an idealize…
We develop a framework for analyzing layered quantum algorithms such as quantum alternating operator ansätze. Our framework relates quantum cost gradient operators, derived from the cost and mixing Hamiltonians, to classical cost difference functions that reflect cost function neighborhood structure. By considering QAOA circuits from the Heisenberg picture, we derive exact general expressions for expectation values as series expansions in the algorithm parameters, cost gradient operators, and cost difference functions. This enables novel interpretability and insight into QAOA behavior in vario…
The Quantum Alternating Operator Ansatz (QAOA+) framework has recently gained attention due to its ability to solve discrete optimization problems on noisy intermediate-scale quantum (NISQ) devices in a manner that is amenable to derivation of worst-case guarantees. We design a technique in this framework to tackle a few problems over maximal matchings in graphs. Even though maximum matching is polynomial-time solvable, most counting and sampling versions are #P-hard. We design a few algorithms that generates superpositions over matchings allowing us to sample from them. In particular, we get …
A Parameter Setting Heuristic for the Quantum Alternating Operator Ansatz
Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center（量子人工智能实验室，NASA艾姆斯研究中心）；USRA Research Institute for Advanced Computer Science (RIACS)（美国大学研究协会高级计算机科学研究所）；Department of Computer Science, University of Chicago（芝加哥大学计算机系）
Approximate counterdiabatic (CD) protocols are a powerful tool to enhance quantum adiabatic processes that allow to reliably manipulate quantum systems on short time scales. However, implementing CD protocols entails the introduction of additional control fields in the Hamiltonian, often associated with highly non-local multi-body interactions. Here, we introduce a novel variational rotated ansatz (RA) to systematically generate experimentally accessible approximate CD protocols. We numerically benchmark our approach on state preparation and adiabatic quantum computing algorithms, and find tha…
We introduce multiple parametrized circuit ansätze and present the results of a numerical study comparing their performance with a standard Quantum Alternating Operator Ansatz approach. The ansätze are inspired by mixing and phase separation in the QAOA, and also motivated by compilation considerations with the aim of running on near-term superconducting quantum processors. The methods are tested on random instances of a weighted quadratic binary constrained optimization problem that is fully connected for which the space of feasible solutions has constant Hamming weight. For the parameter set…
Determining Hamiltonian ground states and energies is a challenging task with many possible approaches on quantum computers. While variational quantum eigensolvers are popular approaches for near term hardware, adiabatic state preparation is an alternative that does not require noisy optimization of parameters. Beyond adiabatic schedules, QAOA is an important method for optimization problems. In this work we modify QAOA to apply to finding ground states of molecules and empirically evaluate the modified algorithm on several molecules. This modification applies physical insights used in classic…
We introduce a variational wavefunction for many-body ground states that involves imaginary time evolution with two different Hamiltonians in an alternating fashion with variable time intervals. We successfully apply the ansatz on the one- and two-dimensional transverse-field Ising model and systematically study its scaling for the one-dimensional model at criticality. We find the total imaginary time required scales logarithmically with system size, in contrast to the linear scaling in conventional Quantum Monte Carlo. We suggest this is due to unique dynamics permitted by alternating imagina…
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we introduce a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good pa…
Variational quantum algorithms involve training parameterized quantum circuits using a classical co-processor. An important variational algorithm, designed for combinatorial optimization, is the quantum approximate optimization algorithm. Realization of this algorithm on any modern quantum processor requires either embedding a problem instance into a Hamiltonian or emulating the corresponding propagator by a gate sequence. For a vast range of problem instances this is impossible due to current circuit depth and hardware limitations. Hence we adapt the variational approach -- using ion native H…
The quantum approximate optimization algorithm (QAOA) has proved to be an effective classical-quantum algorithm serving multiple purposes, from solving combinatorial optimization problems to finding the ground state of many-body quantum systems. Since QAOA is an ansatz-dependent algorithm, there is always a need to design ansatz for better optimization. To this end, we propose a digitized version of QAOA enhanced via the use of shortcuts to adiabaticity. Specifically, we use a counterdiabatic (CD) driving term to design a better ansatz, along with the Hamiltonian and mixing terms, enhancing th…
We describe and demonstrate a method for the computation of quantum dynamics on small, noisy universal quantum computers. This method relies on the idea of `restarting' the dynamics; at least one approximate time step is taken on the quantum computer and then a parameterized quantum circuit ansatz is optimized to produce a state that well approximates the time-stepped results. The simulation is then restarted from the optimized state. By encoding knowledge of the form of the solution in the ansatz, such as ensuring that the ansatz has the appropriate symmetries of the Hamiltonian, the optimize…
Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren…
The variational quantum eigensolver (VQE) is an attracting possible application of near-term quantum computers. Originally, the aim of the VQE is to find a ground state for a given specific Hamiltonian. It is achieved by minimizing the expectation value of the Hamiltonian with respect to an ansatz state by tuning parameters \(\bmθ\) on a quantum circuit which constructs the ansatz. Here we consider an extended problem of the VQE, namely, our objective in this work is to "generalize" the optimized output of the VQE just like machine learning. We aim to find ground states for a given set of Hami…
The quantum approximate optimization algorithm (QAOA) promises to solve classically intractable computational problems in the area of combinatorial optimization. A growing amount of evidence suggests that the originally proposed form of the QAOA ansatz is not optimal, however. To address this problem, we propose an alternative ansatz, which we call QAOA+, that augments the traditional $p = 1$ QAOA ansatz with an additional multiparameter problem-independent layer. The QAOA+ ansatz allows obtaining higher approximation ratios than $p = 1$ QAOA while keeping the circuit depth below that of $p = …
In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N$_2$, H$_2$O, and C$_2$H$_4$, and benchmark systems to illustrate the perfo…
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 a new ansatz dubbed as "Shortcuts to QAOA" (S-QAOA), S-QAOA provides shortcuts to th…
A central component of variational quantum algorithms (VQA) is the state-preparation circuit, also known as ansatz or variational form. This circuit is most commonly designed to respect the symmetries of the problem Hamiltonian and, in this way, constrain the variational search to a subspace of interest. Here, we show that this approach is not always advantageous by introducing ansätze that incorporate symmetry-breaking unitaries. This class of ansätze, that we call Quantum-Optimal-Control-inspired Ansätze (QOCA), is inspired by the theory of quantum optimal control and leads to an improved co…
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, whi…
Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to …
