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arXiv:1701.01450

Practical optimization for hybrid quantum-classical algorithms

Gian Giacomo Guerreschi, Mikhail Smelyanskiy

Published Jan 5, 2017• 99 citations
arXivHTMLPDF

Abstract

A novel class of hybrid quantum-classical algorithms based on the variational approach have recently emerged from separate proposals addressing, for example, quantum chemistry and combinatorial problems. These algorithms provide an approximate solution to the problem at hand by encoding it in the state of a quantum computer. The operations used to prepare the state are not a priori fixed but, quite the opposite, are subjected to a classical optimization procedure that modifies the quantum gates and improves the quality of the approximate solution. While the quantum hardware determines the size of the problem and what states are achievable (limited, respectively, by the number of qubits and by the kind and number of possible quantum gates), it is the classical optimization procedure that determines the way in which the quantum states are explored and whether the best available solution is actually reached. In addition, the quantities required in the optimization, for example the objective function itself, have to be estimated with finite precision in any experimental implementation. While it is desirable to have very precise estimates, this comes at the cost of repeating the state preparation and measurement multiple times. Here we analyze the competing requirements of high precision and low number of repetitions and study how the overall performance of the variational algorithm is affected by the precision level and the choice of the optimization method. Finally, this study introduces quasi-Newton optimization methods in the general context of hybrid variational algorithms and presents quantitative results for the Quantum Approximate Optimization Algorithm.

DOI
10.48550/arxiv.1701.01450

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A novel class of hybrid quantum-classical algorithms based on the variational approach have recently emerged from separate proposals addressing, for example, quantum chemistry and combinatorial problems. These algorithms provide an approximate solution to the problem at hand by encoding it in the state of a quantum computer. The operations used to prepare the state are not a priori fixed but, quite the opposite, are subjected to a classical optimization procedure that modifies the quantum gates and improves the quality of the approximate solution. While the quantum hardware determines the size of the problem and what states are achievable (limited, respectively, by the number of qubits and by the kind and number of possible quantum gates), it is the classical optimization procedure that determines the way in which the quantum states are explored and whether the best available solution is actually reached. In addition, the quantities required in the optimization, for example the objective function itself, have to be estimated with finite precision in any experimental implementation. While it is desirable to have very precise estimates, this comes at the cost of repeating the state preparation and measurement multiple times. Here we analyze the competing requirements of high precision and low number of repetitions and study how the overall performance of the variational algorithm is affected by the precision level and the choice of the optimization method. Finally, this study introduces quasi-Newton optimization methods in the general context of hybrid variational algorithms and presents quantitative results for the Quantum Approximate Optimization Algorithm.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.
ABSTRACT: Even if the quantum computing popularity has recently grown, the currently-available technology is not ready for a fundamental revolution of informatics. In this transition phase, hybrid quantum-classical algorithms can help in understanding the real power that this novel paradigm can reach in the future. In this work, we tested the capability of two such algorithms to solve combinatorial optimization problems: the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). We used them to solve max-cut problems, consisting of separating the vertices of an undirected graph in two sub-groups so that the number of edges connecting vertices of different sub-groups is maximal. To do that, we mapped the problem into a quantum system, using the Ising formalism. In particular, we investigated the effect of changing the number of measurements of the quantum circuit associated with the system on the algorithms convergence. A large number of measurements (or shots) allows a better knowledge of the quantum state, but on the other hand, increases the convergence time, making the algorithms less competitive compared to classical techniques. The results showed that VQE needs a mínimum number of measurements to start converging towards the optimal problem solution. Once this threshold is overcome, the solution improves as 1/√shots. QAOA, on the other hand, does not seem to converge, independently of the number of shots.
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 XXX rotation and for each ZZZZZZ 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.
The computational capabilities of near-term quantum computers are limited by the noisy execution of gate operations and a limited number of physical qubits. Hybrid variational algorithms are well-suited to near-term quantum devices because they allow for a wide range of tradeoffs between the amount of quantum and classical resources used to solve a problem. This paper investigates tradeoffs available at both the algorithmic and hardware levels by studying a specific case – applying the Quantum Approximate Optimization Algorithm (QAOA) to instances of the Maximum Independent Set (MIS) problem. We consider three variants of the QAOA which offer different tradeoffs at the algorithmic level in terms of their required number of classical parameters, quantum gates, and iterations of classical optimization needed. Since MIS is a constrained combinatorial optimization problem, the QAOA must respect the problem constraints. This can be accomplished by using many multi-controlled gate operations which must be decomposed into gates executable by the target hardware. We study the tradeoffs available at this hardware level, combining the gate fidelities and decomposition efficiencies of different native gate sets into a single metric called the gate decomposition cost .
Hybrid quantum/classical variational algorithms can be implemented on noisy intermediate-scale quantum computers and can be used to find solutions for combinatorial optimization problems. Approaches discussed in the literature minimize the expectation of the problem Hamiltonian for a parameterized trial quantum state. The expectation is estimated as the sample mean of a set of measurement outcomes, while the parameters of the trial state are optimized classically. This procedure is fully justified for quantum mechanical observables such as molecular energies. In the case of classical optimization problems, which yield diagonal Hamiltonians, we argue that aggregating the samples in a different way than the expected value is more natural. In this paper we propose the Conditional Value-at-Risk as an aggregation function. We empirically show -- using classical simulation as well as quantum hardware -- that this leads to faster convergence to better solutions for all combinatorial optimization problems tested in our study. We also provide analytical results to explain the observed difference in performance between different variational algorithms.