1,629 research outputs found
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
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.Comment: 51 pages, 2 figures. Revised to match journal pape
Quantum Alternating Operator Ansatz (QAOA) beyond low depth with gradually changing unitaries
The Quantum Approximate Optimization Algorithm and its generalization to
Quantum Alternating Operator Ansatz (QAOA) is a promising approach for applying
quantum computers to challenging problems such as combinatorial optimization
and computational chemistry. In this paper, we study the underlying mechanisms
governing the behavior of QAOA circuits beyond shallow depth in the practically
relevant setting of gradually varying unitaries. We use the discrete adiabatic
theorem, which complements and generalizes the insights obtained from the
continuous-time adiabatic theorem primarily considered in prior work. Our
analysis explains some general properties that are conspicuously depicted in
the recently introduced QAOA performance diagrams. For parameter sequences
derived from continuous schedules (e.g. linear ramps), these diagrams capture
the algorithm's performance over different parameter sizes and circuit depths.
Surprisingly, they have been observed to be qualitatively similar across
different performance metrics and application domains. Our analysis explains
this behavior as well as entails some unexpected results, such as connections
between the eigenstates of the cost and mixer QAOA Hamiltonians changing based
on parameter size and the possibility of reducing circuit depth without
sacrificing performance
Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry
We present the Copupled Cluster (CC) method and the Density matrix
Renormalization Grooup (DMRG) method in a unified way, from the perspective of
recent developments in tensor product approximation. We present an introduction
into recently developed hierarchical tensor representations, in particular
tensor trains which are matrix product states in physics language. The discrete
equations of full CI approximation applied to the electronic Schr\"odinger
equation is casted into a tensorial framework in form of the second
quantization. A further approximation is performed afterwards by tensor
approximation within a hierarchical format or equivalently a tree tensor
network. We establish the (differential) geometry of low rank hierarchical
tensors and apply the Driac Frenkel principle to reduce the original
high-dimensional problem to low dimensions. The DMRG algorithm is established
as an optimization method in this format with alternating directional search.
We briefly introduce the CC method and refer to our theoretical results. We
compare this approach in the present discrete formulation with the CC method
and its underlying exponential parametrization.Comment: 15 pages, 3 figure
A Feasibility-Preserved Quantum Approximate Solver for the Capacitated Vehicle Routing Problem
The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem
(NPO) that arises in various fields including transportation and logistics. The
CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the
most efficient plan for a fleet of vehicles to deliver goods to a set of
customers, subject to the limited carrying capacity of each vehicle. As the
number of possible solutions skyrockets when the number of customers increases,
finding the optimal solution remains a significant challenge. Recently, a
quantum-classical hybrid algorithm known as Quantum Approximate Optimization
Algorithm (QAOA) can provide better solutions in some cases of combinatorial
optimization problems, compared to classical heuristics. However, the QAOA
exhibits a diminished ability to produce high-quality solutions for some
constrained optimization problems including the CVRP. One potential approach
for improvement involves a variation of the QAOA known as the Grover-Mixer
Quantum Alternating Operator Ansatz (GM-QAOA). In this work, we attempt to use
GM-QAOA to solve the CVRP. We present a new binary encoding for the CVRP, with
an alternative objective function of minimizing the shortest path that bypasses
the vehicle capacity constraint of the CVRP. The search space is further
restricted by the Grover-Mixer. We examine and discuss the effectiveness of the
proposed solver through its application to several illustrative examples.Comment: 9 pages, 8 figures, 1 tabl
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