arXiv subjects

Phillip C. Lotshaw

Publications and source records attributed to Phillip C. Lotshaw.

29 records · Page 2Linked to original sources

arXiv2022

Numerical Simulations of Noisy Quantum Circuits for Computational Chemistry

The opportunities afforded by near-term quantum computers to calculate the ground-state properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we investigate the behavior of these noisy quantum circuits using numerical simulations to estimate the accuracy and fidelity of the prepared quantum states relative to the ground truth obtained by conventional means. We implement several different types of ansatz circuits derived from unitary coupled cluster theory for the purposes of estimating the ground-state energy of Sodium Hydride using the variational quantum eigensolver algorithm. We show how relative error in the energy and the fidelity scale with the levels of gate-based noise, the inter-molecular configuration, the ansatz circuit depth, and the parameter optimization methods.

quant-ph↗

arXiv2022

Scaling Quantum Approximate Optimization on Near-term Hardware

The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.

quant-ph↗

arXiv2021

QAOAKit: A Toolkit for Reproducible Study, Application, and Verification of the QAOA

Understanding the best known parameters, performance, and systematic behavior of the Quantum Approximate Optimization Algorithm (QAOA) remain open research questions, even as the algorithm gains popularity. We introduce QAOAKit, a Python toolkit for the QAOA built for exploratory research. QAOAKit is a unified repository of preoptimized QAOA parameters and circuit generators for common quantum simulation frameworks. We combine, standardize, and cross-validate previously known parameters for the MaxCut problem, and incorporate this into QAOAKit. We also build conversion tools to use these parameters as inputs in several quantum simulation frameworks that can be used to reproduce, compare, and extend known results from various sources in the literature. We describe QAOAKit and provide examples of how it can be used to reproduce research results and tackle open problems in quantum optimization.

quant-ph↗

arXiv2021

Multi-angle Quantum Approximate Optimization Algorithm

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 parameters can be found in polynomial time. This new ansatz gives a 33\% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.

quant-ph↗

arXiv2021

Asymmetric temperature equilibration with heat flow from cold to hot in a quantum thermodynamic system

A model computational quantum thermodynamic network is constructed with two variable temperature baths coupled by a linker system, with an asymmetry in the coupling of the linker to the two baths. It is found in computational simulations that the baths come to "thermal equilibrium" at different bath energies and temperatures. In a sense, heat is observed to flow from cold to hot. A description is given in which a recently defined quantum entropy $S^Q_{univ}$ for a pure state "universe" continues to increase after passing through the classical equilibrium point of equal temperatures, reaching a maximum at the asymmetric equilibrium. Thus, a second law account $ΔS^Q_{univ} \ge 0$ holds for the asymmetric quantum process. In contrast, a von Neumann entropy description fails to uphold the entropy law, with a maximum near when the two temperatures are equal, then a decrease $ΔS^{vN} < 0$ on the way to the asymmetric equilibrium.

cond-mat.stat-mech↗

arXiv2021

Globally optimizing QAOA circuit depth for constrained optimization problems

We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.

quant-ph↗

arXiv2021

QuaSiMo: A Composable Library to Program Hybrid Workflows for Quantum Simulation

We present a composable design scheme for the development of hybrid quantum/classical algorithms and workflows for applications of quantum simulation. Our object-oriented approach is based on constructing an expressive set of common data structures and methods that enable programming of a broad variety of complex hybrid quantum simulation applications. The abstract core of our scheme is distilled from the analysis of the current quantum simulation algorithms. Subsequently, it allows a synthesis of new hybrid algorithms and workflows via the extension, specialization, and dynamic customization of the abstract core classes defined by our design. We implement our design scheme using the hardware-agnostic programming language QCOR into the QuaSiMo library. To validate our implementation, we test and show its utility on commercial quantum processors from IBM and Rigetti, running some prototypical quantum simulations.

quant-ph↗

arXiv2021

Empirical performance bounds for quantum approximate optimization

The quantum approximate optimization algorithm (QAOA) is a variational method for noisy, intermediate-scale quantum computers to solve combinatorial optimization problems. Quantifying performance bounds with respect to specific problem instances provides insight into when QAOA may be viable for solving real-world applications. Here, we solve every instance of MaxCut on non-isomorphic unweighted graphs with nine or fewer vertices by numerically simulating the pure-state dynamics of QAOA. Testing up to three layers of QAOA depth, we find that distributions of the approximation ratio narrow with increasing depth while the probability of recovering the maximum cut generally broadens. We find QAOA exceeds the Goemans-Williamson approximation ratio bound for most graphs. We also identify consistent patterns within the ensemble of optimized variational circuit parameters that offer highly efficient heuristics for solving MaxCut with QAOA. The resulting data set is presented as a benchmark for establishing empirical bounds on QAOA performance that may be used to test on-going experimental realizations.

quant-ph↗

arXiv2021

Impact of Graph Structures for QAOA on MaxCut

The quantum approximate optimization algorithm (QAOA) is a promising method of solving combinatorial optimization problems using quantum computing. QAOA on the MaxCut problem has been studied extensively on specific families of graphs, however, little is known about the algorithm on arbitrary graphs. We evaluate the performance of QAOA at depths at most three on the MaxCut problem for all connected non-isomorphic graphs with at most eight vertices and analyze how graph structure affects QAOA performance. Some of the strongest predictors of QAOA success are the existence of odd-cycles and the amount of symmetry in the graph. The data generated from these studies are shared in a publicly-accessible database to serve as a benchmark for QAOA calculations and experiments. Knowing the relationship between structure and performance can allow us to identify classes of combinatorial problems that are likely to exhibit a quantum advantage.

quant-ph↗

arXiv2019

Simulating quantum thermodynamics of a finite system and bath with variable temperature

We construct a finite bath with variable temperature for quantum thermodynamic simulations in which heat flows between a system $\mathcal{S}$ and the bath environment $\mathcal{E}$ in time evolution of an initial $\mathcal{SE}$ pure state. The bath consists of harmonic oscillators that are not necessarily identical. Baths of various numbers of oscillators are considered; a bath with five oscillators is used in the simulations. The bath has a temperature-like level distribution. This leads to definition of a system-environment microcanonical temperature $T_\mathcal{SE}(t)$ which varies with time. The quantum state evolves toward an equilibrium state which is thermal-like, but there is significant deviation from the ordinary energy-temperature relation that holds for an infinite quantum bath, e.g. an infinite system of identical oscillators. There are also deviations from the Einstein quantum heat capacity. The temperature of the finite bath is systematically greater for a given energy than the infinite bath temperature, and asymptotically approaches the latter as the number of oscillators increases. It is suggested that realizations of these finite-size effects may be attained in computational and experimental dynamics of small molecules.

cond-mat.stat-mech↗

arXiv2018

Quantum Thermodynamics, Entropy of the Universe, Free Energy, and the Second Law

We take the view that the standard von Neumann definition, in which the entropy $S^{vN}$ of a pure state is zero, is in evident conflict with the statement of the second law that the entropy of the universe $S_{univ}$ increases in spontaneous processes, $ ΔS_{univ} > 0$. Here we seek an alternative entropy of the universe $S_{univ}$ that is in accord with the second law, in a spirit not dissimilar to von Neumann himself in lesser-known work. We perform simulations of time dependent dynamics for a previously developed \cite{polyadbath} model quantum system becoming entangled with a quantum environment. We test the new definition of the entropy of the system-environment "universe" against the standard thermodynamic relation $ ΔF_{sys} = - T ΔS_{univ} $, calculating system properties using the reduced density matrix and standard von Neumann entropy. Good agreement is obtained, showing the compatibility of an entropy for a pure state of a universe with the statement of the second law and the concept of free energy. Interesting deviation from microcanonical behavior within the zero order energy shell is observed in a context of effectively microcanonical behavior within the much larger total basis of the time dependent universe.

quant-ph↗