This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.This paper discusses the theoretical basis of variational algorithms for thinning degrees of freedom in a lattice Hamiltonian theory. We show that such a thinning can, in principle, be an exact real-space renormalization-group transformation to a lattice with fewer sites. This exact transformation can only be constructed given the exact ground state of the theory. However, this insight teaches us how to significantly improve previous variational algorithms. Further improvements in ability to calculate quantities of interest are achieved by introducing "look-ahead" algorithms which allow us to maintain a renormalization-group interpretation while introducing long-distance physics into the determination of variational parameters. We find that algorithms incorporating both these features are much more powerful than previous truncation algorithms. For pedagogical reasons we present our algorithms using as examples two well-understood theories, namely, free scalar field theory and the Ising model in a transverse field in 1 + 1 dimensions. Discussion of higher dimensions and less trivial theories is included in the final section. An appendix of generalized mean-field theory is also included.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMar 15, 198210 citations

Introduction to variational methods

Springer eBooksJul 16, 20072 citations

Quantum walk informed variational algorithm design

Edric Matwiejew, Jingbo B. Wang

We present a theoretical framework for the analysis of amplitude transfer in Quantum Variational Algorithms (QVAs) for combinatorial optimisation with mixing unitaries defined by vertex-transitive graphs, based on their continuous-time quantum walk (CTQW) representation and the theory of graph automorphism groups. This framework leads to a heuristic for designing efficient problem-specific QVAs. Using this heuristic, we develop novel algorithms for unconstrained and constrained optimisation. We outline their implementation with polynomial gate complexity and simulate their application to the parallel machine scheduling and portfolio rebalancing combinatorial optimisation problems, showing significantly improved convergence over preexisting QVAs. Based on our analysis, we derive metrics for evaluating the suitability of graph structures for specific problem instances, and for establishing bounds on the convergence supported by different graph structures. For mixing unitaries characterised by a CTQW over a Hamming graph on $m$ -tuples of length $n$ , our results indicate that the amplification upper bound increases with problem size like $\mathcal{O}(e^{n \log m})$ .We present a theoretical framework for the analysis of amplitude transfer in Quantum Variational Algorithms (QVAs) for combinatorial optimisation with mixing unitaries defined by vertex-transitive graphs, based on their continuous-time quantum walk (CTQW) representation and the theory of graph automorphism groups. This framework leads to a heuristic for designing efficient problem-specific QVAs. Using this heuristic, we develop novel algorithms for unconstrained and constrained optimisation. We outline their implementation with polynomial gate complexity and simulate their application to the parallel machine scheduling and portfolio rebalancing combinatorial optimisation problems, showing significantly improved convergence over preexisting QVAs. Based on our analysis, we derive metrics for evaluating the suitability of graph structures for specific problem instances, and for establishing bounds on the convergence supported by different graph structures. For mixing unitaries characterised by a CTQW over a Hamming graph on $m$-tuples of length $n$, our results indicate that the amplification upper bound increases with problem size like $\mathcal{O}(e^{n \log m})$.

arXiv:2406.11620Jun 17, 20240 citations

Variational optimization in the AI era

Bryan K. Clark, Dmitrii Kochkov, Di Luo

Bulletin of the American Physical SocietyMar 5, 20200 citations

Schwinger model at finite temperature and density with beta VQE

Akio Tomiya

We investigate a quantum gauge theory at finite temperature and density using a variational algorithm for near-term quantum devices. We adapt $β$ -VQE to evaluate thermal and quantum expectation values and study the phase diagram for massless Schwinger model along with the temperature and density. By compering the exact variational free energy, we find the variational algorithm work for $T>0$ and $μ>0$ for the Schwinger model. No significant volume dependence of the variational free energy is observed in $μ/g \in[0, 1.4]$ . We calculate the chiral condensate and take the continuum extrapolation. As a result, we obtain qualitative picture of the phase diagram for massless Schwinger model.We investigate a quantum gauge theory at finite temperature and density using a variational algorithm for near-term quantum devices. We adapt $β$-VQE to evaluate thermal and quantum expectation values and study the phase diagram for massless Schwinger model along with the temperature and density. By compering the exact variational free energy, we find the variational algorithm work for $T>0$ and $μ>0$ for the Schwinger model. No significant volume dependence of the variational free energy is observed in $μ/g \in[0, 1.4]$. We calculate the chiral condensate and take the continuum extrapolation. As a result, we obtain qualitative picture of the phase diagram for massless Schwinger model.

arXiv:2205.08860May 18, 20223 citations

Stochastic variational method

Springer eBooksJul 16, 20071 citation

m

-tuples of length

n

, our results indicate that the amplification upper bound increases with problem size like

\mathcal{O}(e^{n \log m})

We investigate a quantum gauge theory at finite temperature and density using a variational algorithm for near-term quantum devices. We adapt

β

-VQE to evaluate thermal and quantum expectation values and study the phase diagram for massless Schwinger model along with the temperature and density. By compering the exact variational free energy, we find the variational algorithm work for

T>0

and

μ>0

for the Schwinger model. No significant volume dependence of the variational free energy is observed in

μ/g \in[0, 1.4]

. We calculate the chiral condensate and take the continuum extrapolation. As a result, we obtain qualitative picture of the phase diagram for massless Schwinger model.