Unconventional, special-purpose machines may aid in accelerating the solution of some of the hardest problems in computing, such as large-scale combinatorial optimizations, by exploiting different operating mechanisms than those of standard digital computers. We present a scalable optical processor with electronic feedback that can be realized at large scale with room-temperature technology. Our prototype machine is able to find exact solutions of, or sample good approximate solutions to, a variety of hard instances of Ising problems with up to 100 spins and 10,000 spin-spin connections.

/pdf/a-fully-programmable-100-spin-coherent-ising-machine-with-inaxlu4ff3.pdf

A fully programmable 100-spin coherent Ising machine with all-to-all connections

We assess the prospects for algorithms within the general framework of quantum annealing (QA) to achieve a quantum speedup relative to classical state of the art methods in combinatorial optimization and related sampling tasks. We argue for continued exploration and interest in the QA framework on the basis that improved coherence times and control capabilities will enable the near-term exploration of several heuristic quantum optimization algorithms that have been introduced in the literature. These continuous-time Hamiltonian computation algorithms rely on control protocols that are more advanced than those in traditional ground-state QA, while still being considerably simpler than those used in gate-model implementations. The inclusion of coherent diabatic transitions to excited states results in a generalization called diabatic quantum annealing (DQA), which we argue for as the most promising route to quantum enhancement within this framework. Other promising variants of traditional QA include reverse annealing and continuous-time quantum walks, as well as analog analogues of parameterized quantum circuit ansatzes for machine learning. Most of these algorithms have no known (or likely to be discovered) efficient classical simulations, and in many cases have promising (but limited) early signs for the possibility of quantum speedups, making them worthy of further investigation with quantum hardware in the intermediate-scale regime. We argue that all of these protocols can be explored in a state-of-the-art manner by embracing the full range of novel out-of-equilibrium quantum dynamics generated by time-dependent effective transverse-field Ising Hamiltonians that can be natively implemented by, e.g., inductively-coupled flux qubits, both existing and projected at application scale.

Prospects for Quantum Enhancement with Diabatic Quantum Annealing

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  Users may download and print one copy of any publication from the public portal for the purpose of private study or research.  You may not further distribute the material or use it for any profit-making activity or commercial gain  You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from orbit.dtu.dk on: Jun 04, 2020

Μετακβαντικοί κρυπτογραφικοί αλγόριθμοι (Post-Quantum Cryptography)

In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimization. The new method is based on the physics of domain walls in one dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms. Following on from similar results for the `one hot' method of encoding discrete variables [Hadfield et. al. Algorithms 12.2 (2019): 34] I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz (QAOA). I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.

https://iopscience.iop.org/article/10.1088/2058-9565/ab33c2/pdf

Domain wall encoding of discrete variables for quantum annealing and QAOA

Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. We investigate the performance of continuous-time quantum walks as a tool for finding spin glass ground states, a problem that serves as a useful model for realistic optimization problems. By performing detailed numerics, we uncover significant ways in which solving spin glass problems differs from applying quantum walks to the search problem. Importantly, unlike for the search problem, parameters such as the hopping rate of the quantum walk do not need to be set precisely for the spin glass ground state problem. Heuristic values of the hopping rate determined from the energy scales in the problem Hamiltonian are sufficient for obtaining a better quantum advantage than for search. We uncover two general mechanisms that provide the quantum advantage: matching the driver Hamiltonian to the encoding in the problem Hamiltonian, and an energy redistribution principle that ensures a quantum walk will find a lower energy state in a short timescale. This makes it practical to use quantum walks for solving hard problems, and opens the door for a range of applications on suitable quantum hardware.

https://iopscience.iop.org/article/10.1088/1367-2630/ab5ca2/pdf

Finding spin glass ground states using quantum walks

Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. We investigate the performance of continuous-time quantum walks as a tool for finding spin glass ground states, a problem that serves as a useful model for realistic optimization problems. By performing detailed numerics, we uncover significant ways in which solving spin glass problems differs from applying quantum walks to the search problem. Importantly, unlike for the search problem, parameters such as the hopping rate of the quantum walk do not need to be set precisely for the spin glass ground state problem. Heuristic values of the hopping rate determined from the energy scales in the problem Hamiltonian are sufficient for obtaining a better than square-root scaling. This makes it practical to use quantum walks for solving such problems, and opens the door for a range of applications on suitable quantum hardware.

Finding spin-glass ground states using quantum walks

Quantum amplitude estimation is a key subroutine in a number of powerful quantum algorithms, including quantum-enhanced Monte Carlo simulation and quantum machine learning. Maximum-likelihood quantum amplitude estimation (MLQAE) is one of a number of recent approaches that employ much simpler quantum circuits than the original algorithm based on quantum phase estimation. In this article, we deepen the analysis of MLQAE to put the algorithm in a more prescriptive form, including scenarios where quantum circuit depth is limited. In the process, we observe and explain particular ranges of `exceptional' values of the target amplitude for which the algorithm fails to achieve the desired precision. We then propose and numerically validate a heuristic modification to the algorithm to overcome this problem, bringing the algorithm even closer to being useful as a practical subroutine on near- and mid-term quantum hardware.

Improved maximum-likelihood quantum amplitude estimation

We investigate whether small sized problem instances, which are commonly used in numerical simulations of quantum algorithms for benchmarking purposes, are a good representation of larger instances in terms of hardness. We approach this through a numerical analysis of the hardness of MAX 2-SAT problem instances for various continuous-time quantum algorithms and a comparable classical algorithm. Our results can be used to predict the viability of hybrid approaches that combine multiple algorithms in parallel to take advantage of the variation in the hardness of instances between the algorithms. We find that, while there are correlations in instance hardness between all of the algorithms considered, they appear weak enough that a hybrid strategy would likely be desirable in practice. Our results also show a widening range of hardness of randomly generated instances as the problem size is increased, which demonstrates both the difference in the distribution of hardness at small sizes and the value of a hybrid approach that can reduce the number of extremely hard instances. We identify specific weaknesses that can be overcome with hybrid techniques, such as the inability of these quantum algorithms to efficiently solve satisfiable instances (which is easy classically).

Comparing the hardness of MAX 2-SAT problem instances for quantum and classical algorithms

In the quantum optimisation setting, we build on a scheme introduced by Young et al (2013 Phys. Rev. A 88 062314), where physical qubits in multiple copies of a problem encoded into an Ising spin Hamiltonian are linked together to increase the logical system’s robustness to error. We introduce several innovations that improve the error suppression of this scheme under a special model of control noise, designed to understand how limited precision could be overcome. First, we note that only one copy needs to be correct by the end of the computation, since solution quality can be checked efficiently. Second, we find that ferromagnetic links do not generally help in this ‘one correct copy’ setting, but anti-ferromagnetic links do help on average, by suppressing the chance of the same error being present on all of the copies. Third, we find that minimum-strength anti-ferromagnetic links perform best, by counteracting the spin-flips induced by the errors. We have numerically tested our innovations on small instances of spin glasses from Callison et al (2019 New J. Phys. 21 123022), and we find improved error tolerance for three or more copies in configurations that include frustration. Interpreted as an effective precision increase, we obtain several extra bits of precision on average for three copies connected in a triangle. This provides proof-of-concept of a method for scaling quantum annealing beyond the precision limits of hardware, a step towards fault tolerance in this setting.

/pdf/using-copies-can-improve-precision-in-continuous-time-1mxxjh7h.pdf

Adam Callison

Papers

Finding spin-glass ground states using quantum walks

Improved maximum-likelihood quantum amplitude estimation

Comparing the hardness of MAX 2-SAT problem instances for quantum and classical algorithms

Using copies can improve precision in continuous-time quantum computing