arXiv subjects

Andrew D. King

Publications and source records attributed to Andrew D. King.

At least 37 records · Page 2Linked to original sources

arXiv2021

Quantum annealing simulation of out-of-equilibrium magnetization in a spin-chain compound

Geometrically frustrated spin-chain compounds such as Ca3Co2O6 exhibit extremely slow relaxation under a changing magnetic field. Consequently, both low-temperature laboratory experiments and Monte Carlo simulations have shown peculiar out-of-equilibrium magnetization curves, which arise from trapping in metastable configurations. In this work we simulate this phenomenon in a superconducting quantum annealing processor, allowing us to probe the impact of quantum fluctuations on both equilibrium and dynamics of the system. Increasing the quantum fluctuations with a transverse field reduces the impact of metastable traps in out-of-equilibrium samples, and aids the development of three-sublattice ferrimagnetic (up-up-down) long-range order. At equilibrium we identify a finite-temperature shoulder in the 1/3-to-saturated phase transition, promoted by quantum fluctuations but with entropic origin. This work demonstrates the viability of dynamical as well as equilibrium studies of frustrated magnetism using large-scale programmable quantum systems, and is therefore an important step toward programmable simulation of dynamics in materials using quantum hardware.

quant-ph↗

arXiv2020

Performance benefits of increased qubit connectivity in quantum annealing 3-dimensional spin glasses

An important challenge in superconducting quantum computing is the need to physically couple many devices using quasi-two-dimensional fabrication processes. Recent advances in the design and fabrication of quantum annealing processors have enabled an increase in pairwise connectivity among thousands of qubits. One benefit of this is the ability to minor-embed optimization problems using fewer physical qubits for each logical spin. Here we demonstrate the benefit of this progress in the problem of minimizing the energy of three-dimensional spin glasses. Comparing the previous generation D-Wave 2000Q system to the new Advantage system, we observe improved scaling of solution time and improved consistency over multiple graph embeddings.

quant-ph↗

arXiv2020

Fast clique minor generation in Chimera qubit connectivity graphs

The current generation of D-Wave quantum annealing processor is designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a {\em Chimera} graph $\mathcal C_{M,N,L}$. In order to solve an Ising spin problem with arbitrary pairwise interaction structure, the corresponding graph must be minor-embedded into a Chimera graph. We define a combinatorial class of {\em native clique minors} in Chimera graphs with vertex images of uniform, near minimal size, and provide a polynomial-time algorithm that finds a maximum native clique minor in a given induced subgraph of a Chimera graph. These minors allow improvement over recent work and have immediate practical applications in the field of quantum annealing.

cs.DM↗

arXiv2020

Simulating the Shastry-Sutherland Ising Model using Quantum Annealing

Frustration represents an essential feature in the behavior of magnetic materials when constraints on the microscopic Hamiltonian cannot be satisfied simultaneously. This gives rise to exotic phases of matter including spin liquids, spin ices, and stripe phases. Here we demonstrate an approach to understanding the microscopic effects of frustration by computing the phases of a 468-spin Shastry-Sutherland Ising Hamiltonian using a quantum annealer. Our approach uses mean-field boundary conditions to mitigate effects of finite size and defects alongside an iterative quantum annealing protocol to simulate statistical physics. We recover all phases of the Shastry-Sutherland Ising model -- including the well-known fractional magnetization plateau -- and the static structure factor characterizing the critical behavior at these transitions. These results establish quantum annealing as an emerging method in understanding the effects of frustration on the emergence of novel phases of matter and pave the way for future comparisons with real experiments.

quant-ph↗

arXiv2019

Scaling advantage in quantum simulation of geometrically frustrated magnets

The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of relaxation in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) relaxation timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation over the classical approach of path-integral Monte Carlo (PIMC) fixed-Hamiltonian relaxation with multiqubit cluster updates. The advantage increases with both system size and inverse temperature, exceeding a million-fold speedup over a CPU. This is an important piece of experimental evidence that in general, PIMC does not mimic QA dynamics for stoquastic Hamiltonians. The observed scaling advantage, for simulation of frustrated magnetism in quantum condensed matter, demonstrates that near-term quantum devices can be used to accelerate computational tasks of practical relevance.

quant-ph↗

arXiv2018

Emulating the coherent Ising machine with a mean-field algorithm

The coherent Ising machine is an optical processor that uses coherent laser pulses, but does not employ coherent quantum dynamics in a computational role. Core to its operation is the iterated simulation of all-to-all spin coupling via mean-field calculation in a classical FPGA coprocessor. Although it has been described as "operating at the quantum limit" and a "quantum artificial brain", interaction with the FPGA prevents the coherent Ising machine from exploiting quantum effects in its computations. Thus the question naturally arises: Can the optical portion of the coherent Ising machine be replaced with classical mean-field arithmetic? Here we answer this in the affirmative by showing that a straightforward noisy version of mean-field annealing closely matches CIM performance scaling, while running roughly 20 times faster in absolute terms.

quant-ph↗

arXiv2018

Observation of topological phenomena in a programmable lattice of 1,800 qubits

The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s revealed exotic phases of matter governed by topological properties of low-dimensional materials such as thin films of superfluids and superconductors. Key to this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---due to thermal fluctuations. In the 2D Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT) phase transition has been predicted in the quantum system by theory and simulation. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting flux qubits arranged in a fully-frustrated square-octagonal lattice. Essential to the critical behavior, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We use a simple but previously undemonstrated approach to statistical estimation with an annealing-based quantum processor, performing Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.

quant-ph↗

arXiv2017

Experimental demonstration of perturbative anticrossing mitigation using non-uniform driver Hamiltonians

Perturbative anticrossings have long been identified as a potential computational bottleneck for quantum annealing. This bottleneck can appear, for example, when a uniform transverse driver Hamiltonian is applied to each qubit. Previous theoretical research sought to alleviate such anticrossings by adjusting the transverse driver Hamiltonians on individual qubits according to a perturbative approximation. Here we apply this principle to a physical implementation of quantum annealing in a D-Wave 2000Q system. We use samples from the quantum annealing hardware and per-qubit anneal offsets to produce nonuniform driver Hamiltonians. On small instances with severe perturbative anticrossings, our algorithm yields an increase in minimum eigengaps, ground state success probabilities, and escape rates from metastable valleys. We also demonstrate that the same approach can mitigate biased sampling of degenerate ground states.

quant-ph↗

arXiv2017

Quantum Annealing amid Local Ruggedness and Global Frustration

A recent Google study [Phys. Rev. X, 6:031015 (2016)] compared a D-Wave 2X quantum processing unit (QPU) to two classical Monte Carlo algorithms: simulated annealing (SA) and quantum Monte Carlo (QMC). The study showed the D-Wave 2X to be up to 100 million times faster than the classical algorithms. The Google inputs are designed to demonstrate the value of collective multiqubit tunneling, a resource available to D-Wave QPUs but not to simulated annealing. But the computational hardness in these inputs is highly localized in gadgets, with only a small amount of complexity coming from global interactions, meaning that the relevance to real-world problems is limited. In this study we provide a new synthetic problem class that addresses the limitations of the Google inputs while retaining their strengths. We use simple clusters instead of more complex gadgets and more emphasis is placed on creating computational hardness through frustrated global interactions like those seen in interesting real-world inputs. The logical problems used to generate these inputs can be solved in polynomial time [J. Phys. A, 15:10 (1982)]. However, for general heuristic algorithms that are unaware of the planted problem class, the frustration creates meaningful difficulty in a controlled environment ideal for study. We use these inputs to evaluate the new 2000-qubit D-Wave QPU. We include the HFS algorithm---the best performer in a broader analysis of Google inputs---and we include state-of-the-art GPU implementations of SA and QMC. The D-Wave QPU solidly outperforms the software solvers: when we consider pure annealing time (computation time), the D-Wave QPU reaches ground states up to 2600 times faster than the competition. In the task of zero-temperature Boltzmann sampling from challenging multimodal inputs, the D-Wave QPU holds a similar advantage as quantum sampling bias does not seem significant.

quant-ph↗

arXiv2016

Degeneracy, degree, and heavy tails in quantum annealing

Both simulated quantum annealing and physical quantum annealing have shown the emergence of "heavy tails" in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work comparing the D-Wave Two quantum annealing processor with simulated annealing. On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged. Simulations indicate that perturbative crossings are the primary factor contributing to these heavy tails, while sensitivity to Hamiltonian misspecification error plays a less significant role in this particular setting.

quant-ph↗

arXiv2015

Strongly even-cycle decomposable graphs

A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.

math.CO↗

arXiv2015

Performance of a quantum annealer on range-limited constraint satisfaction problems

The performance of a D-Wave Vesuvius quantum annealer was recently compared to a suite of classical algorithms on a class of constraint satisfaction instances based on frustrated loops. However, the construction of these instances leads the maximum coupling strength to increase with problem size. As a result, larger instances are subject to amplified analog control error, and are effectively annealed at higher temperatures in both hardware and software. We generate similar constraint satisfaction instances with limited range of coupling strength and perform a similar comparison to classical algorithms. On these instances the D-Wave Vesuvius processor, run with a fixed 20$μ$s anneal time, shows a scaling advantage over the software solvers for the hardest regime studied. This scaling advantage opens the possibility of quantum speedup on these problems. Our results support the hypothesis that performance of D-Wave Vesuvius processors is heavily influenced by analog control error, which can be reduced and mitigated as the technology matures.

quant-ph↗

arXiv2014

A superlocal version of Reed's Conjecture

Reed's well-known $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil \frac 12(Δ+1+ω)\rceil$. The second author formulated a {\em local strengthening} of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional relaxation and holds for both quasi-line graphs and graphs with stability number two. We also conjecture that in the fractional version, we can push the locality even further.

cs.DM↗

arXiv2014

Algorithm engineering for a quantum annealing platform

Recent advances bring within reach the viability of solving combinatorial problems using a quantum annealing algorithm implemented on a purpose-built platform that exploits quantum properties. However, the question of how to tune the algorithm for most effective use in this framework is not well understood. In this paper we describe some operational parameters that drive performance, discuss approaches for mitigating sources of error, and present experimental results from a D-Wave Two quantum annealing processor.

cs.DS↗

arXiv2014

A stronger result on fractional strong colourings

Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\leq i \leq r$ every vertex in $V_i$ has at most $\max\{k, |V_i|-k \}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\leq i\leq r$. We believe that this result will be useful as a tool in probabilistic approaches to bounding the chromatic number and fractional chromatic number.

cs.DM↗

arXiv2013

Bounding the fractional chromatic number of $K_Δ$-free graphs

King, Lu, and Peng recently proved that for $Δ\geq 4$, any $K_Δ$-free graph with maximum degree $Δ$ has fractional chromatic number at most $Δ-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $Δ\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $ω$, $Δ$, $χ$ conjecture.

cs.DM↗

arXiv2012

Claw-free graphs, skeletal graphs, and a stronger conjecture on $ω$, $Δ$, and $χ$

The second author's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisties $χ\leq \lceil \frac 12 (Δ+1+ω)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $χ$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called "skeletal" graphs.

cs.DM↗

arXiv2012

A short proof that $χ$ can be bounded $ε$ away from $Δ+1$ towards $ω$

In 1998 the second author proved that there is an $ε>0$ such that every graph satisfies $χ\leq \lceil (1-ε)(Δ+1)+εω\rceil$. The first author recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Hall's Theorem, Brooks' Theorem, the Lovász Local Lemma, and Talagrand's Inequality.

cs.DM↗