Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines

Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a decohering thermal bath. Therefore, we study the finite-temperature critical behavior of the standard benchmark problem used to assess the computational capabilities of these complex machines. We simulate both random-bond Ising models and spin glasses with bimodal and Gaussian disorder on the D-Wave Chimera topology. Our results show that while the worst-case complexity of finding a ground state of an Ising spin glass on the Chimera graph is not polynomial, the finite-temperature phase space is likely rather simple: Spin glasses on Chimera have only a zero-temperature transition. This means that benchmarking optimization methods using spin glasses on the Chimera graph might not be the best benchmark problems to test quantum speedup. We propose alternative benchmarks by embedding potentially harder problems on the Chimera topology. Finally, we also study the (reentrant) disorder-temperature phase diagram of the random-bond Ising model on the Chimera graph and show that a finite-temperature ferromagnetic phase is stable up to 19.85(15)% antiferromagnetic bonds. Beyond this threshold the system only displays a zero-temperature spin-glass phase. Our results therefore show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealing machines.Comment: 8 pages, 5 figures, 1 tabl

Evidence of a glass transition in a 10-state non-mean-field Potts glass

Potts glasses are prototype models that have been used to understand the structural glass transition. However, in finite space dimensions a glass transition remains to be detected in the 10-state Potts glass. Using a one-dimensional model with long-range power-law interactions we present evidence that a glass transition below the upper critical dimension can exist for short-range systems at low enough temperatures. Gaining insights into the structural glass transition for short-range systems using spin models is thus potentially possible, yet difficult.Comment: 4 pages, 1 table, 2 figure

Topological color codes on Union Jack lattices: A stable implementation of the whole Clifford group

We study the error threshold of topological color codes on Union Jack lattices that allow for the full implementation of the whole Clifford group of quantum gates. After mapping the error-correction process onto a statistical mechanical random 3-body Ising model on a Union Jack lattice, we compute its phase diagram in the temperature-disorder plane using Monte Carlo simulations. Surprisingly, topological color codes on Union Jack lattices have similar error stability than color codes on triangular lattices, as well as the Kitaev toric code. The enhanced computational capabilities of the topological color codes on Union Jack lattices with respect to triangular lattices and the toric code demonstrate the inherent robustness of this implementation.Comment: 8 pages, 4 figures, 1 tabl

Erratum: Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines [Phys. Rev. X 4, 021008 (2014)]

Erratum to Phys. Rev. X 4, 021008 (2014): The critical exponent associated with the ferromagnetic susceptibility was computed incorrectly. Furthermore, Ising ferromagnets on the Chimera topology have the same universality class as two-dimensional Ising ferromagnets.Comment: 1 page, 1 figur

Error thresholds for Abelian quantum double models: Increasing the bit-flip stability of topological quantum memory

Current approaches for building quantum computing devices focus on two-level quantum systems which nicely mimic the concept of a classical bit, albeit enhanced with additional quantum properties. However, rather than artificially limiting the number of states to two, the use of d-level quantum systems (qudits) could provide advantages for quantum information processing. Among other merits, it has recently been shown that multi-level quantum systems can offer increased stability to external disturbances - a key problem in current technologies. In this study we demonstrate that topological quantum memories built from qudits, also known as abelian quantum double models, exhibit a substantially increased resilience to noise. That is, even when taking into account the multitude of errors possible for multi-level quantum systems, topological quantum error correction codes employing qudits can sustain a larger error rate than their two-level counterparts. In particular, we find strong numerical evidence that the thresholds of these error-correction codes are given by the hashing bound. Considering the significantly increased error thresholds attained, this might well outweigh the added complexity of engineering and controlling higher dimensional quantum systems.Comment: 7 pages, 3 figure

オオサカダイガク カンゴガク ザッシ 13カン 1ゴウ タイショク キネン トクシュウ

退職記念特

Error tolerance of topological codes with independent bit-flip and measurement errors

Topological quantum error correction codes are currently among the most promising candidates for efficiently dealing with the decoherence effects inherently present in quantum devices. Numerically, their theoretical error threshold can be calculated by mapping the underlying quantum problem to a related classical statistical-mechanical spin system with quenched disorder. Here, we present results for the general fault-tolerant regime, where we consider both qubit and measurement errors. However, unlike in previous studies, here we vary the strength of the different error sources independently. Our results highlight peculiar differences between toric and color codes. This study complements previous results published in New J. Phys. 13, 083006 (2011)

Tricolored Lattice Gauge Theory with Randomness: Fault-Tolerance in Topological Color Codes

We compute the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates, when both qubit and measurement errors are present. By mapping the problem onto a statistical-mechanical three-dimensional disordered Ising lattice gauge theory, we estimate via large-scale Monte Carlo simulations that color codes are stable against 4.5(2)% errors. Furthermore, by evaluating the skewness of the Wilson loop distributions, we introduce a very sensitive probe to locate first-order phase transitions in lattice gauge theories.Comment: 12 pages, 5 figures, 1 tabl

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully-connected infinite-range Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034 (1999)]. Here we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Viana-Bray model. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of a diverging number of neighbors, and is not a generic property of spin-glass models in finite space dimensions.Comment: 5 pages, 4 figures, 1 tabl