Error threshold estimates for surface code with loss of qubits

We estimate optimal thresholds for surface code in the presence of loss via an analytical method developed in statistical physics. The optimal threshold for the surface code is closely related to a special critical point in a finite-dimensional spin glass, which is disordered magnetic material. We compare our estimations to the heuristic numerical results reported in earlier studies. Further application of our method to the depolarizing channel, a natural generalization of the noise model, unveils its wider robustness even with loss of qubits.Comment: 4 pages, 3 figures, 2 tables, title change

Fluctuation Theorems on Nishimori Line

The distribution of the performed work for spin glasses with gauge symmetry is considered. With the aid of the gauge symmetry, which leads to the exact/rigorous results in spin glasses, we find a fascinating relation of the performed work as the fluctuation theorem. The integral form of the resultant relation reproduces the Jarzynski-type equation for spin glasses we have obtained. We show that similar relations can be established not only for the distribution of the performed work but also that of the free energy of spin glasses with gauge symmetry, which provides another interpretation of the phase transition in spin glasses.Comment: 10 pages, and 1 figur

Locations of multicritical points for spin glasses on regular lattices

We present an analysis leading to precise locations of the multicritical points for spin glasses on regular lattices. The conventional technique for determination of the location of the multicritical point was previously derived using a hypothesis emerging from duality and the replica method. In the present study, we propose a systematic technique, by an improved technique, giving more precise locations of the multicritical points on the square, triangular, and hexagonal lattices by carefully examining relationship between two partition functions related with each other by the duality. We can find that the multicritical points of the $\pm J$ Ising model are located at $p_c = 0.890813$ on the square lattice, where $p_c$ means the probability of $J_{ij} = J(>0)$, at $p_c = 0.835985$ on the triangular lattice, and at $p_c = 0.932593$ on the hexagonal lattice. These results are in excellent agreement with recent numerical estimations.Comment: 17pages, this is the published version with some minnor corrections. Previous title was "Precise locations of multicritical points for spin glasses on regular lattices

Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin glass systems. The analysis for the corresponding spin glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be $1-p_c = 0.1096-8$ for the hexagonal lattice and $1-p_c = 0.1092-3$ for the square-octagonal lattice, where $1-p$ denotes the error probability on each qubit. Hence both of them are expected to be slightly lower than the probability $1-p_c = 0.110028$ for the quantum Gilbert-Varshamov bound with a zero encoding rate.Comment: 6 pages, 4 figures, the previous title was "Threshold of topological color code". This is the published version in Phys. Rev.

Multicritical points for the spin glass models on hierarchical lattices

The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry and the replica method. We find that the conjecture does not give the exact answer but leads to locations slightly away from the numerically reliable data. We propose an improved conjecture to give more precise predictions of the multicritical points than the conventional one. This improvement is inspired by a new point of view coming from renormalization group and succeeds in deriving very consistent answers with many numerical data.Comment: 11 pages, 9 figures, 7 tables This is the published versio

Measurement-Based Quantum Computation on Symmetry Breaking Thermal States

We consider measurement-based quantum computation (MBQC) on thermal states of the interacting cluster Hamiltonian containing interactions between the cluster stabilizers that undergoes thermal phase transitions. We show that the long-range order of the symmetry breaking thermal states below a critical temperature drastically enhance the robustness of MBQC against thermal excitations. Specifically, we show the enhancement in two-dimensional cases and prove that MBQC is topologically protected below the critical temperature in three-dimensional cases. The interacting cluster Hamiltonian allows us to perform MBQC even at a temperature an order of magnitude higher than that of the free cluster Hamiltonian.Comment: 8 pages, 7 figure

Jarzynski Equality for an Energy-Controlled System

The Jarzynski equality (JE) is known as an exact identity for nonequillibrium systems. The JE was originally formulated for isolated and isothermal systems, while Adib reported an JE extended to an isoenergetic process. In this paper, we extend the JE to an energy-controlled system. We make it possible to control the instantaneous value of the energy arbitrarily in a nonequilibrium process. Under our extension, the new JE is more practical and useful to calculate the number of states and the entropy than the isoenergetic one. We also show application of our JE to a kind of optimization problems.Comment: 6 pages, 1 figur

Nonequilibrium work on spin glasses in longitudinal and transverse fields

We derive a number of exact relations between equilibrium and nonequilibrium quantities for spin glasses in external fields using the Jarzynski equality and gauge symmetry. For randomly-distributed longitudinal fields, a lower bound is established for the work done on the system in nonequilibrium processes, and identities are proven to relate equilibrium and nonequilibrium quantities. In the case of uniform transverse fields, identities are proven between physical quantities and exponentiated work done to the system at different parts of the phase diagram with the context of quantum annealing in mind. Additional relations are given, which relate the exponentiated work in quantum and simulated (classical) annealing. It is also suggested that the Jarzynski equality may serve as a guide to develop a method to perform quantum annealing under non-adiabatic conditions.Comment: 17 pages, 5 figures, submitted to JPS

Analytical evidence for the absence of spin glass transition on self-dual lattices

We show strong evidence for the absence of a finite-temperature spin glass transition for the random-bond Ising model on self-dual lattices. The analysis is performed by an application of duality relations, which enables us to derive a precise but approximate location of the multicritical point on the Nishimori line. This method can be systematically improved to presumably give the exact result asymptotically. The duality analysis, in conjunction with the relationship between the multicritical point and the spin glass transition point for the symmetric distribution function of randomness, leads to the conclusion of the absence of a finite-temperature spin glass transition for the case of symmetric distribution. The result is applicable to the random bond Ising model with $\pm J$ or Gaussian distribution and the Potts gauge glass on the square, triangular and hexagonal lattices as well as the random three-body Ising model on the triangular and the Union-Jack lattices and the four dimensional random plaquette gauge model. This conclusion is exact provided that the replica method is valid and the asymptotic limit of the duality analysis yields the exact location of the multicritical pointComment: 11 Pages, 4 figures, 1 table. submitted to J. Phys. A Math. Theo