Masayuki Ohzeki

Strong Resilience of Topological Codes to Depolarization

The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates effective error-correction schemes to protect information from noise. We compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase via both large-scale Monte Carlo simulations and the duality method, we are able to demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account. Remarkably, this result agrees within error bars with the result for a different class of codes—topological color codes—where the mapping yields interesting new types of interacting eight-vertex models.

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-pc=0.1096-8 for the hexagonal lattice and 1-pc=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-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.

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 the relationship between two partition functions related with each other by the duality. We can find that the multicritical points of the +/-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.

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

Violation of detailed balance accelerates relaxation.

\pm J

Multicritical points for spin-glass models on hierarchical lattices.

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 point

Sparse modeling approach to analytical continuation of imaginary-time quantum Monte Carlo data

Recent studies have experienced the acceleration of convergence in Markov chain Monte Carlo methods implemented by the systems without detailed balance condition (DBC). However, such advantage of the violation of DBC has not been confirmed in general. We investigate the effect of the absence of DBC on the convergence toward equilibrium. Surprisingly, it is shown that the DBC violation always makes the relaxation faster. Our result implies the existence of a kind of thermodynamic inequality that connects the nonequilibrium process relaxing toward steady state with the relaxation process which has the same probability distribution as its equilibrium state.

Measurement-based quantum computation on symmetry breaking thermal States.

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 different point of view coming from the renormalization group and succeeds in deriving very consistent answers with many numerical data.

Quantum Annealing with the Jarzynski Equality

A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.

Compressing Green's function using intermediate representation between imaginary-time and real-frequency domains

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 enhances 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 1 order of magnitude higher than that of the free cluster Hamiltonian.