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.
Introduction— Against corruption by environmental noise as well as imperfection in implementation, the state of qubits describing quantum information cannot be stable and must be recovered by elaborated procedures, quantum error correction QEC1 ; QEC2 . Quantum error corrections usually work on the computational error on qubits, which do not go out of the basis for computations. Therefore errors come from losses of the physical resource, qubits, can deprive of the performance of error correction. However, if one can detect and identify locations of losses, a modified scheme can recover the original information. Stace and Barret have suggested an error-correcting code, a family of Kitaev’s surface codes Dennis , which is robust against both of the computational errors and losses by a modified scheme according to the location of the lost qubits Loss1 .
In the present paper, we estimate precise values of the error thresholds for the modified error correcting code against both of the computational errors and loss of qubits by use of a systematic theory developed in statistical physics. The key of our analysis is hidden in the disordered magnetic system, spin glasses. Several spin glass models have a special symmetry with exact solvable subspace known as Nishimori line HNbook ; HN81 . The critical point in this subspace, termed as the multicritical point, corresponds to the optimal error threshold in the surface code Dennis . A combination of the duality with the real-space renormalization technique, which are often used to identify the singular points in statistical mechanical models, can derive the precise estimations for the optimal error thresholds NN ; MNN and systematically approach the exact solutions ONB ; Ohzeki . By use of this method, we fill the blank on the analytical study for the optimal error thresholds on several surface codes with loss of qubits. The results reported in this paper provide upper bounds against error rates for any error-correcting schemes. They serve as important benchmarks with which any constructive error correcting procedure as recently proposed in Ref. HM can be compared.
Surface code and spin glass— Let us consider qubits set on each edge of the square lattice embedded on a torus (genus ). We define the star operator for each site , and plaquette operator for each plaquette (site on the dual lattice), where and are Pauli matrices. The product consists of four edges adjacent to each site or plaquette. The stabilizer group is given by the simultaneous eigenstates with the positive eigenvalues for these operators and . Since the star and plaquette operators consist of unit loops on the dual and original square lattices, any contractible loop by and products on each lattice acts trivially on the codespace. On the other hand, any non-contractible loops on the lattice can map the codespace to itself in a nontrivial manner. If we set lattice on a torus, we have qubits and stabilizers. The remaining degrees of freedom of implies existence of two non-contractible loops, winding around the hole of the torus and winding around the body of the torus , and ones and on the dual lattice. These loops can be written in terms of the products of operators as , , , and , which are termed as logical operators. The logical operators can form Pauli algebra of two effective qubits encoded in the topological degrees of freedom on the torus as , and . The combinations of non-contractible loops yield different homology classes for the original and dual square lattices on a single torus. We need to distinguish them for protecting the information from corruption.
In order to evaluate the performance of the error-correcting code, let us define a noise model where each qubit independently gets errors as
Although, if we employ the following analytical method, we can estimate precise values of the error thresholds for “any” cases of , and , we restrict ourselves to two cases: , and (uncorrelated case), and and (depolarizing channel case) for simplicity, where . The error can be regarded as a multiple error and . The errors and can be described as chains and on the original and dual lattices. The endpoints of the error chains and can be detected by applications of star and plaquette operators due to anti-commutation of adjacent errors with operators. From the knowledge of endpoints and without the homology class of the error chains, error syndrome, we infer the most likely homology class of error chains, while considering any reasonable choices. Since and , where and are the contractible loops on both of the lattices, are in equivalent class with the error chains, the probability for the homology class and of the error chains can be written as Dennis
where for the uncorrelated case. The summation is taken over all the possibilities of and , and the product is over all the edges. The parameter stands for the importance/preference to choose the inferred error chain. The quantity in the denominator denotes the probability with the different homology class specified by the logical operators (). We here use to represent the inferred error chains, which takes (, when ), and also for , and . The loop constraints and allow us to use another expression by the Ising variables , and for each lattice on the torus. By use of these expressions, we can find that is written as square of the partition function of the Ising model
where and are the signs of the quenched random couplings in context of spin glasses. When we set , where (Nishimori line), the inference of the error chains is an optimal recovery procedure to identify the most likely homology class HNbook . Each of the quenched random couplings follows the distribution function of the error chains for the uncorrelated case, where
Similarly, we can evaluate the probability for the homology class of the error chains and for the depolarizing channel case as
where we set the parameter as on the Nishimori line . This is written in terms of the partition function of the eight-vertex model with quenched random interaction AHMMH
where and follow the distribution function through as while . We emphasize that, if we tune the probability function appropriately, we can apply our analysis as shown below to inhomogenius case with .
In context of the statistical physics, represents the domain wall. In the low-temperature region implying a small , the order of the degrees of freedom suppresses the fluctuation of the domain wall. The cost for free energy difference due to the domain wall diverges as for . This means that we can infer the equivalent class with the original error chains. On the other hand, in the high-temperature region, the cost vanishes and . This implies that the failure of the recovery occurs at the critical point. Therefore the location of the critical point on the Nishimori line, the multicritical point, identifies the optimal error threshold.
Loss of qubits and bond dilution— Loss of qubits on the lattice implies the modification of the stabilizers as well as the logical operators. However we can reform a complete set of stabilizers even on the damaged lattice due to loss of qubits following the proposed scheme in Ref. Loss1 . The effect of lost qubits appears in the pattern of error chains and , and their weight for the probability, which degrades the performance of error-correcting code. To infer the most likely homology class based on the knowledge of the error chains on the damaged lattice, we reconstruct the original lattice by assigning of weight-zero edges on the lost qubits and irregular weight edges adjacent to the lost qubits as , where is the number of the shared qubits in adjacent edges as in Fig. 1. The weight-zero edges imply that we need to consider a diluted version of the original spin glass system as in Eqs. (3) and (6). It can be achieved by a simple modification of the distribution function for and into, for the uncorrelated case, and , where denotes the ratio of loss of qubits. Similarly, for the depolarizing channel case, . In addition, we have to take into account effects of irregular weight edges adjacent to the lost qubits as carefully discussed in Ref. Loss2 . The effect can be described by highly correlated distribution function depending on the pattern of the lost qubits, although we omit its detailed expression.
Duality analysis for spin glasses— Analyses to clarify the critical phenomena in finite-dimensional spin glasses are intractable in general. However a recent development in the spin glass theory enables us to estimate the precise value of the special critical point on the Nishimori line, which corresponds to the optimal error threshold NN ; MNN ; ONB ; Ohzeki . The method as shown below is based on the duality, which can identify the location of the critical point especially on two-dimensional spin systems WuWang . Let us review the simple pure Ising model case at first. The duality is a symmetry argument by considering the low and high-temperature expansions of the partition function . The painful calculation of both expansions can be replaced by a simple manipulation with the binary Fourier transformation for the local part of the Boltzmann factor, namely edge Boltzmann factor and WuWang . The low-temperature expansion can be expressed by and . On the other hand, the high-temperature expansion is given by the binary Fourier transformation and . We use this fact and find a double expression of the partition function as
where is the normalized partition function and . We here define and . The well-known duality relation is given by rewriting by , which implies a transformation of the temperature. Then the principal Boltzmann factors and with edge spins parallel holds at the critical point .
We employ the replica method, which is often used in theoretical studies on spin glasses, in order to generalize the duality analysis to spin glasses NN ; MNN . Let us consider the duality for the replicated partition function as and simply , where is the configurational average for the quenched randomness according to the distribution functions. The multiple () Fourier transformation again leads us to the double expression of the replicated partition function as
where the subscript of and stands for the number of anti-parallel pair among replicas on each edge. Unfortunately we cannot replace by as the pure case, since the replicated partition function is multivariable. Nevertheless we can estimate the precise location of the critical point even for spin glasses by considering a wider range of the local part of the Boltzmann factor given after the summation of the internal spins. For instance, in the case on the square lattice, we define the cluster Boltzmann factor , where the subscript denotes the configuration of the edge (white-colored) spins as in Fig. 2. We set the equation to lead the location of the critical point as, inspired by the case without quenched randomness, NN ; MNN ; ONB ; Ohzeki ,
The equality even without use of the cluster can give the precise solutions of the critical point for the multicritical point of Ising model as NN ; MNN . Although the above method is not exact, if we increase the size of the used cluster, we can systematically approach the exact solution for the critical points of the Ising model in the higher temperature region than the Nishimori line ONB ; Ohzeki .
Results— In the present study, we consider two clusters as A and B as well as a single edge for the uncorrelated case and a single crossing edges C and two clusters D and E for the depolarizing case as in Fig. 2.
We show several estimations given by Eq. (9) for the uncorrelated case in Table. 1. Although, for the uncorrelated case by B cluster, we have considered the highly correlated distribution function by taking into account the effect of the irregular weight, the results have not been changed from those by a simple distribution function , which is the same as one for the bond-diluted spin glass. As discussed in Ref. Loss2 , the highly correlation between the loss of qubits is found to emerge as a finite-size effect in the numerical investigation (for ). Such the complicated effect does not spoil our analysis. All the results for any does not show drastic changes dependently on the size of the used cluster. It means that our analyses are enough correct to capture the accurate locations of the optimal error threshold. The optimal error thresholds indicate the upper bounds for error threshold by any heuristic methods. As shown in Fig. 3, we compare our results with the inference by use of the matching algorithm namely, the ground state as Loss1 , in which we denote the error thresholds as . We confirm that the heuristic matching algorithm of inference gives , presumably for large . We also give several results for the depolarizing channel case in Table. 2. Similarly to the case without loss of qubits () as have been reported in Ref. AHMMH , the depolarizing channel is more resilient than the uncorrelated case even with loss of qubits. For comparison, let us take an earlier study on an error recovery procedure for the depolarizing channel in Ref. HM . Our result implies that there is still possibility to improve the performance of such a constructive procedure.
All the obtained values are almost stable in the third digits. In a practical sense, our estimations for error thresholds serve as the reference values.
Conclusion— We have estimated the error thresholds for the surface code with loss of qubits, via a finite-dimensional spin glass theory, for both of the uncorrelated and depolarizing channel cases, and shown more resilience of the depolarizing channel even with loss of qubits.
In the sense of study on spin glass, the comparison between the error thresholds by a suboptimal method corresponding to the inference in the ground state Loss1 and optimal ones shows a fascinating feature of the phase boundary of the Ising model as HNbook . The future study will be desired for solving the remaining problem on a realm of spin glasses: or not.
Acknowledgement— The author acknowledges fruitful discussions with and numerical data in Ref. Loss1 from Thomas Stace and Sean Barret. He also thanks hospitality in Rome University during this work. This work was partially supported by MEXT in Japan, Grant-in-Aid for Young Scientists (B) No.20740218.
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