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arXiv:1612.06460 · Physical Review A

Long-time efficacy of the surface code in the presence of a super-Ohmic environment

Daniel A. Lopez-Delgado, E. Novais, Eduardo R. Mucciolo, A. O. Caldeira

Published Jun 23, 2017• 2 citations
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Abstract

We study the long-time evolution of a quantum memory coupled to a bosonic environment on which quantum error correction (QEC) is performed using the surface code. The memory's evolution encompasses $N$ QEC cycles, each of them yielding a nonerror syndrome. This assumption makes our analysis independent of the recovery process. We map the expression for the time evolution of the memory onto the partition function of an equivalent statistical-mechanical spin system. In the super-Ohmic dissipation case the long-time evolution of the memory has the same behavior as the time evolution for just one QEC cycle. For this case we find analytical expressions for the critical parameters of the order-disorder phase transition of an equivalent spin system. These critical parameters determine the threshold value for the system-environment coupling below which it is possible to preserve the memory's state.

DOI
10.1103/physreva.95.062328

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We study the long-time evolution of a quantum memory coupled to a bosonic environment on which quantum error correction (QEC) is performed using the surface code. The memory's evolution encompasses NNN QEC cycles, each of them yielding a nonerror syndrome. This assumption makes our analysis independent of the recovery process. We map the expression for the time evolution of the memory onto the partition function of an equivalent statistical-mechanical spin system. In the super-Ohmic dissipation case the long-time evolution of the memory has the same behavior as the time evolution for just one QEC cycle. For this case we find analytical expressions for the critical parameters of the order-disorder phase transition of an equivalent spin system. These critical parameters determine the threshold value for the system-environment coupling below which it is possible to preserve the memory's state.
We study the fidelity of the surface code in the presence of correlated errors induced by the coupling of physical qubits to a bosonic environment. By mapping the time evolution of the system after one quantum error correction cycle onto a statistical spin model, we show that the existence of an error threshold is related to the appearance of an order-disorder phase transition in the statistical model in the thermodynamic limit. This allows us to relate the error threshold to bath parameters and to the spatial range of the correlated errors.
We study the dependence of the fidelity of the surface code in the presence of a single finite-temperature massless bosonic environment after a quantum error correction cycle. The three standard types of environment are considered: super-Ohmic, Ohmic, and sub-Ohmic. Our results show that, for regimes relevant to current experiments, quantum error correction works well even in the presence of environment-induced, long-range interqubit interactions. A threshold always exists at finite temperatures, although its temperature dependence is very sensitive to the type of environment. For the super-Ohmic case, the critical coupling constant separating high from low fidelity decreases with increasing temperature. For both Ohmic and super-Ohmic cases, the dependence of the critical coupling on temperature is weak. In all cases, the critical coupling is determined by microscopic parameters of the environment. For the sub-Ohmic case, it also depends strongly on the duration of the QEC cycle.
Standard quantum error correction (QEC) models typically assume discrete, Markovian noise, obscuring the continuous quantum nature of physical environments. In this manuscript, we investigate the fundamental limits of an actively corrected surface code coupled to a continuous, un-reset quantum environment at zero and finite temperature. Using the generalized Caldeira-Leggett framework, we map the long-time evolution of the logical qubit to a boundary conformal field theory, establishing an exact equivalence to the anisotropic Kondo model. We evaluate computational times for a finite code distance LLL for all spatial and temporal correlations. Our analysis reveals that a true thermodynamic threshold exists strictly for short-range environments (z>1/(s+1)z>1/(s+1)z>1/(s+1)). In critical or long-range regimes, the macroscopic footprint of the code weaponizes the continuous bath, hindering the topological protection.
We study the resilience of the surface code to decoherence caused by the presence of a bosonic bath. This approach allows us to go beyond the standard stochastic error model commonly used to quantify decoherence and error threshold probabilities in this system. The full quantum mechanical system-bath dynamics is computed exactly over one quantum error correction cycle. Since all physical qubits interact with the bath, space-time correlations between errors are taken into account. We compute the fidelity of the surface code as a function of the quantum error correction time. The calculation allows us to map the problem onto an Ising-like statistical spin model with two-body interactions and a fictitious temperature which is related to the inverse bath coupling constant. The model departs from the usual Ising model in the sense that interactions can be long ranged and can involve complex exchange couplings; in addition, the number of allowed configurations is restricted by the syndrome extraction. Using analytical estimates and numerical calculations, we argue that, in the limit of an infinite number of physical qubits, the spin model sustains a phase transition which can be associated to the existence of an error threshold in the surface code. An estimate of the transition point is given for the case of nearest-neighbor interactions.
We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.
One of the key problems in quantum information processing is to understand the physical limits to quantum computation. Several strategies have been proposed to attenuate errors caused by the interaction of the computer with its surrounding environment and quantum error correction (QEC) is likely the most versatile. A large effort has been devoted to proving that resilience can be achieved by concatenating QEC codes in logical structures. In our work we look into this question at a different angle: we provide an upper bound on the time available to computation given a certain computer, a QEC code, and a decohering environment. We consider a broad class of environments, including those where correlation effects can be induced by gapless modes. Our approach is based on a Hamiltonian formulation where we use coarse graining in time to derive an explicit quantum evolution operator for the logical qubits, taking into account the QEC code. We show that this evolution operator has the same form as that for the original physical qubits, except for a reduced coupling to the environment which can be evaluated systematically for a given geometry and QEC code structure. To quantify the effectiveness of QEC, we compute the trace distance between the real and ideal states of a logical qubit after an arbitrary number of QEC cycles. We derive expressions for the long-time trace distance for several for super-ohmic-, ohmic-, and sub-ohmic-like baths. Given a confidence threshold for the trace distance, we establish the maximum time available for computation in those three cases. This maximum time is controlled by an exponent related to the spatial dimensions and other characteristics of the computer and the environment. For the super-ohmic regime, we find that computation can continue indefinitely, while in the other regimes the maximum time depends strongly on the QEC code, on the number of logical qubits, and on the original environment-computer strength interaction.