The surface code is a promising alternative for implementing fault-tolerant, large-scale quantum information processing. Its high threshold for single-qubit errors under stochastic noise is one of its most attractive features. We develop an exact formulation for the fidelity of the surface code that allows us to probe much further on this promise of strong protection. This formulation goes beyond the stochastic single-qubit error model approximation and can take into account both correlated errors and inhomogeneities in the coupling between physical qubits and the environment. For the case of a bit-flipping environment, we map the complete evolution after one quantum error correction cycle onto the problem of computing correlation functions of a two-dimensional Ising model with boundary fields. Exact results for the fidelity threshold of the surface code are then obtained for several relevant types of noise. Analytical predictions for a representative case are confirmed by Monte Carlo simulations.

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 sustain 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.Comment: 15 pages, 5 figure

We study how the resilience of the surface code is affected by the coupling to a non-Markovian environment at zero temperature. The qubits in the surface code experience an effective dynamics due to the coupling to the environment that induces correlations among them. The range of the effective induced qubit-qubit interaction depends on parameters related to the environment and the duration of the quantum error correction cycle. We show numerically that different interaction ranges set different intrinsic bounds on the fidelity of the code. These bounds are unrelated to the error thresholds based on stochastic error models. We introduce a definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine upper bounds to the fidelity error threshold.

A semiclassical approach that incorporates quantum mechanical behavior of heat capacity in direct caloric effect simulations is proposed. Application of this methodology to study electrocaloric effect in prototypical ferroelectrics ${ \rm PbTiO_3 }$, and ${ \rm BaTiO_3 }$, reveals severe underestimation of electrocaloric response at lowest temperatures by classical simulations. The discrepancy between semiclassical and classical results are found to be largest in ferroics with Debye temperature exceeding the Curie point. A route to enhance electrocaloric effect by tuning the Debye temperature in composite materials is proposed

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The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.

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The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here we present a process for obtaining the eigenspectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational basis. From this projection an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms in order to obtain the eigenvalues. The use case of this approach is demonstrated for ground-sate and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions.