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Projected Entangled Pair States (PEPS) provide a framework for the construction of models where a single tensor gives rise to both Hamiltonian and ground state wavefunction on the same footing. A key problem is to characterize the behavior which emerges in the system in terms of the properties of the tensor, and thus of the Hamiltonian. In this paper, we consider PEPS models with Z2 onsite symmetry and study the occurence of long-range order and spontaneous symmetry breaking. We show how long-range order is connected to a degeneracy in the spectrum of the PEPS transfer operator, and how the latter gives rise to spontaneous symmetry breaking under perturbations. We provide a succinct characterization of the symmetry broken states in terms of the PEPS tensor, and find that using the symmetry broken states we can derive a local entanglement Hamiltonian, thereby restoring locality of the entanglement Hamiltonian for all gapped phases.

The fault-tolerant operation of logical qubits is an important requirement for realizing a universal quantum computer. Spin qubits based on quantum dots have great potential to be scaled to large numbers because of their compatibility with standard semiconductor manufacturing. Here, we show that a quantum error correction code can be implemented using a four-qubit array in germanium. We demonstrate a resonant SWAP gate and by combining controlled-Z and controlled-S−1 gates we construct a Toffoli-like three-qubit gate. We execute a two-qubit phase flip code and find that we can preserve the state of the data qubit by applying a refocusing pulse to the ancilla qubit. In addition, we implement a phase flip code on three qubits, making use of a Toffoli-like gate for the final correction step. Both the quality and quantity of the qubits will require significant improvement to achieve fault-tolerance. However, the capability to implement quantum error correction codes enables co-design development of quantum hardware and software, where codes tailored to the properties of spin qubits and advances in fabrication and operation can now come together to advance semiconductor quantum technology.

Based on numerically optimized real-device gates and parameters we study the performance of the phase-flip (repetition) code on a linear array of gallium arsenide (GaAs) quantum dots hosting singlet-triplet qubits. We first examine the expected performance of the code using simple error models of circuit-level and phenomenological noise, reporting, for example, a circuit-level depolarizing noise threshold of approximately 3%. We then perform density-matrix simulations using a maximum-likelihood and minimum-weight matching decoder to study the effect of real-device dephasing, readout error, and quasistatic as well as fast gate noise. Considering the tradeoff between qubit readout error and dephasing time (T 2) over measurement time, we identify a subthreshold region for the phase-flip code which lies within experimental reach.

We consider Projected Entangled Pair State (PEPS) models with a global ZN symmetry, which are constructed from ZN -symmetric tensors and are thus ZN -invariant wavefunctions, and study the occurence of long-range order and symmetry breaking in these systems. First, we show that longrange order in those models is accompanied by a degeneracy in the so-called transfer operator of the system. We subsequently use this degeneracy to determine the nature of the symmetry broken states, i.e., those stable under arbitrary perturbations, and provide a succinct characterization in terms of the fixed points of the transfer operator (i.e. the different boundary conditions) in the individual symmetry sectors. We verify our findings numerically through the study of a Z3-symmetric model, and show that the entanglement Hamiltonian derived from the symmetry broken states is quasilocal (unlike the one derived from the symmetric state), reinforcing the locality of the entanglement Hamiltonian for gapped phases.

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Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14% under depolarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post-processing methods.