Quantum state tomography-deducing quantum states from measured data-is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. one of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

Quantum tomography is the main method used to assess the quality of quantum information processing devices. However, the amount of resources needed for quantum tomography is exponential in the device size. Part of the problem is that tomography generates much more information than is usually sought. Taking a more targeted approach, we develop schemes that enable (i) estimating the fidelity of an experiment to a theoretical ideal description, (ii) learning which description within a reduced subset best matches the experimental data. Both these approaches yield a significant reduction in resources compared to tomography. In particular, we demonstrate that fidelity can be estimated from a number of simple experiments that is independent of the system size, removing an important roadblock for the experimental study of larger quantum information processing units.

We describe two related methods for reconstructing multi-scale entangled states from a small number of efficiently-implementable measurements and fast post-processing. Both methods only require single-particle measurements and the total number of measurements is polynomial in the number of particles. Data post-processing for state reconstruction uses standard tools, namely matrix diagonalization and conjugate gradient method, and scales polynomially with the number of particles. Both methods prevent the build-up of errors from both numerical and experimental imperfections. The first method is conceptually simpler but requires unitary control. The second method circumvents the need for unitary control but requires more measurements and produces an estimated state of lower fidelity.

We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.PACS numbers: 03.67.Pp, 03.65.Ud, 03.67.Ac A self-correcting quantum memory [1] is a physical system whose quantum state can be preserved over a long period of time without the need for any external intervention. The archetypical self-correcting classical memory is the two-dimensional (2D) Ising ferromagnet. The ground state of this system is two-fold degenerate-all-spin up and all-spin down-so it can store one bit of information. If the memory is put into contact with a heat bath after being initialized in one of these ground states, thermal fluctuations will lead to the creation of small error droplets of inverted spins. The boundary of such droplets are domain walls, i.e., one-dimensional excitations whose energy is proportional to the droplet perimeter. If the temperature is below the critical Curie temperature, the Boltzman factor will prevent the creation of macroscopic error droplets. Thus, when the system is cooled down (either physically or algorithmically) after some macroscopic storage time, it will very likely return to its original ground state: the memory is thermally stable.This behaviour contrasts with the 1D Ising ferromagnet whose domain walls are point-like excitations. Therefore, they can freely diffuse on the chain at no energy cost. As a consequence, arbitrarily large error droplets can form, so this 1D memory is thermally unstable.While the 2D Ising ferromagnet features thermal stability, it is vulnerable to static, local perturbations. Indeed, an arbitrarily weak magnetic field breaks the ground state degeneracy and favours one ground state over the other. When this perturbed system is subject to thermal fluctuations, the bulk contribution of the magnetic field overwhelms the boundary tension of the domain wall, so once error droplets reach a critical size, they rapidly expand to corrupt the memory. This type of instability plagues any systems with a local order parameter, so they cannot be robust quantum memories. Indeed, distinct ground states give different values of this order parameter, so a local field coupling to the order parameter lifts degeneracy.In 2D and higher, there exists quantum systems with no local order parameter and whose spectrum is stable under weak, local perturbations. These systems have a degenerate ground state separated from the other energy levels by a constant energy gap, and perturbations only alter these features by an exponentially vanishing amount as a function of the system size. Kitaev's toric code [2] is the bes...

A two-dimensional topologically ordered quantum memory is well protected against error if the energy gap is large compared to the temperature, but this protection does not improve as the system size increases. We review and critique some recent proposals for improving the memory time by introducing long-range interactions among anyons, noting that instability with respect to small local perturbations of the Hamiltonian is a generic problem for such proposals. We also discuss some broader issues regarding the prospects for scalable quantum memory in two-dimensional systems.