Optimal local unitary encoding circuits for the surface code

The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size L, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has O(L2) time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly 2L time steps to encode an unknown state in a distance L planar code. We further show how an O(L) complexity local unitary encoder for the toric code can be found by enforcing locality in the O(log⁡L)-depth non-local renormalisation encoder. We relate these techniques by providing an O(L) local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

Rectangular surface code under biased noise

To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .

Optimisation of complex integration contours at higher order

We continue our study of contour deformation as a practical tool for dealing with the sign problem using the d-dimensional Bose gas with non-zero chemical potential as a toy model. We derive explicit expressions for contours up to the second order with respect to a natural small parameter and generalise these contours to an ansatz for which the evaluation of the Jacobian is fast (O(1)). We examine the behaviour of the various proposed contours as a function of space-time dimensionality, the chemical potential, and lattice size and geometry and use the mean phase factor as a measure of the severity of the sign problem. In turns out that this method leads to a substantial reduction of the sign problem and that it becomes more efficient as space-time dimensionality is increased. Correlations among contributions to Im 〈S〉 play a key role in determining the mean phase factor and we examine these correlations in detail.

Macroscopic Lattice Boltzmann Method

The lattice Boltzmann method (LBM) is a highly simplified model for fluid flows using a few limited fictitious particles. It has been developed into a very efficient and flexible alternative numerical method in computational physics, demonstrating its great power and potential for resolving more and more challenging physical problems in science and engineering covering a wide range of disciplines such as physics, chemistry, biology, material science and image analysis. The LBM is implemented through the two routine steps of streaming and collision using the three parameters of the lattice size, particle speed and collision operator. A fundamental question is if the two steps are integral to the method or if the three parameters can be reduced to one for a minimal lattice Boltzmann method. In this paper, it is shown that the collision step can be removed and the standard LBM can be reformulated into a simple macroscopic lattice Boltzmann method (MacLAB). This model relies on macroscopic physical variables only and is completely defined by one basic parameter of the lattice size δx, bringing the LBM into a precise “lattice” Boltzmann method. The viscous effect on flows is naturally embedded through the particle speed, making it an ideal automatic simulator for fluid flows. Three additional advantages compared to the existing LBMs are that: (i) physical variables can directly be retained as the boundary conditions; (ii) much less computational memory is required; and (iii) the model is unconditionally stable. The findings are demonstrated and confirmed with numerical tests including flows that are independent of and dependent on fluid viscosity, 2D and 3D cavity flows and an unsteady Taylor–Green vortex flow. This provides an efficient and powerful model for resolving physical problems in various disciplines of science and engineering.

Reversible Switching Between Positive and Negative Thermal Expansion in a Metal-Organic Framework DUT-49

Three-dimensional architectures constructed via coordination of metal ions to organic linkers (broadly termed as metal-organic frameworks, MOFs), are highly interesting for many demanding applications such as gas adsorption, molecular separation, heterogeneous catalysis, molecular sensing etc. Being constructed from heterogeneous components, such framework solids show characteristic features from both of the individual components as well as framework-specific features. One such interesting physicochemical property is thermal expansion, which arises from thermal vibration from the organic linker and metal ions. Herein, we show a very unique example of thermal responsiveness for DUT-49 framework, a MOF well-known for its distinctive negative gas adsorption (NGA) property. In the guest-free form, the framework shows another counter-intuitive phenomenon of negative thermal expansion (NTE), i.e. lattice size increase with decrease of temperature. However, in the solvated state, it shows both NTE and positive thermal expansion (i.e. lattice size decreases with lowering of temperature, PTE) based on a specific temperature range. When the solvent exists in liquid form inside the MOF pore, it retains the pristine NTE nature of the bare framework. But freezing of the solvent inside the pores induces a strain, which causes a structural transformation through in-plane bending of the linker and this squeezes the framework by ~10 % of the unit cell volume. This effect has been verified using 3 different solvents where the structural contraction occurs immediately at the freezing point of individual solvent. Furthermore, studies on a series of DUT-49(M) frameworks with varying metal confirm the general applicability of this mechanism.<br>

Reversible Switching Between Positive and Negative Thermal Expansion in a Metal-Organic Framework DUT-49

Three-dimensional architectures constructed via coordination of metal ions to organic linkers (broadly termed as metal-organic frameworks, MOFs), are highly interesting for many demanding applications such as gas adsorption, molecular separation, heterogeneous catalysis, molecular sensing etc. Being constructed from heterogeneous components, such framework solids show characteristic features from both of the individual components as well as framework-specific features. One such interesting physicochemical property is thermal expansion, which arises from thermal vibration from the organic linker and metal ions. Herein, we show a very unique example of thermal responsiveness for DUT-49 framework, a MOF well-known for its distinctive negative gas adsorption (NGA) property. In the guest-free form, the framework shows another counter-intuitive phenomenon of negative thermal expansion (NTE), i.e. lattice size increase with decrease of temperature. However, in the solvated state, it shows both NTE and positive thermal expansion (i.e. lattice size decreases with lowering of temperature, PTE) based on a specific temperature range. When the solvent exists in liquid form inside the MOF pore, it retains the pristine NTE nature of the bare framework. But freezing of the solvent inside the pores induces a strain, which causes a structural transformation through in-plane bending of the linker and this squeezes the framework by ~10 % of the unit cell volume. This effect has been verified using 3 different solvents where the structural contraction occurs immediately at the freezing point of individual solvent. Furthermore, studies on a series of DUT-49(M) frameworks with varying metal confirm the general applicability of this mechanism.<br>

Reversible Switching Between Positive and Negative Thermal Expansion in a Metal-Organic Framework DUT-49

Three-dimensional architectures constructed via coordination of metal ions to organic linkers (broadly termed as metal-organic frameworks, MOFs), are highly interesting for many demanding applications such as gas adsorption, molecular separation, heterogeneous catalysis, molecular sensing etc. Being constructed from heterogeneous components, such framework solids show characteristic features from both of the individual components as well as framework-specific features. One such interesting physicochemical property is thermal expansion, which arises from thermal vibration from the organic linker and metal ions. Herein, we show a very unique example of thermal responsiveness for DUT-49 framework, a MOF well-known for its distinctive negative gas adsorption (NGA) property. In the guest-free form, the framework shows another counter-intuitive phenomenon of negative thermal expansion (NTE), i.e. lattice size increase with decrease of temperature. However, in the solvated state, it shows both NTE and positive thermal expansion (i.e. lattice size decreases with lowering of temperature, PTE) based on a specific temperature range. When the solvent exists in liquid form inside the MOF pore, it retains the pristine NTE nature of the bare framework. But freezing of the solvent inside the pores induces a strain, which causes a structural transformation through in-plane bending of the linker and this squeezes the framework by ~10 % of the unit cell volume. This effect has been verified using 3 different solvents where the structural contraction occurs immediately at the freezing point of individual solvent. Furthermore, studies on a series of DUT-49(M) frameworks with varying metal confirm the general applicability of this mechanism.<br>

Multi-state Condensation in Berlin–Kac Spherical Models

We consider the Berlin–Kac spherical model for supercritical densities under a periodic lattice energy function which has finitely many non-degenerate global minima. Energy functions arising from nearest neighbour interactions on a rectangular lattice have a unique minimum, and in that case the supercritical fraction of the total mass condenses to the ground state of the energy function. We prove that for any sufficiently large lattice size this also happens in the case of multiple global minima, although the precise distribution of the supercritical mass and the structure of the condensate mass fluctuations may depend on the lattice size. However, in all of these cases, one can identify a bounded number of degrees of freedom forming the condensate in such a way that their fluctuations are independent from the rest of the fluid. More precisely, the original Berlin–Kac measure may be replaced by a factorized supercritical measure where the condensate and normal fluid degrees of freedom become independent random variables, and the normal fluid part converges to the critical Gaussian free field. The proof is based on a construction of a suitable coupling between the two measures, proving that their Wasserstein distance is small enough for the error in any finite moment of the field to vanish as the lattice size is increased to infinity.