Codes and Protocols for Distilling TT, controlled-SS, and Toffoli Gates

We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of 1703.07847. Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal codes, added comments on Clifford circuits for Reed-Muller states (v3) minor chang

Quantization of Hall Conductance For Interacting Electrons on a Torus

We consider interacting, charged spins on a torus described by a gapped Hamiltonian with a unique groundstate and conserved local charge. Using quasi-adiabatic evolution of the groundstate around a flux-torus, we prove, without any averaging assumption, that the Hall conductance of the groundstate is quantized in integer multiples of e^2/h, up to exponentially small corrections in the linear size of the system. In addition, we discuss extensions to the fractional quantization case under an additional topological order assumption on the degenerate groundstate subspace.Comment: 28 pages, 4 figures, This paper significantly simplifies the proof and tightens the bounds previously shown in arXiv:0911.4706 by the same authors. Updated to reflect published versio

Diffusion Processes on Power-Law Small-World Networks

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The results were obtained using self-consistent perturbation theory and can also be understood in terms of a scaling theory, which provides a general framework for understanding processes on small-world networks with different distributions of long-range links.Comment: 4 pages, 3 figures, added references, modified Fig. 2 with added data (PRL, in press

Quantum Codes from High-Dimensional Manifolds

We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on N qubits with logarithmic weight stabilizers and distance N^{1-epsilon} for any epsilon>0. The conjecture is that there is a constant C>0 such that for any n-dimensional torus {mathbb T}^n={mathbb R}^n/Lambda, where Lambda is a lattice, the least volume unoriented n/2-dimensional cycle (using the Euclidean metric) representing nontrivial homology has volume at least C^n times the volume of the least volume n/2-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for Lambda an integral lattice with the cycle restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes. We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split

Finite Size Scaling of Mutual Information: A Scalable Simulation

We develop a quantum Monte Carlo procedure to compute the Renyi mutual information of an interacting quantum many-body system at non-zero temperature. Performing simulations on a spin-1/2 XXZ model, we observe that for a subregion of fixed size embedded in a system of size L, the mutual information converges at large L to a limiting function which displays non-monotonic temperature behavior corresponding to the onset of correlations. For a region of size L/2 embedded in a system of size L, the mutual information divided by L converges to a limiting function of temperature, with apparently nontrivial corrections near critical points.Comment: 4 pages, 4 figure

Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count

We present an infinite family of protocols to distill magic states for $T$-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the $T$-gate depth of the circuit scale linearly in the logarithm of the target error $\delta$ (up to $\log \log 1/\delta$). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of (v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in Quantu