Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length $N$ of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-$k$ segments of the chain, the reduced density matrices of our approximations are $\epsilon$-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like $(k/\epsilon)^{1+o(1)}$, and at the expense of worse but still $\text{poly}(k,1/\epsilon)$ scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension $\exp(O(k/\epsilon))$, which is exponentially worse, but still independent of $N$. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find $O(1)$-accurate local approximations to the ground state in $T(N)$ time implies the ability to estimate the ground state energy to $O(1)$ precision in $O(T(N)\log(N))$ time.Comment: 24 pages, 3 figures. v2: Theorem 1 extended to include construction for general states; Lemma 7 & Theorem 2 slightly improved; figures added; lemmas rearranged for clarity; typos fixed. v3: Reformatted & additional references inserte

A Revised Method of Condensed Tannin Analysis in Leucaena spp.

The proanthocyanidin (butanol-HCl) assay was used to measure condensed tannin (CT) in extracts from Leucaena spp. at the University of Queensland. Inconsistent results were found to be caused by the iron catalyst in the butanol/HCl reagent and the presence of ascorbic acid in the sample matrix which enhanced anthocyanidin development. In addition, preparation of sample extracts by back extraction with organic solvents reduced CT recoveries. An accurate and rapid technique was developed that measures CT directly in 70% aqueous acetone 0.1% (w/v) sodium metabisulphite plant extracts

Peripartum cardiomyopathy: diagnosis and management

No abstract available

Detection of Toxicity in Ruminants Consuming Leucaena (\u3cem\u3eLeucaena leucocephala\u3c/em\u3e) Using a Urine Colorimetric Test

Leucaena (Leucaena leucocephala), a productive leguminous shrub for feeding ruminant livestock, contains the toxic amino acid, mimosine which post- ingestion is converted to 3,4-DHP and 2,3-DHP, isomers of dihydroxy-pyridone. While DHP generally does not exhibit acute toxic symptoms, it has been suggested that it is an appetite suppressant that reduces animal live weight gain (Jones 1994). With no observable symptoms, subclinical toxicity is difficult to detect (Phaikaew et al. 2012). In 1982 the DHP-degrading rumen bacterium named Synergistes jonesii was introduced into Australia as a potential solution to DHP toxicity as it spreads easily throughout cattle herds grazing leucaena (Jones 1994). However, toxicity events reported since the 2003 drought suggest that the toxicity status of herds, previously understood as being protected, may have changed. This may be the result of loss of effective S. jonesii bacteria from the rumen. Widespread subclinical leucaena toxicity has since been confirmed representing a significant economic threat to the beef industry (Dalzell et al. 2012). At present the testing for toxicity requires a sophisticated chemical analysis of urine samples using high performance liquid chromatography (HPLC). Producers, however, require a robust and reliable means to routinely test for toxicity in their herds. A colorimetric urine test protocol is available based on the colour reaction of mimosine and DHP with FeCl3 solution (Jones 1997). When this simpler colorimetric test has been used under a wide range of conditions false negatives have been reported. The aim of this study was to improve the reliability of the FeCL3 urine colour test

Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-\eta)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $\eta$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].Comment: 49 pages, 3 figure

How many qubits are needed for quantum computational supremacy?

Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P $\neq$ NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse assumption. Each version is parameterized by a constant $a$ and asserts that certain specific computational problems with input size $n$ require $2^{an}$ time steps to be solved by a non-deterministic algorithm. Then, we choose a specific value of $a$ for each version that we argue makes the assumption plausible, and based on these conjectures we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and boson sampling circuits (i.e. linear optical networks) with 98 photons are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. In the first two cases, we extend this to constant additive error by introducing an average-case fine-grained conjecture.Comment: 24 pages + 3 appendices, 8 figures. v2: number of qubits calculation updated and conjectures clarified after becoming aware of Ref. [42]. v3: Section IV and Appendix C added to incorporate additive-error simulation

Efficient classical simulation of random shallow 2D quantum circuits

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied

Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-k segments of the chain, the reduced density matrices of our approximations are ϵ-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)^(1+o(1)), and at the expense of worse but still poly(k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp(O(k/ϵ)), which is exponentially worse, but still independent of N. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1)-accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log(N)) time