01951nas a2200217 4500008004100000245008300041210006900124260001500193490000600208520129200214100002001506700002301526700001701549700002201566700002401588700002301612700001301635700002301648700002501671856003701696 2022 eng d00aImplementing a Fast Unbounded Quantum Fanout Gate Using Power-Law Interactions0 aImplementing a Fast Unbounded Quantum Fanout Gate Using PowerLaw c10/27/20220 v43 a
The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as 1/rα in the distance r provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a D-dimensional lattice in time logarithmic in the number of qubits for interactions with α≤D. As a corollary, we show that power-law systems with α≤D are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.
1 aGuo, Andrew, Y.1 aDeshpande, Abhinav1 aChu, Su-Kuan1 aEldredge, Zachary1 aBienias, Przemyslaw1 aDevulapalli, Dhruv1 aSu, Yuan1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2007.0066202380nas a2200181 4500008004100000245007100041210006900112260001400181300000800195490000600203520186000209100001302069700002302082700002402105700001902129700001302148856003702161 2022 eng d00aProvably accurate simulation of gauge theories and bosonic systems0 aProvably accurate simulation of gauge theories and bosonic syste c9/20/2022 a8160 v63 aQuantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1).
1 aTong, Yu1 aAlbert, Victor, V.1 aMcClean, Jarrod, R.1 aPreskill, John1 aSu, Yuan uhttps://arxiv.org/abs/2110.0694201880nas a2200169 4500008004100000022001400041245006100055210006100116260001400177490000600191520140600197100001901603700001301622700001901635700001901654856003701673 2021 eng d a2691-339900aFaster Digital Quantum Simulation by Symmetry Protection0 aFaster Digital Quantum Simulation by Symmetry Protection c2/14/20210 v23 aSimulating the dynamics of quantum systems is an important application of quantum computers and has seen a variety of implementations on current hardware. We show that by introducing quantum gates implementing unitary transformations generated by the symmetries of the system, one can induce destructive interference between the errors from different steps of the simulation, effectively giving faster quantum simulation by symmetry protection. We derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection. In particular, when the symmetry transformations are chosen as powers of a unitary, the error of the algorithm is approximately projected to the so-called quantum Zeno subspaces. We prove a bound on this approximation error, exponentially improving a recent result of Burgarth, Facchi, Gramegna, and Pascazio. We apply our technique to the simulations of the XXZ Heisenberg interactions with local disorder and the Schwinger model in quantum field theory. For both systems, our algorithm can reduce the simulation error by several orders of magnitude over the unprotected simulation. Finally, we provide numerical evidence suggesting that our technique can also protect simulation against other types of coherent, temporally correlated errors, such as the 1/f noise commonly found in solid-state experiments.
1 aTran, Minh, C.1 aSu, Yuan1 aCarney, Daniel1 aTaylor, J., M. uhttps://arxiv.org/abs/2006.1624801759nas a2200169 4500008004100000245008100041210006900122260001500191520123200206100002401438700001301462700002301475700001301498700001801511700002301529856003701552 2021 eng d00aOptimal scaling quantum linear systems solver via discrete adiabatic theorem0 aOptimal scaling quantum linear systems solver via discrete adiab c11/15/20213 aRecently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of log(κ). Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions. We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity. Our O(κlog(1/ε)) complexity is also optimal in terms of the combined scaling in κ and the precision ε. Compared to existing suboptimal methods, our algorithm is simpler and easier to implement. Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.
1 aCosta, Pedro, C. S.1 aAn, Dong1 aSanders, Yuval, R.1 aSu, Yuan1 aBabbush, Ryan1 aBerry, Dominic, W. uhttps://arxiv.org/abs/2111.0815202331nas a2200181 4500008004100000245005200041210005200093260001300145300000700158490000700165520185000172100002302022700001302045700001902058700001802077700001702095856003702112 2021 eng d00aTheory of Trotter Error with Commutator Scaling0 aTheory of Trotter Error with Commutator Scaling c2/1/2021 a490 v113 aThe Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
1 aChilds, Andrew, M.1 aSu, Yuan1 aTran, Minh, C.1 aWiebe, Nathan1 aZhu, Shuchen uhttps://arxiv.org/abs/1912.0885401345nas a2200145 4500008004100000245006600041210006200107260001400169490000600183520092300189100001801112700001301130700001901143856003701162 2020 eng d00aApproximate Quantum Fourier Transform with O(nlog(n)) T gates0 aApproximate Quantum Fourier Transform with Onlogn T gates c3/13/20200 v63 aThe ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer enables the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, and phase estimation. The standard fault-tolerant implementation of an n-qubit QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+t gates, incurring the t-count complexity of O(n log2 (n)). In this paper we show how to obtain approximate QFT with the t-count of O(n log(n)). Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.
1 aNam, Yunseong1 aSu, Yuan1 aMaslov, Dmitri uhttps://arxiv.org/abs/1803.0493301561nas a2200169 4500008004100000245007300041210006900114260001300183490000800196520105300204100001901257700001701276700001301293700002301306700002501329856003701354 2020 eng d00aDestructive Error Interference in Product-Formula Lattice Simulation0 aDestructive Error Interference in ProductFormula Lattice Simulat c6/4/20200 v1243 aQuantum computers can efficiently simulate the dynamics of quantum systems. In this paper, we study the cost of digitally simulating the dynamics of several physically relevant systems using the first-order product formula algorithm. We show that the errors from different Trotterization steps in the algorithm can interfere destructively, yielding a much smaller error than previously estimated. In particular, we prove that the total error in simulating a nearest-neighbor interacting system of n sites for time t using the first-order product formula with r time slices is O(nt/r+nt3/r2) when nt2/r is less than a small constant. Given an error tolerance ε, the error bound yields an estimate of max{O(n2t/ε),O(n2t3/2/ε1/2)} for the total gate count of the simulation. The estimate is tighter than previous bounds and matches the empirical performance observed in Childs et al. [PNAS 115, 9456-9461 (2018)]. We also provide numerical evidence for potential improvements and conjecture an even tighter estimate for the gate count.
1 aTran, Minh, C.1 aChu, Su-Kuan1 aSu, Yuan1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1912.1104701893nas a2200145 4500008004100000245006300041210006300104260001300167490000800180520147500188100001401663700002001677700001301697856003701710 2020 eng d00aEfficiently computable bounds for magic state distillation0 aEfficiently computable bounds for magic state distillation c3/6/20200 v1243 aMagic state manipulation is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to magic state manipulation is the resource theory of magic states, for which one of the goals is to characterize and quantify quantum "magic." In this paper, we introduce the family of thauma measures to quantify the amount of magic in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of magic states. As a first application, we use the min-thauma to bound the regularized relative entropy of magic. As a consequence of this bound, we find that two classes of states with maximal mana, a previously established magic measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of magic states and reveals a fundamental difference between the resource theory of magic states and other resource theories such as entanglement and coherence. As a second application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable magic, which in turn leads to a variety of bounds on the rate at which magic can be distilled, as well as on the overhead of magic state distillation. Finally, we prove that the max-thauma can outperform mana in benchmarking the efficiency of magic state distillation.
1 aWang, Xin1 aWilde, Mark, M.1 aSu, Yuan uhttps://arxiv.org/abs/1812.1014501875nas a2200145 4500008004100000245004100041210003900082260001300121520148600134100001801620700002201638700001301660700001901673856003701692 2020 eng d00aA Sparse Model of Quantum Holography0 aSparse Model of Quantum Holography c8/5/20203 aWe study a sparse version of the Sachdev-Ye-Kitaev (SYK) model defined on random hypergraphs constructed either by a random pruning procedure or by randomly sampling regular hypergraphs. The resulting model has a new parameter, k, defined as the ratio of the number of terms in the Hamiltonian to the number of degrees of freedom, with the sparse limit corresponding to the thermodynamic limit at fixed k. We argue that this sparse SYK model recovers the interesting global physics of ordinary SYK even when k is of order unity. In particular, at low temperature the model exhibits a gravitational sector which is maximally chaotic. Our argument proceeds by constructing a path integral for the sparse model which reproduces the conventional SYK path integral plus gapped fluctuations. The sparsity of the model permits larger scale numerical calculations than previously possible, the results of which are consistent with the path integral analysis. Additionally, we show that the sparsity of the model considerably reduces the cost of quantum simulation algorithms. This makes the sparse SYK model the most efficient currently known route to simulate a holographic model of quantum gravity. We also define and study a sparse supersymmetric SYK model, with similar conclusions to the non-supersymmetric case. Looking forward, we argue that the class of models considered here constitute an interesting and relatively unexplored sparse frontier in quantum many-body physics.
1 aXu, Shenglong1 aSusskind, Leonard1 aSu, Yuan1 aSwingle, Brian uhttps://arxiv.org/abs/2008.0230301668nas a2200169 4500008004100000245006300041210006100104260001400165490000600179520118500185100002301370700002301393700001301416700001401429700001801443856003701461 2020 eng d00aTime-dependent Hamiltonian simulation with L1-norm scaling0 aTimedependent Hamiltonian simulation with L1norm scaling c4/20/20200 v43 aThe difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For the case of sparse Hamiltonian simulation, the gate complexity scales with the L1 norm ∫t0dτ∥H(τ)∥max, whereas the best previous results scale with tmaxτ∈[0,t]∥H(τ)∥max. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. By leveraging the L1-norm information, we obtain polynomial speedups for semi-classical simulations of scattering processes in quantum chemistry.
1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aSu, Yuan1 aWang, Xin1 aWiebe, Nathan uhttps://arxiv.org/abs/1906.0711501284nas a2200145 4500008004100000245004700041210004700088260001500135490000600150520088800156100002301044700002101067700001301088856003701101 2019 eng d00aFaster quantum simulation by randomization0 aFaster quantum simulation by randomization c08/28/20190 v33 aProduct formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that our randomized algorithm may be advantageous even for near-term quantum simulation.
1 aChilds, Andrew, M.1 aOstrander, Aaron1 aSu, Yuan uhttps://arxiv.org/abs/1805.0838500650nas a2200121 4500008004100000245012100041210006900162260001500231490000600246520020700252100001300459856005600472 2019 eng d00aFramework for Hamiltonian simulation and beyond: standard-form encoding, qubitization, and quantum signal processing0 aFramework for Hamiltonian simulation and beyond standardform enc c08/13/20190 v33 aThe propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/rα. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al. [arXiv:1801.03922]. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α>3D (where D is the number of dimensions).
1 aTran, Minh, C.1 aGuo, Andrew, Y.1 aSu, Yuan1 aGarrison, James, R.1 aEldredge, Zachary1 aFoss-Feig, Michael1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1808.0522501490nas a2200133 4500008004100000245005800041210005800099260001500157490000800172520110300180100002301283700001301306856003701319 2019 eng d00aNearly optimal lattice simulation by product formulas0 aNearly optimal lattice simulation by product formulas c12/17/20190 v1233 aProduct formulas provide a straightforward yet surprisingly efficient approach to quantum simulation. We show that this algorithm can simulate an n-qubit Hamiltonian with nearest-neighbor interactions evolving for time t using only (nt)1+o(1) gates. While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and significantly simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity.
1 aChilds, Andrew, M.1 aSu, Yuan uhttps://arxiv.org/abs/1901.0056401888nas a2200145 4500008004100000245004600041210004600087260001400133490000700147520150400154100001401658700002001672700001301692856003701705 2019 eng d00aQuantifying the magic of quantum channels0 aQuantifying the magic of quantum channels c10/8/20190 v213 aTo achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum "magic" or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
1 aWang, Xin1 aWilde, Mark, M.1 aSu, Yuan uhttps://arxiv.org/abs/1903.0448301295nas a2200169 4500008004100000245008000041210006900121260001500190490000600205520078500211100001800996700001901014700001301033700002301046700001901069856003701088 2018 eng d00aAutomated optimization of large quantum circuits with continuous parameters0 aAutomated optimization of large quantum circuits with continuous c2017/10/190 v43 aWe develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers.
1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan1 aChilds, Andrew, M.1 aMaslov, Dmitri uhttps://arxiv.org/abs/1710.0734502587nas a2200157 4500008004100000245011000041210006900151260001500220300001200235520207500247100001902322700001302341700002002354700001802374856003702392 2018 eng d00aQuantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics0 aQuantum singular value transformation and beyond exponential imp c2018/06/05 a193-2043 aQuantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.
1 aGilyen, Andras1 aSu, Yuan1 aLow, Guang, Hao1 aWiebe, Nathan uhttps://arxiv.org/abs/1806.0183801404nas a2200133 4500008004100000245005700041210005500098260001500153490000600168520102800174100001301202700001801215856003701233 2018 eng d00aTime-reversal of rank-one quantum strategy functions0 aTimereversal of rankone quantum strategy functions c2018/01/250 v23 aThe quantum strategy (or quantum combs) framework is a useful tool for reasoning about interactions among entities that process and exchange quantum information over the course of multiple turns. We prove a time-reversal property for a class of linear functions, defined on quantum strategy representations within this framework, that corresponds to the set of rank-one positive semidefinite operators on a certain space. This time-reversal property states that the maximum value obtained by such a function over all valid quantum strategies is also obtained when the direction of time for the function is reversed, despite the fact that the strategies themselves are generally not time reversible. An application of this fact is an alternative proof of a known relationship between the conditional min- and max-entropy of bipartite quantum states, along with generalizations of this relationship.
1 aSu, Yuan1 aWatrous, John uhttps://arxiv.org/abs/1801.0849101648nas a2200169 4500008004100000245006100041210006100102300001400163490000900177520116300186100002301349700001901372700001801391700001901409700001301428856003701441 2018 eng d00aToward the first quantum simulation with quantum speedup0 aToward the first quantum simulation with quantum speedup a9456-94610 v115 3 aWith quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, using diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically infeasible instances of factoring and quantum chemistry, bringing practical quantum computation closer to reality.
1 aChilds, Andrew, M.1 aMaslov, Dmitri1 aNam, Yunseong1 aRoss, Neil, J.1 aSu, Yuan uhttps://arxiv.org/abs/1711.1098001360nas a2200181 4500008004100000022001400041245007000055210006900125260001500194520081800209100001801027700001901045700001801064700001301082700001901095700001601114856004801130 2017 eng d a1871-409900aExtreme learning machines for regression based on V-matrix method0 aExtreme learning machines for regression based on Vmatrix method c2017/06/103 aThis paper studies the joint effect of V-matrix, a recently proposed framework for statistical inferences, and extreme learning machine (ELM) on regression problems. First of all, a novel algorithm is proposed to efficiently evaluate the V-matrix. Secondly, a novel weighted ELM algorithm called V-ELM is proposed based on the explicit kernel mapping of ELM and the V-matrix method. Though V-matrix method could capture the geometrical structure of training data, it tends to assign a higher weight to instance with smaller input value. In order to avoid this bias, a novel method called VI-ELM is proposed by minimizing both the regression error and the V-matrix weighted error simultaneously. Finally, experiment results on 12 real world benchmark datasets show the effectiveness of our proposed methods.
1 aYang, Zhiyong1 aZhang, Taohong1 aLu, Jingcheng1 aSu, Yuan1 aZhang, Dezheng1 aDuan, Yaowu uhttp://dx.doi.org/10.1007/s11571-017-9444-2