اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExponential improvement in precision for simulating sparse Hamiltonians
196
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $epsilon$ using $Obig(tau frac{log(tau/epsilon)}{loglog(tau/epsilon)}big)$ queries and $Obig(tau frac{log^2(tau/epsilon)}{loglog(tau/epsilon)}nbig)$ additional 2-qubit gates, where $tau = d^2 |{H}|_{max} t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of oblivious amplitude amplification that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.
قيم البحث
اقرأ أيضاً
We provide a quantum method for simulating Hamiltonian evolution with complexity polynomial in the logarithm of the inverse error. This is an exponential improvement over existing methods for Hamiltonian simulation. In addition, its scaling with resp
ect to time is close to linear, and its scaling with respect to the time derivative of the Hamiltonian is logarithmic. These scalings improve upon most existing methods. Our method is to use a compressed Lie-Trotter formula, based on recent ideas for efficient discrete-time simulations of continuous-time quantum query algorithms.
We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memor
y (qRAM) which allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve poly-logarithmic dependence on precision. The time complexity of our algorithm, measured in terms of the circuit depth, is $O(tsqrt{N}|H|,mathrm{polylog}(N, t|H|, 1/epsilon))$, where $t$ is the evolution time, $N$ is the dimension of the system, and $epsilon$ is the error in the final state, which we call precision. Our algorithm can be directly applied as a subroutine for unitary implementation and quantum linear systems solvers, achieving $widetilde{O}(sqrt{N})$ dependence for both applications.
Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the Third Law of thermodynamics prohibits perfect cooling; nonet
heless, attainable upper bounds for the asymptotic ground state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive step-wise optimal protocol that outperforms all standard procedures.
It is commonly claimed that only Hamiltonians with a spectrum unbounded both above and below can give purely exponential decay. Because such Hamiltonians have no ground state, they are considered unphysical. Here we show that Hamiltonians which are b
ounded below can give purely exponential decay. This is possible when, instead of looking at the global survival probability, one considers a subsystem only. We conclude that purely exponential decay might not be as unphysical as previously thought.
We develop a framework and give an example for situations where two distinct Hamiltonians living in the same Hilbert space can be used to simulate the same physics. As an example of an analog simulation, we first discuss how one can simulate an infin
ite-range-interaction one-axis twisting Hamiltonian using a short-range nearest-neighbor-interaction Heisenberg XXX model with a staggered field. Based on this, we show how one can build an alternative version of a digital quantum simulator. As a by-product, we present a method for creating many-body maximally entangled states using only short-range nearest-neighbor interactions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
