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Jeongwan Haah - One of the best experts on this subject based on the ideXlab platform.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
Foundations of Computer Science, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error e using O(T polylog(nT/e)) gates with depth O(T polylog(nT/e)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/e. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin Kothari, Guang Hao LowAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using $\mathcal O( nT \mathrm{polylog} (nT/\epsilon))$ gates with depth $\mathcal O(T \mathrm{polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires $\tilde \Omega(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We present a decomposition of the real time evolution operator $e^{-i T H}$ of any local Hamiltonian $H$ on lattices $\Lambda \subseteq \mathbb Z^D$ into local unitaries based on Lieb-Robinson bounds. Combining this with recent quantum simulation algorithms for real time evolution, we find that the resulting quantum simulation algorithm has gate count $\mathcal O( T n ~\mathrm{polylog} (T n/\epsilon))$ and depth $\mathcal O( T ~\mathrm{polylog}(Tn/\epsilon))$, where $n$ is the space volume or the number of qubits, $T$ is the time of evolution, and $\epsilon$ is the accuracy of the simulation in operator norm. In contrast to this, the previous best quantum algorithms have gate count $\mathcal O(Tn^{2} ~\mathrm{polylog} (T n/\epsilon))$. Our approach readily generalizes to time-dependent Hamiltonians as well, and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. Finally, we also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise time-independent bounded local Hamiltonian in one dimension requires $\Omega(Tn / \mathrm{polylog}(Tn) )$ gates in the worst case. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms. Unlike previous Lieb-Robinson bounds, our version gives zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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Energy Landscape of 3D Spin Hamiltonians with Topological Order
Physical Review Letters, 2011Co-Authors: Sergey Bravyi, Jeongwan HaahAbstract:We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.
Matthew B Hastings - One of the best experts on this subject based on the ideXlab platform.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
Foundations of Computer Science, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error e using O(T polylog(nT/e)) gates with depth O(T polylog(nT/e)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/e. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin Kothari, Guang Hao LowAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using $\mathcal O( nT \mathrm{polylog} (nT/\epsilon))$ gates with depth $\mathcal O(T \mathrm{polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires $\tilde \Omega(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We present a decomposition of the real time evolution operator $e^{-i T H}$ of any local Hamiltonian $H$ on lattices $\Lambda \subseteq \mathbb Z^D$ into local unitaries based on Lieb-Robinson bounds. Combining this with recent quantum simulation algorithms for real time evolution, we find that the resulting quantum simulation algorithm has gate count $\mathcal O( T n ~\mathrm{polylog} (T n/\epsilon))$ and depth $\mathcal O( T ~\mathrm{polylog}(Tn/\epsilon))$, where $n$ is the space volume or the number of qubits, $T$ is the time of evolution, and $\epsilon$ is the accuracy of the simulation in operator norm. In contrast to this, the previous best quantum algorithms have gate count $\mathcal O(Tn^{2} ~\mathrm{polylog} (T n/\epsilon))$. Our approach readily generalizes to time-dependent Hamiltonians as well, and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. Finally, we also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise time-independent bounded local Hamiltonian in one dimension requires $\Omega(Tn / \mathrm{polylog}(Tn) )$ gates in the worst case. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms. Unlike previous Lieb-Robinson bounds, our version gives zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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trivial low energy states for commuting Hamiltonians and the quantum pcp conjecture
Quantum Information & Computation, 2013Co-Authors: Matthew B HastingsAbstract:We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have "trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states[1]. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an "interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a 1-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the 1-complex). While this includes the cases considered by Ref. bv, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref. bv but still can be mapped continuously to a 1-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a 1- complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to 1-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere[2], and have useful properties in quantum coding theory.
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trivial low energy states for commuting Hamiltonians and the quantum pcp conjecture
arXiv: Quantum Physics, 2012Co-Authors: Matthew B HastingsAbstract:We consider whether or not Hamiltonians which are sums of commuting projectors have "trivial" ground states which can be constructed by a local quantum circuit of bounded depth and range acting on a product state. While the toric code only has nontrivial ground states, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states. We define an "interaction complex" for a Hamiltonian, generalizing the interaction graph, and we show that if this complex can be continuously mapped to a 1-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonian (this condition holds for all stabilizer Hamiltonians, and we also prove the result for all Hamiltonians under an assumption on the 1-complex). While this includes cases considered by Ref., it also includes other Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref. One motivation for this is the quantum PCP conjecture. Many commonly studied interaction complexes can be mapped to a 1-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, a trivial ground state for the Hamiltonian with those sites removed is a low energy trivial state for the original Hamiltonian. Such states can act as a classical witness to the existence of a low energy state. While this result applies only to commuting Hamiltonians, it suggests that to prove a quantum PCP conjecture one should consider interaction complexes which cannot be mapped to 1-complexes after removing a small fraction of cells. We define this more precisely below; in a sense this generalizes the idea of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere, and have useful properties in quantum coding theory.
Andrew N Jordan - One of the best experts on this subject based on the ideXlab platform.
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optimal adaptive control for quantum metrology with time dependent Hamiltonians
Nature Communications, 2017Co-Authors: Shengshi Pang, Andrew N JordanAbstract:Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is generally necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of T2 time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches T4 in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case. Quantum metrology investigates the improvement given to precision measurements by exploiting quantum mechanics, but it has been mostly limited to systems with static Hamiltonians. Here the authors study it in the general case of time-varying Hamiltonians, showing that optimizing the quantum Fisher information via quantum control provides an advantage.
Sergey Bravyi - One of the best experts on this subject based on the ideXlab platform.
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On Complexity of the Quantum Ising Model
Communications in Mathematical Physics, 2017Co-Authors: Sergey Bravyi, Matthew HastingsAbstract:We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k -local ‘stoquastic’ Hamiltonians with any constant $${k \ge 2}$$ k ≥ 2 . This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class $${\mathsf{StoqMA}}$$ StoqMA —an extension of the classical class $${\mathsf{MA}}$$ MA . As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. Secondly, we study quantum annealing algorithms for finding ground states of classical spin Hamiltonians associated with hard optimization problems. We prove that the quantum annealing with TIM Hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k -local stoquastic Hamiltonians. This subclass includes all Hamiltonians representable as a sum of a k -local diagonal Hamiltonian and a 2-local stoquastic Hamiltonian.
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Energy Landscape of 3D Spin Hamiltonians with Topological Order
Physical Review Letters, 2011Co-Authors: Sergey Bravyi, Jeongwan HaahAbstract:We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.
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quantum simulation of many body Hamiltonians using perturbation theory with bounded strength interactions
Physical Review Letters, 2008Co-Authors: Sergey Bravyi, David P Divincenzo, Daniel Loss, Barbara M TerhalAbstract:We show how to map a given n-qubit target Hamiltonian with bounded-strength k-body interactions onto a simulator Hamiltonian with two-body interactions, such that the ground-state energy of the target and the simulator Hamiltonians are the same up to an extensive error O(epsilon n) for arbitrary small epsilon. The strength of the interactions in the simulator Hamiltonian depends on epsilon and k but does not depend on n. We accomplish this reduction using a new way of deriving an effective low-energy Hamiltonian which relies on the Schrieffer-Wolff transformation of many-body physics.
Robin Kothari - One of the best experts on this subject based on the ideXlab platform.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
Foundations of Computer Science, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses all physically reasonable Hamiltonians. Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error e using O(T polylog(nT/e)) gates with depth O(T polylog(nT/e)). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/e. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires (nT) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin Kothari, Guang Hao LowAbstract:We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and is believed to capture fundamental interactions of physics. Our algorithm simulates the time evolution of such a Hamiltonian on $n$ qubits for time $T$ up to error $\epsilon$ using $\mathcal O( nT \mathrm{polylog} (nT/\epsilon))$ gates with depth $\mathcal O(T \mathrm{polylog} (nT/\epsilon))$. Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in $nT$ and polylogarithmic in $1/\epsilon$. Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires $\tilde \Omega(nT)$ gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
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quantum algorithm for simulating real time evolution of lattice Hamiltonians
arXiv: Quantum Physics, 2018Co-Authors: Jeongwan Haah, Matthew B Hastings, Robin KothariAbstract:We present a decomposition of the real time evolution operator $e^{-i T H}$ of any local Hamiltonian $H$ on lattices $\Lambda \subseteq \mathbb Z^D$ into local unitaries based on Lieb-Robinson bounds. Combining this with recent quantum simulation algorithms for real time evolution, we find that the resulting quantum simulation algorithm has gate count $\mathcal O( T n ~\mathrm{polylog} (T n/\epsilon))$ and depth $\mathcal O( T ~\mathrm{polylog}(Tn/\epsilon))$, where $n$ is the space volume or the number of qubits, $T$ is the time of evolution, and $\epsilon$ is the accuracy of the simulation in operator norm. In contrast to this, the previous best quantum algorithms have gate count $\mathcal O(Tn^{2} ~\mathrm{polylog} (T n/\epsilon))$. Our approach readily generalizes to time-dependent Hamiltonians as well, and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian. Finally, we also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise time-independent bounded local Hamiltonian in one dimension requires $\Omega(Tn / \mathrm{polylog}(Tn) )$ gates in the worst case. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms. Unlike previous Lieb-Robinson bounds, our version gives zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.
