# Guang Hao Lowguang-hao-low-2230

Jan 12 2018 02:00 UTC
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.
Sep 18 2017 14:21 UTC
Jul 19 2017 02:00 UTC
The exponential speedups promised by Hamiltonian simulation on a quantum computer depends crucially on structure in both the Hamiltonian $\hat{H}$, and the quantum circuit $\hat{U}$ that encodes its description. In the quest to better approximate time-evolution $e^{-i\hat{H}t}$ with error $\epsilon$, we motivate a systematic approach to understanding and exploiting structure, in a setting where Hamiltonians are encoded as measurement operators of unitary circuits $\hat{U}$ for generalized measurement. This allows us to define a \emphuniform spectral amplification problem on this framework for expanding the spectrum of encoded Hamiltonian with exponentially small distortion. We present general solutions to uniform spectral amplification in a hierarchy where factoring $\hat{U}$ into $n=1,2,3$ unitary oracles represents increasing structural knowledge of the encoding. Combined with structural knowledge of the Hamiltonian, specializing these results allow us simulate time-evolution by $d$-sparse Hamiltonians using $\mathcal{O}\left(t(d \|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2}\log{(t\|\hat{H}\|/\epsilon)}\right)$ queries, where $\|\hat H\|\le \|\hat H\|_1\le d\|\hat H\|_{\text{max}}$. Up to logarithmic factors, this is a polynomial improvement upon prior art using $\mathcal{O}\left(td\|\hat H\|_{\text{max}}+\frac{\log{(1/\epsilon)}}{\log\log{(1/\epsilon)}}\right)$ or $\mathcal{O}(t^{3/2}(d \|\hat H\|_{\text{max}}\|\hat H\|_{1}\|\hat H\|/\epsilon)^{1/2})$ queries. In the process, we also prove a matching lower bound of $\Omega(t(d\|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2})$ queries, present a distortion-free generalization of spectral gap amplification, and an amplitude amplification algorithm that performs multiplication on unknown state amplitudes.
Apr 18 2017 02:22 UTC
Mar 03 2017 03:26 UTC
Feb 14 2017 23:12 UTC
Guang Hao Low scited Logical Randomized Benchmarking
Feb 14 2017 02:00 UTC
We demonstrate parallel composite quantum logic gates with phases implemented locally through nanoscale movement of ions within a global laser beam of fixed pulse duration. We show that a simple four-pulse sequence suffices for constructing ideal arbitrary single-qubit rotations in the presence of large intensity inhomogeneities across the ion trap due to laser beam-pointing or beam-focusing. Using such sequences, we perform parallel arbitrary rotations on ions in two trapping zones separated by 700 $\mu$m with fidelities comparable to those of our standard laser-controlled gates. Our scheme improves on current transport or zone-dependent quantum gates to include phase modulation with local control of the ion's confinement potential. This enables a scalable implementation of an arbitrary number of parallel operations on densely packed qubits with a single laser modulator and beam path.
Jan 31 2017 02:22 UTC
Jan 20 2017 07:53 UTC
Jan 19 2017 05:23 UTC
Guang Hao Low scited Universal Quantum Hamiltonians
Jan 16 2017 06:34 UTC
Jan 04 2017 20:50 UTC
Dec 15 2016 15:54 UTC
Guang Hao Low scited The surface code with a twist
Dec 06 2016 11:56 UTC
Nov 18 2016 03:23 UTC
Nov 15 2016 03:04 UTC
Guang Hao Low scited Adiabatic Quantum Computing
Nov 09 2016 11:52 UTC
Oct 27 2016 23:53 UTC
Oct 25 2016 02:52 UTC
Guang Hao Low scited Power of one non-clean qubit
Oct 25 2016 02:40 UTC
Oct 21 2016 02:00 UTC
Given a Hermitian operator $\hat{H}=\langle G|\hat{U}|G\rangle$ that is the projection of an oracle $\hat{U}$ by state $|G\rangle$ created with oracle $\hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-i\hat{H}t}$ at time $t$ with error $\epsilon$. We show that this can be done with query complexity $\mathcal{O}\big(t+\frac{\log{(1/\epsilon)}}{\log\log{(1/\epsilon)}}\big)$ to $\hat{G},\hat{U}$ that is optimal, not just in asymptotic limits, but for all values $t,\epsilon$. Furthermore, only $2$ additional ancilla qubits are required in total, together with $\mathcal{O}(1)$ additional single and two-qubit gates per query. Our approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space complexity, as well as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\langle G|\hat{U}|G\rangle$ is a density matrix. A key technical result is `qubitization' which uses controlled-$\hat{U}$ and controlled-$\hat{G}$ to embed $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.