It is well established that simulating a perfect quantum computer with a
classical computer requires computing resources that scale exponentially with
the number of qubits $N$ or the depth $D$ of the circuit. Conversely, a perfect
quantum computer could potentially provide an exponential speed up with respect
to classical hardware. Real quantum computers however are not perfect: they are
characterized by a small error rate $\epsilon$ per operation, so that the
fidelity of the many-qubit quantum state decays exponentially as $ F
(1-\epsilon)^ND$. Here, we discuss a set of classical algorithms based
on matrix product states (MPS) which closely mimic the behavior of actual
quantum computers. These algorithms require resources that scale linearly in
$N$ and $D$ at the cost of making a small error $\epsilon$ per two-qubit gate.
We illustrate our algorithms with simulations of random circuits for qubits
connected in both one and two dimensional lattices. We find that $\epsilon$ can
be decreased at a polynomial cost in computing power down to a minimum error
$\epsilon_ınfty$. Getting below $\epsilon_ınfty$ requires computing resources
that increase exponentially with $\epsilon_ınfty/\epsilon$. For a two
dimensional array of $N=54$ qubits and a circuit with Control-Z gates of depth
$D=20$, a fidelity $F0.002$ can be reached on a single core computer
in a few hours. It is remarkable that such a high fidelity can be obtained
using a variational ansatz that only spans a tiny fraction $(10^-8)$ of
the full Hilbert space. Our results show how the actual computing power
harvested by noisy quantum computers is controlled by the error rate
$\epsilon$.
Description
[2002.07730] What limits the simulation of quantum computers?
%0 Journal Article
%1 zhou2020limits
%A Zhou, Yiqing
%A Stoudenmire, E. Miles
%A Waintal, Xavier
%D 2020
%K physics quantum simulations
%T What limits the simulation of quantum computers?
%U http://arxiv.org/abs/2002.07730
%X It is well established that simulating a perfect quantum computer with a
classical computer requires computing resources that scale exponentially with
the number of qubits $N$ or the depth $D$ of the circuit. Conversely, a perfect
quantum computer could potentially provide an exponential speed up with respect
to classical hardware. Real quantum computers however are not perfect: they are
characterized by a small error rate $\epsilon$ per operation, so that the
fidelity of the many-qubit quantum state decays exponentially as $ F
(1-\epsilon)^ND$. Here, we discuss a set of classical algorithms based
on matrix product states (MPS) which closely mimic the behavior of actual
quantum computers. These algorithms require resources that scale linearly in
$N$ and $D$ at the cost of making a small error $\epsilon$ per two-qubit gate.
We illustrate our algorithms with simulations of random circuits for qubits
connected in both one and two dimensional lattices. We find that $\epsilon$ can
be decreased at a polynomial cost in computing power down to a minimum error
$\epsilon_ınfty$. Getting below $\epsilon_ınfty$ requires computing resources
that increase exponentially with $\epsilon_ınfty/\epsilon$. For a two
dimensional array of $N=54$ qubits and a circuit with Control-Z gates of depth
$D=20$, a fidelity $F0.002$ can be reached on a single core computer
in a few hours. It is remarkable that such a high fidelity can be obtained
using a variational ansatz that only spans a tiny fraction $(10^-8)$ of
the full Hilbert space. Our results show how the actual computing power
harvested by noisy quantum computers is controlled by the error rate
$\epsilon$.
@article{zhou2020limits,
abstract = {It is well established that simulating a perfect quantum computer with a
classical computer requires computing resources that scale exponentially with
the number of qubits $N$ or the depth $D$ of the circuit. Conversely, a perfect
quantum computer could potentially provide an exponential speed up with respect
to classical hardware. Real quantum computers however are not perfect: they are
characterized by a small error rate $\epsilon$ per operation, so that the
fidelity of the many-qubit quantum state decays exponentially as $ {\cal{F}}
\sim (1-\epsilon)^{ND}$. Here, we discuss a set of classical algorithms based
on matrix product states (MPS) which closely mimic the behavior of actual
quantum computers. These algorithms require resources that scale linearly in
$N$ and $D$ at the cost of making a small error $\epsilon$ per two-qubit gate.
We illustrate our algorithms with simulations of random circuits for qubits
connected in both one and two dimensional lattices. We find that $\epsilon$ can
be decreased at a polynomial cost in computing power down to a minimum error
$\epsilon_\infty$. Getting below $\epsilon_\infty$ requires computing resources
that increase exponentially with $\epsilon_\infty/\epsilon$. For a two
dimensional array of $N=54$ qubits and a circuit with Control-Z gates of depth
$D=20$, a fidelity ${\cal F}\ge 0.002$ can be reached on a single core computer
in a few hours. It is remarkable that such a high fidelity can be obtained
using a variational ansatz that only spans a tiny fraction $(\sim 10^{-8})$ of
the full Hilbert space. Our results show how the actual computing power
harvested by noisy quantum computers is controlled by the error rate
$\epsilon$.},
added-at = {2020-02-22T03:04:20.000+0100},
author = {Zhou, Yiqing and Stoudenmire, E. Miles and Waintal, Xavier},
biburl = {https://www.bibsonomy.org/bibtex/2f83adcfd2d4004e5760acc320c1e0db3/kirk86},
description = {[2002.07730] What limits the simulation of quantum computers?},
interhash = {a1b2990abafdd95ffd92586049119083},
intrahash = {f83adcfd2d4004e5760acc320c1e0db3},
keywords = {physics quantum simulations},
note = {cite arxiv:2002.07730Comment: 12 pages, 12 figures},
timestamp = {2020-02-22T03:04:20.000+0100},
title = {What limits the simulation of quantum computers?},
url = {http://arxiv.org/abs/2002.07730},
year = 2020
}