166 lines
2.5 KiB
Markdown
166 lines
2.5 KiB
Markdown
---
|
||
title: "Classical shadow"
|
||
chunk: 1/1
|
||
source: "https://en.wikipedia.org/wiki/Classical_shadow"
|
||
category: "reference"
|
||
tags: "science, encyclopedia"
|
||
date_saved: "2026-05-05T11:06:47.340833+00:00"
|
||
instance: "kb-cron"
|
||
---
|
||
|
||
In quantum computing, classical shadow is a protocol for predicting expectation values of a quantum state using only a logarithmic number of measurements. Given an unknown state
|
||
|
||
|
||
|
||
ρ
|
||
|
||
|
||
{\displaystyle \rho }
|
||
|
||
, a tomographically complete set of gates
|
||
|
||
|
||
|
||
U
|
||
|
||
|
||
{\displaystyle U}
|
||
|
||
(e.g. Clifford gates), a set of
|
||
|
||
|
||
|
||
M
|
||
|
||
|
||
{\displaystyle M}
|
||
|
||
observables
|
||
|
||
|
||
|
||
{
|
||
|
||
O
|
||
|
||
i
|
||
|
||
|
||
}
|
||
|
||
|
||
{\displaystyle \{O_{i}\}}
|
||
|
||
and a quantum channel
|
||
|
||
|
||
|
||
|
||
|
||
E
|
||
|
||
|
||
|
||
|
||
{\displaystyle {\mathcal {E}}}
|
||
|
||
defined by randomly sampling from
|
||
|
||
|
||
|
||
U
|
||
|
||
|
||
{\displaystyle U}
|
||
|
||
, applying it to
|
||
|
||
|
||
|
||
ρ
|
||
|
||
|
||
{\displaystyle \rho }
|
||
|
||
and measuring the resulting state, predict the expectation values
|
||
|
||
|
||
|
||
tr
|
||
|
||
(
|
||
|
||
O
|
||
|
||
i
|
||
|
||
|
||
ρ
|
||
)
|
||
|
||
|
||
{\displaystyle \operatorname {tr} (O_{i}\rho )}
|
||
|
||
. A list of classical shadows
|
||
|
||
|
||
|
||
S
|
||
|
||
|
||
{\displaystyle S}
|
||
|
||
is created using
|
||
|
||
|
||
|
||
ρ
|
||
|
||
|
||
{\displaystyle \rho }
|
||
|
||
,
|
||
|
||
|
||
|
||
U
|
||
|
||
|
||
{\displaystyle U}
|
||
|
||
and
|
||
|
||
|
||
|
||
|
||
|
||
E
|
||
|
||
|
||
|
||
|
||
{\displaystyle {\mathcal {E}}}
|
||
|
||
by running a Shadow generation algorithm. When predicting the properties of
|
||
|
||
|
||
|
||
ρ
|
||
|
||
|
||
{\displaystyle \rho }
|
||
|
||
, a Median-of-means estimation algorithm is used to deal with the outliers in
|
||
|
||
|
||
|
||
S
|
||
|
||
|
||
{\displaystyle S}
|
||
|
||
. Classical shadow is useful for direct fidelity estimation, entanglement verification, estimating correlation functions, and predicting entanglement entropy.
|
||
Recently, researchers have built on classical shadow to devise provably efficient classical machine learning algorithms for a wide range of quantum many-body problems. For example, machine learning models could learn to solve ground states of quantum many-body systems and classify quantum phases of matter.
|
||
|
||
|
||
== References == |