Jump to content

Classical shadow

From Wikipedia, the free encyclopedia

In quantum computing, classical shadow is a protocol for predicting functions of a quantum state using only a logarithmic number of measurements.[1] Given an unknown state , a tomographically complete set of gates (e.g. Clifford gates), a set of observables and a quantum channel defined by randomly sampling from , applying it to and measuring the resulting state, predict the expectation values .[2] A list of classical shadows is created using , and by running a Shadow generation algorithm. When predicting the properties of , a Median-of-means estimation algorithm is used to deal with the outliers in .[3] Classical shadow is useful for direct fidelity estimation, entanglement verification, estimating correlation functions, and predicting entanglement entropy.[1]

Recently, researchers have built on classical shadow to devise provably efficient classical machine learning algorithms for a wide range of quantum many-body problems.[4] For example, machine learning models could learn to solve ground states of quantum many-body systems and classify quantum phases of matter.

Algorithm Shadow generation
Inputs copies of an unknown -qubit state

                  A list of unitaries that is tomographically complete

                  A classical description of a quantum channel

  1. For ranging from to :
    1. Choose a random unitary from
    2. Apply to to get a state
    3. Perform a computational basis measurement on for an outcome
    4. Classically compute and add it to a list
Return


  • "←" denotes assignment. For instance, "largestitem" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the following value.
Algorithm Median-of-means estimation
Inputs A list of observables

                  A classical shadow

                  A positive integer that specifies how many linear estimates of to calculate.

Return A list where
where and where .


  • "←" denotes assignment. For instance, "largestitem" means that the value of largest changes to the value of item.
  • "return" terminates the algorithm and outputs the following value.

References

[edit]
  1. ^ a b Huang, Hsin-Yuan; Kueng, Richard; Preskill, John (2020). "Predicting many properties of a quantum system from very few measurements". Nat. Phys. 16 (10): 1050–1057. arXiv:2002.08953. Bibcode:2020NatPh..16.1050H. doi:10.1038/s41567-020-0932-7. S2CID 211205098.
  2. ^ Koh, D. E.; Grewal, Sabee (2022). "Classical Shadows with Noise". Quantum. 6: 776. arXiv:2011.11580. Bibcode:2022Quant...6..776K. doi:10.22331/q-2022-08-16-776. S2CID 227127118.
  3. ^ Struchalin, G.I.; Zagorovskii, Ya. A.; Kovlakov, E.V.; Straupe, S.S.; Kulik, S.P. (2021). "Experimental Estimation of Quantum State Properties from Classical Shadows". PRX Quantum. 2 (1): 010307. arXiv:2008.05234. doi:10.1103/PRXQuantum.2.010307. S2CID 221103573.
  4. ^ Huang, Hsin-Yuan; Kueng, Richard; Torlai, Giacomo; Albert, Victor A.; Preskill, John (2022). "Provably efficient machine learning for quantum many-body problems". Science. 377 (6613): eabk3333. arXiv:2106.12627. doi:10.1126/science.abk3333. PMID 36137032. S2CID 235624289.