Algorithms
Classical Shadow
An efficient technique for extracting useful classical information from a quantum state using relatively few measurements.
Definition
Classical shadows are a measurement technique that allows researchers to estimate many different properties of a quantum state using a relatively small number of randomized measurements, rather than needing a separate, large set of measurements for each property of interest.
In depth
A classical shadow is a compact classical description of a quantum state, built from a small number of random measurements, that can be used afterward to estimate many properties of the state. Huang, Kueng, and Preskill introduced the method in 2020. Fully reconstructing an n-qubit state, quantum state tomography, needs a number of measurements that grows exponentially with n, which is impractical beyond a few qubits. Classical shadows avoid this by settling for the ability to predict specific properties, not the complete state.
The procedure is short. Apply a random unitary to the state, measure in the computational basis, and record the unitary and the outcome. Each such snapshot gives an unbiased but noisy estimate of the state. Repeating it a modest number of times and averaging gives a shadow, from which the expectation value of any observable can be estimated. The number of snapshots needed to estimate M different observables to precision epsilon grows only logarithmically in M, which makes shadows efficient for predicting many properties from one data set. With random Pauli measurements, local observables on k qubits cost a factor of 3 to the k, and with random Clifford measurements, observables with small Hilbert-Schmidt norm, such as fidelities to stabilizer states, are cheap.
Shadows are used in variational algorithms, where measuring the many terms of a Hamiltonian is a major cost, in estimating entanglement entropies and fidelities, in verifying quantum devices, and in learning from quantum data. Extensions include derandomized shadows, which choose measurements to suit a given list of observables, and shadows that handle noise. The limitation is that global observables with large norm still need exponentially many snapshots, so the method is most powerful for local or structured properties.
Technical Definition
Introduced by Huang, Kueng, and Preskill in 2020, the technique constructs an efficiently storable classical approximation ('shadow') of a quantum state from randomized measurement data, from which many different observable properties can later be estimated without needing to re-run the quantum experiment.
Visual Explanation: An Analogy
Think of classical shadows like taking a few cleverly chosen photographs of a complex 3D object from different random angles, then being able to estimate many different facts about the object's shape later, without needing to physically examine it again from every possible angle.
Key facts
| Item | Detail |
|---|---|
| Introduced | Huang, Kueng, and Preskill, Nature Physics, 2020 |
| Scaling | Number of snapshots grows as log M for M observables |
| Random Pauli shadows | Local k-qubit observables cost a factor of about 3^k |
| Random Clifford shadows | Efficient for observables with small Hilbert-Schmidt norm |
| Comparison | Full tomography needs exponentially many measurements |
Where the idea comes from
Aaronson's 2018 work on shadow tomography laid the theoretical groundwork, and the practical protocol of Huang, Kueng, and Preskill in 2020 made it usable on devices with dozens of qubits.
Real-World Use Cases
- Used to reduce the number of measurements needed when characterizing or verifying quantum states and processes
- An active area of research for making near-term quantum experiments more measurement-efficient
How it differs from related ideas
- A classical shadow predicts selected properties efficiently, while tomography reconstructs the entire state.
- Shadows use random measurements, while standard measurement schemes use fixed bases.
- The classical description is not a full state, but a tool for estimating many observables.
Frequently asked questions
What is a classical shadow?
A compact classical record of a quantum state, from random measurements, that lets you estimate many properties of the state later.
Why is it better than tomography?
It needs far fewer measurements because it only aims to predict certain properties, not reconstruct the whole state.
Where is it used?
In variational algorithms, device verification, entanglement estimation, and learning from quantum data.
Related Terms
Keep exploring
- Algorithm: Variational Quantum Eigensolver (VQE)
- Algorithm: Quantum Error Mitigation Techniques
- Algorithm: SWAP Test
- Hardware: Google Willow