Shadow Tomography of Quantum States

TL;DR

This paper introduces shadow tomography, a method to estimate measurement outcomes on quantum states efficiently, requiring significantly fewer copies of the state than naive approaches, with broad implications for quantum information processing.

Contribution

The paper presents the first efficient procedure for shadow tomography that estimates many measurement outcomes with a polylogarithmic number of quantum state copies.

Findings

Requires only ( ext    ) copies of the state.

Enables learning the behavior of an n-qubit state on all fixed polynomial-size circuits with only n^{O(1)} copies.

Addresses open problems related to private-key quantum money, quantum copy-protected software, and quantum communication.

Abstract

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left( \varepsilon^{-4}\cdot\log^{4} M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O\left( 1\right)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brand\~ao et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.