Shadow Tomography of Quantum States (1711.01053v2)

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.

• The paper’s main contribution demonstrates that shadow tomography can estimate acceptance probabilities with only ~O(ε⁻⁴ log⁴M logD) copies, significantly reducing resource demands.
• The methodology employs gentle measurement and postselected learning techniques to accurately handle high-dimensional observables and mitigate noise.
• The results impact quantum cryptography and computation by providing a scalable approach for estimating quantum state behaviors with limited state copies.

Overview of Shadow Tomography of Quantum States

The paper "Shadow Tomography of Quantum States" by Scott Aaronson introduces the novel concept of 'shadow tomography' and tackles the problem of estimating the acceptance probabilities of a set of two-outcome measurements on an unknown quantum mixed state. The objective is to determine the requisite number of copies of the state to achieve this estimate with high probability, within an additive error margin. The research provides a surprising result: the shadow tomography problem can be solved using only $\widetilde{O}(\varepsilon^{-4} \log^4 M \log D)$ copies of the state, where $M$ is the number of measurements, $D$ is the dimension of the state, and $\varepsilon$ the acceptable error margin.

Main Contributions and Claims

The main contribution of the paper is the development of an efficient procedure for shadow tomography, establishing that polynomially many copies are sufficient even with exponentially large sets of measurements and if the state has exponentially large dimension. The procedure can learn the behavior of a quantum state with respect to an exponential number of observables, especially when dealing with noisy measurements, without exhausting an impractically large number of state copies. This capability resolves an open problem related to applications in quantum cryptography, such as private-key quantum money schemes, and impacts other quantum information realms like quantum advice, copy-protected software, and one-way communication complexities.

Technical Aspects and Methodology

The methodology primarily employs innovative quantum information techniques, especially leveraging concepts like postselected learning and gentle measurement. The paper makes essential use of earlier results related to quantum state tomography, such as those by O'Donnell and Wright and Haah et al., to set a foundational framework. The combination of gentle measurement techniques and the complex 'gentle search' procedure reduces the computational burden of tomography when dealing with a high number of observables.

Numerical and Theoretical Implications

The theoretical implications of shadow tomography extend beyond cryptographic applications, challenging traditional views on the quantum measurement problem itself by offering a non-destructive path to gather meaningful statistical behavior of quantum states more resource-efficiently than full tomography. The efficacious sample complexity of the problem, $\widetilde{O}(\varepsilon^{-4} \log^4 M \log D)$, highlights a stark contrast to previous understandings, aligning the feasibility of understanding quantum systems inexorably tied to quantum computation and uncertain quantum landscapes. Additionally, the elegant handling of measurement and error thresholds benefits practical quantum computation efforts where conservation of state copies is crucial.

Future Directions and Open Problems

The paper concludes by suggesting various avenues for further research, particularly regarding optimizing the current bounds on sample complexity, improving computational efficiency (potentially integrating advanced SDP methods as indicated by follow-up research), and seeking practical quantum information applications. Moreover, questions surrounding computational restrictions and assumptions about the measurement matrices present ongoing challenges to extending the implications and usability of the research beyond theoretical constraints.

In summary, "Shadow Tomography of Quantum States" marks a pivotal step in quantum computational research. It provides clarity on the interplay between quantum measurement complexities and computational resource limitations, offering a blend of deep theoretical insights with potential practical breakthroughs essential for the future of quantum information science.