135 research outputs found
Shadow Tomography of Quantum States
We introduce the problem of *shadow tomography*: given an unknown
-dimensional quantum mixed state , as well as known two-outcome
measurements , estimate the probability that
accepts , to within additive error , for each of the
measurements. How many copies of are needed to achieve this, with high
probability? Surprisingly, we give a procedure that solves the problem by
measuring only copies. This means, for example, that we can learn the behavior of an
arbitrary -qubit state, on all accepting/rejecting circuits of some fixed
polynomial size, by measuring only 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.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018,
revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and
lower bounds with explicit dependence on the dimension
Hamiltonian-Driven Shadow Tomography of Quantum States
Classical shadow tomography provides an efficient method for predicting
functions of an unknown quantum state from a few measurements of the state. It
relies on a unitary channel that efficiently scrambles the quantum information
of the state to the measurement basis. Facing the challenge of realizing deep
unitary circuits on near-term quantum devices, we explore the scenario in which
the unitary channel can be shallow and is generated by a quantum chaotic
Hamiltonian via time evolution. We provide an unbiased estimator of the density
matrix for all ranges of the evolution time. We analyze the sample complexity
of the Hamiltonian-driven shadow tomography. For Pauli observables, we find
that it can be more efficient than the unitary-2-design-based shadow tomography
in a sequence of intermediate time windows that range from an order-1
scrambling time to a time scale of , given the Hilbert space dimension
. In particular, the efficiency of predicting diagonal Pauli observables is
improved by a factor of without sacrificing the efficiency of predicting
off-diagonal Pauli observables.Comment: 4+epsilon pages, 2 figures, with appendix. Add detailed discussion
and numerical evidence in the new version. Add and modify some reference
Collective randomized measurements in quantum information processing
The concept of randomized measurements on individual particles has proven to
be useful for analyzing quantum systems and is central for methods like shadow
tomography of quantum states. We introduce randomized
measurements as a tool in quantum information processing. Our idea is to
perform measurements of collective angular momentum on a quantum system and
actively rotate the directions using simultaneous multilateral unitaries. Based
on the moments of the resulting probability distribution, we propose systematic
approaches to characterize quantum entanglement in a
collective-reference-frame-independent manner. First, we show that existing
spin-squeezing inequalities can be accessible in this scenario. Next, we
present an entanglement criterion based on three-body correlations, going
beyond spin-squeezing inequalities with two-body correlations. Finally, we
apply our method to characterize entanglement between spatially-separated two
ensembles.Comment: 18 pages, 5 figure
Observing Schr\"odinger's Cat with Artificial Intelligence: Emergent Classicality from Information Bottleneck
We train a generative language model on the randomized local measurement data
collected from Schr\"odinger's cat quantum state. We demonstrate that the
classical reality emerges in the language model due to the information
bottleneck: although our training data contains the full quantum information
about Schr\"odinger's cat, a weak language model can only learn to capture the
classical reality of the cat from the data. We identify the quantum-classical
boundary in terms of both the size of the quantum system and the information
processing power of the classical intelligent agent, which indicates that a
stronger agent can realize more quantum nature in the environmental noise
surrounding the quantum system. Our approach opens up a new avenue for using
the big data generated on noisy intermediate-scale quantum (NISQ) devices to
train generative models for representation learning of quantum operators, which
might be a step toward our ultimate goal of creating an artificial intelligence
quantum physicist.Comment: 17 pages, 9 figure
Improvements in Quantum SDP-Solving with Applications
Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case w
Online Learning of Quantum States
Suppose we have many copies of an unknown -qubit state . We measure
some copies of using a known two-outcome measurement , then other
copies using a measurement , and so on. At each stage , we generate a
current hypothesis about the state , using the outcomes of
the previous measurements. We show that it is possible to do this in a way that
guarantees that , the error in our prediction for the next
measurement, is at least at most times. Even in the "non-realizable" setting---where
there could be arbitrary noise in the measurement outcomes---we show how to
output hypothesis states that do significantly worse than the best possible
states at most times on the first
measurements. These results generalize a 2007 theorem by Aaronson on the
PAC-learnability of quantum states, to the online and regret-minimization
settings. We give three different ways to prove our results---using convex
optimization, quantum postselection, and sequential fat-shattering
dimension---which have different advantages in terms of parameters and
portability.Comment: 18 page
Fewer measurements from shadow tomography with -representability conditions
Classical shadow tomography provides a randomized scheme for approximating
the quantum state and its properties at reduced computational cost with
applications in quantum computing. In this Letter we present an algorithm for
realizing fewer measurements in the shadow tomography of many-body systems by
imposing -representability constraints. Accelerated tomography of the
two-body reduced density matrix (2-RDM) is achieved by combining classical
shadows with necessary constraints for the 2-RDM to represent an -body
system, known as -representability conditions. We compute the ground-state
energies and 2-RDMs of hydrogen chains and the N dissociation curve.
Results demonstrate a significant reduction in the number of measurements with
important applications to quantum many-body simulations on near-term quantum
devices
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