3,775 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
Shadow process tomography of quantum channels
Quantum process tomography is a critical capability for building quantum
computers, enabling quantum networks, and understanding quantum sensors. Like
quantum state tomography, the process tomography of an arbitrary quantum
channel requires a number of measurements that scale exponentially in the
number of quantum bits affected. However, the recent field of shadow
tomography, applied to quantum states, has demonstrated the ability to extract
key information about a state with only polynomially many measurements. In this
work, we apply the concepts of shadow state tomography to the challenge of
characterizing quantum processes. We make use of the Choi isomorphism to
directly apply rigorous bounds from shadow state tomography to shadow process
tomography, and we find additional bounds on the number of measurements that
are unique to process tomography. Our results, which include algorithms for
implementing shadow process tomography enable new techniques including
evaluation of channel concatenation and the application of channels to shadows
of quantum states. This provides a dramatic improvement for understanding
large-scale quantum systems.Comment: 12 pages, 5 figures; Added citation to similar work; Errors
corrected. Previous statements of main result first missed and then
miscalculated an exponential cost in system size; Version accepted for
publicatio
Learning Distributions over Quantum Measurement Outcomes
Shadow tomography for quantum states provides a sample efficient approach for
predicting the properties of quantum systems when the properties are restricted
to expectation values of -outcome POVMs. However, these shadow tomography
procedures yield poor bounds if there are more than 2 outcomes per measurement.
In this paper, we consider a general problem of learning properties from
unknown quantum states: given an unknown -dimensional quantum state
and unknown quantum measurements with
outcomes, estimating the probability distribution for applying
on to within total variation distance .
Compared to the special case when , we need to learn unknown distributions
instead of values. We develop an online shadow tomography procedure that solves
this problem with high success probability requiring copies of . We further prove an information-theoretic
lower bound that at least copies of
are required to solve this problem with high success probability. Our
shadow tomography procedure requires sample complexity with only logarithmic
dependence on and and is sample-optimal for the dependence on .Comment: 25 page
Neural-Shadow Quantum State Tomography
Quantum state tomography (QST) is the art of reconstructing an unknown
quantum state through measurements. It is a key primitive for developing
quantum technologies. Neural network quantum state tomography (NNQST), which
aims to reconstruct the quantum state via a neural network ansatz, is often
implemented via a basis-dependent cross-entropy loss function. State-of-the-art
implementations of NNQST are often restricted to characterizing a particular
subclass of states, to avoid an exponential growth in the number of required
measurement settings. To provide a more broadly applicable method for efficient
state reconstruction, we present "neural-shadow quantum state tomography"
(NSQST)-an alternative neural network-based QST protocol that uses infidelity
as the loss function. The infidelity is estimated using the classical shadows
of the target state. Infidelity is a natural choice for training loss,
benefiting from the proven measurement sample efficiency of the classical
shadow formalism. Furthermore, NSQST is robust against various types of noise
without any error mitigation. We numerically demonstrate the advantage of NSQST
over NNQST at learning the relative phases of three target quantum states of
practical interest. NSQST greatly extends the practical reach of NNQST and
provides a novel route to effective quantum state tomography
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
Provable learning of quantum states with graphical models
The complete learning of an -qubit quantum state requires samples
exponentially in . Several works consider subclasses of quantum states that
can be learned in polynomial sample complexity such as stabilizer states or
high-temperature Gibbs states. Other works consider a weaker sense of learning,
such as PAC learning and shadow tomography. In this work, we consider learning
states that are close to neural network quantum states, which can efficiently
be represented by a graphical model called restricted Boltzmann machines
(RBMs). To this end, we exhibit robustness results for efficient provable
two-hop neighborhood learning algorithms for ferromagnetic and locally
consistent RBMs. We consider the -norm as a measure of closeness,
including both total variation distance and max-norm distance in the limit. Our
results allow certain quantum states to be learned with a sample complexity
\textit{exponentially} better than naive tomography. We hence provide new
classes of efficiently learnable quantum states and apply new strategies to
learn them
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