scholarly journals Stabiliser states are efficiently PAC-learnable

2018 ◽  
Vol 18 (7&8) ◽  
pp. 541-552
Author(s):  
Andrea Rocchetto
Keyword(s):  
Wave Function ◽  
Fundamental Property ◽  
Computational Learning Theory ◽  
Quantum Systems ◽  
Quantum States ◽  
Computational Learning ◽  
Sample Complexity ◽  
Classical Simulation ◽  
Probably Approximately Correct ◽  
State Tomography

The exponential scaling of the wave function is a fundamental property of quantum systems with far reaching implications in our ability to process quantum information. A problem where these are particularly relevant is quantum state tomography. State tomography, whose objective is to obtain an approximate description of a quantum system, can be analysed in the framework of computational learning theory. In this model, Aaronson (2007) showed that quantum states are Probably Approximately Correct (PAC)-learnable with sample complexity linear in the number of qubits. However, it is conjectured that in general quantum states require an exponential amount of computation to be learned. Here, using results from the literature on the efficient classical simulation of quantum systems, we show that stabiliser states are efficiently PAC-learnable. Our results solve an open problem formulated by Aaronson (2007) and establish a connection between classical simulation of quantum systems and efficient learnability.

scholarly journals The learnability of quantum states

2007 ◽  
Vol 463 (2088) ◽  
pp. 3089-3114 ◽  
Author(s):  
Scott Aaronson
Keyword(s):  
Quantum Computing ◽  
Probability Distribution ◽  
Quantum State ◽  
Learning Theory ◽  
Communication Protocols ◽  
Computational Learning Theory ◽  
Quantum States ◽  
Computational Learning ◽  
State Tomography ◽  
Long Vectors

Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n . But using ideas from computational learning theory, we show that one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with n . This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice.

scholarly journals Experimental learning of quantum states

2019 ◽  
Vol 5 (3) ◽  
pp. eaau1946 ◽  
Author(s):  
Andrea Rocchetto ◽  
Scott Aaronson ◽  
Simone Severini ◽  
Gonzalo Carvacho ◽  
Davide Poderini ◽  
...  
Keyword(s):  
Learning Theory ◽  
Computational Learning Theory ◽  
Quantum States ◽  
Optical Systems ◽  
Scaling Limits ◽  
Linear Scaling ◽  
Computational Learning ◽  
Experimental Demonstration ◽  
Experimental Learning ◽  
Number Of Particles

The number of parameters describing a quantum state is well known to grow exponentially with the number of particles. This scaling limits our ability to characterize and simulate the evolution of arbitrary states to systems, with no more than a few qubits. However, from a computational learning theory perspective, it can be shown that quantum states can be approximately learned using a number of measurements growing linearly with the number of qubits. Here, we experimentally demonstrate this linear scaling in optical systems with up to 6 qubits. Our results highlight the power of the computational learning theory to investigate quantum information, provide the first experimental demonstration that quantum states can be “probably approximately learned” with access to a number of copies of the state that scales linearly with the number of qubits, and pave the way to probing quantum states at new, larger scales.

scholarly journals Multi-Bloch vector representation of the qutrit

2011 ◽  
Vol 11 (5&6) ◽  
pp. 361-373
Author(s):  
Pawel Kurzynski
Keyword(s):  
Quantum State ◽  
Quantum Systems ◽  
Quantum States ◽  
The Other ◽  
Two Dimensional ◽  
Bloch Vector ◽  
Vector Representation ◽  
Alternative Approach ◽  
Complete Set

An ability to describe quantum states directly by average values of measurement outcomes is provided by the Bloch vector. For an informationally complete set of measurements one can construct unique Bloch vector for any quantum state. However, not every Bloch vector corresponds to a quantum state. It seems that only for two-dimensional quantum systems it is easy to distinguish proper Bloch vectors from improper ones, i.e. the ones corresponding to quantum states from the other ones. I propose an alternative approach to the problem in which more than one vector is used. In particular, I show that a state of the qutrit can be described by the three qubit-like Bloch vectors.

scholarly journals Freezing dynamics of entanglement and nonlocality for qutrit-qutrit (3 ⊗ 3) quantum systems

2019 ◽  
Vol 34 (13) ◽  
pp. 1950102 ◽  
Author(s):  
Mazhar Ali
Keyword(s):  
Finite Time ◽  
Quantum Systems ◽  
Quantum States ◽  
Multipartite Quantum Systems ◽  
Combined Action ◽  
Time Invariant ◽  
Decoherence Free Subspaces

We examine the possibilities of nontrivial phenomena of time-invariant entanglement and freezing dynamics of entanglement for qutrit-qutrit quantum systems. We find no evidence for time-invariant entanglement, however, we do observe that certain quantum states freeze their entanglement after decaying for some time. It is interesting that quantum states are changing whereas their entanglement remains constant. We find that the combined action of decoherence free subspaces and subspaces where quantum states decay, facilitate this phenomenon. This study is an extension of similar phenomena observed for qubit-qubit systems, qubit-qutrit, and multipartite quantum systems. We examine nonlocality of a specific family of states and find the certain instances where the states still remain entangled, however, they can either loose their nonlocality at a finite time or remain nonlocal for all times.

scholarly journals Quantum state space dimension as a quantum resource

2015 ◽  
Vol 13 (06) ◽  
pp. 1550039 ◽  
Author(s):  
A. Plastino ◽  
G. Bellomo ◽  
A. R. Plastino
Keyword(s):  
Quantum Information ◽  
State Space ◽  
Quantum State ◽  
Space Dimension ◽  
Quantum Systems ◽  
Quantum States ◽  
Information Tasks

We argue that the dimensionality of the space of quantum systems’ states should be considered as a legitimate resource for quantum information tasks. The assertion is supported by the fact that quantum states with discord-like capacities can be obtained from classically-correlated states in spaces of dimension large enough. We illustrate things with some simple examples that justify our claim.

Sign in / Sign up

Export Citation Format

Share Document