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How Much Information Is In A Quantum State? Scott Aaronson MIT |
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Page 1: How Much Information Is In A Quantum State? Scott Aaronson MIT |

How Much Information Is In A Quantum State?

Scott AaronsonMIT

|

Page 2: How Much Information Is In A Quantum State? Scott Aaronson MIT |

The Problem

In quantum mechanics, a state of n entangled particles

requires at least 2n complex numbers to specify

nxx x

1,0

To a computer scientist, this is probably the central fact about quantum mechanics

But why should we worry about it?

Page 3: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Answer 1: Quantum State Tomography

Task: Given lots of copies of an unknown quantum state, produce an approximate classical description of it

Not something I made up!“As seen in Science & Nature”

Central problem: To do tomography on an entangled state of n particles, you need ~cn measurements

Current record: 8 particles / ~656,000 experiments (!)

Page 4: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Answer 2: Quantum Computing Skepticism

Some physicists and computer scientists believe quantum computers will be impossible for a fundamental reason

For many of them, the problem is that a quantum computer would “manipulate an exponential amount of information” using only polynomial resources

Levin Goldreich ‘t Hooft Davies Wolfram

But is it really an exponential amount?

Page 5: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Today we’ll tame the exponential beastIdea: “Shrink quantum states down to reasonable

size” by viewing them operationally

Analogy: A probability distribution over n-bit strings also takes ~2n bits to specify. But that fact seems to be “more about the map than the territory”

Holevo’s Theorem (1973): By sending an n-qubit quantum state, Alice can transmit no more than n classical bits to Bob

This talk: New limitations on the information content of quantum states [A. 2004], [A. 2006], [A.-Dechter 2008], [A. Drucker 2009]Lesson: “The linearity of QM helps

tame the exponentiality of QM”

Page 6: How Much Information Is In A Quantum State? Scott Aaronson MIT |

The Absent-Minded Advisor Problem

Can you give your graduate student a quantum state with n qubits (or 10n, or n3, …)—such that by measuring in a suitable basis, the student can learn your answer to any one yes-or-no question of size n?

NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999]Indeed, quantum communication is no better than classical for this problem as n

Page 7: How Much Information Is In A Quantum State? Scott Aaronson MIT |

On the Bright Side…

Theorem (A. 2004): In that case, it suffices for Alice to send Bob only

~n log n log|S|classical bits

Suppose Alice wants to describe an n-qubit quantum state | to Bob … well enough that, for any 2-outcome measurement M in some finite set S, Bob can estimate the probability that M accepts |

Page 8: How Much Information Is In A Quantum State? Scott Aaronson MIT |

|ALL MEASUREMENTSALL MEASUREMENTS PERFORMABLE

USING ≤n2 QUANTUM GATES

Page 9: How Much Information Is In A Quantum State? Scott Aaronson MIT |

How does the theorem work?

Alice is trying to describe the quantum state to Bob

In the beginning, Bob knows nothing about , so he guesses it’s the maximally mixed state 0=I

Then Alice helps Bob improve, by repeatedly telling him a measurement EtS on which his current guess t-1 badly fails

Bob lets t be the state obtained by starting from t-1, then performing Et and postselecting on getting the right outcome

I123

Page 10: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Just two tiny problems with this compression theorem…

1.Computing the classical “compressed representation” of quantum state seems astronomically hard

2.Given the compressed representation, computing the probability some measurement on the state accepts also seems astronomically hard

The “Quantum Occam’s Razor Theorem” [A. 2006] at least addresses the first problem…

Page 11: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Let | be an unknown quantum state of n particles

Suppose you just want to be able to estimate the acceptance probabilities of most measurements E drawn from some probability distribution D

Then it suffices to do the following, for some m=O(n):1.Choose m measurements independently from D

2.Go into your lab and estimate acceptance probabilities of all of them on |

3.Find any “hypothesis state” approximately consistent with all measurement outcomes

Quantum Occam’s Razor Theorem

“Quantum states are PAC-learnable”

Page 12: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Numerical Simulation[A.-Dechter 2008]

We implemented this “pretty-good quantum state tomography” method in MATLAB, using a fast convex programming method developed specifically for this application [Hazan 2008]

We then tested it (on simulated data) using an MIT cluster

We studied how the number of sample measurements m needed for accurate predictions scales with the number of qubits n, for n≤10

Result of experiment: My theorem appears to be true

Page 13: How Much Information Is In A Quantum State? Scott Aaronson MIT |
Page 14: How Much Information Is In A Quantum State? Scott Aaronson MIT |

Recap: Given an unknown n-qubit quantum state , and a set S of two-outcome measurements…

Learning theorem: “Any hypothesis state consistent with a small number of sample points behaves like on most measurements in S”

Postselection theorem: “A particular state T (produced by postselection) behaves like on all measurements in S”

Dream theorem: “Any state that passes a small number of tests behaves like on all measurements in S”

[A.-Drucker 2009]: The dream theorem holdsCaveat: will have more qubits than , and in general be a very different state

Implication: Can get a quantum state that satisfies exp(n) conditions by imposing only poly(n) constraints

Page 15: How Much Information Is In A Quantum State? Scott Aaronson MIT |

SummaryIn many natural scenarios, the “exponentiality” of quantum states is an illusion

That is, there’s a short (though possibly cryptic) classical string that specifies how the quantum state behaves, on any measurement you could actually perform

Applications: Pretty-good quantum state tomography, characterization of quantum computers with “magic initial states”…

Biggest open problem: Find special classes of quantum states that can be learned in a computationally efficient way

“Experimental demonstration” would be nice too