1. How Much Information Is In Entangled Quantum States? Scott Aaronson MIT |

2. The Problem In quantum mechanics, a state of n entangled particles requires at least 2 n complex numbers to specify To a computer scientist, this is probably the central fact about quantum mechanics But why should we worry about it?

3. Answer 1: Quantum State Tomography Task: Given lots of copies of an unknown quantum state, produce an approximate classical description of it Central problem: To do tomography on an entangled state of n particles, you need ~c n measurements Innsbruck group: 8 particles / ~656,000 experiments!

4. 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 LevinGoldreicht HooftDaviesWolfram But is it really an exponential amount?

5. Lets tame the exponential beast Idea: Shrink quantum states down to reasonable size by viewing them operationally Analogy: A probability distribution over n-bit strings also takes ~2 n bits to specify. But that fact seems to be more about the map than the territory Holevos Theorem (1973): By sending an n-qubit quantum state, Alice can transmit no more than n classical bits to Bob This talk: Limitations on the information content of quantum states [A. 2004], [A. 2006], [A. Drucker 2010] Lesson: The linearity of QM helps tame the exponentiality of QM

6. First, where does the exponentiality of quantum states manifest itself? Quantum Proofs and Advice: Let G be a finite (but exponentially-large) group G, and let H G be an exponentially- large subgroup. Then the following highly-entangled state (if you have it) can be used to decide whether a given element x G is in H or not, and to prove statements of the form x H: The trick: Measure the first qubit in [A.-Kuperberg 2007]: For the Group Non- Membership problem, there might also be short classical proofs that are quickly verifiable by a QC But at least relative to a quantum oracle, there are also cases where quantum proofs are provably exponentially more compact than classical proofs Outstanding challenge to show such a separation relative to a classical oracle

7. The Absent-Minded Advisor Problem Can you give your graduate student a quantum state | with ~n qubits, 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

8. 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 |

9. | ALL MEASUREMENTS ALL MEASUREMENTS PERFORMABLE USING n 2 QUANTUM GATES

10. 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 its the maximally mixed state 0 =I Then Alice helps Bob improve, by repeatedly telling him a measurement E t S on which his current guess t-1 badly fails Bob lets t be the state obtained by starting from t-1, then performing E t and postselecting on getting the right outcome I 1 2 3

11. 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 Occams Razor Theorem [A. 2006] at least addresses the first problem…

12. Let | be an unknown entangled 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 Occams Razor Theorem Quantum states are PAC-learnable

13. 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 We studied how the number of sample measurements m needed for accurate predictions scales with the number of qubits n, for n10 Result of experiment: My theorem appears to be true

15. Combining My Postselection and Quantum Learning Results [A.-Drucker 2010]: Let | be an n-qubit state and let T be a complexity bound. Then there exists a local Hamiltonian H on poly(n,T) qubits, such that any 2-outcome measurement on | performable by a circuit of size T can be simulated in poly(n,T) time by a suitable measurement M on the unique ground state of H Application: Trusted quantum advice is equivalent to trusted classical advice + untrusted quantum advice

16. Summary In many natural scenarios, the exponentiality of quantum states is an illusion That is, theres 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 [Aaronson-Gottesman, in preparation] Learnability of stabilizer states