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How Much Information Is In A Quantum State? Scott Aaronson MIT

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Computer Scientist / Physicist Nonaggression Pact You accept that, for this talk: Polynomial vs. exponential is the basic dichotomy of the universe For all x means for all x In return, I will not inflict the following computational complexity classes on you: #P AM AWPP BQP BQP/qpoly MA NP P/poly PH PostBQP PP PSPACE QCMA QIP QMA SZK YQP

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An infinite amount, of course, if you want to specify the state exactly… Life is too short for infinite precision So, how much information is in a quantum state?

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A More Serious Point In general, a state of n possibly-entangled qubits takes ~2 n bits to specify, even approximately To a computer scientist, this is arguably the central fact about quantum mechanics But why should we worry about it? Spin-½ particles

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Answer 1: Quantum State Tomography Task: Given lots of copies of an unknown quantum state, produce an approximate classical description of Not something I just made up! As seen in Science & Nature Well-known problem: To do tomography on an entangled state of n spins, you need ~c n measurements Current record: 8 spins / ~656,000 experiments (!) This is a conceptual problemnot just a practical one!

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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?

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Today well tame the exponential beast Setting the stage: Holevos Theorem and random access codes Describing a state by postselected measurements [A. 2004] Pretty good tomography using far fewer measurements [A. 2006] - Numerical simulation [A.-Dechter, in progress] Encoding quantum states as ground states of simple Hamiltonians [A.-Drucker 2009] 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

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Theorem [Holevo 1973]: By sending an n-qubit state, Alice can communicate no more than n classical bits to Bob (or 2n bits assuming prior entanglement) How can that be? Well, Bob has to measure, and measuring makes most of the wavefunction go poof… Lesson: The linearity of QM helps tame the exponentiality of QM Alice Bob

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The Absent-Minded Advisor Problem Can you give your graduate student a quantum state with n qubits (or 10n, or n 3, …)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

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Then shell need to send ~c n bits, in the worst case. But… suppose Bob only needs to be able to estimate Tr(E ) for every measurement E in a finite set S. On the Bright Side… Theorem (A. 2004): In that case, it suffices for Alice to send ~n log n log|S| bits Suppose Alice wants to describe an n-qubit state to Bob, well enough that for any 2-outcome measurement E, Bob can estimate Tr(E )

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| ALL MEASUREMENTS ALL MEASUREMENTS PERFORMABLE USING n 2 QUANTUM GATES

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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 his guess, by repeatedly telling him a measurement E t S on which his guess t-1 badly fails Bob lets t be the state obtained by starting from t-1, then performing E t and postselecting on the right outcome I 1 2 3

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Claim: After only O(n) of these learning steps, Bob gets a state T such that Tr(E T ) Tr(E ) for all measurements E S. Proof Sketch: For simplicity, assume =| | is pure and Tr(E ) is 1/n 2 or 1-1/n 2 for all E S. Let p be the probability that E 1,E 2,…,E T all yield the desired outcomes. By assumption, p is at most (say) (2/3) T On the other hand, if Bob had made the lucky guess 0 =| |, then p wouldve been at least (say) 0.9 But we can decompose I as an equal mixture of | and 2 n -1 other states orthogonal to | ! Hence p 0.9/2 n 0.9/2 n (2/3) T T=O(n) Conclusion: Alice can describe to Bob by telling him its behavior on E 1,E 2,…,E T. This takes O(n log|S|) bits

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Weve shown that for any n-qubit state and set S of observables, one can compress the measurement data {Tr(E )} E S into a classical string x of only Õ(nlog|S|) bits Just two tiny problems… 1.Computing x seems astronomically hard 2.Given x, estimating Tr(E ) also seems astronomically hard Ill now state the Quantum Occams Razor Theorem, which at least addresses the first problem… Discussion

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Let be an unknown quantum state of n spins Suppose you just want to be able to estimate Tr(E ) for most measurements E drawn from some probability measure D Then it suffices to do the following, for some m=O(n): 1.Choose E 1,…,E m independently from D 2.Go into your lab and estimate Tr(E i ) for each 1im 3.Find any hypothesis state such that Tr(E i ) Tr(E i ) for all 1im Quantum Occams Razor Theorem

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and with probability at least 1- over the choice of E 1,…,E m. Theorem [A. 2006]: Provided (C a constant) for all i, youll be guaranteed that Quantum states are PAC-learnable Proof combines Ambainis et al.s result on the impossibility of quantum compression with some power tools from classical computational learning theory

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Remark 1: To do this pretty good tomography, you dont need any prior assumptions about ! (No Bayesian nuthin...) Removes a lot of conceptual problems... Instead, you assume a distribution D over measurements Might be preferableafter all, you can control which measurements to apply, but not what is Remark 2: Given the measurement data Tr(E 1 ),…,Tr(E m ), finding a hypothesis state consistent with it could still be an exponentially hard computational problem Semidefinite / convex programming in 2 n dimensions But this seems unavoidable: even finding a classical hypothesis consistent with data is conjectured to be hard!

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Numerical Simulation [A.-Dechter, in progress] We implemented the pretty-good tomography algorithm in MATLAB, using a fast convex programming method developed specifically for this application [Hazan 2008] We then tested it (on simulated data) using MITs computing cluster 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

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Recap: Given an unknown n-qubit entangled 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 holds Proof combines Quantum Occams Razor Theorem with a new classical result about isolatability of function s Caveat: will have more qubits than, and in general be a very different state

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A Practical Implication Its the year 2500. Everyone and her grandfather has a personal quantum computer. Youre a software vendor who sells magic initial states that extend quantum computers problem-solving abilities. Amazingly, you only need one kind of state in your store: ground states of 1D nearest-neighbor Hamiltonians! Reason: Finding ground states of 1D spin systems is known to be universal for quantum constraint satisfaction problems [Aharonov-Gottesman-Irani-Kempe 2007], building on [Kitaev 1999] UNIVERSAL RESOURCE STATE

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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 a 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

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Postselection theorem: quant-ph/0402095 Learning theorem: quant-ph/0608142 Ground state theorem, numerical simulations: in preparation www.scottaaronson.com (/papers /talks /blog)

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