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

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

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?

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

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!

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?

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

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

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

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 )

| ALL MEASUREMENTS ALL MEASUREMENTS PERFORMABLE USING n 2 QUANTUM GATES

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

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

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

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

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

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!

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

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

A Practical Implication Its the year 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

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

Postselection theorem: quant-ph/ Learning theorem: quant-ph/ Ground state theorem, numerical simulations: in preparation (/papers /talks /blog)