1. Quantum Double Feature Scott Aaronson (MIT) The Learnability of Quantum States Quantum Software Copy-Protection

2. Starting Point For This Work: A Practical Problem In Experimental Physics (!) (well, actually the starting point was whether BQP/qpoly QMA/poly … but lets say it was experimental physics) We have an unknown quantum state (possibly involving many entangled particles) We can reliably produce as many copies of as we want, and measure each copy in a different basis Our goal is to learn an (approximate) classical description of The physicists call this quantum state tomography. There are whole books, conferences, etc. about it

3. But theres a problem… The number of measurements needed grows exponentially with the number of particles (Indeed, even just writing down the state takes exponentially many bits) Even the physicists know this is a problem Current record: Tomography of an 8-qubit state [Häffner et al., Nature, 2005] Required 656,000 measurements, each repeated 100 times So, can a generic state of 10,000 particles never be learned within the lifetime of the universe? (One can hear the QC skeptics crowing: Its just like how we said!)

4. A completely irrelevant detour into quantum coding lower bounds Berkeley. 1999. Ambainis, Nayak, Ta-Shma, and Vazirani want to encode a classical string x 1 …x n into a quantum state | with o(n) qubits, such that by measuring | in an appropriate basis, you can recover any bit x i of your choice They prove this is impossible: quantum random access codes do no better than classical codes Upshot: An n-qubit state has ~2 n degrees of freedom, but only ~n independent and reliably-measurable degrees of freedom

5. All I did: turned Ambainis et al.s qulemon into qulemonade Suppose we have a probability distribution D over two- outcome measurements, and we only care about (approximately) predicting the outcomes of most measurements drawn from D We can do that, with high probability, using a number of sample measurements from D that increases only linearly with the number of qubits

6. Let be an n-qubit mixed state. Let D be a distribution over two-outcome measurements. Suppose we draw m measurements E 1,…,E m independently from D, and then output a hypothesis state such that |Tr(E i )-Tr(E i )| for all i. Then provided /10 and well have with probability at least 1- over E 1,…,E m The Quantum Occams Razor Theorem Result says nothing about the computational complexity of preparing a hypothesis state that agrees with measurement results I can make the dependence and and more reasonable, at the cost of replacing n by n log 2 n

7. Proof Idea Interpret Ambainis et al.s result as proving an O(n) upper bound on the fat-shattering dimension of n-qubit quantum states, considered as a concept class Use results from computational learning theory (e.g. [Bartlett-Long 95]), which say that every concept class has sample complexity linear in its fat-shattering dimension

8. f(x,y) Simple Application to Communication Complexity f: Boolean function mapping Alices N-bit string x and Bobs M-bit string y to a binary output D 1 (f), R 1 (f), Q 1 (f): Deterministic, randomized, and quantum one-way communication complexities of f How much can quantum communication save? In 2004 I showed that for all f (partial or total), D 1 (f)=O(M Q 1 (f)logQ 1 (f)) x y GBUSTERS L AliceBob

9. Theorem: R 1 (f)=O(M Q 1 (f)) for all f, partial or total Proof: Fix Alices input x By Yaos minimax principle, Alice can consider a worst- case distribution D over Bobs input y Alices classical message will consist of y 1,…,y T drawn from D, together with f(x,y 1 ),…,f(x,y T ), where T= (Q 1 (f)) Bob searches for a quantum message that yields the right answers on y 1,…,y T By the learning theorem, with high probability such a yields the right answers on most y drawn from D

10. Another cute application You buy a state | at the quantum software store The vendor says, just feed | to your quantum computer as advice, and itll be deciding f in no time! But you dont trust | to work as expected Theorem: For any distribution D over inputs, theres a small (poly-size) set of test inputs x 1,…,x t, such that if you try | on the test inputs and it works, then whp it will also work on most inputs drawn from D

11. Quantum Double Feature Scott Aaronson (MIT) The Learnability of Quantum States Quantum Software Copy-Protection

12. Classically: Giving someone a program that they can use but not copy is fundamentally impossible (tell that to Sony/BMG…) Quantumly: Well, its called the No-Cloning Theorem for a reason… Question: Given a Boolean function f:{0,1} n {0,1}, can you give your customers a state | f that lets them evaluate f, but doesnt let them prepare more states from which f can be evaluated? Can they use the state more than once? Answer: Sure, without loss of generality Note: Were going to have to make computational assumptions

13. Example where quantum copy-protection seems possible Consider the class of point functions: f s (x)=1 if x=s, f s (x)=0 otherwise Encode s by a permutation such that 2 =e. Choose 1,…, k uniformly at random. Then give your customers the following state: Given any permutation, I claim one can use | to test whether = with error probability 2 -k On the other hand, | doesnt seem useful for preparing additional states with the same property Theorem: This scheme is provably secure under the assumption that it cant be broken. (Assumption is related to, but stronger than, the hardness of the Hidden Subgroup Problem over S n )

14. Example where quantum copy- protection is not possible Let G be a finite group, for which we can efficiently prepare |G (a uniform superposition over the elements) Let H be a subgroup with |H| |G|/polylog|G| Given |H, Watrous showed we can efficiently decide membership in H Check whether |H and |Hx are equal or orthogonal Furthermore: given a program to decide membership in H, we can efficiently prepare |H First prepare |G, then postselect on membership in H Conclusion: Any program to decide membership in H can be pirated But apparently, only by a fully quantum pirate

15. Speculation: Every class of functions can be quantumly copy-protected, except the ones that cant for trivial reasons (i.e., the ones that are quantumly learnable from inputs and outputs) Main Result [A. 2034]: There exists a quantum oracle relative to which this speculation is correct Thus, even if it isnt, we wont be able to prove that by any quantumly relativizing technique Second application of my proof techniques [Mosca-Stebila]: Provably unforgeable quantum money (Provided theres a quantum oracle at the cash register)

16. For each circuit C, choose a meaningless quantum label | C uniformly at random Our quantum oracle will map | C |x |0 to | C |x |C(x) (and also |C |0 to |C | C ) Intuitively, then, having | C is just the same as having a black box for C Goal: Show that if C is not learnable, then | C cant be pirated To prove this, we need to construct a simulator, which takes any quantum algorithm that pirates | C, and converts it into an algorithm that learns C Handwaving Proof Idea

17. Ingredient #1 in the simulator construction: a Complexity-Theoretic No-Cloning Theorem Theorem: Suppose a quantum algorithm is given an n- qubit state |, and can also access a quantum oracle U that recognizes | (i.e., maps | to -| and every | with | =0 to itself). Then the algorithm still needs (2 n/2 ) queries to U to prepare the state | | or anything close to it Note: Contains both the No-Cloning Theorem and the optimality of Grover search as special cases Proof Idea: Generalization of Ambainiss adversary method

18. Ingredient #2: Pseudorandom States Clearly the | p s can be prepared in polynomial time Lemma: If p is chosen uniformly at random, then | p looks like a completely random n-qubit state - Even if we get polynomially many copies of | p - Even if we query the quantum oracle, which depends on | p So the simulator can use | p s in place of | C s where p is a degree-d univariate polynomial over GF(2 n ) for some d=poly(n), and p 0 (x) is the leading bit of p(x)

19. Future Directions Computationally-efficient learning algorithms Efficient algorithm to reconstruct an unknown stabilizer state after O(n) n-qubit measurements: [A., Gottesman], in pre-preparation Experimental implementation! Simulation of pretty-good tomography in MATLAB: [A., Dechter], in progress Quantum copy-protection: get rid of the oracle! DUNCE DUNCE