Quantum Double Feature The Learnability of Quantum States

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

Starting Point For This Work: A Practical Problem In Experimental Physics (!) (well, actually the starting point was whether BQP/qpoly QMA/poly … but let’s 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

But there’s 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: “It’s just like how we said!”)

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…xn into a quantum state | with o(n) qubits, such that by measuring | in an appropriate basis, you can recover any bit xi 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

All I did: turned Ambainis et al. ’s qulemon into qulemonade Suppose we have a probability distribution D over twooutcome 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

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

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 fatshattering dimension

Simple Application to Communication Complexity x y Alice Bob f(x, y) f: Boolean function mapping Alice’s N-bit string x and Bob’s 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? TERS D (f)=O(M Q L (f)log. Q (f)) US GB In 2004 I showed that for all f (partial or total), 1 1 1

Theorem: R 1(f)=O(M Q 1(f)) for all f, partial or total Proof: Fix Alice’s input x By Yao’s minimax principle, Alice can consider a worstcase distribution D over Bob’s input y Alice’s 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

Another cute application You buy a state | at the quantum software store The vendor says, “just feed | to your quantum computer as advice, and it’ll be deciding f in no time!” But you don’t trust | to work as expected Theorem: For any distribution D over inputs, there’s a small (poly-size) set of “test inputs” x 1, …, xt, such that if you try | on the test inputs and it works, then whp it will also work on most inputs drawn from D

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

Classically: Giving someone a program that they can use but not copy is fundamentally impossible (tell that to Sony/BMG…) Quantumly: Well, it’s 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 doesn’t 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: We’re going to have to make computational assumptions

Example where quantum copy-protection seems possible Consider the class of point functions: fs(x)=1 if x=s, fs(x)=0 otherwise. This scheme is provably secure Theorem: under the assumption that 2=e. broken. Encode s by a permutation such it can’t be Choose 1, …, k uniformly at random. Then give your customers (Assumption is the following state: related to, but stronger than, the hardness of the Hidden Subgroup Problem over Sn) Given any permutation ’, I claim one can use | to test whether ’= with error probability 2 -k On the other hand, | doesn’t seem useful for preparing additional states with the same property

Example where quantum copyprotection 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”

Speculation: Every class of functions can be quantumly copy-protected, except the ones that can’t 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 isn’t, we won’t be able to prove that by any “quantumly relativizing technique” Second application of my proof techniques [Mosca-Stebila]: Provably unforgeable quantum money (Provided there’s a quantum oracle at the cash register)

Handwaving Proof Idea 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 can’t 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

Ingredient #1 in the simulator construction: a “Complexity-Theoretic No-Cloning Theorem” Theorem: Suppose a quantum algorithm is given an nqubit 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 Ambainis’s adversary method

Ingredient #2: Pseudorandom States 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) 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

D U N C E Future Directions D U N C E 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!