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Quantum Software Copy-Protection Scott Aaronson (MIT) |

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Many people have a legitimate interest in keeping their intellectual property from being copied… But if quantum mechanics isnt physics in the usual senseif its not about matter, or energy, or waves then what is it about? Well, from my perspective, its about information, probabilities, and observables, and how they relate to each other.

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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: Certainly, if they buy poly(n) copies of it Note: Were going to have to make computational assumptions

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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 )

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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 one can efficiently decide membership in H Given an element x G, check whether H|Hx is 0 or 1 Furthermore: given a program to decide membership in H, one 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

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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)

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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

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Ingredient #1 in the simulator construction: 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., U| = -| and U| = | for all | =0). Then the algorithm still needs ~2 n/2 queries to U to prepare any state having non-negligible overlap with | | Observation: Contains both the No-Cloning Theorem and the optimality of Grover search as special cases! Proof Idea: A new generalization of Ambainiss quantum adversary method, to the case where the starting state already has some information about the answer

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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)

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Future Directions Get rid of the oracle! Clarify the relationship between copy-protection and obfuscation The constant error regime: what is information-theoretically possible? DUNCE DUNCE

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

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

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