Scott Aaronson ( UT Austin ) Joint work with Xinyi Chen, Elad Hazan, Satyen Kale, and Ashwin Nayak

1. Online Learning of Quantum States  Scott Aaronson (UT Austin) Joint work with Xinyi Chen, Elad Hazan, Satyen Kale, and Ashwin Nayak arXiv:1802.09025 / NeurIPS 2018

2. An n-qubit pure state requires 2n complex numbers to specify, even approximately: Yet measuring yields at most n bits (Holevo’s Theorem) So should we say that the 2n complex numbers are “really there” in a single copy of |—or “just in our heads”? A probability distribution over n-bit strings also involves 2n real numbers. But probabilities don’t interfere!

3. Quantum State Tomography Task: Given lots of copies of an unknown D-dimensional quantum mixed state , produce an approximate classical description of  O’Donnell and Wright and Haah et al., STOC’2016: ~(D2) copies of  are necessary and sufficient for this Experimental Record (Song et al. 2017): 10 qubits, millions of measurement settings!Keep in mind: D = 2n

4. Quantum Occam’s Razor Theorem (A. 2006) • Let be an unknown D-dimensional state • Suppose you just want to be able to estimate the acceptance probabilities of most measurements E drawn from some probability distribution • Then it suffices to do the following, for some m=O(log D): • Choose m measurements independently from  • Go into your lab and estimate acceptance probabilities of all of them on  • Find any “hypothesis state” approximately consistent with all measurement outcomes “Quantum states are PAC-learnable”

5. Can prove by combining two facts: (1) The class of [0,1]-valued functions of the form f(E)=Tr(E), where  is a D-dimensional mixed state, has -fat-shattering dimension O((log D)/2) Largest k for which we can find inputs x1,…,xk, and values a1,…,ak[0,1], such that all 2k possible behaviors involving f(xi) exceeding ai by  or vice versa are realized by some f in the class (2) Any class of [0,1]-valued functions is PAC-learnable using a number of samples linear in its fat-shattering dimension  [Alon et al., Bartlett-Long]

6. To upper-bound the fat-shattering dimension of quantum states: Use the lower bound for quantum random access codes [Nayak 1999]. Namely: You need (n) qubits to encode n bits x1,…,xn into a state , so that any xi of your choice can later be recovered w.h.p. by measuring  Then turn this lemon into lemonade!

7. Here’s one way: let b1,…,bm be the outcomes of measurements E1,…,Em Then choose a hypothesis state  to minimize How do we find the hypothesis state? This is a convex programming problem, which can be solved in time polynomial in D=2n (good enough in practice for n15 or so) Optimized, linear-time iterative method for this problem: [Hazan 2008]

8. You know, PAC-learning is so 1990s. Who wants to assume a fixed, unchanging distribution  over the sample data? These days all the cool kids prove theorems about online learning. What’s online learning?

9. Online Learning Two-outcome measurements E1,E2,… on a D-dimensional state arrive one by one—chosen by an adaptive adversary No more fixed distribution over measurements, no independence, no nothin’! For each Et, you—the learner—are challenged to guess Tr(Et). If you’re off by more than /3, you’re then told the true value, or at least an /3-approximation to it Goal: Give a learning strategy that upper bounds the total number of times your guess will ever be more than  off from the truth (can’t know in advance which times those will be…)

10. Postselected learning Sequential fat-shattering dimension Online Convex Optimization / Matrix Multiplicative Weights

11. Theorem 1: There’s an explicit strategy, for online learning of an n-qubit quantum state, that’s wrong by > at most O(n/2) times (and this is tight) Theorem 2: Even if the data the adversary gives you isn’t consistent with any n-qubit quantum state, there’s still an explicit strategy such that your total regret, after T iterations, is at most Tight for L1, possibly not for L2 Regret = (Your total error) – (Total error if you’d started with the best hypothesis state from the very beginning) Error can be either L1 or L2

12. My Way: Postselected Learning 2 3 1 I/2n In the beginning, the learner knows nothing about , so he guesses it’s the maximally mixed state0= I/2n Each time the learner encounters a measurement Et on which his current hypothesis t-1 badly fails, he tries to improve—by letting t be the state obtained by starting from t-1, then performing Et and postselecting on getting the right outcome Amplification + Gentle Measurement Lemma are used to bound the damage caused by these measurements

13. Let  = const for simplicity Crucial Claim: The iterative learning procedure must converge to log(n), after at most T=O(n log(n)) serious mistakes Proof: Let pt = Pr[first t postselections all succeed]. Then If pt wasn’t less than, say, (2/3)pt-1, learning would’ve ended! Solving, we find that t = O(n log(n))

14. Ashwin’s Way: Sequential Fat-Shattering Dimension New generalization of Ashwin’s random access codes lower bound from 1999: You can store at most m=O(n) bits x=(x1,…,xm) in an n-qubit state x, in such a way that any one xi of your choice can later be read out w.h.p. by measuring x—even if the measurement basis is allowed to depend on x1,…,xi-1 Implies an O(n/2) upper bound for the “online / sequential” version of -fat-shattering dimension Combined with a general result of [Rakhlin et al. 2015], automatically implies online learning alg + regret bound

15. Elad, Xinyi, Satyen’s Way:Online Convex Optimization Regularized Follow-the-Leader [Hazan 2015] Gradient descent using von Neumann entropy Matrix Multiplicative Weights [Arora and Kale 2016] Technical work: Generalize power tools that already existed for real matrices to complex Hermitian ones Yields optimal mistake bound and regret bound

16. Application: Shadow Tomography The Task (A. 2016): Let  be an unknown D-dimensional mixed state. Let E1,…,EM be known 2-outcome POVMs. Estimate Pr[Ei accepts ] to within  for all i[M]—the “shadows” that  casts on E1,…,EM—with high probability, by measuring as few copies of  as possible Clearly k=O(D2) copies suffice (do ordinary tomography) Clearly k=O(M) suffice (apply each Ei to separate copies) But what if we wanted to know, e.g., the behavior of an n-qubit state on all accept/reject circuits with n2 gates? Could we have

17. Theorem (A., STOC’2018): It’s possible to do Shadow Tomography using only copies Proof (in retrospect): Just combine two ingredients! • The “Quantum OR Bound” [A. 2006, corrected by Harrow-Montanaro-Lin 2017], to repeatedly search for a measurement Ei such that Tr(Eit) is far from Tr(Ei), without much damaging the ’s. Here t is our current hypothesis state (initially, 0 = maximally mixed) • Our online learning theorem, to upper-bound the number of updates to t until we converge on a hypothesis state Ts.t.Tr(EiT)Tr(Ei) for everyi[M]

18. New Shadow Tomography Protocol![A.-Rothblum 2019, coming any day to an arXiv near you] Exploits a new connection that Guy and I discovered between gentle measurement of quantum states, and differential privacy (an area of classical CS) Based on the “Private Multiplicative Weights” algorithm [Hardt-Rothblum 2010] Uses this many copies of : while also being online and gentle (properties we didn’t have before) Explicitly uses online learning theorem as a key ingredient

19. Open Problems • Generalize to k-outcome measurements • Optimal regret for L2 loss • What special cases of online learning of quantum states can be done with only poly(n) computation? • What’s the true sample complexity of shadow tomography?