1. PAC-Learning and Reconstruction of Quantum States
Scott Aaronson (University of Texas at Austin)
COLT’2017, Amsterdam

2. What Is A Quantum (Pure) State?
A unit vector of complex numbers
Infinitely many bits in the amplitudes ,—but when you measure, the state “collapses” to just 1 bit!
(0 with probability ||2, 1 with probability ||2)
Holevo’s Theorem (1973): By sending n qubits, Alice can communicate at most n classical bits to Bob (or 2n if they pre-share entanglement)

3. The Problem for CS
A state of n entangled qubits requires 2n complex numbers to specify, even approximately:
To a computer scientist, this is probably the central fact about quantum mechanics
Why should we worry about it?

4. Answer 1: Quantum State Tomography
Task: Given lots of copies of an unknown quantum state, produce an approximate classical description of it
Central problem: To do tomography on a general state of n particles, you clearly need ~cn measurements Innsbruck group (2005): 8 particles / ~656,000 experiments!

5. Answer 2: Quantum Computing Skepticism
Levin
Goldreich
‘t Hooft
Davies
Wolfram
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
But is it really an exponential amount?

6. Can we tame the exponential beast?
Idea: “Shrink quantum states down to reasonable size” by viewing them operationally
Analogy: A probability distribution over n-bit strings also takes ~2n bits to specify. But seeing a sample only provides n bits
In this talk, I’ll survey 14 years of results showing how the standard tools from COLT can be used to upper-bound the “effective size” of quantum states
[A. 2004], [A. 2006], [A.-Drucker 2010], [A. 2017], [A.-Hazan-Nayak 2017]
Lesson: “The linearity of QM helps tame the exponentiality of QM”

7. The Absent-Minded Advisor Problem
|
Can you hand all your grad students the same nO(1)-qubit quantum state |, so that by measuring their copy of | in a suitable basis, each student can learn the {0,1} answer to their n-bit thesis question?
NO [Nayak 1999, Ambainis et al. 1999]
Indeed, quantum communication is no better than classical for this task as n

8. classical bits (trivial bounds: exp(n), |S|)
On the Bright Side…
Suppose Alice wants to describe an n-qubit quantum state | to Bob, well enough that, for any 2-outcome measurement E in some finite set S, Bob can estimate Pr[E(|) accepts] to constant additive error
Theorem (A. 2004): In that case, it suffices for Alice to send Bob only
~n log n  log|S|
classical bits (trivial bounds: exp(n), |S|)

9. | ALL YES/NO MEASUREMENTS PERFORMABLE USING ≤n2 GATES

10. Interlude: Mixed States
DD Hermitian psd matrix with trace 1
Encodes (everything measurable about) a probability distribution where each superposition |i occurs with probability pi
A yes-or-no measurement can be represented by a DD Hermitian psd matrix E with all eigenvalues in [0,1]
The measurement “accepts”  with probability Tr(E), and “rejects”  with probability 1-Tr(E)

11. How does the theorem work?1 3 2 I
Alice is trying to describe the n-qubit state =|| to Bob
In the beginning, Bob knows nothing about , so he guesses it’s the maximally mixed state 0=I (actually I/2n = I/D)
Then Alice helps Bob improve, by repeatedly telling him a measurement EtS on which his current guess t-1 badly fails
Bob lets t be the state obtained by starting from t-1, then performing Et and postselecting on getting the right outcome

12. Question: Why can Bob keep reusing the same state?
To ensure that, we actually need to “amplify” | to |log(n), slightly increasing the number of qubits from n to n log n
Gentle Measurement / Almost As Good As New Lemma:
Suppose a measurement of a mixed state  yields a certain outcome with probability 1-
Then after the measurement, we still have a state ’ that’s -close to  in trace distance

13. Solving, we find that t = O(n log(n))
Crucial Claim: Bob’s iterative learning procedure will “converge” on a state T that behaves like |log(n) on all measurements in the set S, after at most T=O(n log(n)) iterations
Proof: Let pt = Pr[first t postselections all succeed]. Then
Solving, we find that t = O(n log(n))
So it’s enough for Alice to tell Bob about T=O(n log(n)) measurements E1,…,ET, using log|S| bits per measurement
If pt wasn’t less than, say, (2/3)pt-1, learning would’ve ended!
Complexity theory consequence: BQP/qpoly  PostBQP/poly
(Open whether BQP/qpoly=BQP/poly)

14. Quantum Occam’s Razor Theorem “Quantum states are PAC-learnable”
Let | be an unknown entangled state of n particles
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(n):
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”

15. Fat-Shattering Dimension
To prove the theorem, we use Kearns and Schapire’s
Fat-Shattering Dimension
Let C be a class of functions from S to [0,1]. We say a set {x1,…,xk}S is -shattered by C if there exist reals a1,…,ak such that, for all 2k possible statements of the form
f(x1)a1-  f(x2)a2+  …  f(xk)ak-,
there’s some fC that satisfies the statement.
Then fatC(), the -fat-shattering dimension of C, is the size of the largest set -shattered by C.

16. Small Fat-Shattering Dimension Implies Small Sample Complexity
Let C be a class of functions from S to [0,1], and let fC. Suppose we draw m elements x1,…,xm independently from some distribution , and then output a hypothesis hC such that |h(xi)-f(xi)| for all i. Then provided /7 and
we’ll have
with probability at least 1- over x1,…,xm.
Bartlett-Long 1996, building on Alon et al. 1993, building on Blumer et al. 1989

17. Upper-Bounding the Fat-Shattering Dimension of Quantum States
Nayak 1999: If we want to encode k classical bits into n qubits, in such a way that any bit can be recovered with probability 1-p, then we need n(1-H(p))k
Corollary (“turning Nayak’s result on its head”): Let Cn be the set of functions that map an n-qubit measurement E to Tr(E), for some . Then
No need to thank me!
Quantum Occam’s Razor Theorem follows easily…

18. How do we find the hypothesis state?
Here’s one way: let b1,…,bm be the outcomes of measurements E1,…,Em
Then choose a hypothesis state  to minimize
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]

19. Numerical Simulation [A.-Dechter 2008]
We implemented Hazan’s algorithm in MATLAB, and tested it on simulated quantum state data
We studied how the number of sample measurements m needed for accurate predictions scales with the number of qubits n, for n≤10
Result of experiment: My theorem appears to be true…

20. Now tested on real lab data as well. [Rocchetto et al
Now tested on real lab data as well! [Rocchetto et al. 2017, in preparation]

21. Combining My Postselection and Quantum Learning Results
[A.-Drucker 2010]: Given an n-qubit state  and complexity bound T, there exists an efficient measurement V on poly(n,T) qubits, such that any state that “passes” V can be used to efficiently estimate ’s response to any yes-or-no measurement E that’s implementable by a circuit of size T
Application: Trusted quantum advice is equivalent to trusted classical advice + untrusted quantum advice
In complexity terms: BQP/qpoly  QMA/poly
Proof uses boosting-like techniques, together with results on -covers and fat-shattering dimension

22. New Result [A. 2017, in preparation]: “Shadow Tomography”
Theorem: Let  be an unknown D-dimensional state, and let E1,…,EM be known 2-outcome measurements. Suppose we want to know Tr(Ei) to within additive error , for all i[M]. We can achieve this, with high probability, given only k copies of , where
Open Problem: Dependence on D removable?
Challenge: How to measure E1,…,EM without destroying our few copies of  in the process!

23. Proof Idea
Theorem [A. 2006, fixed by Harrow-Montanaro-Lin 2016]:
Let  be an unknown D-dimensional state, and let E1,…,EM be known 2-outcome measurements. Suppose we’re promised that either there exists an i such that Tr(Ei)c, or else Tr(Ei)c- for all i[M]. We can decide which, with high probability, given only k copies of , where
Indeed, can find an i with Tr(Ei)c-, if given O(log4 M / 2) copies of 
Now run my postselected learning algorithm [A. 2004], but using the above to find the Ei’s to postselect on!

24. Online Learning of Quantum States
Answers a question of Elad Hazan (2015)
Corollary of my Postselected Learning Algorithm: Let  be an unknown D-dimensional state. Suppose we’re given 2-outcome measurements E1,E2,… in sequence, each followed by the value of Tr(Et) to within /2. Each time, we’re challenged to guess a hypothesis state  such that
A.-Hazan-Nayak 2017: Can also use general convex optimization tools, or an upper bound on “sequential fat-shattering dimension” of quantum states, to get a regret bound for the agnostic case, now involving a t term
We can do this so that we fail at most k times, where

25. Summary
The tools of COLT let us show that in many scenarios, the “exponentiality” of quantum states is more bark than bite
I.e. there’s a short classical string that specifies how the quantum state behaves, on any 2-outcome measurement you could actually perform
Applications from complexity theory to experimental quantum state tomography…
Alas, these results don’t generalize to many-outcome measurements, or to learning quantum processes
Biggest future challenge: Find subclasses of states and measurements for which these learning procedures are computationally efficient (Stabilizer states: Rocchetto 2017)