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The Learnability of Quantum States Scott Aaronson University of Waterloo
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The Learnability of Quantum States Scott Aaronson University of Waterloo.

Mar 26, 2015

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Page 1: The Learnability of Quantum States Scott Aaronson University of Waterloo.

The Learnability of Quantum States

Scott Aaronson

University of Waterloo

Page 2: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Quantum State TomographySuppose we have a physical process that produces a quantum mixed state

By applying the process repeatedly, we can prepare as many copies of as we want

To each copy, we then apply a binary measurement E, obtaining ‘1’ with probability Tr(E) and ‘0’ otherwise

Our goal is to learn an approximate description of

EXPERIMENTALISTS ACTUALLY DO THIS

To learn about chemical reactions (Skovsen et al. 2003), test equipment (D’Ariano et al. 2002), study decoherence mechanisms (Resch et al. 2005), …

Page 3: The Learnability of Quantum States Scott Aaronson University of Waterloo.

But there’s a problem…

To do tomography on an entangled state of n qubits, we need (4n) measurements

The current record: 8 qubits (Häffner et al. 2005), requiring 656,100 experiments (!)

Does this mean that a generic 10,000-particle state can never be “learned” within the lifetime of the universe?

If so, would call into question the operational status of quantum states themselves (and make quantum computing skeptics extremely happy)…

Fear not! Why would he be raising this

“problem” if he wasn’t gonna demolish it?

Page 4: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Let be an n-qubit mixed state. Let D be a distribution over two-outcome measurements. Suppose we draw m measurements E1,…,Em independently from D, and then output a “hypothesis state” such that|Tr(Ei)-Tr(Ei)|≤ for all i. Then provided /10 and

we’ll have

with probability at least 1- over E1,…,Em

The Quantum Occam’s Razor Theorem

,1

log1

log1

2222

n

m

1TrTrPr EEDE

Page 5: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Remarks

Can make dependence and and more reasonable, at the cost of a log2n factor:

1

loglog1 2

2

nnOm

1

log1

2

nmThe above bound is nearly tight:

Result says nothing about the computational complexity of preparing a hypothesis state that agrees with measurement results

Implies that we can do “pretty good tomography,” using a number of measurements that grows only linearly (!) with the number of qubits n

Page 6: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Fat-Shattering DimensionLet 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 fC that satisfies the statement.

To prove the theorem, we need a notion introduced by Kearns and Schapire called

Then fatC(), the -fat-shattering dimension of C, is the size of the largest set -shattered by C.

Page 7: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Let C be a class of functions from S to [0,1], and let fC. Suppose we draw m elements x1,…,xm independently from some distribution D, and then output a hypothesis hC such that |h(xi)-f(xi)| for all i. Then provided /7 and

we’ll have

with probability at least 1- over x1,…,xm.

Small Fat-Shattering Dimension Implies Small Sample Complexity

1Pr xfxhDx

,1

log1

log35

fat1 222

Cm

Proof uses a 1996 result of Bartlett and Long—building on Alon et al., building on Blumer et al., building on Valiant

Page 8: The Learnability of Quantum States Scott Aaronson University of Waterloo.

.fat2

n

OnC

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

Quantum Occam’s Razor Theorem follows easily…

Upper-Bounding the Fat-Shattering Dimension of Quantum States

No need to thank me!

Page 9: The Learnability of Quantum States Scott Aaronson University of Waterloo.

f(x,y)

Simple Application of Quantum Occam’s Razor Theorem to Communication Complexity

f: Boolean function mapping Alice’s N-bit string x and Bob’s M-bit string y to a binary output

D1(f), R1(f), Q1(f): Deterministic, randomized, and quantum one-way communication complexities of f

How much can quantum communication save?

• It’s known that D1(f)=O(M Q1(f)) for all total f

• In 2004 I showed that for all f,D1(f)=O(M Q1(f)logQ1(f))

x y

GBUSTERSL

Alice Bob

Page 10: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Theorem: R1(f)=O(M Q1(f)) for all f, partial or total

Proof: Fix Alice’s input x

By Yao’s minimax principle, Alice can consider a worst-case distribution D over Bob’s input y

Alice’s classical message will consist of y1,…,yT drawn from D, together with f(x,y1),…,f(x,yT), where T=(Q1(f))

Bob searches for a quantum message that yields the right answers on y1,…,yT

By the Quantum Occam’s Razor Theorem, with high probability such a yields the right answers on most y drawn from D

Page 11: The Learnability of Quantum States Scott Aaronson University of Waterloo.

What about computational complexity?

BQP/qpoly: Class of problems solvable in quantum polynomial time, with help from poly-size “quantum advice state” that depends only on input length n

Can this result be improved to BQP/qpoly QMA/poly?(QMA: Quantum Merlin-Arthur)Theorem: HeurBQP/qpoly HeurQMA/polyOr in English: We can use trusted classical advice to verify that untrusted quantum advice will work on most inputs

A. 2004: BQP/qpoly PP/poly“Classical advice can always simulate quantum advice, provided we use exponentially more computation”

Page 12: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Proof Idea: The classical advice to the HeurQMA/poly verifier will consist of “training inputs” x1,…,xm where m=poly(n), as well as whether xiL for all i

Given a purported quantum advice state |, the verifier first checks that | yields the right answers on the training inputs, and only then uses it on its real input x

Stronger Result: HeurBQP/qpoly = HeurYQP/polyHere YQP (“Yoda Quantum Polynomial-Time”) is like

QMAcoQMA, except that a single witness must work for all inputs of length n

By the Quantum Occam’s Razor Theorem, if | passes the initial test, then w.h.p. it works on most inputs

Technical part is to do the verification without destroying |

Page 13: The Learnability of Quantum States Scott Aaronson University of Waterloo.

Open Problems

Computationally-efficient learning algorithms

Experimental implementation!

Tighter bounds on number of measurements

Does BQP/qpoly = YQP/poly?

Is D1(f) = O(M Q1(f))?

DUNCE

DUNCE