How Much Information Is In A Quantum State?

Scott AaronsonMIT

Computer Scientist / Physicist Nonaggression Pact

You accept that, for this talk:

• Polynomial vs. exponential is the basic dichotomy of the universe

• “For all x” means “for all x”

In return, I will not inflict the following computational complexity classes on you:

#P AM AWPP BQP BQP/qpoly MA NP P/poly PH PostBQP PP PSPACE QCMA QIP QMA SZK YQP

An infinite amount, of course, if you want to specify the state exactly…

Life is too short for infinite precision

02C0

1

So, how much information is in a quantum state?

A More Serious Point

In general, a state of n possibly-entangled qubits takes

~2n bits to specify, even approximately

nxx x

1,0

To a computer scientist, this is arguably the central fact about quantum mechanics

But why should we worry about it?

Spin-½ particles

Answer 1: Quantum State Tomography

Task: Given lots of copies of an unknown quantum state , produce an approximate classical description of

Not something I just made up!“As seen in Science & Nature”

Well-known problem: To do tomography on an entangled state of n spins, you need ~cn measurements

Current record: 8 spins / ~656,000 experiments (!)

This is a conceptual problem—not just a practical one!

Answer 2: Quantum Computing Skepticism

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

Levin Goldreich ‘t Hooft Davies Wolfram

But is it really an exponential amount?

Today we’ll tame the exponential beast

• Setting the stage: Holevo’s Theorem and random access codes

• Describing a state by postselected measurements [A. 2004]

• “Pretty good tomography” using far fewer measurements [A. 2006]

- Numerical simulation [A.-Dechter, in progress]

• Encoding quantum states as ground states of simple Hamiltonians [A.-Drucker 2009]

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 that fact seems to be “more about the map than the territory”

Theorem [Holevo 1973]: By sending an n-qubit state , Alice can communicate no more than n classical bits to Bob (or 2n bits assuming prior entanglement)

How can that be? Well, Bob has to measure , and measuring makes most of the wavefunction go poof…

Lesson: “The linearity of QM helps tame the exponentiality of QM”

Alice

Bob

The Absent-Minded Advisor Problem

Can you give your graduate student a quantum state with n qubits (or 10n, or n3, …)—such that by measuring in a suitable basis, the student can learn your answer to any one yes-or-no question of size n?

NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999]Indeed, quantum communication is no better than classical for this problem as n

Then she’ll need to send ~cn bits, in the worst case.

But… suppose Bob only needs to be able to estimate Tr(E) for every measurement E in a finite set S.

On the Bright Side…

Theorem (A. 2004): In that case, it suffices for

Alice to send ~n log n log|S| bits

Suppose Alice wants to describe an n-qubit state to Bob, well enough that for any 2-outcome measurement E, Bob can estimate Tr(E)

|ALL MEASUREMENTSALL MEASUREMENTS PERFORMABLE

USING ≤n2 QUANTUM GATES

How does the theorem work?

Alice is trying to describe the quantum state to Bob

In the beginning, Bob knows nothing about , so he guesses it’s the maximally mixed state 0=I

Then Alice helps Bob improve his guess, by repeatedly telling him a measurement EtS on which his guess t-1 badly fails

Bob lets t be the state obtained by starting from t-1, then performing Et and postselecting on the right outcome

I123

Claim: After only O(n) of these learning steps, Bob gets a state T such that Tr(ET)Tr(E) for all measurements ES.

Proof Sketch: For simplicity, assume =|| is pure and Tr(E) is ≤1/n2 or 1-1/n2 for all ES.

Let p be the probability that E1,E2,…,ET all yield the desired outcomes. By assumption, p is at most (say) (2/3)T

On the other hand, if Bob had made the lucky guess 0=||, then p would’ve been at least (say) 0.9

But we can decompose I as an equal mixture of | and 2n-1 other states orthogonal to |! Hence p 0.9/2n

0.9/2n ≤ (2/3)T T=O(n)

Conclusion: Alice can describe to Bob by telling him its behavior on E1,E2,…,ET. This takes O(n log|S|) bits

We’ve shown that for any n-qubit state and set S of observables, one can “compress” the measurement data {Tr(E)} ES into a classical string x of only Õ(nlog|S|) bits

Just two tiny problems…

1.Computing x seems astronomically hard

2.Given x, estimating Tr(E) also seems astronomically hard

I’ll now state the “Quantum Occam’s Razor Theorem,” which at least addresses the first problem…

Discussion

Let be an unknown quantum state of n spins

Suppose you just want to be able to estimate Tr(E) for most measurements E drawn from some probability measure D

Then it suffices to do the following, for some m=O(n):

1.Choose E1,…,Em independently from D

2.Go into your lab and estimate Tr(Ei) for each 1≤i≤m

3.Find any “hypothesis state” such that Tr(Ei)Tr(Ei) for all 1≤i≤m

Quantum Occam’s Razor Theorem

and

,1TrTrPr~

EEDE

with probability at least 1- over the choice of E1,…,Em.

Theorem [A. 2006]: Provided

1

log1

log244

nCm (C a constant)

for all i, you’ll be guaranteed that

7

TrTr2 ii EE

“Quantum states are PAC-learnable”

Proof combines Ambainis et al.’s result on the impossibility of quantum compression with some power tools from classical computational learning theory

Remark 1: To do this “pretty good tomography,” you don’t need any prior assumptions about ! (No Bayesian nuthin’...)

Removes a lot of conceptual problems...

Instead, you assume a distribution D over measurements

Might be preferable—after all, you can control which measurements to apply, but not what isRemark 2: Given the measurement data Tr(E1),…,Tr(Em), finding a hypothesis state consistent with it could still be an exponentially hard computational problem

Semidefinite / convex programming in 2n dimensions

But this seems unavoidable: even finding a classical hypothesis consistent with data is conjectured to be hard!

Numerical Simulation[A.-Dechter, in progress]

We implemented the “pretty-good tomography” algorithm in MATLAB, using a fast convex programming method developed specifically for this application [Hazan 2008]

We then tested it (on simulated data) using MIT’s computing cluster

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

Recap: Given an unknown n-qubit entangled quantum state , and a set S of two-outcome measurements…

Learning theorem: “Any hypothesis state consistent with a small number of sample points behaves like on most measurements in S”

Postselection theorem: “A particular state T (produced by postselection) behaves like on all measurements in S”

Dream theorem: “Any state that passes a small number of tests behaves like on all measurements in S”

[A.-Drucker 2009]: The dream theorem holds

Proof combines Quantum Occam’s Razor Theorem with a new classical result about “isolatability” of functions

Caveat: will have more qubits than , and in general be a very different state

A “Practical” ImplicationIt’s the year 2500. Everyone and her grandfather has a personal quantum computer.

You’re a software vendor who sells “magic initial states” that extend quantum computers’ problem-solving abilities.

Amazingly, you only need one kind of state in your store: ground states of 1D nearest-neighbor Hamiltonians!

Reason: Finding ground states of 1D spin systems is known to be “universal” for quantum constraint satisfaction problems[Aharonov-Gottesman-Irani-Kempe 2007], building on [Kitaev 1999]

UNIVERSAL RESOURCE STATE

SummaryIn many natural scenarios, the “exponentiality” of quantum states is an illusion

That is, there’s a short (though possibly cryptic) classical string that specifies how a quantum state behaves, on any measurement you could actually perform

Applications: Pretty-good quantum state tomography, characterization of quantum computers with “magic initial states”…

Biggest open problem: Find special classes of quantum states that can be learned in a computationally efficient way

“Experimental demonstration” would be nice too

Postselection theorem: quant-ph/0402095

Learning theorem: quant-ph/0608142

Ground state theorem, numerical simulations: “in preparation”

www.scottaaronson.com(/papers /talks /blog)