how much information is in a quantum state? scott aaronson mit
TRANSCRIPT
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How Much Information Is In A Quantum State?
Scott AaronsonMIT
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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
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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?
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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
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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!
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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?
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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”
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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
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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
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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)
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|ALL MEASUREMENTSALL MEASUREMENTS PERFORMABLE
USING ≤n2 QUANTUM GATES
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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
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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
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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
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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
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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
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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!
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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
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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
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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
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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
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Postselection theorem: quant-ph/0402095
Learning theorem: quant-ph/0608142
Ground state theorem, numerical simulations: “in preparation”
www.scottaaronson.com(/papers /talks /blog)