So You Think Quantum Computing Is Bunk? Scott Aaronson (MIT) | You measurin ME?

So You Think Quantum Computing Is Bunk?

Scott Aaronson (MIT)

| You

measurin’ ME?

Quantum Computing

When I first heard about QC (around 1996), I was certain it was bunk!

But to find the “catch,” I’d first have to figure out what the deal was with quantum mechanics itself…

“Like probability theory, but over the complex numbers”

Quantum Mechanics in 1 Slide

n

1

nnn

n

uu

uu

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111

n

1

1,1

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iii C

Quantum Mechanics:

Linear transformations that conserve 2-norm of

amplitude vectors:Unitary matrices

np

p

1

nnn

n

ss

ss

1

111

nq

q

1

1,01

n

iii pp

Probability Theory:

Linear transformations that conserve 1-norm of

probability vectors:Stochastic matrices

“The source of all quantum weirdness”Interference

1 1 112 2 2

1 1 0 1

2 2 2

0

10 1

2

0 1

2

1 1 102 2 2

1 1 1 1

2 2 2

Possible states of a single quantum bit, or qubit:

If you ask |0+|1 whether it’s |0 or |1, it answers |0 with probability ||2 and |1 with probability ||2.And it sticks with its answer from then on!

Measurement

Measurement is a “destructive” process:

The “deep mystery” of QM: Who decides when a “measurement” happens? An “outsider’s view”:

10

Unitary

World1World0World10

“Many Worlds? Or Many Words?”

Product state of two qubits:

1010

Entangled state (can’t be written as product state):

2

1100 11011000

The qubit simply gets entangled with your own body (and lots of other stuff), so that it collapses to |0 or |1 “relative to you”

A general entangled state of n qubits requires ~2n amplitudes to specify:

Quantum Computing“Quantum Mechanics on Steroids”

nxx x

1,0

Presents an obvious practical problem when using conventional computers to simulate quantum mechanics

Feynman 1981: So then why not turn things around, and build computers that themselves exploit superposition?

Shor 1994: Such a computer could do more than simulate QM—e.g., it could factor integers in polynomial time

Interesting

Where we are: A QC has now factored 21 into 37, with high probability (Martín-López et al. 2012)

Why is scaling up so hard? Decoherence!

The famous Fault-Tolerance Theorem suggests we only need to get decoherence down to some finite level (~1% per qubit per gate time?) to do arbitrarily long quantum computations

Many discussions of the feasibility of QC focus entirely on the Fault-Tolerance Theorem and its assumptions

My focus is different! For I take it as obvious that, if QC is impossible, there must exist a deeper explanation than “such-and-such error-correction schemes might not work against every conceivable kind of noise”

A few physicists and computer scientists remain vocally skeptical that scalable QC is possible…

(And perhaps a much larger number are “silently skeptical”?)

‘t Hooft Kalai Goldreich Wolfram Alicki Dyakanov Levin

One historical analogy: People thought Charles Babbage’s idea was cute but would never work in practice

And they were right—for ~130 years!

3 “Skeptical Positions” and My Responses

1. The difficulties are immense! QC might not be practical for a very long time (and will have limited applications even if built)

My response: Agreement

2. QC will fail because quantum mechanics itself is wrong

My response: Awesome! A revolution in physics—even better than QC. Count me in

3. Quantum mechanics is fine, but QC won’t work because of some “principle of unavoidable noise” (?) on top of QM

My response: Also wonderful! Explain your principle, why it’s true, and why it kills QC. Does it imply a fast classical simulation of “realistic” quantum systems?

Skeptical position I won’t address in this talk: BPP=BQP

Common Reasons for QC Skepticism1. “Sounds too good to be true / like science fiction”

Response: Would any science-fiction writer have imagined a computer that solved factoring, discrete log, and a few other special problems, but not NP-complete problems?

2. Annoyance at hype/misrepresentations in popular press

Response: Tell me about it…

3. The Extended Church-Turing Thesis rules out QC

Response: The ECT was a CS encroachment onto physics’ turf … we can’t cry foul if physics counterattacks us!

4. “n qubits couldn’t possibly encode 2n bits”

5. Underlying skepticism of QM itself (or modern physics in general?)

The “2n Bits Is Too Many” ArgumentShouldn’t we search for a more “reasonable” theory that agrees with QM on existing experiments, but:

Lets us feasibly prepare only a singly-exponential number of states, not a doubly-exponential number?

Predicts that in a volume of size n, only poly(n) bits can be reliably stored and retrieved, not exp(n) bits?

Lets us summarize the results of exp(n) possible measurements on an n-qubit state using only poly(n) classical bits?

Predicts that n-qubit states should be “PAC-learnable” with only poly(n) samples, not exp(n)?

Such a theory exists! It’s called quantum mechanics

OK, but suppose QC is impossible.Obvious question: What’s the criterion that tells us

which quantum-mechanical experiments can be done, and which ones can’t?

Possibility 1: Precision in Amplitudes. “The major problem is the requirement that basic quantum equations hold to multi-hundredth if not millionth decimal positions where the significant digits of the relevant quantum amplitudes reside. We have never seen a physical law valid to over a dozen decimals … Are quantum amplitudes still complex numbers to such accuracies or do they become quaternions, colored graphs, or sick-humored gremlins?” —Leonid Levin

Obvious Response:

Possibility 2: OK, small amplitudes might be fine for separable states—but entanglement is an illusion.

Obvious Response: The Bell Inequality(and its experimental violation)

Possibility 3: Fine, 2 or 3 particles might be entangled, but a thousand particles could never be entangled!

That doesn’t work either…

High-temperature superconductors

Buckyball double-slit experiment

Needed: A “Sure/Shor separator” (A. 2004), between the many-particle quantum states

we’re sure we can create and those that suffice for things like Shor’s algorithm

PRINCIPLED

LINE

+

|01 |12

++

|01 |11 |02 |12

1

2

1

2

1

2

1

2

2

7

3

7

My Candidate: “Tree Size”

Symmetrized states of n identical fermions/bosons can be shown to have tree size n(log n)

(Using the breakthrough lower bound of [Raz 2004] on the multilinear formula size of the permanent and determinant)

n(log n) lower bound probably also holds for 2D and 3D spin lattices

(Indeed, in all these cases, the true tree size is probably exp(n))

But this doesn’t work either!

“God, Dice, Yadda Yadda”

A completely different way quantum mechanics might be “not the whole story”: What if there were “deeper, underlying” physical laws, and quantum mechanics was “merely a statistical tool” derivable from those laws?

Recently, I became interested in -epistemic theories, an attempt to formalize the above “Einsteinian impulse”…

Note: If quantum mechanics were exactly derivable, this still wouldn’t kill QC! But maybe it could tell us where to look for a breakdown?

A set of “ontic states” (ontic = philosopher-speak for “real”)

For each pure state |Hd, a probability measure over ontic states

For each orthonormal basis B=(v1,…,vd) and i[d], a “response function” Ri,B:[0,1], satisfying

A d-dimensional -Epistemic Theory is defined by:

(Conservation of Probability)

(Born Rule)

Can trivially satisfy these axioms by setting =Hd, = the point measure concentrated on

| itself, and Ri,B()=|vi||2

Gives a completely uninteresting restatement of quantum mechanics (called the “Beltrametti-

Bugajski theory”)

Accounts beautifully for one qubit -epistemically!

(One qutrit: Already a problem…)

More Interesting Example: Kochen-Specker Theory

Observation: If |=0, then and can’t overlap

Call the theory maximally nontrivial if (as above) and overlap whenever | and | are not orthogonal

Response functions Ri,B(): deterministically return basis vector closest to |

Suppose we assume = (“-epistemic theories must behave well under tensor product”)

Then there’s a 2-qubit entangled measurement M, such that the only way to explain M’s behavior on the 4 states

PBR (Pusey-Barrett-Rudolph 2011) No-Go Theorem

is using a “trivial” theory that doesn’t mix 0 and +.

(Can be generalized to any pair of states, not just |0 and |+)

Bell’s Theorem: Can’t “locally” simulate all separable measurements on a fixed entangled state

PBR Theorem: Can’t “locally” simulate a fixed entangled measurement on all separable states (at least nontrivially so)

But suppose we drop PBR’s tensor assumption. Then:Theorem (A.-Bouland-Chua-Lowther ‘13): There’s a maximally-nontrivial -epistemic theory in any finite dimension d

Cover Hd with -nets, for all =1/n

Mix the states in pairs of small balls (B,B), where |,| both belong to some -net(“Mix” = make their ontic distributions overlap)

To mix all non-orthogonal states, take a “convex combination” of countably many such theories

Albeit an extremely weird one!Solves the main open problem of Lewis et al. ‘12

Ideas of the construction:

Theorem (ABCL’13): There’s no symmetric, maximally-nontrivial -epistemic theory in dimensions d3

Our proof, in the general case, uses some measure theory and differential geometry (and strangely, currently works only with complex amplitudes, not real ones)

On the other hand, suppose we want our theory to be symmetric—meaning that

and

If scalable QC is indeed possible, are there any experiments that could help demonstrate that—short

of actually building a general-purpose QC?

Some possibilities:

- Keep 1 qubit coherent for an extremely long time(Current record: ~15 minutes in ion traps)

- Quantum adiabatic optimization(the “D-Wave approach”)

- BosonSampling(and other restricted QC proposals)

BosonSampling [A.-Arkhipov 2011] “For when you only need your QC to overthrow the

Extended Church-Turing Thesis, not do anything useful”

n identical photons are generated, sent through a network of beamsplitters, then measured to see where they are

The result: A sample from a distribution {px}, such that each probability px equals |Per(Ax)|2, for some known nn complex matrix Ax (Permanent: Famous #P-complete problem)

Theorem: A classical computer can’t sample the same distribution in polynomial time, unless P#P=BPPNP.

We conjecture that this extends even to approximate/noisy classical simulations. Leads to a beautiful complexity-theoretic open problem! Is it #P-complete to approximate Per(A), with high probability over an nn matrix A of independent N(0,1) Gaussians?

Recent BosonSampling demonstrations with 3-4 photons [Broome et al., Tillmann et al., Walmsley et al., Crespi et al.]

If this could be scaled to ~20-30 photons, it would probably BosonSample faster than a classical simulation of itself…

Main engineering challenge: Deterministic generation of single photons, for synchronized arrival at the detectors

ConclusionI don’t know for sure that scalable QC is possible

But I do know that the popular framing of the question gets it exactly backwards!

Believing that QC can work doesn’t make you a starry-eyed visionary, but a scientific conservative

Doubting that QC can work doesn’t make you a cautious realist, but a scientific radical