Quantum Computing and the Limits of the Efficiently Computable

Scott AaronsonMIT

Things we never see…

Warp drive Perpetuum mobile

GOLDBACH CONJECTURE: TRUE

NEXT QUESTION

Übercomputer

The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively

So what about the third one? What are the ultimate physical limits on what can be feasibly computed? And do those limits have any implications for physics?

PEfficiently solvable

NPEfficiently verifiable

NP-complete

NP-hardAll NP problems are efficiently

reducible to these

Graph connectivityPrimality testingMatrix determinantLinear programming…

Matrix permanentHalting problem…

Hamilton cycleSteiner treeGraph 3-coloringSatisfiabilityMaximum clique… Factoring

Graph isomorphism…

“OUR STANDARD MODEL”

Does P=NP?The (literally) $1,000,000 question

If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude.—Gödel to von Neumann, 1956

The Extended Church-Turing Thesis (ECT)

“Any physically-realistic computing device can be simulated by a

deterministic or probabilistic Turing machine, with at most polynomial

overhead in time and memory”

But how sure are we of this thesis?What would a challenge to it look like?

An important presupposition underlying P vs. NP is the

Jordan-Lee-Preskill 2012: Simple interacting quantum field theories (e.g., 4 theory) can be simulated efficiently using a “garden-variety” quantum computer

The LHC Computer?

Old proposal: Dip two glass plates with pegs between them into soapy water.

Let the soap bubbles form a minimum Steiner tree connecting the pegs—thereby solving a known NP-hard problem “instantaneously”

Relativity Computer

DONE

Zeno’s Computer

STEP 1

STEP 2

STEP 3STEP 4

STEP 5

Tim

e (s

econ

ds)

Time Travel Computer

R CTC R CR

C

0 0 0

Answer

“Causality-Respecting Register”

“Closed Timelike

Curve Register”

Polynomial Size Circuit

S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669.

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

Quantum Computing

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

But Can QCs Actually Be Built?Where we are now: A quantum computer has factored 21 into 37, with high probability (Martín-López et al. 2012)

Why is scaling up so hard? Because of decoherence: unwanted interaction between a QC and its external environment, “prematurely measuring” the quantum state

A few skeptics, in CS and physics, even argue that building a QC will be fundamentally impossible

I don’t expect them to be right, but I hope they are! If so, it would be a revolution in physics

And for me, putting quantum mechanics to the test is the biggest reason to build QCs—the applications are icing!

Key point: factoring is not believed to be NP-complete!

And today, we don’t believe quantum computers can solve NP-complete problems in polynomial time in general

(though not surprisingly, we can’t prove it)

Bennett et al. 1997: “Quantum magic” won’t be enough

If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one

(That bound is actually achievable, using Grover’s algorithm!)

If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow

Quantum Adiabatic Algorithm(Farhi et al. 2000)

HiHamiltonian with easily-prepared ground state

HfGround state encodes solution

to NP-complete problem

Problem: “Eigenvalue gap” can be exponentially small

Includes PNP as a special case, but is stronger

No longer a purely mathematical conjecture, but also a claim about the laws of physics

Could be invoked to “explain” why adiabatic systems have small spectral gaps, why protein folding gets stuck in metastable states, why the Schrödinger equation is linear, why time only flows in one direction…

“The No-SuperSearch Postulate”There is no physical means to solve NP-complete

problems in polynomial time.

OK, but can computational complexity engage even more

deeply with the content of modern physics? What other new insights

has it given the physicists?

Thanks for asking! I’ll give several examples, drawn from my own work and others’

Quantum Computing and the Interpretation of Quantum Mechanics?

“To those who still cling to a single-universe world-view, I issue this challenge: explain how Shor's algorithm works … When Shor's algorithm has factorized a number, using 10⁵⁰⁰ or so times the computational resources that can be seen to be present, where was the number factorized? … How, and where, was the computation performed?”

David Deutsch’s argument for Many Worlds:

Possible response: “To those who cling to a many-universe world-view, explain why the NP-complete problems still seem to be hard”

Schrödinger vs. Heisenberg vs. Feynman?Schrödinger and Heisenberg pictures of quantum mechanics: Require exponential time and exponential space to simulate using a classical computer

Bohmian mechanics?Postulates “real” trajectories for particles, which are guided along by the quantum state to reproduce the predictions of quantum mechanicsA. 2005: Calculating Bohmian trajectories is probably intractable even for a quantum computer!If we could do it, then we could also solve Graph Isomorphism in polynomial time, and break arbitrary collision-resistant hash functions

Feynman picture: Still exponential time, but only polynomial space

Two of my favorite functions

nS

n

iiiaA

1,Per

nS

n

iiiaA

1,

sgn1Det

Easily computable #P-complete [Valiant]

BOSONSFERMIONS

Two Basic Types of Particle in Nature

Free fermions can be simulated easily by a

classical computer[Valiant, Terhal-DiVincenzo]

Free bosons probably can’t be easily simulated by a classical computer

[A.-Arkhipov]2012: Experimental demonstrations of “BosonSampling” with 3-4 photons!

Computational Complexity and the Black-Hole Information Loss Problem

Maybe the single most striking application so far of complexity to fundamental physics

Hawking 1970s: Black holes radiate!

The radiation seems thermal (uncorrelated with whatever fell in)—but if quantum mechanics is true, then it can’t be

Susskind et al. 1990s: “Black-hole complementarity.” In string theory / quantum gravity, the Hawking radiation should just be a scrambled re-encoding of the same quantum states that are also inside the black hole

The Firewall Paradox [Almheiri et al. 2012]If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits (after waiting ~1070 years for enough to come out), then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior?

Entanglement among Hawking photons detected!

Harlow-Hayden 2013: Sure, there’s some unitary transformation that Alice could apply to the Hawking radiation, that would generate a “firewall” inside the event horizon. But how long would it take her to apply it?

Plausible answer: Exponential in the number of qubits inside the black hole! Or for an astrophysical black hole,

70102~ yearsShe wouldn’t have made a dent before the black hole had already evaporated anyway! So … problem solved?

HH’s argument: If Alice could achieve (a plausible formalization of) her decoding task, then she could also break collision-resistant hash functions—beyond what even QCs seem able to doRecently, I strengthened the HH argument, to show that Alice could even invert arbitrary one-way functions

Any one of these would make me as happy as low-energy SUSY would make you:

Prove P≠NP

Prove that not even quantum computers can solve NP-complete problems

Build a scalable quantum computer(or even more interesting, show that it’s impossible)

Clarify whether all of known physics can be simulated by a quantum computer

Use No-SuperSearch or related impossibility principles to make progress in quantum gravity

Conclusion